Poisson Consulting

What are s-values?

Mon, 25 May 2026 00:00:00 +0000

P-values are a common measure of statistical significance that are often misunderstood.

One issue is that the scale of p-values can be difficult to interpret. For example, small p-values such as 0.0001 and 0.00001 are easily confused, and many readers often convert p-values to fractions ($\frac1{100}, \frac1{1,000}, \frac1{10,000}$), rounding them in the process (e.g., $0.0103 \approx 0.01 = \frac1{100}$).

S-values address the issue of interpretability by converting p-values into a more tangible representation of probability: the number of heads in a row on a fair coin.

Figure 1 shows the relationship between s-values and p-values. A p-value of 0.5 corresponds to an s-value of 1, since a probability of 0.5 corresponds to a single successful coin flip. Similarly, the common significance cutoff of 0.05 is equivalent to flipping 4.32 heads in a row.

Figure 1: The relationship between s-values and p-values.

S-values are particularly convenient when comparing or evaluating p-values that are very small or correspond to difficult fractions. See Table 1 below for some examples.

Table 1: Example conversions from p-values to their equivalent simplest fractions and (rounded) s-values.
P-value	Fraction	S-value
0.5000000	1 / 2	1.00
0.2500000	1 / 4	2.00
0.1000000	1 / 10	3.32
0.0625000	1 / 16	4.00
0.0500000	1 / 20	4.32
0.0100000	1 / 100	6.64
0.0058366	3 / 514	7.42
0.0010000	1 / 1000	9.97
0.0001000	1 / 10000	13.30
0.0000010	1 / 1e+06	19.90

How many fish are in the river?

Thu, 05 Feb 2026 00:00:00 +0000

On February 28th, 2025 Joe Thorley, presented on “How many trout are in the river?” Rethinking the logic of riverine population abundance assessments at the Alberta Native Trout Science Workshop & Gathering II

The abstract was as follows

“Obtaining a reliable answer to the question of ‘how many trout are in the river?’ plays an important role in monitoring threats to fish populations. Although approaches to answering this question are often formulated in terms of statistical procedures, we present the underlying logic and rethink some misconceptions. If a method was 100% efficient at capturing fish and was applied to the entire river, the population abundance would simply be the total number of fish caught. However, typically, the capture efficiency is less than 100% and can be highly variable which necessitates methods to estimate the efficiency. Currently, depletion-removal methods estimate the number of fish based on the rate of decline of the catch under the often-erroneous assumption that the capture efficiency does not change between passes. Mark-recapture makes the more reasonable assumption that marked and unmarked fish have the same probability of recapture. As a method can rarely be applied to the entire river it is also necessary to estimate the number of fish at the unsampled sites. Index sites with high fish densities do not allow the number of fish at the other sites to be estimated and may even be decoupled from changes in the population abundance. At the other extreme totally random site selection introduces unnecessary uncertainty. A stratified approach requires more thought but produces more reliable estimates. Fish densities are often calculated in terms of the number of trout by wetted area but this assumes that doubling the stream width is equivalent to doubling the stream length and means that trout densities change with discharge. Lineal fish densities, in contrast, facilitate comparisons within and among systems while also allowing the effect of stream width to be estimated. We provide examples from the literature and via thought experiments of both logically valid and invalid estimates.”

The presentation is available here.

Environmental variability shapes biodiversity and protected area priorities in Canada

Thu, 01 Jan 2026 00:00:00 +0000

Decision analysis rooted in Indigenous and Western scientific knowledge identifies cost-effective strategies for managing hyperabundant deer to restore keystone places

Thu, 13 Nov 2025 00:00:00 +0000

Spatial Stream Network Analysis of Skeena Watershed Stream Temperatures 2025

Wed, 10 Sep 2025 00:00:00 +0000

The suggested citation for this analytic appendix is:

Hill, N.E., Thorley, J.L., & Irvine, A. (2025) Spatial Stream Network Analysis of Skeena Watershed Stream Temperatures 2025. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1130667589.

Background

The primary goal of the current analyses is to answer the following questions:

How can we model stream temperature to include spatial correlation through a stream network, and predict growing season degree days?

Data Preparation

Hourly water temperature data collected in the Skeena watershed in northern British Columbia between 2015 and 2025 were downloaded from water-temp-bc, which stores a combination of historic and recent real-time hydrometric data from Environment and Climate Change Canada.

Hourly air temperature data (at two metres above ground level) for the Skeena watershed for the years 2015-2025 were downloaded from the ERA-5-Land simulation using the Copernicus Climate Change Service (C3S) Climate Data Store as NetCDF files (Muñoz Sabater 2019).

The data were extracted, cleaned (i.e., checked for errors and corrected if possible), and tidied (i.e., manipulated into a consistent format) using using R version 4.5.1 (R Core Team 2022).

Key assumptions of the data preparation included:

For each site, the time series of stream temperature values were classified as “unreasonable” if they were $<$ 0˚C or $>$ 30˚C, changed more than 2˚C from one hour to the next, were adjacent to another unreasonable value, or within a 5-hour gap between two unreasonable value; these were excluded from the analysis.
The simulated air temperature from the modeled simulation is a reasonable approximation of the truth.
Discharge data were not included in this analysis because Pacific Climate Impacts Consortium’s Gridded Hydrologic Model Output does not currently have output for the Skeena watershed.
Seven sites with few observed data points proved insufficient to capture the annual-scale fluctuations in stream temperature; these sites (station numbers 08EC014, 08ED004, 08EE008, 08EE025, 08EF001, 08EG012, and 08EG018) were excluded from the analysis.

Statistical Analysis

Model parameters were estimated using Bayesian methods. Where sufficient information was available, causal parameters were estimated accounting for biases due to correlation by embedding the assumed causal pathways into the model in the form of a causal network (Arif and MacNeil 2023) which is represented visually as a directed acyclic graph (DAG). The estimates were produced using Stan (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

Unless stated otherwise, the Bayesian analyses used weakly informative prior distributions (Gelman et al. 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor $\hat{R} \leq 1.05$ (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) $\textrm{ESS} \geq 150$ for each of the monitored parameters (Kery and Schaub 2011, 61).

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the proportion of zeros in the data and the first four central moments (mean, variance, skewness, and kurtosis) of the deviance residuals were compared to the expected values by simulating new data based on the posterior distribution and assumed sampling distribution and calculating the deviance residuals.

The sensitivity of the posteriors to the choice of prior distributions was evaluated by separately power-scaling the priors and likelihood, estimating the properties of the perturbed posteriors using Pareto smoothed importance sampling (Vehtari et al. 2017), and evaluating the distance between the base and perturbed posteriors using the cumulative Jensen-Shannon (CJS) distance (Kallioinen et al. 2023). A threshold CJS distance of 0.05 was used to indicate sensitivity; a prior sensitivity above this value suggests a strong prior, and a likelihood sensitivity below this value suggests that the likelihood is non-informative (Kallioinen et al. 2023). For hierarchical effects, only the top-level parameter was power-scaled to avoid perturbing the priors multiple times (Kallioinen et al. 2023).

The parameters are summarized in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020), and the surprisal s-value (Greenland 2019). Together a pair of lower and upper compatibility limits (CLs) are referred to as a compatibility interval (CI). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.32 bits, which is equivalent to a p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row on a fair coin.

Variable selection was based on the heuristic of directional certainty (Kery and Schaub 2011; Murtaugh 2014; Castilho and Prado 2021). Fixed effects were included if their s-value was $>$ 4.32 bits (Kery and Schaub 2011). Based on a similar argument, random effects were included if their standard deviation had a lower 95% CL $>$ 5% of the median estimate.

The results are displayed graphically by plotting the modeled relationships between individual variables and the response with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their arithmetic mean and first level values, respectively, while random effects are held constant at their typical value (Kery and Schaub 2011, 77–82). Unless stated otherwise the typical value is the arithmetic mean. When informative the influence of a particular variable is expressed in terms of the effect size (i.e., relative change in the response variable) with the 95% CI (Bradford et al. 2005).

The analyses were implemented using R version 4.5.1 (R Core Team 2022) and the embr family of packages.

Model Descriptions

Stream Temperature

The data were analyzed using a Spatial Stream Network model (Ver Hoef and Peterson 2010; Peterson and Hoef 2010), with code adapted from the SSNbayes package (Santos-Fernandez et al. 2022). The necessary stream network distances and connectivity were calculated using the BC Freshwater Atlas and the SSNbler (Peterson et al. 2024) and SSN2 (Dumelle et al. 2024) R packages. Air and stream temperature data were averaged by site and week; modeling was done on this weekly time scale.

The expected stream temperatures were modeled using the 3-parameter version of the air2stream model (Toffolon and Piccolroaz 2015). The average stream temperature (in ˚C) in the first week, $W_{s,j=1}$ was estimated by the model and assumed to be the same for all sites.

For all subsequent weeks (i.e., $j > 1$), the change in the stream temperature (in ˚C) between week $j - 1$ and week $j$ for the $s^{th}$ site, $\Delta W_{s,j}$, was modeled as follows:

\[\begin{equation} \Delta W_{s,j} = a1_s + a2_s A_{s,j} - a3_s W_{s,j - 1}, \end{equation}\]

where $a1_s$, $a2_s$, and $a3_s$ are the parameters of the air2stream model for the $s^{th}$ site, $A_{s,j}$ is the air temperature (in ˚C) for the $s^{th}$ site in the $j^{th}$ week and $W_{s, j - 1}$ is the expected stream temperature at the $s^{th}$ site in the previous week.

The expected stream temperature for the $s^{th}$ site in the $j^{th}$ week was then calculated:

\[\begin{equation} W_{s,j} = W_{s,j - 1} + \Delta W_{s,j}. \end{equation}\]

Growing Season Degree Days (GSDD) are the accumulated thermal units (in ˚C) during the growing season based on the mean daily water temperature values, which is a useful predictor of age-0 rainbow and westslope cutthroat trout size at the beginning of winter. The start and end of the growing season were based on the definitions of Coleman and Fausch (2007):

Start: the beginning of the first week that average stream temperatures exceeded and remained above 5˚C for the season.
End: the last day of the first week that average stream temperature dropped below 4˚C.

GSDD were derived for each site and year by assuming that the daily stream temperatures at each site were the predicted weekly mean stream temperature for every day in the given week.

Key assumptions of the model include:

The stream network is dendritic, not braided.
The expected stream temperatures were set to 0˚C if they were estimated to be negative by the model.
The stream temperature in the first week is the same for all sites.
The parameters of the air2stream model ($a1$, $a2$, and $a3$) vary randomly by site.
The residual variation is multivariate normally distributed.
The covariance structure of the residual variation combines the following covariance components:
- Nugget (allows for variation at a single location)
- Exponential tail-down (allows for spatial dependence between flow-connected and flow-unconnected locations)

Preliminary analysis found that:

Allowing the initial stream temperature to vary randomly by site produced unrealistic stream temperatures for January (> 10˚C).

Model Templates

Stream Temperature

data {
  int nsite;
  int nweek;

  int <lower=0> N_y_obs; // number observed values
  int <lower=0> N_y_mis; // number missing values
  array[N_y_obs] int<lower=1> i_y_obs;
  array[N_y_mis] int<lower=1> i_y_mis;
  vector [N_y_obs] y_obs;  // matrix[N_y_obs,1] y_obs[T];
  array[nsite * nweek] real air_temp;
  array[nsite * nweek] int<lower=0> site;
  array[nsite * nweek] int<lower=0> week;

  matrix [nsite, nsite] D;
  matrix [nsite, nsite] I;
  matrix [nsite, nsite] H;
  matrix [nsite, nsite] flow_con_mat;

parameters {
  vector<lower=0, upper=30>[N_y_mis] y_mis; // declaring the missing y

  real<lower=0> sigma_nug; // sd of nugget effect
  real<lower=0> sigma_td; // sd of tail-down
  real<lower=0> alpha_td; // range of the tail-down model
  real<lower=0> bInitialTemp;
  real<lower=0> s1;
  real<lower=0> s2;
  real<lower=0> s3;
  real<lower=-5, upper=15> m1;
  real<lower=-5, upper=1.5> m2;
  real<lower=-5, upper=5> m3; 
  array[nsite] real<lower=-5, upper=15> a1;
  array[nsite] real<lower=-5, upper=1.5> a2;
  array[nsite] real<lower=-5, upper=5> a3;

transformed parameters {
  vector[nsite * nweek] y; // long vector of y
  array[nweek] vector[nsite] Y;
  matrix[nsite, nsite] C_td; // tail-down cov
  real <lower=0> var_nug; // nugget
  real <lower=0> var_td; // partial sill tail-down

  vector[nsite * nweek] eTempDiff;
  vector<lower=0, upper=30>[nsite * nweek] eTemp; 
  array[nweek] vector[nsite] mu;
  y[i_y_obs] = y_obs;
  y[i_y_mis] = y_mis;
  var_nug = sigma_nug^2; // variance nugget
  var_td = sigma_td^2; // variance tail-down
  // Place observations into matrices
  for (t in 1:nweek){
    Y[t] = y[((t - 1) * nsite + 1):(t * nsite)];
  }
  eTemp[1:nsite] = rep_vector(bInitialTemp, nsite);
  for (i in (nsite + 1):(nweek * nsite)) {
    eTempDiff[i] = (a1[site[i]] + a2[site[i]] * air_temp[i] - a3[site[i]] * eTemp[i - nsite]);
    
    eTemp[i] = eTemp[i - nsite] + eTempDiff[i];
    if (eTemp[i] < 0) {
      eTemp[i] = 0.0;
    }
  }
  // Define 1st mu
  mu[1] = eTemp[1:nsite];
  // Define rest of mu; ----
  for (t in 2:nweek){
    mu[t] = eTemp[((t - 1) * nsite + 1):(t * nsite)];
  }
  // Covariance matrices ----
  // Tail-down exponential model
    for (i in 1:nsite) {
    for (j in 1:nsite) {
      if (flow_con_mat[i, j] == 1) { // if points are flow connected
        C_td[i, j] = var_td * exp(-3 * H[i, j] / alpha_td);
      }
      else{ // if points are flow unconnected
        C_td[i, j] = var_td * exp(-3 * (D[i, j] + D[j, i]) / alpha_td);
      }
    }
  }

model {
  sigma_nug ~ exponential(0.05); // sd nugget
  sigma_td ~ exponential(2); // sd tail-down
  alpha_td ~ normal(20000, 20000); // range tail-down
  bInitialTemp ~ normal(1, 1);

  s1 ~ exponential(1);
  s2 ~ exponential(1);
  s3 ~ exponential(1);
  m1 ~ normal(0.8, 1);
  m2 ~ normal(0.4, 1);
  m3 ~ normal(0.4, 1);
  a1 ~ normal(m1, s1);
  a2 ~ normal(m2, s2);
  a3 ~ normal(m3, s3);

  for (t in 1:nweek) {
    target += multi_normal_cholesky_lpdf(Y[t] | mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6));
  }

generated quantities {
  array[nweek * nsite] real ePrediction;
  array[nweek] real log_lik;
  real lprior;
  for (t in 1:nweek) {
    log_lik[t] = multi_normal_cholesky_lpdf(Y[t] | mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6));
    ePrediction[((t - 1) * nsite + 1):(t * nsite)] = to_array_1d(multi_normal_cholesky_rng(mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6)));
  }
  lprior = exponential_lpdf(sigma_nug | 0.05) +
           exponential_lpdf(sigma_td | 2) +
           (normal_lpdf(alpha_td | 20000, 20000) - normal_lccdf(0 | 20000, 20000)) + // truncated normal
           (normal_lpdf(bInitialTemp | 1, 1) - normal_lccdf(0 | 1, 1)) + // truncated normal
           normal_lpdf(m1 | 0.8, 1) +
           normal_lpdf(m2 | 0.4, 1) +
           normal_lpdf(m3 | 0.4, 1) +
           exponential_lpdf(s1 | 1) +
           exponential_lpdf(s2 | 1) +
           exponential_lpdf(s3 | 1);

Block 1. Model description.

Results

Tables

Locations

Table 1. Stream temperature site locations.

site	station name	longitude	latitude
08EB003	SKEENA RIVER AT GLEN VOWELL	-127.673	55.3011
08EB004	KISPIOX RIVER NEAR HAZELTON	-127.716	55.4338
08EB005	SKEENA RIVER ABOVE BABINE RIVER	-127.687	55.7171
08EB007	KITWANGA RIVER NEAR KITWANGA	-128.054	55.1164
08EC001	BABINE RIVER AT BABINE	-126.63	55.3225
08EC004	PINKUT CREEK NEAR TINTAGEL	-125.43	54.4054
08EC013	BABINE RIVER AT OUTLET OF NILKITKWA LAKE	-126.698	55.4265
08ED001	NANIKA RIVER AT OUTLET OF KIDPRICE LAKE	-127.452	53.9303
08ED002	MORICE RIVER NEAR HOUSTON	-127.427	54.1168
08EE003	BULKLEY RIVER NEAR HOUSTON	-126.719	54.3994
08EE004	BULKLEY RIVER AT QUICK	-126.9	54.6186
08EE005	BULKLEY RIVER NEAR SMITHERS	-127.133	54.7697
08EE012	SIMPSON CREEK AT THE MOUTH	-127.204	54.8099
08EE013	BUCK CREEK AT THE MOUTH	-126.65	54.3961
08EE020	TELKWA RIVER BELOW TSAI CREEK	-127.497	54.6074
08EF005	ZYMOETZ RIVER ABOVE O.K. CREEK	-128.325	54.4936
08EG019	KITSUMKALUM RIVER BELOW ALICE CREEK	-128.744	54.6793

Stream Temperature

Table 2. Parameter descriptions.

Parameter	Description
`C_td`	Covariance matrix of the tail-down exponential model
`D`	Downstream hydrologic distance matrix
`H`	Total hydrologic distance matrix
`I`	The identity matrix
`N_y_mis`	Number of missing water temperature values
`N_y_obs`	Number of observed water temperature values
`Y[t]`	Vector of water temperature values for all sites in the `t`^th week
`a1[i]`	Intercept-type parameter of the air2stream model for the `i`^th site
`a2[i]`	Effect of `air_temp[i]` on `eTempDiff[i]` for the `i`^th site
`a3[i]`	Effect of the previous week’s expected water temperature (`eTemp[i - nsite]`) on `eTempDiff[i]`, for the `i`^th site
`air_temp[i]`	The `i`^th air temperature value (˚C)
`alpha_td`	The variance of spatially independent points
`bInitialTemp`	Expected average water temperature for the week starting 01-01-2015 for all sites
`eTempDiff[i]`	Expected difference in average water temperature from the previous week
`eTemp[i]`	Expected value of `water_temp[i]`
`flow_con_mat`	Site connectivity matrix
`i_y_mis`	Indexes of missing water temperature values
`i_y_obs`	Indexes of observed water temperature values
`m1`	Mean of the site-wise random effect for the `a1` parameter
`m2`	Mean of the site-wise random effect for the `a2` parameter
`m3`	Mean of the site-wise random effect for the `a3` parameter
`mu[t]`	Vector of `eTemp` values for all sites in the `t`^th week
`nsite`	Number of sites
`nweek`	Number of weeks
`s1`	Standard deviation of the site-wise random effect for the `a1` parameter
`s2`	Standard deviation of the site-wise random effect for the `a2` parameter
`s3`	Standard deviation of the site-wise random effect for the `a3` parameter
`sigma_nug`	Standard deviation of the nugget effect
`sigma_td`	Standard deviation of the exponential tail-down covariance model
`site[i]`	The `i`^th site
`var_nug`	Variance of the nugget effect
`var_td`	Variance of the exponential tail-down covariance model
`week[i]`	The `i`^th week
`y[i]`	The `i`^th water temperature value (˚C)
`y_mis`	Vector of missing water temperature values
`y_obs`	Vector of observed water temperature values

Table 3. Model coefficients.

term	estimate	lower	upper	svalue
alpha_td	1.93e+04	1.03e+04	4.35e+04	10.6
bInitialTemp	1.68	3.33e-01	2.78	10.6
m1	9.40e-01	7.51e-01	1.13	10.6
m2	2.60e-01	2.01e-01	3.24e-01	10.6
m3	3.19e-01	2.65e-01	3.77e-01	10.6
s1	3.56e-01	2.57e-01	5.23e-01	10.6
s2	1.29e-01	9.34e-02	1.91e-01	10.6
s3	1.10e-01	7.93e-02	1.63e-01	10.6
sigma_nug	8.06e-01	7.87e-01	8.24e-01	10.6
sigma_td	8.35e-01	7.79e-01	8.95e-01	10.6

Table 4. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged	num_divergent	max_treedepth	ebfmi
9401	3802	3	500	2	482	1.014	TRUE	0	0	1.13

Table 5. Model sensitivity.

information	prior_cjs	lik_cjs	nterms	insensitive
strong prior	0.088	0.611	2	FALSE
weak prior & informative data	0.032	0.224	13250	TRUE

Table 6. Model sensitivity by parameter.

parameter	information	prior_cjs	lik_cjs	nterms	insensitive
a1	weak prior & informative data	0.006	0.295	17	TRUE
a2	weak prior & informative data	0.007	0.281	17	TRUE
a3	weak prior & informative data	0.008	0.292	17	TRUE
alpha_td	strong prior	0.078	1.016	1	FALSE
bInitialTemp	strong prior	0.088	0.611	1	FALSE
ePrediction	weak prior & informative data	0.032	0.224	9401	TRUE
m1	weak prior & informative data	0.006	0.34	1	TRUE
m2	weak prior & informative data	0.006	0.431	1	TRUE
m3	weak prior & informative data	0.006	0.598	1	TRUE
s1	weak prior & informative data	0.017	0.665	1	TRUE
s2	weak prior & informative data	0.007	0.822	1	TRUE
s3	weak prior & informative data	0.013	0.558	1	TRUE
sigma_nug	weak prior & informative data	0.012	4.76	1	TRUE
sigma_td	weak prior & informative data	0.006	0.846	1	TRUE
y_mis	weak prior & informative data	0.029	0.239	3792	TRUE

Table 7. Mean GSDD by year (with 95% prediction intervals).

year	estimate	lower	upper
2015	2031	1944	2117
2016	2120	2034	2211
2017	1906	1829	1992
2018	2044	1958	2125
2019	2042	1962	2137
2020	1815	1741	1896
2021	1925	1847	2016
2022	1984	1910	2060
2023	2233	2157	2315
2024	2006	1922	2093

Table 8. Mean GSDD by site (with 95% prediction intervals).

site	estimate	lower	upper
08EE020	1029	964	1098
08EE012	1365	1310	1418
08EB005	1653	1605	1706
08EF005	1743	1683	1804
08EB003	1869	1817	1921
08EB007	1910	1841	1978
08ED001	1937	1884	1987
08EE013	2030	1983	2077
08EG019	2056	1992	2119
08EB004	2062	2011	2113
08ED002	2139	2079	2200
08EE005	2309	2251	2367
08EE003	2312	2265	2359
08EC001	2330	2272	2385
08EE004	2364	2313	2416
08EC013	2449	2401	2499
08EC004	2639	2588	2688

Figures

Stream Temperature

Figure 1. Tail-down covariance by hydrologic distance (with 95% CIs).

Figure 2. Predicted water temperature by date (with 95% prediction intervals). The points are the observed data.

Figure 3. Predicted GSDD by year and site (with 95% prediction intervals).

Figure 4. GSDD median estimate and width of 95% prediction interval (PI) in 2015 by site. The black lines are the stream network.

Figure 5. GSDD median estimate and width of 95% prediction interval (PI) in 2016 by site. The black lines are the stream network.

Figure 6. GSDD median estimate and width of 95% prediction interval (PI) in 2017 by site. The black lines are the stream network.

Figure 7. GSDD median estimate and width of 95% prediction interval (PI) in 2018 by site. The black lines are the stream network.

Figure 8. GSDD median estimate and width of 95% prediction interval (PI) in 2019 by site. The black lines are the stream network.

Figure 9. GSDD median estimate and width of 95% prediction interval (PI) in 2020 by site. The black lines are the stream network.

Figure 10. GSDD median estimate and width of 95% prediction interval (PI) in 2021 by site. The black lines are the stream network.

Figure 11. GSDD median estimate and width of 95% prediction interval (PI) in 2022 by site. The black lines are the stream network.

Figure 12. GSDD median estimate and width of 95% prediction interval (PI) in 2023 by site. The black lines are the stream network.

Figure 13. GSDD median estimate and width of 95% prediction interval (PI) in 2024 by site. The black lines are the stream network.

Acknowledgements

The organizations and individuals whose contributions have made this analytic appendix possible include:

Hillcrest Geographics
- Simon Norris
Poisson Consulting
- Priscilla Durojaiye

References

Arif, Suchinta, and M. Aaron MacNeil. 2023. “Applying the Structural Causal Model Framework for Observational Causal Inference in Ecology.” Ecological Monographs 93 (1): e1554. https://doi.org/10.1002/ecm.1554.

Bradford, Michael J, Josh Korman, and Paul S Higgins. 2005. “Using Confidence Intervals to Estimate the Response of Salmon Populations (Oncorhynchus Spp.) to Experimental Habitat Alterations.” Canadian Journal of Fisheries and Aquatic Sciences 62 (12): 2716–26. https://doi.org/10.1139/f05-179.

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Taylor & Francis.

Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, et al. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.

Castilho, Leonardo Braga, and Paulo Inácio Prado. 2021. “Towards a Pragmatic Use of Statistics in Ecology.” PeerJ 9 (September): e12090. https://doi.org/10.7717/peerj.12090.

Coleman, Mark A., and Kurt D. Fausch. 2007. “Cold Summer Temperature Limits Recruitment of Age-0 Cutthroat Trout in High-Elevation Colorado Streams.” Transactions of the American Fisheries Society 136 (5): 1231–44. https://doi.org/10.1577/T05-244.1.

Dumelle, Michael, Erin E. Peterson, Jay M. Ver Hoef, Alan Pearse, and Daniel J. Isaak. 2024. “SSN2: The Next Generation of Spatial Stream Networkmodeling in R.” Journal of Open Source Software 9 (99): 6389. https://doi.org/10.21105/joss.06389.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Greenland, Sander. 2019. “Valid p -Values Behave Exactly as They Should: Some Misleading Criticisms of p -Values and Their Resolution With s -Values.” The American Statistician 73 (sup1): 106–14. https://doi.org/10.1080/00031305.2018.1529625.

Greenland, Sander, and Charles Poole. 2013. “Living with p Values: Resurrecting a Bayesian Perspective on Frequentist Statistics.” Epidemiology 24 (1): 62–68. https://doi.org/10.1097/EDE.0b013e3182785741.

Kallioinen, Noa, Topi Paananen, Paul-Christian Bürkner, and Aki Vehtari. 2023. “Detecting and Diagnosing Prior and Likelihood Sensitivity with Power-Scaling.” Statistics and Computing 34 (1): 57. https://doi.org/10.1007/s11222-023-10366-5.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Taylor; Francis, CRC Press.

Muñoz Sabater, J. 2019. ERA5-Land Hourly Data from 1950 to Present. Copernicus Climate Change Service (C3S) Climate Data Store (CDS). https://doi.org/10.24381/cds.e2161bac.

Murtaugh, Paul A. 2014. “In Defense of p Values.” Ecology 95 (3): 611–17. https://doi.org/10.1890/13-0590.1.

Peterson, Erin E., and Jay M. Ver Hoef. 2010. “A Mixed‐model Moving‐average Approach to Geostatistical Modeling in Stream Networks.” Ecology 91 (3): 644–51. https://doi.org/10.1890/08-1668.1.

Peterson, Erin, Michael Dumelle, Alan Pearse, Dan Teleki, and Jay M. Ver Hoef. 2024. SSNbler: Assemble SSN Objects in R.

R Core Team. 2022. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

Rafi, Zad, and Sander Greenland. 2020. “Semantic and Cognitive Tools to Aid Statistical Science: Replace Confidence and Significance by Compatibility and Surprise.” BMC Medical Research Methodology 20 (1): 244. https://doi.org/10.1186/s12874-020-01105-9.

Santos-Fernandez, Edgar, Jay M. Ver Hoef, Erin E. Peterson, James McGree, Daniel Isaak, and Kerrie Mengersen. 2022. “Bayesian Spatio-Temporal Models for Stream Networks.” Computational Statistics & Data Analysis 170 (June): 107446. https://doi.org/10.1016/j.csda.2022.107446.

Toffolon, Marco, and Sebastiano Piccolroaz. 2015. “A Hybrid Model for River Water Temperature as a Function of Air Temperature and Discharge.” Environmental Research Letters 10 (11): 114011. https://doi.org/10.1088/1748-9326/10/11/114011.

Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. “Practical Bayesian Model Evaluation Using Leave-One-Out Cross-Validation and WAIC.” Statistics and Computing 27 (5): 1413–32. https://doi.org/10.1007/s11222-016-9696-4.

Ver Hoef, Jay M., and Erin E. Peterson. 2010. “A Moving Average Approach for Spatial Statistical Models of Stream Networks.” Journal of the American Statistical Association 105 (489): 6–18. https://doi.org/10.1198/jasa.2009.ap08248.

bisonpicsuite: A set of R packages to estimate wood bison population parameters from remote camera data

Thu, 17 Jul 2025 00:00:00 +0000

The Human Shield Hypothesis: Does Predator Avoidance of Humans Create Refuges for Prey?

Sun, 01 Jun 2025 00:00:00 +0000

bbousuite: A set of R packages to facilitate analysis of boreal caribou survival and recruitment data

Fri, 30 May 2025 00:00:00 +0000

How climate change and population growth will shape attendance and human-wildlife interactions at British Columbia parks

Tue, 01 Apr 2025 00:00:00 +0000

Effects of vasectomy on breeding-related movement and activity in free-ranging white-tailed deer

Sat, 01 Mar 2025 00:00:00 +0000

ssdtools: Open-Source Software for Water Quality Guidelines

Sat, 01 Mar 2025 00:00:00 +0000

Clients: Province of BC, Environment and Climate Change Canada, Australian Government Department of Climate Change, Energy, the Environment and Water

Species sensitivity distributions (SSDs) are the standard statistical method for deriving water quality and environmental guidelines. In 2018, Poisson Consulting released ssdtools, a peer-reviewed R package that fits SSDs using multiple distributions and model averaging, published in the Journal of Open Source Software (Thorley & Schwarz, 2018).

To make the method accessible to non-R users, Poisson Consulting followed up with shinyssdtools (Dalgarno, 2021, Journal of Open Source Software), a Shiny web application that provides a point-and-click interface for the same analyses. In 2025, ssdtools v2 (Thorley, Fisher, Fox & Schwarz, Journal of Open Source Software) added expanded distributions, censored data support, and improved model-averaging methods.

The tools are now used by state regulators in Australia, Canada, and New Zealand. This project demonstrates our commitment to building open, transparent, and reproducible tools that raise the standard for evidence-based environmental protection.

How resource abundance and resource stochasticity affect organisms' range sizes

Sat, 01 Feb 2025 00:00:00 +0000

Long-Term Fisheries Stewardship in the Kootenay Basin

Sat, 01 Feb 2025 00:00:00 +0000

Clients: BC Hydro, Fish and Wildlife Compensation Program, BC Ministry of Forests, Ktunaxa Nation Council

Since 2008, Poisson Consulting has contributed to many fisheries studies across the Kootenay and Columbia basins in southeastern British Columbia. These studies combine acoustic telemetry, mark-recapture studies (PIT and reward tags), snorkel surveys, and boat electrofishing to estimate population abundance, natural and fishing mortality, growth, and spatial distribution for key species including Bull Trout, Rainbow Trout, Kokanee, Mountain Whitefish, and Walleye.

Using Bayesian hierarchical models, Poisson Consulting provides annual estimates that directly inform fishing regulations and hydro dam operating decisions. The Lower Columbia River Fish Population Indexing program, for example, evaluates multi-species responses to discharge regimes at BC Hydro’s HLK and Brilliant dams, while the Kootenay Lake exploitation study uses acoustic receiver arrays to partition total mortality between harvest and natural causes.

ssdtools v2: An R package to fit Species Sensitivity Distributions

Tue, 28 Jan 2025 00:00:00 +0000

Boreal Caribou and Wood Bison Population Monitoring

Wed, 01 Jan 2025 00:00:00 +0000

Clients: Environment and Climate Change Canada, Province of Alberta

Reliable estimates of survival and recruitment are essential for managing species at risk, yet the analytical methods have historically been inconsistent across jurisdictions. Poisson Consulting developed two peer-reviewed R package suites to address this gap.

bbousuite (Dalgarno et al., 2025, Journal of Open Source Software) provides a standardized Bayesian workflow for estimating boreal caribou adult female survival and calf recruitment from hunter-based monitoring data. The suite handles common data-quality challenges, including small sample sizes, missing observations, and variable survey effort, and produces the estimates required for federal recovery planning.

bisonpicsuite (Hill, Choquette & Kortello, 2025, Journal of Open Source Software) applies similar principles to estimate wood bison population parameters from remote camera data, building on earlier state-space population modelling of the Mackenzie Wood Bison herd in the Northwest Territories.

Together, these tools demonstrate how purpose-built, open-source software can improve the consistency and transparency of wildlife monitoring across agencies and jurisdictions.

Genetic connectivity of wolverines in western North America

Fri, 15 Nov 2024 00:00:00 +0000

Spatial Stream Network Analysis of Nechako Watershed Stream Temperatures 2022b

Fri, 25 Oct 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Hill, N.H., Thorley, J.L., & Irvine, A. (2024) Spatial Stream Network Analysis of Nechako Watershed Stream Temperatures 2022b. A Poisson Consulting Analysis Appendix. URL: https://www.poissonconsulting.ca/f/1295467017.

Background

The primary goal of the current analyses is to answer the following question:

How can we model stream temperature to include spatial correlation through a stream network?

Data Preparation

Sub-hourly water temperature data collected in the Nechako Watershed in northern British Columbia between 2019 and 2021 were downloaded as csv files from Zenodo (Morris et al. 2022).

Hourly air temperature data (at two metres above ground level) for the Nechako Watershed for the years 2019-2021 were downloaded from the ERA-5-Land simulation using the Copernicus Climate Change Service (C3S) Climate Data Store as NetCDF files (Muñoz Sabater 2019).

Daily baseflow and surface runoff data for the Nechako Watershed for the years 2019-2021 using the ACCESS1-0_rcp85 climate scenario were downloaded from the Pacific Climate Impacts Consortium’s Gridded Hydrologic Model Output as NetCDF files (Victoria Pacific Climate Impacts Consortium 2020).

The data were prepared for analysis using R version 4.4.1 (R Core Team 2022).

Key assumptions of the data preparation included:

All stream temperature data are correct, except those flagged “Fail”, which were excluded from analysis (see Gilbert et al. (2022) for details).
The simulated air temperature and discharge data from their respective modeled simulations are reasonable approximations of the truth.
The daily discharge at each stream temperature site was calculated by taking the weighted average of the sum of the baseflow and runoff for the watershed area upstream of the site; these were then averaged over weekly periods for each site.
One site that corresponded to just four consecutive observed data points proved insufficient to capture the annual-scale fluctuations in stream temperature; this site (WHC) was dropped from the analysis.
There were 146 observations with negative stream temperatures measurements, which were all set to 0˚C.

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using Stan (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

The sensitivity of the posteriors to the choice of prior distributions was evaluated by doubling the standard deviations of the priors and then using $\hat{R}$ to evaluate whether the samples were drawn from the same posterior distribution (Thorley and Andrusak 2017).

The parameters are summarized in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). Together a pair of lower and upper compatibility limits (CLs) are referred to as a compatibility interval (CI). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.32 bits, which is equivalent to a p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row on a fair coin.

The analyses were implemented using R version 4.4.1 (R Core Team 2022) and the embr family of packages.

Model Descriptions

Stream Temperature

The data were analyzed using a Spatial Stream Network model (Ver Hoef and Peterson 2010; Peterson and Hoef 2010), with code adapted from the SSNbayes package (Santos-Fernandez et al. 2022). The necessary stream network distances and connectivity were calculated using the BC Freshwater Atlas. Air and stream temperature data were averaged by site and week; modeling was done on this weekly time scale.

The expected stream temperatures were modeled using the 4-parameter version of the air2stream model (Toffolon and Piccolroaz 2015). The average stream temperature (in ˚C) in the first week, $W_{s,j=1}$ was estimated by the model and assumed to be the same for all sites.

For all subsequent weeks (i.e., $j > 1$), the change in the stream temperature (in ˚C) between week $j - 1$ and week $j$ for the $s^{th}$ site, $\Delta W_{s,j}$, was modeled as follows:

\[\begin{equation} \Delta W_{s,j} = \frac{1}{(\theta_{s,j})^{a4_s}}(a1_s + a2_s A_{s,j} - a3_s W_{s,j - 1}), \end{equation}\]

where $a1_s$, $a2_s$, $a3_s$, and $a4_s$ are the parameters of the air2stream model for the $s^{th}$ site, $A_{s,j}$ is the air temperature (in ˚C) for the $s^{th}$ site in the $j^{th}$ week, $W_{s, j - 1}$ is the expected stream temperature at the $s^{th}$ site in the previous week, and $\theta_{s,j}$ is the dimensionless discharge for the $s^{th}$ site in the $j^{th}$ week. $\theta_{s,j}$ was calculated as follows:

\[\begin{equation} \theta_{s,j} = \frac{d_{s,j}}{\bar{d_s}} \end{equation}\]

where $d_{s,j}$ is the discharge for the $s^{th}$ site in the $j^{th}$ week, and $\bar{d_s} = \frac{\sum_{j = 1}^{J}(d_{s,j})}{J}$ is the mean discharge across all $J$ weeks for the $s^{th}$ site.

The expected stream temperature for the $s^{th}$ site in the $j^{th}$ week was then calculated:

\[\begin{equation} W_{s,j} = W_{s,j - 1} + \Delta W_{s,j}. \end{equation}\]

Start: the beginning of the first week that average stream temperatures exceeded and remained above 5˚C for the season.
End: the last day of the first week that average stream temperature dropped below 4˚C.

GSDD were derived for each site and year by assuming that the daily stream temperatures at each site were the predicted weekly mean stream temperature for every day in the given week.

Key assumptions of the model include:

The stream network is dendritic, not braided.
The expected stream temperatures were set to 0˚C if they were estimated to be negative by the model.
The stream temperature in the first week is the same for all sites.
The parameters of the air2stream model ($a1$, $a2$, $a3$, and $a4$) vary randomly by site.
The residual variation is multivariate normally distributed.
The covariance structure of the residual variation combines the following covariance components:
- Nugget (allows for variation at a single location)
- Exponential tail-down (allows for spatial dependence between flow-connected and flow-unconnected locations)

Preliminary analysis found that:

The exponential tail-down model was better at explaining the spatial correlation in the data than exponential tail-up or euclidean distance models (Ver Hoef and Peterson 2010; Peterson and Hoef 2010).
The full 8-parameter air2stream model did not converge.
Preliminary analysis found that allowing the initial stream temperature to vary randomly by site produced unrealistic stream temperatures for January (> 10˚C).

Model Templates

Stream Temperature

.data {
  int nsite;
  int nweek;

  int <lower=0> N_y_obs; // number observed values
  int <lower=0> N_y_mis; // number missing values
  int <lower=1> i_y_obs [N_y_obs] ;  // [N_y_obs,T]
  int <lower=1> i_y_mis [N_y_mis] ;  // [N_y_mis,T]
  vector [N_y_obs] y_obs;  // matrix[N_y_obs,1] y_obs[T];
  real discharge [nsite * nweek];
  real air_temp [nsite * nweek];
  int<lower=0> site[nsite * nweek];
  int<lower=0> week[nsite * nweek];

  matrix [nsite, nsite] D;
  matrix [nsite, nsite] I;
  matrix [nsite, nsite] H;
  matrix [nsite, nsite] flow_con_mat;

parameters {
  vector<lower=0, upper=30>[N_y_mis] y_mis; // declaring the missing y

  real<lower=0> sigma_nug; // sd of nugget effect
  real<lower=0> sigma_td; // sd of tail-down
  real<lower=0> alpha_td; // range of the tail-down model
  real<lower=0> bInitialTemp;
  real<lower=0> s1;
  real<lower=0> s2;
  real<lower=0> s3;
  real<lower=0> s4;
  real<lower=-5, upper=15> m1;
  real<lower=-5, upper=1.5> m2;
  real<lower=-5, upper=5> m3; 
  real<lower=-1, upper=1> m4;
  real<lower=-5, upper=15> a1[nsite];
  real<lower=-5, upper=1.5> a2[nsite]; 
  real<lower=-5, upper=5> a3[nsite]; 
  real<lower=-1, upper=1> a4[nsite];

transformed parameters {
  vector[nsite * nweek] y; // long vector of y
  vector[nsite] Y[nweek]; // array of y
  matrix[nsite, nsite] C_td; // tail-down cov
  real <lower=0> var_nug; // nugget
  real <lower=0> var_td; // partial sill tail-down

  vector[nsite * nweek] eTempDiff;
  vector<lower=0, upper=30>[nsite * nweek] eTemp;
  vector[nsite] mu [nweek];
  y[i_y_obs] = y_obs;
  y[i_y_mis] = y_mis;
  var_nug = sigma_nug^2; // variance nugget
  var_td = sigma_td^2; // variance tail-down
  // Place observations into matrices
  for (t in 1:nweek){
    Y[t] = y[((t - 1) * nsite + 1):(t * nsite)];
  }
  eTemp[1:nsite] = rep_vector(bInitialTemp, nsite);
  for (i in (nsite + 1):(nweek * nsite)) {
    eTempDiff[i] = (1/(discharge[i]^a4[site[i]])) * (a1[site[i]] + a2[site[i]] * air_temp[i] - a3[site[i]] * eTemp[i - nsite]);
    
    eTemp[i] = eTemp[i - nsite] + eTempDiff[i];
    if (eTemp[i] < 0) {
      eTemp[i] = 0.0;
    }
  }
  // Define 1st mu
  mu[1] = eTemp[1:nsite];
  // Define rest of mu; ----
  for (t in 2:nweek){
    mu[t] = eTemp[((t - 1) * nsite + 1):(t * nsite)];
  }
  // Covariance matrices ----
  // Tail-down exponential model
    for (i in 1:nsite) {
    for (j in 1:nsite) {
      if (flow_con_mat[i, j] == 1) { // if points are flow connected
        C_td[i, j] = var_td * exp(-3 * H[i, j] / alpha_td);
      }
      else{ // if points are flow unconnected
        C_td[i, j] = var_td * exp(-3 * (D[i, j] + D[j, i]) / alpha_td);
      }
    }
  }

model {
  sigma_nug ~ exponential(0.05); // sd nugget
  sigma_td ~ exponential(2); // sd tail-down
  alpha_td ~ normal(0, 20000) T[0, ]; // range tail-down
  bInitialTemp ~ normal(0, 0.1) T[0, ];

  s1 ~ exponential(50);
  s2 ~ exponential(50);
  s3 ~ exponential(50);
  s4 ~ exponential(50);
  m1 ~ normal(0.8, 1);
  m2 ~ normal(0.4, 1);
  m3 ~ normal(0.4, 1);
  m4 ~ normal(0.1, 1);
  a1 ~ normal(m1, s1);
  a2 ~ normal(m2, s2);
  a3 ~ normal(m3, s3);
  a4 ~ normal(m4, s4);

  for (t in 1:nweek) {
    target += multi_normal_cholesky_lpdf(Y[t] | mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6));
  }

Block 1. Model description.

Results

Tables

Stream Temperature

Table 1. Parameter descriptions.

Parameter	Description
`C_td`	Covariance matrix of the tail-down exponential model
`D`	Downstream hydrologic distance matrix
`H`	Total hydrologic distance matrix
`I`	The identity matrix
`N_y_mis`	Number of missing water temperature values
`N_y_obs`	Number of observed water temperature values
`Y[t]`	Vector of water temperature values for all sites in the `t`^th week
`a1[i]`	Intercept-type parameter of the air2stream model for the `i`^th site
`a2[i]`	Effect of `air_temp[i]` on `eTempDiff[i]` for the `i`^th site
`a3[i]`	Effect of the previous week’s expected water temperature (`eTemp[i - nsite]`) on `eTempDiff[i]`, for the `i`^th site
`a4[i]`	Effect of `discharge[i]` on `eTempDiff[i]` for the `i`^th site
`air_temp[i]`	The `i`^th air temperature value (˚C)
`alpha_td`	The variance of spatially independent points
`bInitialTemp`	Expected average water temperature for the week starting 01-01-2019 for all sites
`discharge[i]`	Dimensionless discharge for the `i`^th observation (discharge for that observation divided by the mean discharge across all observations for that site)
`eTempDiff[i]`	Expected difference in average water temperature from the previous week
`eTemp[i]`	Expected value of `water_temp[i]`
`flow_con_mat`	Site connectivity matrix
`i_y_mis`	Indexes of missing water temperature values
`i_y_obs`	Indexes of observed water temperature values
`m1`	Mean of the site-wise random effect for the `a1` parameter
`m2`	Mean of the site-wise random effect for the `a2` parameter
`m3`	Mean of the site-wise random effect for the `a3` parameter
`m4`	Mean of the site-wise random effect for the `a4` parameter
`mu[t]`	Vector of `eTemp` values for all sites in the `t`^th week
`nsite`	Number of sites
`nweek`	Number of weeks
`s1`	Standard deviation of the site-wise random effect for the `a1` parameter
`s2`	Standard deviation of the site-wise random effect for the `a2` parameter
`s3`	Standard deviation of the site-wise random effect for the `a3` parameter
`s4`	Standard deviation of the site-wise random effect for the `a4` parameter
`sigma_nug`	Standard deviation of the nugget effect
`sigma_td`	Standard deviation of the exponential tail-down covariance model
`site[i]`	The `i`^th site
`var_nug`	Variance of the nugget effect
`var_td`	Variance of the exponential tail-down covariance model
`week[i]`	The `i`^th week
`y[i]`	The `i`^th water temperature value (˚C)
`y_mis`	Vector of missing water temperature values
`y_obs`	Vector of observed water temperature values

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
alpha_td	9.77e+03	6.16e+03	1.69e+04	10.6
bInitialTemp	7.14e-02	4.01e-03	2.36e-01	10.6
m1	6.30e-01	4.65e-01	7.96e-01	10.6
m2	3.22e-01	2.67e-01	3.76e-01	10.6
m3	3.21e-01	2.73e-01	3.74e-01	10.6
m4	2.79e-01	1.80e-01	3.93e-01	10.6
s1	2.67e-01	2.07e-01	3.45e-01	10.6
s2	1.18e-01	9.08e-02	1.59e-01	10.6
s3	1.08e-01	8.46e-02	1.45e-01	10.6
s4	2.01e-01	1.55e-01	2.66e-01	10.6
sigma_nug	4.20e-01	3.85e-01	4.58e-01	10.6
sigma_td	1.37e+00	1.19e+00	1.59e+00	10.6

Table 3. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
3297	2126	3	500	5	543	1.013	TRUE

Table 4. Model sensitivity.

all	analysis	sensitivity	bound
all	1.013	1.076	1.132

Figures

Stream Temperature

Figure 1. Tail-down covariance by hydrologic distance (with 95% CIs).

Figure 2. Predicted water temperature by date (with 95% CIs). The points are the observed data.

Figure 3. Predicted GSDD by year and site (with 95% CIs).

Figure 4. GSDD median estimate and width of 95% CI by year and site. The black lines are the stream network.

Acknowledgements

The organisations and individuals whose contributions have made this analytic appendix possible include:

Hillcrest Geographics
- Simon Norris

References

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Taylor & Francis.

Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, et al. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.

Castilho, Leonardo Braga, and Paulo Inácio Prado. 2021. “Towards a Pragmatic Use of Statistics in Ecology.” PeerJ 9 (September): e12090. https://doi.org/10.7717/peerj.12090.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Gilbert, Derek E., Jeremy E. Morris, Anna R. Kaveney, and Stephen J. Déry. 2022. “Sub-Hourly Water Temperature Data Collected Across the Nechako Watershed, 2019-2021.” Data in Brief 43 (August): 108425. https://doi.org/10.1016/j.dib.2022.108425.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Taylor; Francis, CRC Press.

Morris, Jeremy, Derek Gilbert, Anna Kaveney, and Stephen Déry. 2022. Sub-Hourly Water Temperature Collected by UNBC’s Northern Hydrometeorology Group (NHG) Across the Nechako Watershed, 2019-2021. https://doi.org/10.5281/zenodo.6426023.

Muñoz Sabater, J. 2019. ERA5-Land Hourly Data from 1950 to Present. Copernicus Climate Change Service (C3S) Climate Data Store (CDS). https://doi.org/10.24381/cds.e2161bac.

Murtaugh, Paul A. 2014. “In Defense of p Values.” Ecology 95 (3): 611–17. https://doi.org/10.1890/13-0590.1.

R Core Team. 2022. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

Thorley, Joseph L., and Greg F. Andrusak. 2017. “The Fishing and Natural Mortality of Large, Piscivorous Bull Trout and Rainbow Trout in Kootenay Lake, British Columbia (2008–2013).” PeerJ 5 (January): e2874. https://doi.org/10.7717/peerj.2874.

Victoria Pacific Climate Impacts Consortium, University of. 2020. Gridded Hydrologic Model Output. https://data.pacificclimate.org/portal/hydro_model_out/map/.

Duncan Lardeau Juvenile Rainbow Trout Abundance 2024

Wed, 02 Oct 2024 00:00:00 +0000

The suggested citation for this analytic report is:

Thorley, J.L., Andrusak, G.F. and Amies-Galonski, E. (2024) Duncan Lardeau Juvenile Rainbow Trout Abundance 2024. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/435610556.

Background

Rainbow Trout rear in the Lardeau and Lower Duncan rivers. Since 2006 (with the exception of 2015) annual spring snorkel surveys have been conducted to estimate the abundance and distribution of age-1 Rainbow Trout. From 2006 to 2010 the surveys were conducted at fixed index sites. Since 2011 fish observations have been mapped to the river based on their spatial coordinates as recorded by GPS.

The primary aims of the current analyses were to:

Estimate the egg deposition.
Estimate the spring abundance of age-1 fish by year.
Estimate the stock-recruitment relationship between the egg deposition and the abundance of age-1 recruits the following spring.
Estimate the survival from age-1 to age-2.
Estimate the inlake survival by year and cohort.
Estimate the expected number of spawners without in river and/or in lake variation.

Methods

Data Preparation

The data were provided by the Ministry of Forests, Lands and Natural Resource Operations (MFLNRO). The historical and current snorkel count data were manipulated using R version 4.4.1 (R Core Team 2022) and organised in an SQLite database.

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2003) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman et al. 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor $\hat{R} \leq 1.05$ (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) $\textrm{ESS} \geq 150$ for each of the monitored parameters (Kery and Schaub 2011, 61).

The parameters are summarised in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). Together a pair of lower and upper compatibility limits (CLs) are referred to as a compatibility interval (CI). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.32 bits, which is equivalent to a p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row on a fair coin.

Variable selection was based on the heuristic of directional certainty Castilho and Prado (2021). Fixed effects were included if their s-value was $>$ 4.32 bits (Kery and Schaub 2011). Based on a similar argument, random effects were included if their standard deviation had a lower 95% CL $>$ 5% of the median estimate.

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the number of zeros and the first four central moments (mean, variance, skewness and kurtosis) for the deviance residuals were compared to the expected values by simulating new residuals. In this context the s-value indicates how surprising each observed metric is given the estimated posterior probability distribution for the residual variation.

Where computationally practical, the sensitivity of the parameters to the choice of prior distributions was evaluated by increasing the standard deviations of all normal, half-normal and log-normal priors by an order of magnitude and then using $\hat{R}$ to evaluate whether the samples were drawn from the same posterior distribution (Thorley and Andrusak 2017).

Unless stated otherwise the typical value is the arithmetic mean. When informative the influence of a particular variables is expressed in terms of the effect size (i.e., relative change in the response variable) with the 95% CI (Bradford et al. 2005).

The analyses were implemented using R version 4.4.1 (R Core Team 2022) and the embr family of packages.

Model Descriptions

Length Correction

The annual bias (inaccuracy) and error (imprecision) in observer’s fish length estimates when spotlighting (standing) and snorkeling were quantified from the divergence of their length distribution from the length distribution for all observers (including measured fish) in that year. More specifically, the length correction that minimised the Jensen-Shannon divergence (Lin 1991) between the two distributions provided a measure of the inaccuracy while the minimum divergence (the Jensen-Shannon divergence was calculated with log to base 2 which means it lies between 0 and 1) provided a measure of the imprecision.

After correcting the fish lengths, age-1 individuals were assumed to be those with a fork length $\leq$ 100 mm.

Abundance

The abundance was estimated from the count data using an overdispersed Poisson model (Kery and Schaub 2011, 55–56). The annual abundance estimates represent the total number of fish in the study area.

Key assumptions of the abundance model include:

The lineal fish density varies with year, useable width and river kilometer as a polynomial, and randomly with site.
The observer efficiency at marking sites varies by study design (GPS versus Index).
The observer efficiency also varies by visit type (marking versus count) within study design and randomly by snorkeller.
The expected count at a site is the expected lineal density multiplied by the site length, the observer efficiency and the proportion of the site surveyed.
The residual variation in the actual count is gamma-Poisson distributed.

Condition

The condition of fish with a fork length $\geq$ 500 mm was estimated via an analysis of mass-length relations (He et al. 2008).

More specifically the model was based on the allometric relationship

\[ W = \alpha_c L^{\beta_c}\]

where $W$ is the weight (mass), $\alpha_c$ is the coefficent, $\beta_c$ is the exponent and $L$ is the length.

To improve chain mixing the relation was log-transformed, i.e.,

\[ \log(W) = \log(\alpha_c) + \beta_c \log(L).\]

Key assumptions of the condition model include:

$\alpha_c$ can vary randomly by year.
The residual variation in weight is log-normally distributed.

Fecundity

The fecundity of females with a fork length $\geq$ 500 mm was estimated via an analysis of fecundity-mass relations.

More specifically the model was based on the allometric relationship

\[ F = \alpha_f W^{\beta_f}\]

where $F$ is the fecundity, $\alpha_f$ is the coefficent, $\beta_f$ is the exponent and $W$ is the weight.

To improve chain mixing the relation was log-transformed.

Key assumptions of the fecundity model include:

The residual variation in fecundity is log-normally distributed.

Spawner Size

The average length of the spawners in each year (for years for which it was unavailable) was estimated from the mean weight of Rainbow Trout in the Kootenay Lake Rainbow Trout Mailout Survey (KLRT) using a linear regression. This approach was suggested by Rob Bison.

Egg Deposition

The egg deposition in each year was estimated by

converting the average length of spawners to the average weight using the condition relationship for a typical year
adjusting the average weight by the annual condition effect (interpolating where unavailable)
converting the average weight to the average fecundity using the fecundity relationship
multiplying the average fecundity by the AUC based estimate of the number of females (assuming a sex ratio of 1:1)

Stock-Recruitment

The relationship between the number of eggs ($E$) and the abundance of age-1 individuals the following spring ($R$) was estimated using a Beverton-Holt stock-recruitment model (Walters and Martell 2004):

\[ R = \frac{\alpha_s \cdot E}{1 + \beta_s \cdot E} \quad,\]

where $\alpha_s$ is the maximum number of recruits per egg (egg survival), and $\beta_s$ is the density dependence.

Key assumptions of the stock-recruitment model include:

The residual variation in the number of recruits is log-normally distributed with the standard deviation scaling with the uncertainty in the number of recruits.

The age-1 maximum carrying capacity ($K$) is given by:

\[ K = \frac{\alpha_s}{\beta_s}\]

and the $E_{K/2}$ Limit Reference Point (Mace 1994) ($E_{0.5 R_{max}}$), which corresponds to the stock (number of eggs) that produce 50% of the maximum recruitment ($K$), by \[E_{K/2} = \frac{1}{\beta_s}\]

The LRP was also converted into a number of spawners in a typical year (assuming 6,000 eggs per spawner and a sex ratio of 1:1).

Age-1 to Age-2 Survival

The relationship between the number of age-1 individuals and the number of age-2 individuals the following year was estimated using a linear regression through the origin where the slope was constrained to lie between 0 and 1 by a logistic transformation.

Key assumptions of the survival rate model include:

The residual variation in the number of age-2 individuals is log-normally distributed.

Calculated In-lake Survival

The in-lake survival of each recruit cohort was calculated by dividing the number of spawners in a given year by the number of age-1 recruits 6 years previous, assuming all spawners return at age-7.

Modelled In-lake Survival

The in-lake survival was estimated by modelling the relationship between the number of spawners in each year and the number of age-1 recruits four, five, and six years prior. The number of years indexed back is a key assumption and input to the model implying different ages of return. A proportional value was then assigned to each age class describing the proportion of recruits adopting each return strategy, for the pre and post collapse periods separately.

The survival in each year was fit as the fractional multiplier required to produce the expected number of spawners given the assumed proportions of each return strategy. Each estimate of year survival was effectively informed by the age-1 to spawner survival of all sets of recruits that passed through a given year. The sensitivity of the survival estimates to the proportion of fish adopting each return strategy was evaluated by refitting the model assuming different sets of proportions for each strategy in the pre and post collapse periods separately. The cohort survival was estimated by summing the proportional subsets of spawners returning at the three different ages assumed to originate from a single recruit cohort, and dividing that sum by the number of recruits in the original cohort.

An additional Beverton-Holt stock-recruitment model was used to analyse the relationship between spawners ($E$) and subsequent age-1 recruits ($R$), facilitating the estimation of recruit numbers prior to the availability of actual recruit data, and enabling the estimation of survival rates throughout the entire time-series of spawner counts.

Key assumptions of the in-lake survival model include:

Spawning fish return at age five, six, or seven.
Spawning fish fish return in some fixed proportion of age classes across all years in each period.
Each spawning fish produces the same number of eggs in each year.
The collapse of Kootenay Lake Kokanee occured in 2013.
The residual variation in the number of spawners is log-normally distributed.

Expected Spawners

The expected spawners without in river and/or in lake variation was calculated from the stock-recruitment relationship and assuming an age-1 to spawner survival of 0.79%.

Reproductive Rate

The maximum reproductive rate (the number of spawners per spawner at low density) not accounting for fishing mortality was calculated by multiplying $\alpha_s$ (number of recruits per egg at low density) from the stock-recruitment relationship by the inlake survival by the estimated eggs per spawner in each year (assuming a sex ratio of 1:1). The inlake survival from age-1 to spawning was calculated by dividing the subsequent number of returning spawners by the number of age-1 recruits assuming that equal numbers of fish adopt the strategy of intending to return at age 5, 6 and 7.

Results

Model Templates

Abundance

  data {

    int<lower=0> nMarked;
    int<lower=0> Marked[nMarked];
    int<lower=0> Resighted[nMarked];
    int<lower=0> IndexMarked[nMarked];

    int<lower=0> nObs;
    int Marking[nObs];
    int Index[nObs];
    int<lower=0> nSwimmer;
    int<lower=0> Swimmer[nObs];
    int<lower=0> nYear;
    int<lower=0> Year[nObs];
    real Rkm[nObs];
    int<lower=0> nSite;
    int<lower=0> Site[nObs];

    real SiteLength[nObs];
    real SurveyProportion[nObs];

    int Count[nObs];
  }
  parameters {
    real bEfficiency;
    real bEfficiencyIndex;

    real bDensity;
    real<lower=0> sDensityYear;
    vector[nYear] bDensityYear;

    vector[4] bDensityRkm;
    real<lower=0> sDensitySite;
    vector[nSite] bDensitySite;

    real bEfficiencyMarking;
    real bEfficiencyMarkingIndex;

    real<lower=0,upper=5> sEfficiencySwimmer;
    vector[nSwimmer] bEfficiencySwimmer;

    real<lower=0> sDispersion;
  }
  model {

    vector[nObs] eDensity;
    vector[nObs] eEfficiency;
    vector[nObs] eAbundance;
    vector[nObs] eCount;

    sDispersion ~ gamma(0.01, 0.01);

    bDensity ~ normal(0, 2);
    bDensityRkm ~ normal(0, 2);
    sDensitySite ~ uniform(0, 5);
    bDensitySite ~ normal(0, sDensitySite);
    sDensityYear ~ uniform(0, 5);
    bDensityYear ~ normal(0, sDensityYear);

    bEfficiency ~ normal(0, 5);
    bEfficiencyIndex ~ normal(0, 5);
    bEfficiencyMarking ~ normal(0, 5);
    bEfficiencyMarkingIndex ~ normal(0, 5);
    sEfficiencySwimmer ~ uniform(0, 5);
    bEfficiencySwimmer ~ normal(0, sEfficiencySwimmer);

    for (i in 1:nMarked) {
      target += binomial_lpmf(Resighted[i] | Marked[i],
        inv_logit(
          bEfficiency +
          bEfficiencyIndex * IndexMarked[i] +
          bEfficiencyMarking +
          bEfficiencyMarkingIndex * IndexMarked[i]
        ));
    }

    for (i in 1:nObs) {
      eDensity[i] = exp(bDensity +
                        bDensityRkm[1] * Rkm[i] +
                        bDensityRkm[2] * pow(Rkm[i], 2.0) +
                        bDensityRkm[3] * pow(Rkm[i], 3.0) +
                        bDensityRkm[4] * pow(Rkm[i], 4.0) +
                        bDensitySite[Site[i]] +
                        bDensityYear[Year[i]]);

      eEfficiency[i] = inv_logit(
        bEfficiency +
        bEfficiencyIndex * Index[i] +
        bEfficiencyMarking * Marking[i] +
        bEfficiencyMarkingIndex * Index[i] * Marking[i] +
        bEfficiencySwimmer[Swimmer[i]]);

      eAbundance[i] = eDensity[i] * SiteLength[i];

      eCount[i] = eAbundance[i] * eEfficiency[i] * SurveyProportion[i];
    }

    target += neg_binomial_2_lpmf(Count | eCount, sDispersion);
  }

Block 1. Abundance model description.

Condition

 data {
  int nYear;
  int nObs;

  vector[nObs] Length;
  vector[nObs] Weight;
  int Year[nObs];

parameters {
  real bWeight;
  real bWeightLength;
  real sWeightYear;

  vector[nYear] bWeightYear;
  real sWeight;

model {

  vector[nObs] eWeight;

  bWeight ~ normal(-10, 5);
  bWeightLength ~ normal(3, 2);

  sWeightYear ~ normal(-2, 5);

  for (i in 1:nYear) {
    bWeightYear[i] ~ normal(0, exp(sWeightYear));
  }

  sWeight ~ normal(-2, 5);
  for(i in 1:nObs) {
    eWeight[i] = bWeight + bWeightLength * log(Length[i]) + bWeightYear[Year[i]];
    Weight[i] ~ lognormal(eWeight[i], exp(sWeight));
  }

Block 2.

Fecundity

 data {
  int nObs;

  vector[nObs] Weight;
  vector[nObs] Fecundity;

parameters {
  real bFecundity;
  real bFecundityWeight;
  real sFecundity;

model {

  vector[nObs] eFecundity;

  bFecundity ~ uniform(0, 5);
  bFecundityWeight ~ uniform(0, 2);
  sFecundity ~ uniform(0, 1);

  for(i in 1:nObs) {
    eFecundity[i] = log(bFecundity) + bFecundityWeight * log(Weight[i]);
    Fecundity[i] ~ lognormal(eFecundity[i], sFecundity);
  }

Block 3.

Stock-Recruitment

model {
  a ~ dunif(0, 1)
  b ~ dunif(0, 0.1)
  sScaling ~ dunif(0, 5)

  eRecruits <- a * Stock / (1 + Stock * b)

  for(i in 1:nObs) {
    esRecruits[i] <- SDLogRecruits[i] * sScaling
    Recruits[i] ~ dlnorm(log(eRecruits[i]), esRecruits[i]^-2)
  }

Block 4. Stock-Recruitment model description.

Age-1 to Age-2 Survival

model {
  bSurvival ~ dnorm(0, 2^-2)
  sRecruits ~ dnorm(0, 2^-2) T(0,)

  for(i in 1:nObs) {
    logit(eSurvival[i]) <- bSurvival
    eRecruits[i] <- Stock[i] * eSurvival[i]
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

Block 5. In-river survival model description.

Modelled In-lake Survival

.model{
  a ~ dnorm(300, 50^-2) T(0,)
  b ~ dexp(1)
  sRecruits ~ dexp(1)
  sSpawnersReturning ~ dexp(1)
  bSurv ~ dnorm(0, 2^-2)
  sYearSurv ~ dexp(1)
  bPostCollapse ~ dnorm(0, 1^-2)

  for (i in 1:nObs) {
    eRecruits[i] <- a * SpawnersActive[i] / (1 + SpawnersActive[i] * b)
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

  for (i in 1:nObs) {
    bYearSurv[i] ~ dnorm(0, sYearSurv^-2)
    logit(eYearSurv[i]) <- bSurv + bYearSurv[i] + bPostCollapse * PostCollapse[i]
  }

  for (i in 1:(nObs - return - 1)) {
    eSpawnersCohort[i, 1] <- Recruits[i] * prod(eYearSurv[i:(i + return - 1)])
    eSpawnersCohort[i, 2] <- Recruits[i] * prod(eYearSurv[i:(i + return - 1 + 1)])
    eSpawnersCohort[i, 3] <- Recruits[i] * prod(eYearSurv[i:(i + return - 1 + 2)])
  }

  for (i in 3:(nObs - return - 1)) {
    eSpawnersReturning[i] <-
      (1 - PostCollapse[i]) * eSpawnersCohort[i, 1] * PropPre1[i] + PostCollapse[i] * eSpawnersCohort[i, 1] * PropPost1[i] +
      (1 - PostCollapse[i]) * eSpawnersCohort[i - 1, 2] * PropPre2[i] + PostCollapse[i] * eSpawnersCohort[i - 1, 2] * PropPost2[i] +
      (1 - PostCollapse[i]) * eSpawnersCohort[i - 2, 3] * PropPre3[i] + PostCollapse[i] * eSpawnersCohort[i - 2, 3] * PropPost3[i]
  }

  for (i in (return + 2 + 1):(nObs - 1)) {
    SpawnersReturning[i] ~ dlnorm(log(eSpawnersReturning[i - return]), sSpawnersReturning^-2)
  }

Block 6. Model description.

Tables

Abundance

Table 1. Parameter descriptions.

Parameter	Description
`Index`	Whether the `i`^th visit was to an index site
`Marking[i]`	Whether the `i`^th visit was to a site with marked fish
`Rkm[i]`	River kilometer of `i`^th visit
`SiteLength[i]`	Length of site of `i`^th visit
`Site[i]`	Site of `i`^th visit
`SurveyProportion[i]`	Proportion of site surveyed on `i`^th visit
`Swimmer[i]`	Snorkeler on `i`^th site visit
`Year[i]`	Year of `i`^th site visit
`bDensityRkm[i]`	`i`^th-order polynomial coefficients of effect of river kilometer on `bDensity`
`bDensitySite[i]`	Effect of `i`^th `Site` on `bDensity`
`bDensityYear[i]`	Effect of `i`^th `Year` on `bDensity`
`bDensity`	Intercept for `log(eDensity)`
`bEfficiencyIndex`	Effect of `Index` on `bEfficiency`
`bEfficiencyMarkingIndex`	Effect of `Marking` and `Index` on `bEfficiency`
`bEfficiencyMarking`	Effect of `Marking` on `bEfficiency`
`bEfficiencySwimmer[i]`	Effect of `i`^th `Swimmer` on `bEfficiency`
`bEfficiency`	Intercept of `logit(eEfficiency)`
`eAbundance[i]`	Expected abundance of fish at site of `i`^th visit
`eCount[i]`	Expected total number of fish at site of `i`^th visit
`eDensity[i]`	Expected lineal density of fish at site of `i`^th visit
`eEfficiency[i]`	Expected observer efficiency on `i`^th visit
`sDensitySite`	SD of `bDensitySite`
`sDispersion`	Overdispersion of `Count[i]`
`sEfficiencySwimmer`	SD of `bEfficiencySwimmer`

Age-1

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	-1.3743281	-1.8258968	-0.9303509	10.551708
bDensityRkm[1]	-0.1612247	-0.3034501	-0.0151078	4.823788
bDensityRkm[2]	0.5698486	0.3659929	0.7817977	10.551708
bDensityRkm[3]	-0.1297692	-0.2034507	-0.0631312	10.551708
bDensityRkm[4]	-0.2589458	-0.3258539	-0.1919248	10.551708
bEfficiency	-1.7790018	-2.0489025	-1.5103778	10.551708
bEfficiencyIndex	0.3945270	-0.2418596	1.0143952	2.270938
bEfficiencyMarking	0.5344403	0.3276782	0.7408086	10.551708
bEfficiencyMarkingIndex	0.8102509	0.1937737	1.4419331	6.464245
sDensitySite	0.6983107	0.6340298	0.7680758	10.551708
sDensityYear	0.7144574	0.4966376	1.0965765	10.551708
sDispersion	1.3162959	1.1995853	1.4580268	10.551708
sEfficiencySwimmer	0.3650904	0.2157020	0.6900803	10.551708

Table 3. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
4111	13	3	500	5	644	1.006	TRUE

Table 4. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.3763075	0.4307954	0.4110922	0.4528156	10.551708
mean	-0.2985711	-0.3666153	-0.3945797	-0.3370116	10.551708
variance	0.6005416	0.8397831	0.7980553	0.8790226	10.551708
skewness	0.2654207	0.5745270	0.5012478	0.6489050	10.551708
kurtosis	-0.4022754	-0.1511623	-0.3188455	0.0498346	7.381783

Table 5. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.007	1.004

Age-2

Table 6. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	-2.2703833	-2.7037196	-1.8402817	10.5517083
bDensityRkm[1]	-0.2507536	-0.4379654	-0.0550451	7.0922766
bDensityRkm[2]	0.0871299	-0.1737306	0.3337300	0.9573837
bDensityRkm[3]	0.0485288	-0.0448802	0.1357759	1.8001642
bDensityRkm[4]	-0.1006657	-0.1820998	-0.0170527	5.5975120
bEfficiency	-1.9405264	-2.2940156	-1.5696199	10.5517083
bEfficiencyIndex	0.5477041	-0.0646702	1.2117753	3.5630236
bEfficiencyMarking	0.8973363	0.6060298	1.1943134	10.5517083
bEfficiencyMarkingIndex	1.9251162	0.8944422	3.0229240	10.5517083
sDensitySite	0.8599432	0.7779852	0.9455527	10.5517083
sDensityYear	0.5162494	0.3607655	0.7999322	10.5517083
sDispersion	1.2125268	1.0491808	1.3894261	10.5517083
sEfficiencySwimmer	0.2981136	0.1732462	0.5238337	10.5517083

Table 7. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
4111	13	3	500	5	934	1.007	TRUE

Table 8. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.5876916	0.6108003	0.5897531	0.6311177	4.851268
mean	-0.3036252	-0.3340637	-0.3621187	-0.3080274	5.796821
variance	0.5537440	0.7006562	0.6585497	0.7430052	10.551708
skewness	0.7617150	0.9273396	0.8443825	1.0259777	10.551708
kurtosis	-0.0730003	0.4754988	0.2153444	0.7970010	10.551708

Table 9. Model sensitivity.

all	analysis	sensitivity	bound
all	1.007	1.005	1.004

Condition

Table 10. Parameter descriptions.

Parameter	Description
`Length[i]`	Fork length of `i`^th fish
`Weight[i]`	Recorded weight of `i`^th fish
`Year[i]`	Year `i`^th fish was captured
`bWeightLength`	Intercept of effect of `log(Length)` on `bWeight`
`bWeightYear[i]`	Effect of `i`^th `Year` on `bWeight`
`bWeight`	Intercept of `log(eWeight)`
`eWeight[i]`	Expected `Weight` of `i`^th fish
`sWeightYear`	Log standard deviation of `bWeightYear`
`sWeight`	Log standard deviation of residual variation in `log(Weight)`

Table 11. Model coefficients.

term	estimate	lower	upper	svalue
bWeight	-12.745405	-13.133834	-12.338893	10.55171
bWeightLength	3.213730	3.152203	3.272080	10.55171
sWeight	-1.926025	-1.966230	-1.885349	10.55171
sWeightYear	-1.991494	-2.264444	-1.672420	10.55171

Table 12. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1201	4	3	500	10	903	1.006	TRUE

Table 13. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0000771	0.0005379	-0.0568825	0.0549890	0.0213019
variance	0.9779704	1.0005631	0.9235082	1.0876789	0.8254901
skewness	-0.4265371	-0.0017194	-0.1288944	0.1332694	10.5517083
kurtosis	2.0182810	-0.0107774	-0.2525675	0.3011076	10.5517083

Table 14. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.007	1.005

Fecundity

Table 15. Parameter descriptions.

Parameter	Description
`Fecundity[i]`	Fecundity of `i`^th fish (eggs)
`Weight[i]`	Weight of `i`^th fish (mm)
`bFecundityWeight`	Effect of `log(Weight)` on `log(bFecundity)`
`bFecundity`	Intercept of `eFecundity`
`eFecundity[i]`	Expected `Fecundity` of `i`^th fish
`sFecundity`	SD of residual variation in `log(Fecundity)`

Table 16. Model coefficients.

term	estimate	lower	upper	svalue
bFecundity	3.8437989	1.5652072	4.9608686	10.55171
bFecundityWeight	0.8620380	0.8312411	0.9644906	10.55171
sFecundity	0.1283548	0.0946227	0.1842704	10.55171

Table 17. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
22	3	3	500	10	981	1.006	TRUE

Table 18. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0097217	0.0001140	-0.4089293	0.4168115	0.0508664
variance	0.9147370	0.9500014	0.4922016	1.7079577	0.1647680
skewness	-2.1716992	0.0107975	-0.9538329	0.9134493	10.5517083
kurtosis	6.6817124	-0.4269790	-1.2364339	1.9763747	10.5517083

Table 19. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1	1.005

Stock-Recruitment

Table 20. Parameter descriptions.

Parameter	Description
`Recruits[i]`	Number of recruits from `i`^th spawn year
`SDLogRecruits[i]`	Standard deviation of uncertainty in `log(Recruits[i])`
`Stock[i]`	Number of eggs in `i`^th spawn year
`a`	`Recruits` per `Stock` at low density
`b`	Density-dependence
`eRecruits[i]`	Expected number of recruits from `i`^th spawn year
`esRecruits[i]`	Expected SD of residual variation in `Recruits`
`sScaling`	Scaling term for SD of residual variation in `log(eRecruits)`

Age-1

Table 21. Estimated reference points (with 80% CRIs).

Metric	estimate	lower	upper
eggs	297000	115000.00000	593000.0000
spawners	99	38.33333	197.6667

Table 22. Model coefficients.

term	estimate	lower	upper	svalue
a	0.3573298	0.1800010	0.9239662	10.55171
b	0.0000034	0.0000012	0.0000121	10.55171
sScaling	2.6643747	1.8981738	4.0219737	10.55171

Table 23. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
18	3	3	500	100	1383	1.001	TRUE

Table 24. Estimated carry capacity (with 95% CRIs).

estimate	lower	upper	Carrying Capacity
106000	65400	165000	Maximum

Table 25. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0537591	-0.0066176	-0.4866942	0.4824889	0.2399599
variance	0.8565058	0.9476934	0.4403791	1.7753230	0.3705560
skewness	0.4543657	0.0037044	-0.9824065	1.0366028	1.4510459
kurtosis	-0.7661141	-0.4631965	-1.3116462	1.8005312	0.8254901

Table 26. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1	1

Table 27. Summary of the key stock-recruitment values and estimates for ‘observed’ and ‘no cull’ scenarios. The estimated number of recruits in the ‘no cull’ scenario assumes the same residual variation as in the estimate for the observed recruits.

SpawnYear	Spawners Observed	Spawners No Cull	Stock Observed	Stock No Cull	Recruits Observed	Recruits No Cull
2022	464	609	423045	555303	56272	62919
2023	331	772	329957	769891	27325	47845
2024	502	1228	571796	1399395	NA	NA

Age-1 to Age-2 Survival

Table 28. Parameter descriptions.

Parameter	Description
`Recruits[i]`	Number of age-2 juveniles from `i`^th spawn year
`Stock[i]`	Number of age-1 juveniles from `i`^th spawn year
`bSurvival`	`logit(eSurvival)`
`eSurvival[i]`	Expected annual survival for `i`^th spawn year
`sRecruits`	SD of residual variation in `Recruits`

Age-2

Table 29. Model coefficients.

term	estimate	lower	upper	svalue
bSurvival	-0.8334961	-1.1793013	-0.4337028	10.55171
sRecruits	0.4634771	0.3368493	0.7012649	10.55171

Table 30. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
16	2	3	500	1	646	1	TRUE

Table 31. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0224818	-0.0207777	-0.4987503	0.4750927	0.0096437
variance	0.9048126	0.9385264	0.4092355	1.8833568	0.1518963
skewness	-0.2015127	0.0378957	-0.9644045	0.9613019	0.7204010
kurtosis	-0.9277552	-0.5264601	-1.3537739	1.5303327	1.1572456

Table 32. Model sensitivity.

all	analysis	sensitivity	bound
all	1	1	1

Modelled In-lake Survival

Table 33. Parameter descriptions.

Parameter	Description
	`bPostCollapse`
	The effect of the collapse on `eYearSurv`.
`PostCollapse[i]`	The `i`^th value for the state (pre or post collapse) of Kootenay Lake.
`PropPre1[i]`	The `i`^th value of the assumed proportion of spawners in the 1st age class returning.
`PropPre2[i]`	The `i`^th value of the assumed proportion of spawners in the 2nd age class returning.
`PropPre3[i]`	The `i`^th value of the assumed proportion of spawners in the 3rd age class returning.
`Recruits[i]`	The `i`^th Recruits value
`SpawnersActive[i]`	The `i`^th SpawnersActive value
`SpawnersReturning[i]`	The `i`^th SpawnersReturning value
`a`	`Recruits` per `Spawner` at low density.
`bSurv`	The intercept for the effect of year on `eYearSurv[i]`.
`bYearSurv[i]`	The effect of year on `eYearSurv[i]`.
`b`	Density-dependence.
`eCohortSurv[i]`	The expected cohort survival `Recruits[i]`.
`eRecruits[i]`	Expected value of `Recruits[i]`
`eSpawnersReturning[i]`	Expected value of `SpawnersReturning[i]`
`eYearSurv`	Expected value of eYearSurv[i].
`sRecruits`	SD of residual variation in `eRecruits`
`sSpawnersReturning`	SD of residual variation in `eSpawnersReturning`
`sYearSurv`	SD of residual variation in `eYearSurv`.

Returning Spawners

All 5

Table 34. Model coefficients.

term	estimate	lower	upper	svalue
a	281.00000	208.00000	376.00000	10.600
b	0.00209	0.00101	0.00376	10.600
bPostCollapse	-0.11600	-0.42700	0.30100	0.957
bSurv	-0.83200	-0.97500	-0.68200	10.600
sRecruits	0.37400	0.30000	0.48600	10.600
sSpawnersReturning	0.34700	0.16600	0.56300	10.600
sYearSurv	0.42100	0.18400	0.67800	10.600

Table 35. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0585474	0.0101621	-0.6354573	0.6227919	0.1821109
variance	1.1320100	0.9263430	0.3160870	2.1790993	0.5644442
skewness	-0.4466541	0.0192519	-1.1142246	1.1491123	1.1788432
kurtosis	-0.1869997	-0.7026992	-1.5239490	1.4774435	1.0028864

All 6

Table 36. Model coefficients.

term	estimate	lower	upper	svalue
a	282.00000	2.10e+02	376.00000	10.6
b	0.00197	9.67e-04	0.00348	10.6
bPostCollapse	-0.12600	-4.34e-01	0.23500	1.2
bSurv	-0.46900	-6.03e-01	-0.33000	10.6
sRecruits	0.37200	3.00e-01	0.47800	10.6
sSpawnersReturning	0.38300	2.49e-01	0.54600	10.6
sYearSurv	0.39500	2.10e-01	0.65200	10.6

Table 37. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0666896	-0.0106073	-0.6496175	0.5831963	0.2651505
variance	1.8505189	0.9282416	0.2833162	2.0650615	3.3918369
skewness	-0.1194776	0.0210291	-1.1144232	1.1648746	0.3313299
kurtosis	-1.1441895	-0.7008306	-1.5504887	1.3433113	1.2825816

All 7

Table 38. Model coefficients.

term	estimate	lower	upper	svalue
a	286.00000	207.00000	379.00000	10.60
b	0.00207	0.00101	0.00348	10.60
bPostCollapse	-0.18000	-0.52100	0.20700	1.72
bSurv	-0.17100	-0.31700	-0.00142	4.32
sRecruits	0.36300	0.29200	0.45800	10.60
sSpawnersReturning	0.32300	0.16600	0.47500	10.60
sYearSurv	0.44800	0.25300	0.77300	10.60

Table 39. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0493682	0.0154623	-0.6062108	0.6355902	0.2129719
variance	1.2259044	0.9169064	0.2873991	2.1803832	0.8357463
skewness	-0.5246227	-0.0041322	-1.0976921	1.1662647	1.4405726
kurtosis	-0.8303292	-0.7116310	-1.5488373	1.2693171	0.2063030

All Equal

Table 40. Model coefficients.

term	estimate	lower	upper	svalue
a	281.00000	1.99e+02	376.00000	10.600
b	0.00211	9.03e-04	0.00378	10.600
bPostCollapse	-0.11000	-4.45e-01	0.32000	0.924
bSurv	-0.56800	-7.12e-01	-0.40800	10.600
sRecruits	0.39900	3.10e-01	0.52100	10.600
sSpawnersReturning	0.28000	1.55e-01	0.42600	10.600
sYearSurv	0.45500	2.51e-01	0.72500	10.600

Table 41. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0584431	0.0028843	-0.6527444	0.5904975	0.2559393
variance	1.1820976	0.9309631	0.3109260	2.1219486	0.6921735
skewness	-0.9465507	0.0106024	-1.1107880	1.1557899	3.4748927
kurtosis	-0.3122755	-0.7150041	-1.5350465	1.2903107	0.8085569

Pre 7 Post 6

Table 42. Model coefficients.

term	estimate	lower	upper	svalue
a	285.00000	2.08e+02	374.00000	10.60
b	0.00201	9.02e-04	0.00352	10.60
bPostCollapse	-0.39000	-6.75e-01	-0.08310	7.38
bSurv	-0.18600	-3.03e-01	-0.05120	6.64
sRecruits	0.36300	2.92e-01	0.46200	10.60
sSpawnersReturning	0.36200	2.51e-01	0.51300	10.60
sYearSurv	0.35500	1.88e-01	0.56800	10.60

Table 43. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0873372	-0.0018749	-0.6202202	0.6093356	0.3705560
variance	2.0115994	0.9294763	0.3055676	2.1138667	3.7572924
skewness	0.2553128	0.0021376	-1.1324240	1.1407068	0.6523513
kurtosis	-1.0365229	-0.7206386	-1.5176846	1.2930266	0.7851794

Pre 7 Post Half 6

Table 44. Model coefficients.

term	estimate	lower	upper	svalue
a	284.0000	2.09e+02	371.00000	10.60
b	0.0019	9.15e-04	0.00331	10.60
bPostCollapse	-0.3330	-6.32e-01	0.00389	4.18
bSurv	-0.1840	-3.12e-01	-0.03500	5.69
sRecruits	0.3620	2.90e-01	0.46600	10.60
sSpawnersReturning	0.2890	1.52e-01	0.43500	10.60
sYearSurv	0.4040	2.36e-01	0.64800	10.60

Table 45. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0756881	-0.0070924	-0.5563411	0.6390384	0.3508097
variance	1.5676145	0.9480466	0.3097392	2.1288103	2.1212557
skewness	-0.1323705	-0.0055186	-1.1255508	1.0668854	0.3168908
kurtosis	-1.0676600	-0.7293763	-1.5156546	1.2827296	0.9314884

Active Spawners Only

All 6

Table 46. Model coefficients.

term	estimate	lower	upper	svalue
a	282.00000	2.07e+02	370.00000	10.60
b	0.00192	8.86e-04	0.00337	10.60
bPostCollapse	-0.20000	-4.74e-01	0.10400	2.36
bSurv	-0.47000	-5.95e-01	-0.34300	10.60
sRecruits	0.36600	2.95e-01	0.46500	10.60
sSpawnersReturning	0.34000	1.94e-01	0.50200	10.60
sYearSurv	0.38100	2.05e-01	0.63300	10.60

Table 47. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0412512	0.0033332	-0.6560916	0.6067165	0.1712472
variance	1.5558683	0.9145400	0.3002060	2.0641426	2.0558532
skewness	-0.4580048	0.0281382	-1.1177285	1.1677583	1.4045033
kurtosis	-1.2149941	-0.7320847	-1.5349819	1.4565607	1.5772937

Figures

Length Correction

Figure 1. Measured length-frequency histogram by year.

Figure 2. Corrected length-frequency histogram by year and observation type.

Abundance

Age-1

Figure 3. Predicted abundance of age-1 Rainbow Trout in the Duncan and Lardeau Rivers by year (with 95% CRIs).

Figure 4. The Coefficient of Variation by year.

Figure 5. Predicted lineal density of age-1 Rainbow Trout in 2010 by river kilometre (with 95% CRIs).

Figure 6. Predicted observer efficiency for age-1 Rainbow Trout by visit type and study design (with 95% CRIs).

Age-2

Figure 7. Predicted abundance of age-2 Rainbow Trout in the Duncan and Lardeau Rivers by year (with 95% CRIs).

Figure 8. The Coefficient of Variation by year.

Figure 9. Predicted lineal density of age-2 Rainbow Trout in 2010 by river kilometre (with 95% CRIs).

Figure 10. Predicted observer efficiency for age-2 Rainbow Trout by visit type and study design (with 95% CRIs).

Condition

Figure 11. The percent change in the body condition for an average length fish relative to a typical year by year (with 95% CRIs).

Fecundity

Figure 12. The fecundity-weight relationship (with 95% CRIs).

Spawner Size

Figure 13. The mean length of spawning Rainbow Trout by the mean weight of Rainbow Trout in the KLRT.

Figure 14. The mean weight of Adult Rainbow Trout in the KLRT by year.

Egg Deposition

Figure 15. The spawner abundance by year.

Figure 16. The estimated spawner fecundity by year.

Figure 17. The egg deposition by year.

Stock-Recruitment

Age-1

Figure 18. Predicted stock-recruitment relationship between spawners and age-1 recruits (with 95% CRIs). The estimated number of recruits if no spawners were culled is also shown in red, assuming the same residual variation from the expected value as with the recruits actually observed in each year.

Figure 19. Predicted egg survival to age-1 by egg deposition (with 95% CRIs). The predicted egg survival if no spawners were culled is also shown in red, assuming the same residual variation from the expected value as with the number of recruits actually observed in each year.

Figure 20. Predicted percent of age-1 recruits carry capacity by egg deposition (with 80% CRIs).

Age-1 to Age-2 Survival

Age-2

Figure 21. Estimated age-1 and age-2 survival by spawn year.

Figure 22. Predicted relationship between age-1 and age-2 abundance (with 95% CRIs).

Calculated In-lake Survival

Figure 23. The number of spawners by return year.

Figure 24. The estimated egg deposition by return year.

Figure 25. The number of expected age-1 recruits by spawn year based on the estimated egg deposition.

Figure 26. Apparent survival from expected age-1 (with a moving average of 3 years) to spawners by return year assuming all individuals return at age-6.

Expected Spawners

Figure 27. Expected spawners by return year and in river and in lake variation based on age-1 recruitment.

Figure 28. Expected spawners by return year and in river and in lake variation based on age-2 recruitment.

Modelled In-lake Survival

Stock-Recruitment

Figure 29. Predicted stock-recruitment relationship between spawners and age-1 recruits (with 95% CRIs).

Returning Spawners

All 5

Figure 30. Estimated survival in each year given the assumed proportions of age classes returning.

Figure 31. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes.

All 6

Figure 32. Estimated survival in each year given the assumed proportions of age classes returning.

Figure 33. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes.

All 7

Figure 34. Estimated survival in each year given the assumed proportions of age classes returning.

Figure 35. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes.

All Equal

Figure 36. Estimated survival in each year given the assumed proportions of age classes returning.

Figure 37. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes.

Pre 7 Post 6

Figure 38. Estimated survival in each year given the assumed proportions of age classes returning.

Figure 39. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes.

Pre 7 Post Half 6

Figure 40. Estimated survival in each year given the assumed proportions of age classes returning.

Figure 41. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes.

Active Spawners Only

All 6

Figure 42. Estimated survival in each year given the assumed proportions of age classes returning and considering active spawners only.

Figure 43. Estimated cohort survival by the spawn year each cohort originated from given the assumed proportions of spawner age classes and considering active spawners only.

Reproductive Rate (age-1)

Figure 44. Survival from age-1 to spawning by return year and by the type of recruits used to calculate survival.

Reproductive Rate (age-2)

Figure 45. Survival from age-2 to spawning by return year.

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

Habitat Conservation Trust Foundation (HCTF) and the anglers, hunters, trappers and guides who contribute to the Trust.
Fish and Wildlife Compensation Program (FWCP) and its program partners BC Hydro, the Province of BC and Fisheries and Oceans Canada.
Ministry of Forests, Lands and Natural Resource Operations (MFLNRO)
- Greg Andrusak
- Matt Neufeld
- Jeff Burrows
- Tyler Weir
- Rob Bison
Poisson Consulting
- Evan Amies-Galonski
- Seb Dalgarno
- Robyn Irvine
Stefan Himmer
Vicky Lipinski
John Hagen
Scott Decker
Jody Schick
Gillian Sanders
Jeremy Baxter
Jeff Berdusco
Dave Derosa

References

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Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, et al. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.

Castilho, Leonardo Braga, and Paulo Inácio Prado. 2021. “Towards a Pragmatic Use of Statistics in Ecology.” PeerJ 9 (September): e12090. https://doi.org/10.7717/peerj.12090.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

He, Ji X., James R. Bence, James E. Johnson, David F. Clapp, and Mark P. Ebener. 2008. “Modeling Variation in Mass-Length Relations and Condition Indices of Lake Trout and Chinook Salmon in Lake Huron: A Hierarchical Bayesian Approach.” Transactions of the American Fisheries Society 137 (3): 801–17. https://doi.org/10.1577/T07-012.1.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

Lin, J. 1991. “Divergence Measures Based on the Shannon Entropy.” IEEE Transactions on Information Theory 37 (1): 145–51. https://doi.org/10.1109/18.61115.

Mace, Pamela M. 1994. “Relationships Between Common Biological Reference Points Used as Thresholds and Targets of Fisheries Management Strategies.” Canadian Journal of Fisheries and Aquatic Sciences 51 (1): 110–22. https://doi.org/10.1139/f94-013.

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Plummer, Martyn. 2003. “JAGS: A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling.” In Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), edited by Kurt Hornik, Friedrich Leisch, and Achim Zeileis. Vienna, Austria.

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Kaslo Bull Trout Productivity 2023

Thu, 11 Jul 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Amies-Galonski, E., Andrusak, G.F., Stephens, C.R.A., and Thorley, J.L. (2024) Kaslo Bull Trout Productivity 2023. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1402985678.

Background

The Kaslo River and Keen Creek, which is a tributary of the Kaslo River, are important Bull Trout spawning and rearing tributaries on Kootenay Lake. From 2012 to 2023, field crews have night-snorkeled these systems in the fall and recorded all Bull Trout less than 350 mm in length. Keen Creek has not been snorkelled since 2019 due to visibility issues. Snorkel and electrofishing marking crews have also captured and tagged juvenile Bull Trout for the snorkel crews to resight. Redd counts have been conducted in both systems since 2006, with the exception of 2020, when Keen Creek was not surveyed. The primary goal of the current analyses is to answer the following questions:

What is the observer efficiency when night-snorkeling for juvenile Bull Trout in the Kaslo River and Keen Creek?
What are the numbers of age-1 Bull Trout in the Kaslo River and Keen Creek?
What is the relationship between the stock (number of redds and eggs) and the resultant numbers of age-1 Bull Trout two years later?

Data Preparation

The data were cleaned, tidied and databased using R version 4.4.1 (R Core Team 2024).

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2003). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

The parameters are summarized in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). The coefficient of variation is also shown for juvenile abundance to understand sampling efforts. The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.3 bits, which is equivalent to a significant p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row.

Variable selection was based on the heuristic of directional certainty (Kery and Schaub 2011). Fixed effects were included if their s-value was $>$ 4.32 bits (Kery and Schaub 2011). Based on a similar argument, random effects were included if their standard deviation had lower 95% CLs $>$ 5% of the median estimate.

The results are displayed graphically by plotting the modeled relationships between individual variables and the response with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their average values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% confidence/credible intervals (CIs, Bradford et al. 2005).

The analyses were implemented using R version 4.4.1 (R Core Team 2024) and the mbr family of packages.

Model Descriptions

Length Correction

The annual bias (inaccuracy) and error (imprecision) in observer’s fish length estimates when spotlighting (standing) and snorkeling were quantified from the divergence of their length distribution from the length distribution for JLT and SH (the two most experience snorkelers) in that year. More specifically, the length correction that minimized the Jensen-Shannon divergence (Lin 1991) provided a measure of the inaccuracy while the minimum divergence (the Jensen-Shannon divergence was calculated with log to base 2 which means it lies between 0 and 1) provided a measure of the imprecision.

Observer Efficiency

All resighted fish with a tag were allocated to the closest unallocated marked fish (with the same colour tag) by fork length and distance. The marked fish were analysed using a Bayesian logistic regression model. The key assumption of the logistic regression model is that:

The observer efficiency varies by the fork length.

The preliminary analysis for observer efficiency indicated that system, observer, gradient, sinuosity, and river kilometre were not informative predictors.

Lineal Density

Both systems were broken into 100 m sites by bank. The lineal density at each site was estimated using an over-dispersed Poisson Generalized Linear Mixed Model.

Key assumptions of the Bayesian GLMM include:

The lineal density varies by system and stream distance.
The lineal density varies randomly by site and system within year.
The observer efficiency for each system is as estimated by the observer efficiency model.

The preliminary analysis for density indicated that site sinuosity and gradient were not informative predictors of lineal density.

Condition

The condition of adults was estimated using an allometric weight-length relationship. Key assumptions of the condition model include:

Weight varies by length.
Weight varies randomly by year.
The residual variation in weight is log normally distributed.

Fecundity

Female spawner fecundity was estimated using an allometric egg-weight relationship. The key assumptions of the fecundity model include:

Fecundity varies with weight.
The residual variation in fecundity is log normally distributed.

Spawner Length

The average length of spawners in each system and year was estimated using a linear regression. Key assumptions of the spawner length model include:

Length varies randomly with system, year, and system within year.
Length varies with sex and catch type.
The residual variation is normally distributed.

Egg Deposition

The total egg deposition in each year was estimated by

Converting the average spawner length to the average weight using the condition relationship for a typical year.
Adjusting the average weight by the annual condition effect (interpolating where unavailable)
Converting the average weight to the average fecundity using the fecundity relationship
Multiplying the average fecundity by the number of females (assuming 1 female per redd)

Eggs Stock-Recruitment

The stock-recruitment relationship between the total number eggs and the abundance of age-1 individuals was estimated using a Bayesian Beverton-Holt stock-recruitment curve. Key assumptions of the final BH SR model include:

The prior uncertainty in the egg to age-1 survival is described by a Beta distribution with an alpha of 4 and beta of 49 which has a mean of 0.075 and a standard deviation of 0.036. This is based off the assumption that a recruit has a ~50% chance of surviving through each summer or winter and must pass through two winters and two summers to survive to age-1.
The carrying capacity varies between systems.
The residual variation in the recruits is log normally distributed.

The $E_{K/2}$ Limit Reference Point (Mace 1994) ($E_{0.5 R_{max}}$) was calculated, corresponding to the stock (number of eggs per 100 meters) that produce 50% of the maximum recruitment ($K$).

Model Templates

Observer Efficiency

.model{
  bIntercept ~ dnorm(0, 2^-2)
  bLength ~ dnorm(0, 2^-2)

  for(i in 1:length(Observed)){
    logit(eObserved[i]) <-  bIntercept + bLength * Length[i]
    Observed[i] ~ dbern(eObserved[i])
  }

Block 1. Final model.

Lineal Density

.model{
  bEfficiency ~ dnorm(Efficiency[1], EfficiencySD[1]^-2) T(0, 1)

  b0 ~ dnorm(0, 5^-2)
  bRkm ~ dnorm(0, 2^-2)
  bSystem[1] <- 0
  for(i in 2:nSystem) {
    bSystem[i] ~ dnorm(0, 2^2)
  }

  sSystemYear ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:nSystem) {
    for(j in 1:nYear) {
      bSystemYear[i,j] ~ dnorm(0, sSystemYear^-2)
    }
  }

  sSite ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:nSite) {
    bSite[i] ~ dnorm(0, sSite^-2)
  }

  sDispersion ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:length(Count)) {
    log(eDensity[i]) <- b0 + bSystem[System[i]] + bRkm * Rkm[i] + bSite[Site[i]] + bSystemYear[System[i],Year[i]]
    eDispersion[i] ~ dgamma(sDispersion^-2, sDispersion^-2)
    Count[i] ~ dpois(eDensity[i] * Length[i] * Coverage[i] * bEfficiency * eDispersion[i])
  }

Block 2. The final model.

Condition

.model{
  b0 ~ dnorm(6, 2^-2)
  bLength ~ dnorm(3, 1^-2)
  sWeight ~ dnorm(0, 1^-2) T(0,)
  sYear ~ dnorm(0, 1^-2) T(0,)

  for (i in 1:nYear) {
    bYear[i] ~ dnorm(0, sYear^-2)
  }

  for (i in 1:nObs) {
    log(eWeight[i]) <- b0 + (log(Length[i]) - log(500)) * bLength + bYear[Year[i]]
    Weight[i] ~ dlnorm(log(eWeight[i]), sWeight^-2)
  }

Block 3. Model description.

Fecundity

.model{
  b0 ~ dnorm(0, 2^-2)
  bWeight ~ dnorm(1, 2^-2)
  sFecundity ~ dnorm(0, 2^-2) T(0,)

  for (i in 1:nObs) {
    log(eFecundity[i]) <- b0 + (log(Weight[i]) - log(2300)) * bWeight
    Fecundity[i] ~ dlnorm(log(eFecundity[i]), sFecundity^-2)
  }

Block 4. Model description.

Spawner Length

.model{
  bLength ~ dnorm(650, 100^-2)
  sLength ~ dnorm(0, 100^-2) T(0,)
  sYear ~ dnorm(0, 100^-2) T(0,)
  sSystem ~ dnorm(0, 100^-2) T(0,)
  sYearSystem ~ dnorm(0, 100^-2) T(0,)
  bSex ~ dbeta(1, 1)
  bFemale ~ dnorm(0, 100^-2)
  bBias ~ dnorm(0, 100^-2)

  for (i in 1:nYear) {
    bYear[i] ~ dnorm(0, sYear^-2)
  }

  for (i in 1:nSystem) {
    bSystem[i] ~ dnorm(0, sSystem^-2)
  }

  for (i in 1:nYear) {
    for (j in 1:nSystem) {
        bYearSystem[i, j] ~ dnorm(0, sYearSystem^-2)
    }
  }

  bCatchType[1] <- 0
  for(i in 2:nCatchType) {
    bCatchType[i] ~ dnorm(0, 100^-2)
  }

  for (i in 1:nObs) {
    Female[i] ~ dbern(bSex)
    eLength[i] <- bLength + bSystem[System[i]] + bFemale * Female[i] + bCatchType[CatchType[i]] + bYear[Year[i]] + bYearSystem[Year[i], System[i]] + bBias * Bias[i]
    Length[i] ~ dnorm(eLength[i], sLength^-2)
  }

Block 5. Model description.

Stock-Recruiment

.model {
  bAlpha ~ dbeta(4, 49)

  bK ~ dnorm(3, 1^-2)
  bKPopulation ~ dnorm(0, 1^-2)
  sRecruits ~ dexp(1)

  for(i in 1:nObs){
    log(eK[i]) <- bK + (Kaslo[i] * bKPopulation - (1-Kaslo[i]) * bKPopulation)
    eBeta[i] <-  bAlpha/eK[i]
    eRecruits[i] <- bAlpha * Stock[i] / (1 + eBeta[i] * Stock[i])
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

Block 6. Final model.

Results

Tables

Coverage

Table 1. Total length of river bank counted (including replicates) by system and year.

System	Year	Length (km)
Kaslo River	2012	10.57 [km]
Kaslo River	2013	13.43 [km]
Kaslo River	2014	11.45 [km]
Kaslo River	2015	7.32 [km]
Kaslo River	2016	11.59 [km]
Kaslo River	2017	9.79 [km]
Kaslo River	2018	8.44 [km]
Kaslo River	2019	7.19 [km]
Kaslo River	2020	10.42 [km]
Kaslo River	2021	9.56 [km]
Kaslo River	2022	8.05 [km]
Kaslo River	2023	8.11 [km]
Keen Creek	2012	1.44 [km]
Keen Creek	2013	0.95 [km]
Keen Creek	2014	0.67 [km]
Keen Creek	2015	0.72 [km]
Keen Creek	2016	0.85 [km]
Keen Creek	2017	1.68 [km]
Keen Creek	2018	3.37 [km]
Keen Creek	2019	1.48 [km]

Observer Efficiency

Table 2. Parameter descriptions.

Parameter	Description
`Length`	The standardized fork length
`Observed`	The number of individuals observed (`0` or `1`)
`Tagged`	The number of tagged individuals (`1`)
`bIntercept`	The intercept for `logit(eObserved)`
`bLength2`	The effect of `Length` on the effect of `Length` on `bIntercept`
`bLength`	The effect of `Length` on `bIntercept`
`eObserved`	The expected probability of observing an individual

Table 3. Final parameter estimates.

term	estimate	lower	upper	svalue
bIntercept	-1.5593445	-1.897939	-1.2421253	10.55171
bLength	0.6079224	0.280856	0.9477344	10.55171

Table 4. Observer Efficiency estimates for a 123 mm Bull Trout.

System	Efficiency	EfficiencyLower	EfficiencyUpper	EfficiencySD
Kaslo River	0.142078	0.0987725	0.1929706	0.0240301

Table 5. Final model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
268	2	3	500	1	786	1.002	TRUE

Table 6. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.8097015	0.8059701	0.7425373	0.8694030	0.0588536
mean	-0.1679152	-0.1683389	-0.2759600	-0.0498453	0.0096437
variance	0.8968665	0.8908402	0.6664427	1.0912734	0.0291267
skewness	1.5306960	1.5222512	1.0894278	2.1194824	0.0310896
kurtosis	0.5983035	0.6100963	-0.6093740	2.8574118	0.0174054

Table 7. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.003	1.002

Lineal Density

Table 8. Estimated abundance (fish) of age-1 Bull Trout with lower and upper 95% credible limits and coefficient of variation by year and river.

Year	System	estimate	lower	upper	cv
2012	Kaslo River	5107.869	3545.7143	8173.021	0.2176262
2012	Keen Creek	2173.276	1306.0407	3987.646	0.2915600
2013	Kaslo River	4216.892	2903.2104	6608.208	0.2123758
2013	Keen Creek	1648.513	911.1797	3272.413	0.3318413
2014	Kaslo River	3721.962	2591.4833	5913.252	0.2150330
2014	Keen Creek	2206.908	1157.3256	4503.746	0.3556031
2015	Kaslo River	6374.147	4320.4885	9995.018	0.2157109
2015	Keen Creek	3371.943	1844.0110	6965.817	0.3537431
2016	Kaslo River	4134.967	2865.8228	6609.196	0.2170935
2016	Keen Creek	937.128	428.2800	1922.398	0.3722731
2017	Kaslo River	8001.216	5538.6897	12621.272	0.2140733
2017	Keen Creek	2028.217	1233.1551	3677.365	0.2856634
2018	Kaslo River	5810.582	3997.3898	9078.355	0.2117757
2018	Keen Creek	1512.254	961.5201	2590.865	0.2569295
2019	Kaslo River	5154.482	3555.1868	8309.843	0.2204419
2019	Keen Creek	1475.379	821.9809	2684.820	0.3007438
2020	Kaslo River	5770.862	4103.0407	9155.816	0.2110122
2021	Kaslo River	2776.786	1811.7895	4536.370	0.2367343
2022	Kaslo River	2250.079	1442.8266	3708.818	0.2429176
2023	Kaslo River	6866.100	4769.7541	11006.612	0.2192135

Table 9. Parameter descriptions.

Parameter	Description
`Count`	Number of fish counted
`Coverage`	Proportion of site surveyed
`Dispersion`	Factor for random effect of overdispersion
`Efficiency`	The observer efficiency from the observer efficiency model
`Length`	Length of site (m)
`Site`	The site
`System`	The system
`Year`	The year
`b0`	Intercept of log(eDensity)
`bDispersion`	The random effect of overdispersion
`bEfficiency`
`bSite`	The random effect of `Site` on `bSystemYear`
`bSystemYear`	The effect of `System` and `Year` on `log(eDensity)`
`bSystem`	The effect of `System` on `log(eDensity)`
`eCount`	The expected `Count`
`eDensity`	The expected lineal density
`log_sDispersion`	`log(sDispersion)`
`log_sSite`	`log(sSite)`
`sDispersion`	The SD of `bDispersion`
`sSite`	The SD of `bSite`
`sSystemYear`	The SD of `bSystemYear`

Table 10. Parameter estimates.

term	estimate	lower	upper	svalue
b0	-2.5429964	-2.9606769	-2.0320938	10.551708
bEfficiency	0.1401128	0.0897090	0.1885600	10.551708
bRkm	0.4561438	0.3645865	0.5499002	10.551708
bSystem[1]	0.0000000	0.0000000	0.0000000	0.000000
bSystem[2]	1.0624896	0.5476039	1.5231357	8.966746
sDispersion	0.5331838	0.4453158	0.6258223	10.551708
sSystemYear	0.4413003	0.2834421	0.7050285	10.551708

Table 11. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1388	6	3	500	50	292	1.008	TRUE

Table 12. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.4149856	0.4445245	0.4149856	0.4751621	4.229780
mean	-0.2603045	-0.2945708	-0.3468069	-0.2435131	2.234296
variance	0.7463757	0.9183465	0.8557615	0.9895602	10.551708
skewness	0.3036967	0.5251026	0.4192433	0.6476973	10.551708
kurtosis	-0.7289565	-0.3985195	-0.6267938	-0.0560726	10.551708

Table 13. Model sensitivity.

all	analysis	sensitivity	bound
all	1.008	1.009	1.01

Condition

Table 14. Parameter descriptions.

Parameter	Description
`Length[i]`	The `i`^th Length value
`Weight[i]`	The `i`^th Weight value
`Year[i]`	The `i`^th Year value
`b0`	Intercept for `eLength`
`bLength`	The effect of Length on `eWeight`
`bYear[i]`	The effect of year on `eWeight`
`eWeight[i]`	Expected value of `Weight[i]`
`sWeight`	SD of residual variation in `Weight`
`sYear`	SD of Year

Table 15. Model coefficients.

term	estimate	lower	upper	svalue
b0	7.0887436	6.9931843	7.1787403	10.55171
bLength	2.9170071	2.8631538	2.9746733	10.55171
sWeight	0.1288644	0.1227500	0.1355313	10.55171
sYear	0.1059902	0.0623674	0.2286522	10.55171

Table 16. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
795	4	3	500	50	206	1.002	TRUE

Table 17. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0021031	0.0006858	-0.0708325	0.0726021	0.0994670
variance	0.9863351	0.9987187	0.9014857	1.0985697	0.3410369
skewness	-0.4416248	0.0042388	-0.1689356	0.1608926	10.5517083
kurtosis	1.7325035	-0.0227240	-0.3036803	0.3510825	10.5517083

Table 18. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.016	1.008

Fecundity

Table 19. Parameter descriptions.

Parameter	Description
`Fecundity[i]`	The `i`^th Fecundity value
`b0`	Intercept for `eFecundity`
`bWeight`	Effect of `Weight` on `b0`
`eFecundity[i]`	Expected value of `Fecundity[i]`
`sFecundity`	SD of residual variation in `Fecundity`

Table 20. Model coefficients.

term	estimate	lower	upper	svalue
b0	8.4863227	8.4411741	8.5291575	10.55171
bWeight	1.0349493	0.9126504	1.1579293	10.55171
sFecundity	0.1217978	0.0938748	0.1646431	10.55171

Table 21. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
28	3	3	500	1	738	1.004	TRUE

Table 22. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0149816	-0.0056209	-0.3676026	0.3941117	0.1391384
variance	0.9063396	0.9712021	0.5350564	1.5870858	0.3434739
skewness	0.3363233	0.0105872	-0.8998687	0.8876153	1.3544916
kurtosis	-0.5690242	-0.3638803	-1.1306175	1.7538485	0.5224210

Table 23. Model sensitivity.

all	analysis	sensitivity	bound
all	1.004	1.009	1.006

Spawner Length

Table 24. Parameter descriptions.

Parameter	Description
`Bias[i]`	The `i`^th Bias Value
`CatchType[i]`	The `i`^th CatchType value
`Length[i]`	The `i`^th Length value
`Sex[i]`	The `i`^th Sex value
`System[i]`	The `i`^th System value
`Year[i]`	The `i`^th Year value
`bBias`	The effect of Bias on `bLength`
`bCatchType`	Effect of `CatchType` on `bLength`
`bLength`	Intercept for `eLength`
`bSex`	Effect of `Sex` on `bLength`
`bSystem`	Effect of `System` on `bLength`
`bYearSystem`	The effect of `Year` and `System` on `bLength`
`bYear`	Effect of `Year` on `bLength`
`eLength[i]`	Expected value of `Length[i]`
`sLength`	SD of residual variation in `Length`
`sYearSystem`	SD of `bYearSystem`

Table 25. Model coefficients.

term	estimate	lower	upper	svalue
bBias	84.545793	29.906526	133.058571	7.744353
bCatchType[2]	-33.647301	-56.126633	-10.011362	7.744353
bFemale	-82.867408	-91.951305	-73.276573	10.551708
bLength	644.506086	591.137885	694.249642	10.551708
bSex	0.369632	0.345423	0.392889	10.551708
sLength	83.507098	80.483588	86.770467	10.551708
sSystem	60.136039	37.585775	94.831880	10.551708
sYear	54.069704	33.003196	85.434732	10.551708
sYearSystem	18.164616	2.243049	43.196284	10.551708

Table 26. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
1434	9	3	500	50	354	1.009	TRUE

Table 27. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0075484	0.0010037	-0.0528173	0.0541317	0.3289134
variance	0.9504014	0.9999625	0.9277660	1.0771834	2.4172819
skewness	-0.3036651	-0.0012658	-0.1300370	0.1273369	10.5517083
kurtosis	2.5317646	-0.0157129	-0.2323216	0.2729959	10.5517083

Table 28. Model sensitivity.

all	analysis	sensitivity	bound
all	1.009	1.006	1.007

Egg Deposition

Table 29. The estimated total egg deposition by system and year.

Year	System	Length	Condition	Weight	Fecundity	Redds	Eggs
2012	Kaslo River	627.1446	1.1067112	2568.647	5434.889	431	2342437.26
2012	Keen Creek	663.2136	1.1067112	3022.792	6430.141	80	514411.28
2013	Kaslo River	672.0069	1.0832110	3074.479	6545.393	305	1996344.73
2013	Keen Creek	722.1255	1.0832110	3793.250	8132.355	50	406617.77
2014	Kaslo River	579.3941	1.0597109	1952.104	4092.341	113	462434.52
2014	Keen Creek	626.3260	1.0597109	2450.204	5176.398	16	82822.36
2015	Kaslo River	547.1266	1.0362108	1614.142	3361.202	136	457123.45
2015	Keen Creek	563.9368	1.0362108	1763.984	3686.532	80	294922.57
2016	Kaslo River	551.7009	1.0172306	1623.679	3381.519	340	1149716.57
2016	Keen Creek	589.9201	1.0172306	1974.635	4140.692	34	140783.54
2017	Kaslo River	562.3425	1.0176310	1718.155	3585.721	360	1290859.39
2017	Keen Creek	595.1184	1.0176310	2026.420	4252.501	113	480532.58
2018	Kaslo River	534.8907	0.9612682	1402.167	2904.966	267	775625.89
2018	Keen Creek	576.6260	0.9612682	1746.061	3647.457	51	186020.30
2019	Kaslo River	569.5035	0.8516374	1491.965	3098.048	131	405844.25
2019	Keen Creek	607.8237	0.8516374	1804.146	3773.151	33	124514.00
2020	Kaslo River	541.8591	0.9358603	1417.422	2937.192	110	323091.11
2020	Keen Creek	581.3008	0.9358603	1740.433	3634.836	NA	NA
2021	Kaslo River	492.0802	0.9358603	1070.478	2195.629	180	395213.26
2021	Keen Creek	536.0975	0.9358603	1374.042	2844.426	23	65421.81
2022	Kaslo River	509.0207	0.9358603	1181.474	2432.177	290	705331.44
2022	Keen Creek	551.6551	0.9358603	1493.430	3101.200	44	136452.80
2023	Kaslo River	523.9773	0.9358603	1285.488	2653.859	148	392771.10
2023	Keen Creek	566.2185	0.9358603	1612.006	3356.887	10	33568.87

Stock-Recruiment

Table 30. Parameter descriptions.

Parameter	Description
`Recruits`	Age-1 Bull Trout density
`Stock`	Bull Trout egg density
`bK`	Density Carrying Capacity
`eBeta[i]`	Density-dependence for the `i`^th population
`eRecruits`	Expected density of `Recruits`
`bKPopulation`	Population effect on `bK`
`bAlpha`	Recruits per stock per 100 meters at low stock density
`sRecruits`	SD of residual variation in `Recruits`

Table 31. Density Carrying Capacity (K).

System	estimate	upper	lower	svalue
Kaslo River	19.90592	29.60411	14.27755	10.55171
Keen Creek	33.71936	58.80922	21.98614	10.55171

Table 32. Model coefficients.

term	estimate	lower	upper	svalue
bAlpha	0.0497795	0.0185864	0.1317834	10.551708
bK	3.2557089	2.9732293	3.6700660	10.551708
bKPopulation	-0.2644685	-0.5164776	-0.0285350	4.936998
sRecruits	0.3894936	0.2745541	0.6124351	10.551708

Table 33. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
16	4	3	500	100	1472	1.001	TRUE

Table 34. Ek/2 Limit Reference Point estimates.

Kaslo	Recruits	Stock	estimate	lower	upper	svalue
TRUE	1	1	396.6708	130.1094	1473.079	10.55171
FALSE	1	1	682.2082	212.5405	2524.130	10.55171

Table 35. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0093564	-0.0026889	-0.5033992	0.4800748	0.0232542
variance	0.8317671	0.9586407	0.4161975	1.8426092	0.4629200
skewness	0.0648623	0.0119448	-0.9838751	0.9895700	0.1412569
kurtosis	-0.8849219	-0.4945978	-1.3248555	1.6770664	1.0301078

Table 36. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.002	1

Figures

Systems

Figure 1. Spatial distribution of fish-bearing channel.

Figure 2. Channel elevation by river kilometre and system.

Figure 3. Catchment area by river kilometre and system.

Coverage

Figure 4. Snorkel count coverage by year and bank.

Sites

Figure 5. Site gradient by river kilometre and system.

Figure 6. Site sinuosity by river kilometre and system.

Length Correction

Figure 7. Corrected length-frequency histogram by year and observation type.

Fish

Figure 8. Length-frequency plot of marked Bull Trout by year and system, coloured by tag colour.

Figure 9. Corrected length-frequency plot of observed Bull Trout by year and age class.

Observer Efficiency

Figure 10. Distribution of marked juvenile Bull Trout by year and tag color.

Figure 11. Estimated observer efficiency by fork length and system (with 95% CIs).

Lineal Density

Figure 12. Percent coverage by year and system.

Figure 13. Estimated density by year and system (with 95% CIs).

Figure 14. The estimated abundance by year and system (with 95% CIs).

Figure 15. The estimated density by site and system (with 95% CIs).

Figure 16. The Coefficient of Variation by system and year.

Redds

Figure 17. Complete redds by system and spawn year.

Condition

Figure 18. The weight-length relationship (with 95% CIs).

Figure 19. The percent change in the body condition for an average length fish relative to a typical year by year (with 95% CIs).

Fecundity

Figure 20. The predicted relationship between Fecundity and Weight for Bull Trout (with 95% CIs), data from Brunson (1952).

Spawner Length

Figure 21. Length frequency of Bull Trout spawners by sex.

Figure 22. Length frequency of Bull Trout spawners by catch type in 2018 and 2019.

Figure 23. Average Length of Bull Trout spawners in Keen Creek compared to all other surveyed tributaries.

Figure 24. The expected length of female spawners by year for Kaslo River and Keen Creek (with 95% CIs).

Egg Deposition

Figure 25. The estimated spawner fecundity by year uncorrected for condition (with 95% CIs).

Figure 26. The estimated spawner fecundity by year corrected for condition.

Figure 27. The estimated total egg deposition by system and year.

Stock-Recruiment

Figure 28. Estimated stock-recruitment relationship (with 95% CIs). Additional modeled relationships (grey lines) derived from randomly sampled parameter values are also displayed. The points are labelled by spawn year.

Figure 29. Predicted egg survival to age-1 by egg deposition (with 95% CRIs).

Conclusions

Observer efficiency is approximately 14% for age-1 Bull Trout.
Age-1 Bull Trout are relatively evenly distributed with respect to mesohabitat.
Lineal densities of age-1 Bull Trout increase with river kilometre in both systems.
The age-1 carrying capacity is estimated to be 199 fish per km in Kaslo River and 337 fish per km in Keen Creek.

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

Habitat Conservation Trust Foundation
Ministry of Forest, Lands and Natural Resource Operations
- Greg Andrusak
- Emmanuel Abecia
Ministry of Environment
- Jen Sarchuk
BC Fish and Wildlife
- Trina Radford
Stephan Himmer
Gillian Sanders
Jeff Berdusco
Vicky Lipinski
Jimmy Robbins
Jason Bowers
Seb Dalgarno

References

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Taylor & Francis.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

Lin, J. 1991. “Divergence Measures Based on the Shannon Entropy.” IEEE Transactions on Information Theory 37 (1): 145–51. https://doi.org/10.1109/18.61115.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Taylor; Francis, CRC Press.

R Core Team. 2024. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

Lower Columbia River Fish Population Indexing 2023

Fri, 28 Jun 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Thorley, J.L. (2024) Lower Columbia River Fish Population Indexing 2023. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/25988200.

Background

In the mid 1990s BC Hydro began operating Hugh L. Keenleyside (HLK) Dam to reduce dewatering of Mountain Whitefish and Rainbow Trout eggs.

The primary goal of the Lower Columbia River Fish Population Indexing program is to answer two key management questions:

What are the abundance, growth rate, survival rate, body condition, age distribution, and spatial distribution of subadult and adult Whitefish, Rainbow Trout, and Walleye in the Lower Columbia River?

What is the effect of inter-annual variability in the Whitefish and Rainbow Trout flow regimes on the abundance, growth rate, survival rate, body condition, and spatial distribution of subadult and adult Whitefish, Rainbow Trout, and Walleye in the Lower Columbia River?

The inter-annual variability in the Whitefish and Rainbow Trout flow regimes was quantified in terms of the percent egg dewatering as greater flow variability is associated with more egg stranding.

Methods

Data Preparation

The fish indexing data were provided by Okanagan Nation Alliance and Golder Associates in the form of an Access database. The discharge and temperature data were obtained from the Columbia Basin Hydrological Database maintained by Poisson Consulting. The Rainbow Trout egg dewatering estimates were provided by CLBMON-46 (Irvine et al. 2015) and the Mountain Whitefish egg stranding estimates by Golder Associates (2013).

The data were prepared for analysis using R version 4.4.1 (R Core Team 2018).

Discharge

Missing hourly discharge values for Hugh-Keenleyside Dam (HLK), Brilliant Dam (BRD) and Birchbank (BIR) were estimated by first leading the BIR values by 2 hours to account for the lag. Values missing at just one of the dams were then estimated assuming $HLK + BRD = BIR$. Negative values were set to be zero. Next, missing values spanning $\leq$ 28 days were estimated at HLK and BRD based on linear interpolation. Finally any remaining missing values at BIR were set to be $HLK + BRD$.

Data Analysis

Model parameters were estimated using hierarchical Bayesian methods. The parameters were produced using JAGS (Plummer 2015) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2016).

The one exception is the length-at-age estimates which were produced using the mixdist R package (Macdonald 2012) which implements Maximum Likelihood with Expectation Maximization.

The parameters are summarised in terms of the point estimate, lower and upper 95% credible limits (CLs) and the surprisal s-value (Greenland 2019). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value can be considered a test of directionality. More specifically it indicates how surprising (in bits) it would be to discover that the true value of the parameter is in the opposite direction to the estimate. An s-value (Chow and Greenland 2019) is the Shannon transform (-log to base 2) of the corresponding p-value (Kery and Schaub 2011; Greenland and Poole 2013). A surprisal value of 4.3 bits, which is equivalent to a p-value of 0.05 indicates that the surprise would be equivalent to throwing 4.3 heads in a row. The condition that non-essential explanatory variables have s-values $\geq$ 4.3 bits provides a useful model selection heuristic (Kery and Schaub 2011).

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the number of zeros and the first four central moments (mean, variance, skewness and kurtosis) for the deviance residuals were compared to the expected values by simulating new residuals. In this context the s-value indicates how surprising each metric is given the estimated posterior probability distribution for the residual variation.

In the case of the survival analysis, the posterior predictive check used the Freeman-Tukey statistic for the discrepancy with the expected number of recaptures for each tagging cohort and year (Kery and Schaub 2011).

Where computationally practical, the sensitivity of the parameters to the choice of prior distributions was evaluated by increasing the standard deviations of all normal, half-normal and log-normal priors by an order of magnitude and then using $\hat{R}$ to test whether the samples where drawn from the same posterior distribution (Thorley and Andrusak 2017).

The results are displayed graphically by plotting the modeled relationships between particular variables and the response(s) with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent or n-fold change in the response variable) with 95% credible intervals (CIs, Bradford et al. 2005).

The analyses were implemented using R version 4.4.1 (R Core Team 2020) and the mbr family of packages.

Model Descriptions

Condition

The expected weight of fish of a given length were estimated from the data using an allometric mass-length model (He et al. 2008).

\[W = \alpha L^{\beta}\]

Key assumptions of the condition model include:

The expected weight is allowed to vary with length and date.
The expected weight is allowed to vary randomly with year.
The relationship between weight and length is allowed to vary with date.
The relationship between weight and length is allowed to vary randomly with year.
The residual variation in weight is log-skew-normally distributed.

Only previously untagged fish were included in models to avoid potential effects of tagging on body condition. Preliminary analyses indicated that the annual variation in weight was not correlated with the annual variation in the relationship between weight and length.

Growth

Annual growth of fish were estimated from the inter-annual recaptures using the Fabens method (Fabens 1965) for estimating the von Bertalanffy growth curve (von Bertalanffy 1938). This curve is based on the premise that:

\[ \frac{\text{d}L}{\text{d}t} = k (L_{\infty} - L)\]

where $L$ is the length of the individual, $k$ is the growth coefficient and $L_{\infty}$ is the maximum length.

Integrating the above equation gives:

\[ L_t = L_{\infty} (1 - e^{-k(t - t_0)})\]

where $L_t$ is the length at time $t$ and $t_0$ is the time at which the individual would have had zero length.

The Fabens form allows

\[ L_r = L_c + (L_{\infty} - L_c) (1 - e^{-kT})\]

where $L_r$ is the length at recapture, $L_c$ is the length at capture and $T$ is the time between capture and recapture.

Key assumptions of the growth model include:

The mean maximum length $L_{\infty}$ is constant.
The growth coefficient $k$ is allowed to vary randomly with year.
The residual variation in growth is normally distributed.

The growth model was only fitted to Walleye with a fork length at release less than 450 mm.

Site Fidelity

The extent to which sites are closed, i.e., fish remain at the same site between sessions, was evaluated with a logistic ANCOVA (Kery 2010). The model estimates the probability that intra-annual recaptures were caught at the same site versus a different one. Key assumptions of the site fidelity model include:

The expected site fidelity is allowed to vary with fish length.
Observed site fidelity is Bernoulli distributed.

Length as a second-order polynomial was not found to be a significant predictor for site fidelity.

The estimated probability of being caught at the same site versus a different site was then converted into the site fidelity by assuming that those fish which were recaught at a different site represented just 32 % of those that left the site. The correction factor corresponds to the proportion of the river bank that belongs to index sites.

Observer Length Correction

The annual bias (inaccuracy) and error (imprecision) in observer’s fish length estimates were quantified from the divergence of the length distribution of their observed fish from the length distribution of the measured fish. More specifically, the length correction that minimized the Jensen-Shannon divergence (Lin 1991) between the two distributions provided a measure of the inaccuracy while the minimum divergence (the Jensen-Shannon divergence was calculated with log to base 2 which means it lies between 0 and 1) provided a measure of the imprecision.

Length-At-Age

The expected length-at-age of Mountain Whitefish and Rainbow Trout were estimated from annual length-frequency distributions using a finite mixture distribution model (Macdonald and Pitcher 1979).

There were assumed to be three distinguishable normally-distributed age-classes for Mountain Whitefish (Age-0, Age-1, Age-2 and Age-3+) two for Rainbow Trout (Age-0, Age-1, Age-2+). Initially the model was fitted to the data from all years combined. The model was then fitted to the data for each year separately with the initial values set to be the estimates from the combined values. The only constraints were that the standard deviations of the MW age-classes were identical in the combined analysis and fixed at the initial values in the individual years. Models which did not converge were excluded and the length cutoff was calculated based on linear interpolation of the estimates for the neighboring years.

Rainbow Trout and Mountain Whitefish were categorized as Fry (Age-0), Juvenile (Age-1) and Adult (Age-2+) based on their length-based ages. All Walleye were considered to be Adults.

Survival

The annual adult survival rate was estimated by fitting a Cormack-Jolly-Seber model (Kery and Schaub 2011, 220–31) to inter-annual recaptures of adults.

Key assumptions of the survival model include:

Survival varies randomly with year.
The encounter probability for adults is allowed to vary with the total bank length sampled.

Preliminary analysis indicated that only including visits to index sites did not substantially change the results.

Recapture Efficiency

The probability of recapture was estimated using a recapture-based binomial model (Kery and Schaub 2011, 134–36, 384–88).

Key assumptions of the recapture efficiency model include:

The recapture probability varies randomly by session within year.
The number of recaptures is described by a binomial distribution.

Preliminary analyses indicated that the direction of effect of the frequency of the electrofishing current (30, 60 or 120 Hz) was uncertain.

Abundance

The abundance was estimated from the catch and bias-corrected observer count data using an overdispersed Poisson model (Kery and Schaub 2011, 55–56).

Key assumptions of the abundance model include:

The fish density varies randomly with site, year and site within year.
The capture efficiency at a typical fish density is the recapture efficiency in a typical session divided by the site fidelity.
The count efficiency varies from the capture efficiency.
The overdispersion varies by visit type (count or catch).
The catches and counts are described by a gamma-Poisson distribution.

Distribution

The distribution was calculated in terms of the Shannon index of evenness in each year for each species and life-stage. The index was calculated using the following formula where $S$ is the number of sites and $p_i$ is the proportion of the total density belonging to the ith site

\[ E = \frac{-\sum p_i \log(p_i)}{\log(S)}\]

Survival (Abundance-based)

The subadult ($S_t$) and adult ($A_t$) abundance estimates were used to calculate the subadult and adult survival ($\phi_t$) in year $t$ based on the relationship

\[\phi_t = \frac{A_t}{S_{t-1} + A_{t-1}}\]

Weight

The weight ($W_t$) in year $t$ was estimated using the condition model from the expected adult length from the length-at-age model.

Fecundity

Mountain Whitefish

The fecundity-weight relationship for Mountain Whitefish was estimated from data collected by Boyer et al. (2017) for the Madison River, Montana. The data were analysed using an allometric model of the form

\[F = \alpha W^{\beta}\]

Key assumptions of the fecundity model include:

The residual variation in fecundity is log-normally distributed.

Rainbow Trout

Following (Andrusak and Thorley 2019) the fecundity ($F_t$) in year $t$ of an adult female Rainbow Trout was calculated from the expected weight ($W_t$) in grams using the equation:

\[F_t = 3.8 \cdot W_t^{0.9}\]

Egg Deposition

The total egg deposition ($E_t$) in year t was calculated from the estimated fecundity ($F_t$) and adult abundance ($A_t$), assuming that the population was 50% female according to the equation \[E_t = F_t * \frac{A_t}{2}\]

Stock-Recruitment

The relationship between the total number of eggs deposited ($E_t$) and the resultant number of subadults (age-1 recruits) ($S_{t+1}$) was estimated using a Beverton-Holt stock-recruitment model (Walters and Martell 2004):

\[S_{t+1} = \frac{\alpha \cdot E_t}{1 + \beta \cdot E_t}\]

where $\alpha$ is the egg to age-1 survival at low density and $\beta$ is the density-dependence.

Key assumptions of the stock-recruitment model include:

The egg to recruit survival at low density ($\alpha$) was likely less than 1% (the prior distribution for $\alpha$ was a zero truncated normal with standard deviation of 0.003.
The expected log number of recruits varies with the proportional egg loss.
The residual variation in the number of recruits is log-normally distributed.

The expected egg survival for a given egg deposition is $S / E_t$ which is given by the equation

\[\phi_E = \frac{\alpha}{1 + \beta * E}\]

Age-Ratios

The proportion of Age-1 Mountain Whitefish $r^1_t$ from a given spawn year $t$ is calculated from the proportion of Age-1 & Age-2 fish $P^1_t$ & $P^2_t$ respectively in the length-at-age analysis, which were lead or lagged so that all values were with respect to the spawn year:

\[r^1_t = \frac{P^1_{t+2}}{P^1_{t+2} + P^2_{t+2}}\]

As the proportion of Age-2 fish might be expected to be influenced by the percentage egg loss $Q_t$ three years prior, the predictor variable $\Pi_t$ used is:

\[\Pi_t = \textrm{log}(Q_t/Q_{t-1})\]

The ratio was logged to ensure it was symmetrical about zero (Tornqvist et al. 1985).

The relationship between $r^1_t$ and $\Pi_t$ was estimated using a Bayesian regression (Kery 2010) loss model.

Key assumptions of the final model include:

The log odds of the proportion of Age-1 fish varies with the log of the ratio of the percent egg losses.
The residual variation is normally distributed.

The relationship between percent dewatering and subsequent recruitment is expected to depend on stock abundance (Subbey et al. 2014) which might be changing over the course of the study. Consequently, preliminary analyses allowed the slope of the regression line to change by year. However, year was not a significant predictor and was therefore removed from the final model. The effect of dewatering on Mountain Whitefish abundance was expressed in terms of the predicted percent change in Age-1 Mountain Whitefish abundance by egg loss in the spawn year relative to 10% egg loss in the spawn year. The egg loss in the previous year was fixed at 10%. The percent change could not be calculated relative to 0% in the spawn or previous year as $\Pi_t$ is undefined in either case.

Adjusted Recruitment

The abundance of Age-1 Rainbow Trout was estimated based on the proportion of the Rainbow Trout eggs dewatered the previous year and the abundance of age-1 Mountain Whitefish caught in the same year to account for inter-annual variation in age-1 salmonid abundance and/or capture efficiency.

The relationship(s) were estimated using a Generalized Linear Model (GLM).

Key assumptions of the final model include:

The abundance of Age-1 Rainbow Trout varies with proportion of the eggs dewatered and then number of Age-1 Mountain Whitefish caught in the same year.
The residual variation is log-normally distributed.

Model Templates

Condition

 data {
  int nYear;
  int nObs;

  vector[nObs] Length;
  vector[nObs] Weight;
  vector[nObs] Dayte;
  int Year[nObs];

parameters {
  real bWeight;
  real bWeightLength;
  real bWeightDayte;
  real bWeightLengthDayte;
  real<lower=0> sWeightYear;
  real<lower=0> sWeightLengthYear;

  vector[nYear] bWeightYear;
  vector[nYear] bWeightLengthYear;
  real bShape;
  real<lower=0> sWeight;

model {

  vector[nObs] eWeight;
  vector[nObs] eMu;

  bWeight ~ normal(5, 4);
  bWeightLength ~ normal(3, 1);

  bWeightDayte ~ normal(0, 1);
  bWeightLengthDayte ~ normal(0, 1);

  sWeightYear ~ normal(0, 1);
  sWeightLengthYear ~ normal(0, 1);

  for (i in 1:nYear) {
    bWeightYear[i] ~ normal(0, sWeightYear);
    bWeightLengthYear[i] ~ normal(0, sWeightLengthYear);
  }

  sWeight ~ normal(0, 5);
  bShape ~ normal(0, 2);
  for(i in 1:nObs) {
    eWeight[i] = exp(bWeight + bWeightDayte * Dayte[i] + bWeightYear[Year[i]] + (bWeightLength + bWeightLengthDayte * Dayte[i] + bWeightLengthYear[Year[i]]) * Length[i]);
    eMu[i] <- eWeight[i] - sWeight * (bShape / sqrt(1 + bShape^2)) * sqrt(2 / pi());
    log(Weight[i]) ~ skew_normal(log(eMu[i]), sWeight, bShape);
  }

Block 1. Model description.

Growth

.model {
  bK ~ dnorm (0, 5^-2)
  sKYear ~ dnorm(0, 2^-2) T(0,)

  for (i in 1:nYear) {
    bKYear[i] ~ dnorm(0, sKYear^-2)
    log(eK[i]) <- bK + bKYear[i]
  }

  bLinf ~ dunif(200, 1000)
  sGrowth ~ dnorm(0, 25^-2) T(0,)
  for (i in 1:length(Year)) {
    eGrowth[i] <- max(0, (bLinf - LengthAtRelease[i]) * (1 - exp(-sum(eK[Year[i]:(Year[i] + dYears[i] - 1)]))))
    Growth[i] ~ dnorm(eGrowth[i], sGrowth^-2)
  }

Block 2.

Site Fidelity

.model {

  bFidelity ~ dnorm(0, 1^-2)
  bLength ~ dnorm(0, 1^-2)

  for (i in 1:length(Fidelity)) {
    logit(eFidelity[i]) <- bFidelity + bLength * Length[i]
    Fidelity[i] ~ dbern(eFidelity[i])
  }

Block 3.

Survival

.model{
  bEfficiency ~ dnorm(-3, 2^-2)
  bEfficiencySampledLength ~ dnorm(0, 1^-2)

  bSurvival ~ dnorm(0, 2^-2)

  sSurvivalYear ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:nYear) {
    bSurvivalYear[i] ~ dnorm(0, sSurvivalYear^-2)
  }

  for(i in 1:(nYear-1)) {
    logit(eEfficiency[i]) <- bEfficiency + bEfficiencySampledLength * SampledLength[i]
    logit(eSurvival[i]) <- bSurvival + bSurvivalYear[i]

    eProbability[i,i] <- eSurvival[i] * eEfficiency[i]
    for(j in (i+1):(nYear-1)) {
      eProbability[i,j] <- prod(eSurvival[i:j]) * prod(1-eEfficiency[i:(j-1)]) * eEfficiency[j]
    }
    for(j in 1:(i-1)) {
      eProbability[i,j] <- 0
    }
  }
  for(i in 1:(nYear-1)) {
    eProbability[i,nYear] <- 1 - sum(eProbability[i,1:(nYear-1)])
  }

  for(i in 1:(nYear - 1)) {
    Marray[i, 1:nYear] ~ dmulti(eProbability[i,], Released[i])
  }

  for (i in 1:(nYear - 1)) {
    for (j in 1:nYear) {
      eMarray[i, j] <- Released[i] * eProbability[i, j]
      eOrg[i, j] <- pow((pow(Marray[i, j], 0.5) - pow(eMarray[i, j], 0.5)), 2)
    }
  }

  # Generate replicate data and compute fit stats from them
  for (i in 1:(nYear - 1)) {
    eMarray2[i, 1:nYear] ~ dmulti(eProbability[i, ], Released[i])
    for (j in 1:nYear) {
      eNew[i, j] <- pow((pow(eMarray2[i, j], 0.5) - pow(eMarray[i, j], 0.5)), 2)
    }
  }
  bFit <- sum(eOrg[, ])
  bFitNew <- sum(eNew[, ])

Block 4.

Recapture Efficiency

data {
  int<lower=1> nObs;
  int<lower=1> nAnnual;
  int<lower=1> nSession;

  int <lower=0> Recaptures[nObs];
  int <lower=0> Tagged[nObs];
  int <lower=1> Annual[nObs];
  int <lower=1> Session[nObs];

parameters {
  real bEfficiency;

  real <lower = 0> sEfficiencyAnnualSession;
  matrix[nAnnual,nSession] bEfficiencyAnnualSession;

model {
  vector[nObs] eEfficiency;

  bEfficiency ~ normal(-5, 2);
  sEfficiencyAnnualSession ~ normal(0, 1);

  for(i in 1:nAnnual) {
    bEfficiencyAnnualSession[i,] ~ normal(0, sEfficiencyAnnualSession);
  }

  for (i in 1:nObs) {
    eEfficiency[i] = inv_logit(bEfficiency + bEfficiencyAnnualSession[Annual[i],Session[i]]);

    Recaptures[i] ~ binomial(Tagged[i], eEfficiency[i]);
  }

Block 5.

Abundance

.model {
  bDensity ~ dnorm(5, 4^-2)

  sDensityAnnual ~ dnorm(0, 1^-2) T(0,)
  for (i in 1:nAnnual) {
    bDensityAnnual[i] ~ dnorm(0, sDensityAnnual^-2)
  }

  sDensitySite ~ dnorm(0, 1^-2) T(0,)
  sDensitySiteAnnual ~ dnorm(0, 1^-2) T(0,)
  for (i in 1:nSite) {
    bDensitySite[i] ~ dnorm(0, sDensitySite^-2)
    for (j in 1:nAnnual) {
      bDensitySiteAnnual[i, j] ~ dnorm(0, sDensitySiteAnnual^-2)
    }
  }

  bEfficiencyVisitType[1] <- 0
  for (i in 2:nVisitType) {
    bEfficiencyVisitType[i] ~ dnorm(0, 2^-2)
  }

  for(i in 1:nVisitType) {
    sDispersionVisitType[i] ~ dnorm(0, 2^-2) T(0,)
  }

  bEfficiency ~ dnorm(Efficiency, EfficiencySD^2) T(EfficiencyLower, EfficiencyUpper)
  bFidelity ~ dnorm(Fidelity, FidelitySD^2) T(FidelityLower, FidelityUpper)

  for (i in 1:length(Fish)) {
    log(eDensity[i]) <- bDensity + bDensitySite[Site[i]] + bDensityAnnual[Annual[i]] + bDensitySiteAnnual[Site[i],Annual[i]]

    eAbundance[i] <- eDensity[i] * SiteLength[i]

    logit(eEfficiency[i]) <- logit(bEfficiency/ bFidelity) + bEfficiencyVisitType[VisitType[i]]

    eFish[i] <- eAbundance[i] * ProportionSampled[i] * eEfficiency[i]

    eDispersion[i] ~ dgamma(sDispersionVisitType[VisitType[i]]^-2, sDispersionVisitType[VisitType[i]]^-2)
    Fish[i] ~ dpois(eFish[i] * eDispersion[i])
  }

Block 6.

Fecundity

model {
  bFecundity ~ dnorm(0, 5^-2)
  bFecundityWeight ~ dnorm(1, 1^-2) T(0,)

  sFecundity ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(Weight)) {
    eFecundity[i] = bFecundity + bFecundityWeight * log(Weight[i])
    Fecundity[i] ~ dlnorm(eFecundity[i], sFecundity^-2)
  }

Block 7.

Stock-Recruitment

.model {
  bAlpha ~ dnorm(0, 0.003^-2) T(0,)
  bBeta ~ dnorm(0, 0.007^-2) T(0, )
  bEggLoss ~ dnorm(0, 100^-2)

  sRecruits ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(Recruits)){
    log(eRecruits[i]) <- log(bAlpha * Eggs[i] / (1 + bBeta * Eggs[i])) + bEggLoss * EggLoss[i]
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

Block 8.

Age-Ratios

.model{
  bProbAge1 ~ dnorm(0, 1^-2)
  bProbAge1Loss ~ dnorm(0, 1^-2)

  sProbAge1 ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(Age1Prop)){
    eAge1Prop[i] <- bProbAge1 + bProbAge1Loss * LossLogRatio[i]
    Age1Prop[i] ~ dnorm(eAge1Prop[i], sProbAge1^-2)
  }

Block 9.

Adjusted Recruitment

.model{
  b0 ~ dnorm(0, 10^-2)
  bMw ~ dnorm(0, 2^-2)
  bEggLoss ~ dnorm(0, 2^-2)

  sRb ~ dnorm(0, 2^-2) T(0,)
  for (i in 1:nObs) {
    log(eRb[i]) <- b0 + bMw * Mw[i] + bEggLoss * EggLoss[i]
    Rb[i] ~ dlnorm(log(eRb[i]), sRb^-2)
  }

Block 10. Model description.

Results

Tables

Condition

Table 1. Parameter descriptions.

Parameter	Description
`Dayte[i]`	Standardised day of year `i`^th fish was captured
`Length[i]`	Log-transformed and centered fork length of `i`^th fish
`Weight[i]`	Recorded weight of `i`^th fish
`Year[i]`	Year `i`^th fish was captured
`bShape`	Shape parameter for log-skew-normal distribution
`bWeightDayte`	Effect of `Dayte` on `bWeight`
`bWeightLengthDayte`	Effect of `Dayte` on `bWeightLength`
`bWeightLengthYear[i]`	Effect of `i`^th `Year` on `bWeightLength`
`bWeightLength`	Intercept of effect of `Length` on `bWeight`
`bWeightYear[i]`	Effect of `i`^th `Year` on `bWeight`
`bWeight`	Intercept of `log(eWeight)`
`eWeight[i]`	Expected `Weight` of `i`^th fish
`sWeightLengthYear`	Log standard deviation of `bWeightLengthYear`
`sWeightYear`	Log standard deviation of `bWeightYear`
`sWeight`	Log standard deviation of residual variation in `log(Weight)`

Mountain Whitefish

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
bShape	-1.7596751	-1.8500044	-1.6688337	10.551708
bWeight	5.6173566	5.6000370	5.6334255	10.551708
bWeightDayte	-0.0200314	-0.0232514	-0.0168029	10.551708
bWeightLength	3.1859257	3.1484523	3.2185760	10.551708
bWeightLengthDayte	-0.0111731	-0.0203732	-0.0022353	5.693727
sWeight	0.2014886	0.1978942	0.2050174	10.551708
sWeightLengthYear	0.0940325	0.0682514	0.1363724	10.551708
sWeightYear	0.0422773	0.0322774	0.0569640	10.551708

Table 3. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
17296	8	3	500	2	309	1.006	TRUE

Table 4. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.000000
mean	-0.1128971	-0.1223685	-0.1386956	-0.1050678	1.889930
variance	1.0211375	1.0243502	1.0020963	1.0448302	0.355721
skewness	-0.0397195	-0.0067123	-0.0425378	0.0279700	3.496426
kurtosis	1.4054382	-0.0640597	-0.1354137	0.0118392	10.551708

Table 5. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.03	1.02

Rainbow Trout

Table 6. Model coefficients.

term	estimate	lower	upper	svalue
bShape	-1.8134605	-1.9040578	-1.7329943	10.55171
bWeight	6.1356949	6.1259289	6.1473366	10.55171
bWeightDayte	-0.0048284	-0.0068816	-0.0028156	10.55171
bWeightLength	2.9261288	2.9088066	2.9430533	10.55171
bWeightLengthDayte	0.0321489	0.0260910	0.0383218	10.55171
sWeight	0.1390545	0.1366838	0.1412970	10.55171
sWeightLengthYear	0.0427004	0.0306240	0.0603598	10.55171
sWeightYear	0.0252598	0.0195504	0.0342267	10.55171

Table 7. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
18832	8	3	500	2	280	1.029	TRUE

Table 8. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.1150896	-0.1274497	-0.1436177	-0.1117753	2.9296564
variance	1.0227191	1.0254480	1.0046993	1.0450937	0.3313299
skewness	-0.0598224	-0.0069170	-0.0409158	0.0257597	7.3817833
kurtosis	1.8036221	-0.0688194	-0.1356438	0.0033283	10.5517083

Table 9. Model sensitivity.

all	analysis	sensitivity	bound
all	1.029	1.012	1.017

Walleye

Table 10. Model coefficients.

term	estimate	lower	upper	svalue
bShape	-0.6098789	-0.8259877	0.5380519	2.2163179
bWeight	6.3357136	6.2590198	6.3564311	10.5517083
bWeightDayte	0.0153619	0.0129046	0.0177614	10.5517083
bWeightLength	3.2313514	3.1992024	3.2640784	10.5517083
bWeightLengthDayte	-0.0044429	-0.0189185	0.0090671	0.9024521
sWeight	0.1009029	0.0918703	0.1065800	10.5517083
sWeightLengthYear	0.0709277	0.0508560	0.1024622	10.5517083
sWeightYear	0.0345052	0.0261870	0.0483973	10.5517083

Table 11. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
10827	8	3	500	2	238	1.01	TRUE

Table 12. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0120476	-0.0126798	-0.0375836	0.0145177	0.0608574
variance	0.9981422	1.0021291	0.9760856	1.0297085	0.4211377
skewness	0.0329149	0.0014472	-0.0464042	0.0469951	2.5460837
kurtosis	0.9485097	-0.0017916	-0.0920452	0.0938867	10.5517083

Table 13. Model sensitivity.

all	analysis	sensitivity	bound
all	1.01	1.017	1.011

Growth

Table 14. Parameter descriptions.

Parameter	Description
`Growth[i]`	Observed growth between release and recapture of `i`^th recapture
`LengthAtRelease[i]`	Length at previous release of `i`^th recapture
`Year[i]`	Release year of `i`^th recapture
`bKYear[i]`	Effect of `i`^th `Year` on `bK`
`bK`	Intercept of `log(eK)`
`bLinf`	Mean maximum length
`dYears[i]`	Years between release and recapture of `i`^th recapture
`eGrowth`	Expected `Growth` between release and recapture
`eK[i]`	Expected von Bertalanffy growth coefficient from `i-1`^th to `i`^th year
`sGrowth`	Log standard deviation of residual variation in `Growth`
`sKYear`	Log standard deviation of `bKYear`

Mountain Whitefish

Table 15. Model coefficients.

term	estimate	lower	upper	svalue
bK	-0.9342808	-1.1362380	-0.7395660	10.55171
bLinf	398.6324411	392.7648431	403.8259587	10.55171
sGrowth	11.6040088	10.7009614	12.5620281	10.55171
sKYear	0.3728239	0.2491282	0.5778401	10.55171

Table 16. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
306	4	3	500	50	960	1.002	TRUE

Table 17. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0261438	0.0000000	0.0000000	0.0000000	10.551708
mean	0.0487974	-0.0005565	-0.1102029	0.1122585	1.183202
variance	0.9318759	0.9982148	0.8364365	1.1650087	1.315694
skewness	-0.2259691	-0.0042705	-0.2777083	0.2611181	3.084103
kurtosis	0.3864349	-0.0726978	-0.4678129	0.5855705	2.725160

Table 18. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.004	1.001

Rainbow Trout

Table 19. Model coefficients.

term	estimate	lower	upper	svalue
bK	-0.1110219	-0.2538574	0.0287311	3.220791
bLinf	480.7136533	475.8816857	485.2194152	10.551708
sGrowth	30.1483751	29.1340717	31.1531233	10.551708
sKYear	0.2984256	0.2193773	0.4200256	10.551708

Table 20. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1550	4	3	500	50	942	1.006	TRUE

Table 21. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0045161	0.0000000	0.0000000	0.0000000	10.5517083
mean	0.0108420	-0.0000465	-0.0489351	0.0515086	0.5729978
variance	0.9855958	1.0007767	0.9330846	1.0735097	0.6077283
skewness	0.2760629	0.0002324	-0.1197702	0.1225306	10.5517083
kurtosis	0.7943234	-0.0187792	-0.2350125	0.2594113	10.5517083

Table 22. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.003	1.004

Walleye

Table 23. Model coefficients.

term	estimate	lower	upper	svalue
bK	-2.5922812	-3.0576221	-2.1550169	10.55171
bLinf	772.9157257	643.1441704	972.2909076	10.55171
sGrowth	17.8470733	16.5218847	19.4358664	10.55171
sKYear	0.3051209	0.1891197	0.4833352	10.55171

Table 24. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
297	4	3	500	50	183	1.005	TRUE

Table 25. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0067340	0.0000000	0.0000000	0.0000000	10.551708
mean	0.0523269	-0.0001739	-0.1138791	0.1110633	1.521041
variance	0.9378333	0.9957646	0.8461986	1.1639991	1.178843
skewness	0.1632857	-0.0011728	-0.2684422	0.2663497	2.039956
kurtosis	1.4633707	-0.0598180	-0.4625812	0.6098552	10.551708

Table 26. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.01	1.007

Site Fidelity

Table 27. Parameter descriptions.

Parameter	Description
`Fidelity[i]`	Whether the `i`^th recapture was encountered at the same site as the previous encounter
`Length[i]`	Length at previous encounter of `i`^th recapture
`bFidelity`	Intercept of `logit(eFidelity)`
`bLength`	Effect of length on `logit(eFidelity)`
`eFidelity[i]`	Expected site fidelity of `i`^th recapture

Mountain Whitefish

Table 28. Model coefficients.

term	estimate	lower	upper	svalue
bFidelity	-0.3114539	-0.6466032	0.0323559	3.7062182
bLength	-0.0680504	-0.4301916	0.2753351	0.5279539

Table 29. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
137	2	3	500	1	990	1.001	TRUE

Table 30. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.5766423	0.5766423	0.4598540	0.6934307	0.0271665
mean	-0.0506163	-0.0430528	-0.2450518	0.1638717	0.0749621
variance	1.3687436	1.3612875	1.2064254	1.4070262	0.2129719
skewness	0.3102094	0.3098254	-0.1610066	0.8008200	0.0252090
kurtosis	-1.9011828	-1.8836885	-1.9971703	-1.3484905	0.1328015

Table 31. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.008	1.005

Rainbow Trout

Table 32. Model coefficients.

term	estimate	lower	upper	svalue
bFidelity	0.6693461	0.5318184	0.8082597	10.55171
bLength	-0.3262309	-0.4606331	-0.1875010	10.55171

Table 33. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
891	2	3	500	1	924	1.001	TRUE

Table 34. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.3423120	0.3423120	0.2985410	0.3866723	0.0009615
mean	0.0998627	0.1013383	0.0241068	0.1714192	0.0310896
variance	1.2528997	1.2486188	1.1719672	1.3178394	0.1181228
skewness	-0.6554811	-0.6649579	-0.8576742	-0.4643063	0.1098016
kurtosis	-1.5105410	-1.4983362	-1.7358715	-1.1962634	0.1243954

Table 35. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.004	1.002

Walleye

Table 36. Model coefficients.

term	estimate	lower	upper	svalue
bFidelity	0.6520479	0.4134291	0.9321331	10.5517083
bLength	-0.0834509	-0.3621558	0.2025802	0.8845967

Table 37. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
237	2	3	500	1	720	1.001	TRUE

Table 38. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.3417722	0.3417722	0.2658228	0.4219409	0.0115802
mean	0.1003464	0.1017318	-0.0458675	0.2405510	0.0193523
variance	1.2778804	1.2737647	1.1039010	1.3742060	0.0689003
skewness	-0.6673052	-0.6658715	-1.0847475	-0.3154871	0.0648732
kurtosis	-1.5510030	-1.5506840	-1.8893902	-0.8200443	0.0000000

Table 39. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.003	1.002

Length-At-Age

Mountain Whitefish

Table 40. The estimated upper length cutoffs (mm) by age and year.

Year	Age0	Age1	Age2
1990	NA	NA	NA
1991	NA	NA	NA
2001	144	263	346
2002	158	260	344
2003	151	261	345
2004	158	255	343
2005	161	255	346
2006	160	263	340
2007	158	271	334
2008	167	252	343
2009	163	257	344
2010	165	259	342
2011	154	264	341
2012	156	265	342
2013	162	263	344
2014	167	261	346
2015	153	265	344
2016	135	269	338
2017	150	264	343
2018	165	254	343
2019	162	259	343
2020	149	270	338
2021	155	270	339
2022	155	268	340
2023	154	264	346

Rainbow Trout

Table 41. The estimated upper length cutoffs (mm) by age and year.

Year	Age0	Age1
1990	153	355
1991	133	349
2001	153	353
2002	152	352
2003	157	348
2004	143	342
2005	157	349
2006	165	358
2007	160	365
2008	146	346
2009	147	345
2010	144	345
2011	155	348
2012	149	349
2013	164	354
2014	152	344
2015	153	345
2016	146	342
2017	139	338
2018	140	340
2019	160	333
2020	152	350
2021	163	350
2022	141	352
2023	140	351

Survival

Table 42. Parameter descriptions.

Parameter	Description
`SampledLength`	Total standardised length of river sampled
`bEfficiencySampledLength`	Effect of `SampledLength` on `bEfficiency`
`bEfficiency`	Intercept for `logit(eEfficiency)`
`bSurvivalYear[i]`	Effect of `Year` on `bSurvival`
`bSurvival`	Intercept for `logit(eSurvival)`
`eEfficiency[i]`	Expected recapture probability in `i`^th year
`eSurvival[i]`	Expected survival probability from `i-1`^th to `i`^th year
`sSurvivalYear`	Log SD of `bSurvivalYear`

Mountain Whitefish

Table 43. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-4.3658541	-4.5391772	-4.1896696	10.55171
bEfficiencySampledLength	0.3581081	0.1439954	0.5603659	10.55171
bSurvival	1.0393816	0.4773641	1.8240747	8.22978
sSurvivalYear	1.1932298	0.6806415	2.2638286	10.55171

Table 44. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
22	4	3	500	200	1308	1.004	TRUE

Table 45. Model posterior predictive checks.

statistic	observed	median	lower	upper	svalue
Freeman-Tukey	78.7629	66.43571	54.51944	79.40329	4.01255

Table 46. Model sensitivity.

all	analysis	sensitivity	bound
all	1.004	1.004	1.007

Rainbow Trout

Table 47. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-2.5348632	-2.6976239	-2.3975046	10.5517083
bEfficiencySampledLength	0.0109202	-0.1221177	0.1472788	0.2085225
bSurvival	-0.4351481	-0.6186928	-0.2290185	10.5517083
sSurvivalYear	0.3536312	0.1848044	0.5716120	10.5517083

Table 48. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
22	4	3	500	200	1164	1.002	TRUE

Table 49. Model posterior predictive checks.

statistic	observed	median	lower	upper	svalue
Freeman-Tukey	31.73302	37.17956	28.64017	46.9776	1.806874

Table 50. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.005	1.003

Walleye

Table 51. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-3.5610762	-3.7582896	-3.3756827	10.551708
bEfficiencySampledLength	0.1895840	0.0483894	0.3522460	6.303781
bSurvival	0.1996499	-0.0823731	0.5277372	2.803515
sSurvivalYear	0.5040962	0.2281748	0.9031388	10.551708

Table 52. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
22	4	3	500	200	1206	1.003	TRUE

Table 53. Model posterior predictive checks.

statistic	observed	median	lower	upper	svalue
Freeman-Tukey	51.0737	48.40455	38.37937	60.25303	0.7299343

Table 54. Model sensitivity.

all	analysis	sensitivity	bound
all	1.003	1.003	1.002

Recapture Efficiency

Table 55. Parameter descriptions.

Parameter	Description
`Annual[i]`	Year of `i`^th visit
`Recaptures[i]`	Number of marked fish recaught during `i`^th visit
`Session[i]`	Session of `i`^th visit
`Tagged[i]`	Number of marked fish tagged prior to `i`^th visit
`bEfficiencySessionAnnual`	Effect of `Session` within `Annual` on `logit(eEfficiency)`
`bEfficiency`	Intercept for `logit(eEfficiency)`
`eEfficiency[i]`	Expected efficiency on `i`^th visit
`sEfficiencySessionAnnual`	SD of `bEfficiencySessionAnnual`

Mountain Whitefish

Subadult

Table 56. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-5.7620545	-6.2826521	-5.40581	10.55171
sEfficiencyAnnualSession	0.4970555	0.0391616	1.17356	10.55171

Table 57. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1624	2	3	500	50	237	1.024	TRUE

Table 58. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.9766010	0.9766010	0.9655172	0.9858374	0.0458967
mean	-0.1278424	-0.1282420	-0.1519067	-0.1023167	0.0409441
variance	0.1334132	0.1390853	0.0880788	0.1996072	0.2744209
skewness	5.8476550	5.7990684	4.7632142	7.2456816	0.1077287
kurtosis	36.8230177	38.1666333	25.2567511	61.7827476	0.1930571

Table 59. Model sensitivity.

all	analysis	sensitivity	bound
all	1.024	1.003	1.026

Adult

Table 60. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-6.0641718	-6.413151	-5.8062494	10.55171
sEfficiencyAnnualSession	0.2408184	0.025413	0.7186576	10.55171

Table 61. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1884	2	3	500	50	270	1.005	TRUE

Table 62. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.9708068	0.9702760	0.9591295	0.9798301	0.2074124
mean	-0.1414915	-0.1406473	-0.1631875	-0.1188303	0.0953538
variance	0.1672060	0.1637029	0.1188376	0.2163874	0.1433785
skewness	5.3197412	5.0555762	4.3284844	5.9996428	0.8426244
kurtosis	32.2384706	28.9403270	20.5381706	42.3069631	0.7687101

Table 63. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.012	1.01

Rainbow Trout

Subadult

Table 64. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-3.6626366	-3.7880177	-3.5409193	10.55171
sEfficiencyAnnualSession	0.4014269	0.2958902	0.5336311	10.55171

Table 65. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1916	2	3	500	50	1174	1.005	TRUE

Table 66. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.7583507	0.7359081	0.7139875	0.7588727	4.2118583
mean	-0.2513067	-0.2252739	-0.2603076	-0.1883351	2.8306091
variance	0.6110336	0.6629370	0.6043968	0.7269269	3.7062182
skewness	1.3335599	1.2816570	1.1595880	1.4210940	1.0861419
kurtosis	1.6968170	1.6438549	1.1572838	2.2963281	0.2422319

Table 67. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.006	1.003

Adult

Table 68. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-4.3251133	-4.4636779	-4.198694	10.55171
sEfficiencyAnnualSession	0.2393776	0.0562689	0.416635	10.55171

Table 69. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1981	2	3	500	50	536	1.005	TRUE

Table 70. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.8722867	0.8616860	0.8414942	0.8798587	1.9150836
mean	-0.2340084	-0.2213594	-0.2521103	-0.1923925	1.1788432
variance	0.4827108	0.4802973	0.4211323	0.5427237	0.0729386
skewness	2.4794726	2.0784822	1.8865309	2.2831684	10.5517083
kurtosis	7.8309307	4.2660846	3.2026853	5.5193150	10.5517083

Table 71. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.007	1.004

Walleye

Table 72. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-5.0343472	-5.2715878	-4.8374121	10.55171
sEfficiencyAnnualSession	0.5731814	0.3593516	0.8376875	10.55171

Table 73. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
2016	2	3	500	50	1203	1.005	TRUE

Table 74. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.9176587	0.9097222	0.8930928	0.9250992	1.5376878
mean	-0.1966210	-0.1918068	-0.2193781	-0.1644100	0.4549931
variance	0.3272155	0.3443901	0.2921400	0.4033556	0.8564800
skewness	2.8354783	2.6829015	2.4267462	2.9742310	1.8340318
kurtosis	9.1988550	8.3904728	6.5377912	10.7211466	1.1275420

Table 75. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.009	1.004

Abundance

Table 76. Parameter descriptions.

Parameter	Description
`Annual`	Year
`Efficiency`	Capture efficiency
`Efficiency`	Site Fidelity
`Fish`	Number of fish captured or counted
`ProportionSampled`	Proportion of site surveyed
`SiteLength`	Length of site
`Site`	Site
`VisitType`	Survey type (catch versus count)
`bDensityAnnual`	Effect of `Annual` on `bDensity`
`bDensitySiteAnnual`	Effect of `Site` within `Annual` on `bDensity`
`bDensitySite`	Effect of `Site` on `bDensity`
`bDensity`	Intercept for `log(eDensity)`
`eDensity`	Expected density
`sDensityAnnual`	Log SD of effect of `Annual` on `bDensity`
`sDensitySiteAnnual`	Log SD of effect of `Site` within `Annual` on `bDensity`
`sDensitySite`	Log SD of effect of `Site` on `bDensity`
`sDispersionVisitType`	Overdispersion of `Fish` by `VisitType`

Mountain Whitefish

Subadult

Table 77. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	4.7766411	3.6954948	5.6678402	10.55171
bEfficiency	0.0031482	0.0019304	0.0044115	10.55171
bEfficiencyVisitType[2]	1.2527497	1.1169802	1.3966145	10.55171
bFidelity	0.2598266	0.1083878	0.4099776	10.55171
sDensityAnnual	0.6582065	0.4983769	0.9338448	10.55171
sDensitySite	0.8445964	0.7011586	1.0304544	10.55171
sDensitySiteAnnual	0.4429457	0.3939745	0.4907322	10.55171
sDispersionVisitType[1]	0.4456816	0.4066538	0.4842042	10.55171
sDispersionVisitType[2]	0.9061228	0.8041829	1.0109641	10.55171

Table 78. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3296	9	3	500	500	81	1.048	FALSE

Table 79. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.2539442	0.2809466	0.2654733	0.2962758	8.966746
mean	-0.2229342	-0.2722898	-0.3069464	-0.2386065	6.851268
variance	0.7068123	0.9847646	0.9408738	1.0283427	10.551708
skewness	0.1000454	0.2761074	0.2028399	0.3466265	10.551708
kurtosis	-0.4361872	-0.3311090	-0.4709171	-0.1791040	2.712505

Table 80. Model sensitivity.

all	analysis	sensitivity	bound
all	1.048	1.014	1.037

Adult

Table 81. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	5.5734503	5.0363909	6.1111673	10.55171
bEfficiency	0.0024256	0.0016845	0.0029647	10.55171
bEfficiencyVisitType[2]	1.5214040	1.3974230	1.6419940	10.55171
bFidelity	0.1965414	0.1469993	0.2533608	10.55171
sDensityAnnual	0.2856437	0.2071506	0.4204352	10.55171
sDensitySite	1.0652588	0.9068060	1.2728537	10.55171
sDensitySiteAnnual	0.3928856	0.3512841	0.4359668	10.55171
sDispersionVisitType[1]	0.5016559	0.4726264	0.5328033	10.55171
sDispersionVisitType[2]	0.7573845	0.6841052	0.8430933	10.55171

Table 82. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3296	9	3	500	500	220	1.02	TRUE

Table 83. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1796117	0.2038835	0.1893204	0.2183025	10.5517083
mean	-0.2185147	-0.2717877	-0.3058549	-0.2379291	8.2297802
variance	0.6707343	1.0092724	0.9627892	1.0540117	10.5517083
skewness	-0.0053635	0.1826797	0.1084906	0.2539325	10.5517083
kurtosis	-0.3117070	-0.3081454	-0.4337810	-0.1535883	0.0668854

Table 84. Model sensitivity.

all	analysis	sensitivity	bound
all	1.02	1.014	1.033

Rainbow Trout

Subadult

Table 85. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	4.5542968	4.2510803	4.8266173	10.55171
bEfficiency	0.0252564	0.0223034	0.0280455	10.55171
bEfficiencyVisitType[2]	1.3156510	1.2000190	1.4381275	10.55171
bFidelity	0.5556421	0.4761171	0.6354068	10.55171
sDensityAnnual	0.2770909	0.1988649	0.3983033	10.55171
sDensitySite	0.6403094	0.5357897	0.7776025	10.55171
sDensitySiteAnnual	0.4199177	0.3823397	0.4593491	10.55171
sDispersionVisitType[1]	0.3705302	0.3439481	0.3964911	10.55171
sDispersionVisitType[2]	0.7073705	0.6326097	0.7860101	10.55171

Table 86. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3296	9	3	500	500	596	1.006	TRUE

Table 87. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1395631	0.1495752	0.1371359	0.1617112	2.959251
mean	-0.1886641	-0.2413146	-0.2768634	-0.2063297	7.744353
variance	0.6372557	1.0309866	0.9833930	1.0787123	10.551708
skewness	-0.1545148	0.1074275	0.0307790	0.1773514	10.551708
kurtosis	-0.1210253	-0.2653999	-0.3849576	-0.1173612	4.142317

Table 88. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.009	1.005

Adult

Table 89. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	5.0230046	4.7642483	5.2750643	10.55171
bEfficiency	0.0131666	0.0114610	0.0147164	10.55171
bEfficiencyVisitType[2]	1.1625061	1.0672149	1.2564530	10.55171
bFidelity	0.4200889	0.3849790	0.4600874	10.55171
sDensityAnnual	0.3030419	0.2272273	0.4252834	10.55171
sDensitySite	0.6163643	0.5276485	0.7251566	10.55171
sDensitySiteAnnual	0.2492769	0.2156547	0.2846138	10.55171
sDispersionVisitType[1]	0.3556273	0.3275170	0.3822441	10.55171
sDispersionVisitType[2]	0.5915920	0.5264653	0.6599515	10.55171

Table 90. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3296	9	3	500	500	687	1.007	TRUE

Table 91. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1137743	0.1298544	0.1183252	0.1422937	7.381783
mean	-0.1729971	-0.2334843	-0.2695229	-0.1984358	8.966746
variance	0.6450050	1.0379050	0.9941665	1.0852058	10.551708
skewness	-0.1355976	0.0805613	0.0045261	0.1531176	10.551708
kurtosis	-0.0580298	-0.2480747	-0.3715394	-0.0943592	5.693727

Table 92. Model sensitivity.

all	analysis	sensitivity	bound
all	1.007	1.009	1.006

Walleye

Table 93. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	4.7399074	4.1576782	5.2840277	10.55171
bEfficiency	0.0064691	0.0051858	0.0077981	10.55171
bEfficiencyVisitType[2]	0.9654252	0.8588251	1.0835466	10.55171
bFidelity	0.3532851	0.2083278	0.4969209	10.55171
sDensityAnnual	0.4062081	0.3029777	0.5853731	10.55171
sDensitySite	0.3717362	0.2950688	0.4659394	10.55171
sDensitySiteAnnual	0.2915298	0.2537621	0.3304803	10.55171
sDispersionVisitType[1]	0.4203297	0.3903645	0.4492400	10.55171
sDispersionVisitType[2]	0.6816581	0.5990381	0.7722126	10.55171

Table 94. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3296	9	3	500	500	195	1.024	TRUE

Table 95. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1316748	0.1662621	0.1520024	0.1808252	10.5517083
mean	-0.1893318	-0.2537669	-0.2894480	-0.2193394	10.5517083
variance	0.6916338	1.0388336	0.9975987	1.0846841	10.5517083
skewness	-0.1925823	0.1118355	0.0400116	0.1849212	10.5517083
kurtosis	-0.3187531	-0.3496027	-0.4760385	-0.1888757	0.6224499

Table 96. Model sensitivity.

all	analysis	sensitivity	bound
all	1.024	1.014	1.017

Fecundity

Table 97. Parameter descriptions.

Parameter	Description
`Fecundity[i]`	Fecundity of `i`^th fish (eggs)
`Weight[i]`	Weight of `i`^th fish (g)
`bFecundityWeight`	Effect of `log(Weight)` on `log(bFecundity)`
`bFecundity`	Intercept of `eFecundity`
`eFecundity[i]`	Expected `Fecundity` of `i`^th fish
`sFecundity`	SD of residual variation in `log(Fecundity)`

Mountain Whitefish

Table 98. Model coefficients.

term	estimate	lower	upper	svalue
bFecundity	2.9079204	2.0939813	3.7431218	10.55171
bFecundityWeight	0.9999442	0.8737297	1.1248463	10.55171
sFecundity	0.1309071	0.0998381	0.1809365	10.55171

Table 99. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
28	3	3	500	500	784	1.006	TRUE

Table 100. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0023890	-0.0028296	-0.3707738	0.3628603	0.0271665
variance	0.9070264	0.9764472	0.5380645	1.6197364	0.3361753
skewness	0.0276272	0.0051170	-0.8620544	0.7803337	0.0689003
kurtosis	-0.7063561	-0.3584825	-1.1195985	1.5822523	1.0780025

Table 101. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.002	1.004

Stock-Recruitment

Table 102. Parameter descriptions.

Parameter	Description
`EggLoss`	Proportional egg loss
`Eggs`	Total egg deposition
`Recruits`	Number of Age-1 recruits
`bAlpha`	`eRecruits` per `Stock` at low `Stock` density
`bBeta`	Expected density-dependence
`bEggLoss`	Effect of `EggLoss` on `log(eRecruits)`
`eRecruits`	Expected `Recruits`
`sRecruits`	SD of residual variation in `log(Recruits)`

Mountain Whitefish

Table 103. Model coefficients.

term	estimate	lower	upper	svalue
bAlpha	0.0036395	0.0009591	0.0084292	10.5517083
bBeta	0.0000002	0.0000000	0.0000005	10.5517083
bEggLoss	-0.3803164	-2.3578308	1.5938972	0.5279539
sRecruits	0.5584720	0.4001685	0.8289082	10.5517083

Table 104. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
19	4	3	500	50	970	1.001	TRUE

Table 105. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0242352	-0.0110520	-0.4455285	0.4336415	0.1864794
variance	0.8812918	0.9543527	0.4463960	1.6987197	0.2814129
skewness	-0.2280838	0.0106186	-0.9239293	0.9040852	0.6705943
kurtosis	-0.9094830	-0.4482496	-1.2872101	1.4276061	1.4829300

Table 106. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.001	1.599

Rainbow Trout

Table 107. Model coefficients.

term	estimate	lower	upper	svalue
bAlpha	0.0047213	0.0021731	0.0089056	10.551708
bBeta	0.0000003	0.0000001	0.0000006	10.551708
bEggLoss	14.2753182	-2.5579377	31.9166845	3.453676
sRecruits	0.2590153	0.1905266	0.3818405	10.551708

Table 108. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
21	4	3	500	50	576	1.004	TRUE

Table 109. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0327049	0.0051164	-0.4138982	0.4258180	0.1370230
variance	0.8714455	0.9728617	0.4740156	1.6931031	0.3981562
skewness	-0.0783743	0.0174654	-0.9038562	0.9036716	0.2720977
kurtosis	0.5276555	-0.4361493	-1.2685061	1.4783804	2.0719280

Table 110. Model sensitivity.

all	analysis	sensitivity	bound
all	1.004	1.014	1.635

Age-Ratios

Table 111. Parameter descriptions.

Parameter	Description
`Age1[i]`	The number of Age-1 fish in the `i`^th year
`Age1and2[i]`	The number of Age-1 and Age-2 fish in the `i`^th year
`LossLogRatio[i]`	The `log` of the ratio of the percent egg losses
`bProbAge1Loss`	Effect of `LossLogRatio` on `bProbAge1`
`bProbAge1`	Intercept for `logit(eProbAge1)`
`eProbAge1[i]`	The expected proportion of Age-1 fish in the `i`^th year
`sDispersion`	SD of extra-binomial variation

Table 112. Model coefficients.

term	estimate	lower	upper	svalue
bProbAge1	0.2262727	-0.0862555	0.5611520	2.626896
bProbAge1Loss	-0.2500217	-0.6702924	0.2167366	1.940684
sProbAge1	0.7203736	0.5513726	1.0336407	10.551708

Table 113. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
21	3	3	500	1	567	1.003	TRUE

Table 114. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0011832	-0.0072092	-0.4351058	0.4140140	0.0271665
variance	0.9192619	0.9626508	0.4887891	1.7419409	0.1799316
skewness	-0.4114957	-0.0008554	-0.9483289	0.9541953	1.5601864
kurtosis	-0.9624045	-0.3939409	-1.2366711	1.7658854	1.9856542

Table 115. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.004	1.006

Adjusted Recruitment

Table 116. Parameter descriptions.

Parameter	Description
`Eggloss[i]`	Proportional RB egg loss for the `i`^th spawn year
`Mw[i]`	Abundance of age-1 MW caught in the same year as age-1 RB from the `i`^th spawn year
`Rb[i]`	Abundance of age-1 RB for the `i`^th spawn year
`b0`	Intercept for `eRb[i]`
`bEggLoss`	Effect of `EggLoss` on `eRb[i]`
`bMw`	Effect of `Mw` on `eRb[i]`
`eRb[i]`	Expected value of `Rb[i]`
`sRb`	SD of residual variation in `eRb[i]`

Table 117. Model coefficients.

term	estimate	lower	upper	svalue
b0	9.7076923	9.6224759	9.7948390	10.551708
bEggLoss	0.0363123	-0.0621380	0.1356059	1.073950
bMw	0.1563436	0.0602859	0.2624106	7.744353
sRb	0.1985673	0.1473766	0.2801007	10.551708

Table 118. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
23	4	3	500	50	1436	1	TRUE

Table 119. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0019992	0.0038183	-0.4080401	0.3835868	0.0271665
variance	0.8505956	0.9796990	0.5071463	1.6355639	0.5902586
skewness	0.8047524	0.0048387	-0.8771124	0.8957869	3.7315293
kurtosis	1.6305717	-0.3898895	-1.2054447	1.8408086	4.0125495

Table 120. Model sensitivity.

all	analysis	sensitivity	bound
all	1	1.001	1.001

Figures

Condition

Figure 1. Predicted length-mass relationship by species.

Subadult

Mountain Whitefish

Figure 2. Estimated change in condition relative to a typical year for a 200 mm Mountain Whitefish by year (with 95% CRIs).

Rainbow Trout

Figure 3. Estimated change in condition relative to a typical year for a 250 mm Rainbow Trout by year (with 95% CRIs).

Adult

Mountain Whitefish

Figure 4. Estimated change in condition relative to a typical year for a 350 mm Mountain Whitefish by year (with 95% CRIs).

Rainbow Trout

Figure 5. Estimated change in condition relative to a typical year for a 500 mm Rainbow Trout by year (with 95% CRIs).

Walleye

Figure 6. Estimated change in condition relative to a typical year for a 400 mm Walleye by year (with 95% CRIs).

Growth

Figure 7. Estimated von Bertalanffy growth curve by species. The growth curve for Walleye is not shown because the growth parameters were not representative for the younger age-classes.

Mountain Whitefish

Figure 8. Estimated change in von Bertalanffy growth coefficient (k) relative to a typical year by year (with 95% CIs).

Figure 9. Predicted maximum growth by year (with 95% CIs).

Rainbow Trout

Figure 10. Estimated change in von Bertalanffy growth coefficient (k) relative to a typical year by year (with 95% CIs).

Figure 11. Predicted maximum growth by year (with 95% CIs).

Walleye

Figure 12. Estimated change in von Bertalanffy growth coefficient (k) relative to a typical year by year (with 95% CIs).

Figure 13. Predicted maximum growth by year (with 95% CIs).

Site Fidelity

Figure 14. Probability of recapture at the same site versus a different site by fish length (with 95% CRIs).

Figure 15. Site fidelity by fish length (with 95% CRIs).

Observer Length Correction

Figure 16. Length inaccuracy and imprecision by observer, year and species.

Figure 17. Uncorrected length density plots by species, year and observer.

Figure 18. Corrected length density plots by species, year and observer.

Length-At-Age

Mountain Whitefish

Figure 19. Length-frequency histogram with length-at-age predictions.

Figure 20. Average length of an age-0 individual by year.

Figure 21. Average length of an age-1 individual by year.

Figure 22. Average length of an age-2 individual by year.

Figure 23. Average length of an age-3+ individual by year.

Rainbow Trout

Figure 24. Length-frequency histogram with length-at-age predictions.

Figure 25. Average length of an age-0 individual by year.

Figure 26. Average length of an age-1 individual by year.

Figure 27. Average length of an age-2+ individual by year.

Survival

Adult

Mountain Whitefish

Figure 28. Predicted annual survival for an adult Mountain Whitefish.

Figure 29. Predicted annual efficiency for an adult Mountain Whitefish.

Rainbow Trout

Figure 30. Predicted annual survival for an adult Rainbow Trout.

Figure 31. Predicted annual efficiency for an adult Rainbow Trout.

Walleye

Figure 32. Predicted annual survival for an adult Walleye.

Figure 33. Predicted annual efficiency for an adult Walleye.

Recapture Efficiency

Figure 34. Predicted recapture efficiency by species and life stage (with 95% CRIs).

Mountain Whitefish

Subadult

Figure 35. Predicted recapture efficiency for a subadult Mountain Whitefish by session and year (with 95% CRIs).

Adult

Figure 36. Predicted recapture efficiency for an adult Mountain Whitefish by session and year (with 95% CRIs).

Rainbow Trout

Subadult

Figure 37. Predicted recapture efficiency for a subadult Rainbow Trout by session and year (with 95% CRIs).

Adult

Figure 38. Predicted recapture efficiency for an adult Rainbow Trout by session and year (with 95% CRIs).

Walleye

Figure 39. Predicted recapture efficiency for an adult Walleye by session and year (with 95% CRIs).

Abundance

Figure 40. Effect of counting (versus capture) on encounter efficiency at typical density by species and stage (with 95% CIs).

Figure 41. Effect of counting (versus capture) on overdispersion by species and stage (with 95% CIs).

Mountain Whitefish

Subadult

Figure 42. Estimated abundance of subadult Mountain Whitefish by year (with 95% CIs).

Figure 43. Estimated lineal river count density of subadult Mountain Whitefish by site in a typical year (with 95% CIs).

Figure 44. Estimated evenness of subadult Mountain Whitefish at index sites by year (with 95% CIs).

Figure 45. Estimated density of subadult Mountain Whitefish at non-index relative to index sites by year.

Adult

Figure 46. Estimated abundance of adult Mountain Whitefish by year (with 95% CIs).

Figure 47. Estimated lineal river count density of adult Mountain Whitefish by site in a typical year (with 95% CIs).

Figure 48. Estimated evenness of adult Mountain Whitefish at index sites by year (with 95% CIs).

Figure 49. Estimated density of adult Mountain Whitefish at non-index relative to index sites by year.

Rainbow Trout

Subadult

Figure 50. Estimated abundance of subadult Rainbow Trout by year (with 95% CIs).

Figure 51. Estimated lineal river count density of subadult Rainbow Trout by site in a typical year (with 95% CIs).

Figure 52. Estimated evenness of subadult Rainbow Trout at index sites by year (with 95% CIs).

Figure 53. Estimated density of subadult Rainbow Trout at non-index relative to index sites by year.

Adult

Figure 54. Estimated abundance of adult Rainbow Trout by year (with 95% CIs).

Figure 55. Estimated lineal river count density of adult Rainbow Trout by site in a typical year (with 95% CIs).

Figure 56. Estimated evenness of adult Rainbow Trout at index sites by year (with 95% CIs).

Figure 57. Estimated density of adult Rainbow Trout at non-index relative to index sites by year.

Walleye

Figure 58. Estimated abundance of adult Walleye by year (with 95% CIs).

Figure 59. Estimated lineal river count density of adult Walleye by site in a typical year (with 95% CIs).

Figure 60. Estimated evenness of adult Walleye at index sites by year (with 95% CIs).

Figure 61. Estimated density of adult Walleye at non-index relative to index sites by year.

Survival (Abundance-based)

Mountain Whitefish

Figure 62. Predicted annual survival for adult and subadult Mountain Whitefish.

Rainbow Trout

Figure 63. Predicted annual survival for adult and subadult Rainbow Trout.

Weight

Mountain Whitefish

Figure 64. Predicted weight of an adult Mountain Whitefish by year (with 95% CIs).

Rainbow Trout

Figure 65. Predicted weight of an adult Rainbow Trout by year (with 95% CIs).

Fecundity

Mountain Whitefish

Figure 66. The fecundity-weight relationship for Mountain Whitefish (with 95% CRIs). The data are from Boyer et al (2017).

Figure 67. Predicted fecundity of an adult female Mountain Whitefish by year (with 95% CIs).

Rainbow Trout

Figure 68. Predicted fecundity of an adult female Rainbow Trout by year (with 95% CIs).

Egg Deposition

Mountain Whitefish

Figure 69. Predicted total egg deposition by Mountain Whitefish by year (with 95% CIs).

Rainbow Trout

Figure 70. Predicted total egg deposition by Rainbow Trout by year (with 95% CIs).

Stock-Recruitment

Mountain Whitefish

Figure 71. Predicted stock-recruitment relationship by spawn year (with 95% CRIs).

Figure 72. Predicted egg to age-1 survival by total egg deposition (with 95% CRIs).

Figure 73. Predicted effect of egg loss on the number of age-1 recruits (with 95% CRIs).

Rainbow Trout

Figure 74. Predicted stock-recruitment relationship by spawn year (with 95% CRIs).

Figure 75. Predicted egg to age-1 survival by total egg deposition (with 95% CRIs).

Figure 76. Predicted effect of egg loss on the number of age-1 recruits (with 95% CRIs).

Age-Ratios

Figure 77. Proportion of Age-1 Mountain Whitefish by spawn year.

Figure 78. Percentage Mountain Whitefish egg loss by spawn year.

Figure 79. Proportion of Age-1 Mountain Whitefish by percentage egg loss ratio, labelled by spawn year. The predicted relationship is indicated by the solid black line (with 95% CRIs).

Figure 80. Predicted effect of egg loss on the number of age-1 Mountain Whitefish recruits by egg loss relative to 10% egg loss (with 95% CRIs).

Adjusted Recruitment

Figure 81. Relationship between age-1 Rainbow Trout and age-1 Mountain Whitefish in the same year of capture by Rainbow Trout spawn year (with 95% CIs).

Figure 82. Predicted effect of egg loss on the number of age-1 Rainbow Trout recruits (with 95% CRIs).

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

BC Hydro
- Matt Casselman
- Guy Martel
- Teri Neighbour
- Margo Sadler
- Gillian Kong
Okanagan Nation Alliance
- Evan Smith
- Amy Duncan
WSP
- Brad Hildebrand
- David Roscoe
- Dustin Ford
- Sima Usvyatsov
- Dana Schmidt
- Demitria Burgoon
- Chris King
Poisson Consulting
- Seb Dalgarno
- Robyn Irvine
- Sarah Lyons
- Nadine Hussein
Bronwen Lewis
Blaine Cook
Geoff Sawatzky
Paul Snow
Eleanor Duifhuis
Ross Zeleznik

References

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Boyer, Jan K., Christopher S. Guy, Molly A. H. Webb, Travis B. Horton, and Thomas E. McMahon. 2017. “Reproductive Ecology, Spawning Behavior, and Juvenile Distribution of Mountain Whitefish in the Madison River, Montana.” Transactions of the American Fisheries Society 146 (5): 939–54. https://doi.org/10.1080/00028487.2017.1313778.

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Taylor & Francis.

Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, et al. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.

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Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Golder Associates Ltd. 2013. Lower Columbia River Whitefish Spawning Ground Topography Survey: Year 3 Summary Report. A {Golder} {Associates} {Ltd}. {Report} Nos. 10-1492-0142F. BC Hydro.

Irvine, R. L., J. T. A. Baxter, and J. L. Thorley. 2015. Lower Columbia River Rainbow Trout Spawning Assessment: Year 7 (2014 Study Period). A {Mountain} {Water} {Research} and {Poisson} {Consulting} {Ltd}. {Report}. BC Hydro. http://www.bchydro.com/content/dam/BCHydro/customer-portal/documents/corporate/environment-sustainability/water-use-planning/southern-interior/clbmon-46-yr7-2015-02-04.pdf.

Kery, Marc. 2010. Introduction to WinBUGS for Ecologists: A Bayesian Approach to Regression, ANOVA, Mixed Models and Related Analyses. Elsevier. http://public.eblib.com/EBLPublic/PublicView.do?ptiID=629953.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

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Macdonald, P. D. M., and T. J. Pitcher. 1979. “Age-Groups from Size-Frequency Data: A Versatile and Efficient Method of Analyzing Distribution Mixtures.” Journal of the Fisheries Research Board of Canada 36 (8): 987–1001. https://doi.org/10.1139/f79-137.

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Plummer, Martyn. 2015. JAGS Version 4.0.1 User Manual. http://sourceforge.net/projects/mcmc-jags/files/Manuals/4.x/.

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R Core Team. 2020. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

Subbey, S., J. A. Devine, U. Schaarschmidt, and R. D. M. Nash. 2014. “Modelling and Forecasting Stock-Recruitment: Current and Future Perspectives.” ICES Journal of Marine Science 71 (8): 2307–22. https://doi.org/10.1093/icesjms/fsu148.

Tornqvist, Leo, Pentti Vartia, and Yrjo O. Vartia. 1985. “How Should Relative Changes Be Measured?” The American Statistician 39 (1): 43. https://doi.org/10.2307/2683905.

von Bertalanffy, L. 1938. “A Quantitative Theory of Organic Growth (Inquiries on Growth Laws Ii).” Human Biology 10: 181–213.

Walters, Carl J., and Steven J. D. Martell. 2004. Fisheries Ecology and Management. Princeton University Press.

Cowichan Lake Cuththroat Trout Exploitation Analysis 2023

Tue, 23 Apr 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Dalgarno, S. & Thorley, J.L. (2024) Cowichan Lake Cuththroat Trout Exploitation Analysis 2023. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1638637752.

Background

Cowichan Lake supports an important Cutthroat Trout fishery. To provide information on the natural and fishing mortality, Cutthroat Trout were caught by angling and tagged with acoustic transmitters and/or a $100 reward tag. Some acoustic transmitters were equipped with depth and temperature sensors.

A fixed receiver array was deployed in the lake to monitor fish movements and location year-round. Additional fixed receivers were deployed during the spawning season at known spawning tributaries. A lake census was completed 4 times per year using mobile receivers.

The objectives of this analysis are to:

Estimate fishing and natural mortality.
Estimate annual variation in fishing and natural mortality.
Estimate seasonal variation in fishing and natural mortality.
Estimate effect of handling on natural mortality.
Estimate variation in fishing and natural mortality by fork length.
Estimate spawning probability.
Estimate variation in spawning probability by fork length.
Estimate spawning mortality.

Data Preparation

The data were provided to Poisson Consulting Ltd. by Aswea Porter following extensive data screening.

Statistical Analysis

Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman, Simpson, and Betancourt 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor $\hat{R} \leq 1.05$ (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) $\textrm{ESS} \geq 150$ for each of the monitored parameters (Kery and Schaub 2011, 61).

The condition that parameters describing the effects of secondary (nuisance) explanatory variable(s) have significant p-values was used as a model selection heuristic (Kery and Schaub 2011). Based on a similar argument, the condition that random effects have a standard deviation with a lower 95% CL $>$ 5% of the estimate was used as an additional model selection heuristic. Primary explanatory variables are evaluated based on their estimated effect sizes with 95% CLs (Bradford, Korman, and Higgins 2005)

The analyses were implemented using R version 4.3.3 (R Core Team 2021) and the mbr family of packages.

Model Descriptions

Survival

The natural mortality and recapture probability were estimated using a Bayesian individual state-space survival model (Thorley and Andrusak 2017) with monthly intervals. The survival model incorporated natural, handling and spawning mortality, acoustic detections and movements from a fixed receiver array and quarterly lake census, vertical movements from acoustic tags equipped with depth sensors, FLOY tag loss, recapture, reporting and release.

The expected fork length at monthly period $t$ was estimated from length at capture ($L_{i,fi}$) plus the length increment expected under a Von Bertalanffy Growth Curve (Walters and Martell 2004) \[L_{i,t} = L_{i,fi} + (L_{∞} − L_{i,fi} )(1 − e^{(−\frac{1}{12} \cdot k(t −fi))})\] where $fi$ is the period at capture and the $\frac{1}{12}$ exponent adjusts for the fact that there are 12 periods per year.Linf was fixed to a biologically plausible value of 775 mm based on prior information (personal communication, R. Ptolemy); k of 0.23 was estimated from 88 aged Cutthroat Trout in Cowichan Lake (personal communication, Brendan Andersen).

The probability of spawning at estimated fork length $L$ is determined by

\[S = \frac{L^{S_p}}{L_s^{S_p} + L^{S_p}} \cdot {E_s}\]

where $L_s$ is the length at which 50% of fish are mature, $E_s$ is the probability of a mature fish spawning and $S_p$ is the maturity (as a function of length) power.

Additional key assumptions of the survival model include all but two of the assumptions in Thorley and Andrusak (2017) – the exceptions being 3 (no loss of FLOY tags) and 11 (anglers report all FLOY tags). The survival model also assumes that:

The monthly interval probability of FLOY tag loss is 0.001.
All recaptured fish that are released have their FLOY tags removed.
Reporting of a recaptured fish with a FLOY tag is likely greater than 90%.
The effect of handling mortality is restricted to the first two monthly periods after initial capture or recapture.
The effect of spawning mortality is restricted to the spawning period.
Acoustic tags are functional over their estimated battery lifespan and are not shed.
Recapture probability varies by month and size class.
Natural mortality varies by month, study year and size class.
The probability of detection depends on whether a fish is alive or has died near or far from a receiver and whether a census has occurred.

Study years span from October 01 to September 30 of the following year (i.e., study year 1 is 2019-10-01 to 2020-09-30). Since data collection started in October, use of study year instead of calendar year ensured that all months were represented in each year. Observations after study year 4 (i.e., in October 2023) were removed from this year’s analysis.

Size classes are defined as small ($\le$ 350 mm), medium (351 $-$ 500 mm) and large (> 500 mm). Size class membership can change in each period depending on the fork length predicted by the growth model.

Preliminary analyses indicated lack of support for variation in recapture probability and spawning probability by study year.

We assumed that the monthly interval probability of FLOY tag loss was 0.001 (0.012 annual probability) based on tag loss estimates from double-tagged Lake Trout in Horse Lake.

Preliminary attempts to estimate tag loss in the model using an uninformative prior confirmed that this assumption was consistent with the available information, although we had to fix tag loss in the model to avoid poor convergence.

We assumed that fish detected in monitored spawning tributaries represents all spawning tagged fish in the lake. The ‘spawning window’ was defined as February to May given that there was consistent receiver coverage in spawning tributaries across years. Receivers were also placed in spawning tributaries in January in study years 3 and 4. Four fish that were detected in a spawning tributary only in January (all in study year 4) were not assigned a status of ‘spawning’. Two study years had partial coverage in February (starting Feb 10-11 and Feb 17-19). These spawning detections were included because we assumed that most spawning events would occur toward the end of the month.

Model Templates

Survival

.model{
  bMortality ~ dnorm(-3, 3^-2)
  bMortalityHandling ~ dnorm(0, 2^-2)
  bMonthsHandling <- 2
  bMortalitySpawning ~ dnorm(0, 2^-2)
  bLs ~  dunif(100, 1000)
  bSp ~ dexp(1/20)
  bEs ~ dbeta(1,1)

  bDetectedAlive ~ dbeta(1, 1)
  bDieNear ~ dbeta(1, 1)
  bDetectedDeadNear ~ dbeta(10, 1)
  bDetectedDeadFar ~ dbeta(1, 10)
  bDetectedCensus ~ dbeta(1, 1)
  bMovedH ~ dbeta(1, 1)
  bMovedV ~ dbeta(1, 1)
  bRecaptured ~ dnorm(0, 2^-2)
  bReported ~ dbeta(1, 1)
  bReleased ~ dbeta(1, 1)
  bTagLoss <- 0.001

  bMortalityAnnual[1] <- 0
  for (i in 2:nAnnual) {
    bMortalityAnnual[i] ~ dnorm(0, 2^-2)
  }
  bRecapturedSize[1] <- 0
  for (i in 2:nSize) {
    bRecapturedSize[i] ~ dnorm(0, 2^-2)
  }
  bMortalitySize[1] <- 0
  for (i in 2:nSize) {
    bMortalitySize[i] ~ dnorm(0, 2^-2)
  }
  sRecapturedMonth ~ dexp(1)
  for (i in 1:nMonth) {
    bRecapturedMonth[i] ~ dnorm(0, sRecapturedMonth^-2)
  }
  sMortalityMonth ~ dexp(1)
  for (i in 1:nMonth) {
    bMortalityMonth[i] ~ dnorm(0, sMortalityMonth^-2)
  }
  for (i in 1:nCapture){
    eMortalityHandling[i, PeriodCapture[i]] <- ilogit(bMortalityHandling)
    eMonthsSinceCapture[i,PeriodCapture[i]] <- 1-Recaptured[i,PeriodCapture[i]]
    eMortalitySpawning[i, PeriodCapture[i]] <- ilogit(bMortalitySpawning) * SpawningPeriod[PeriodCapture[i]]
    
    eDetectedAlive[i] <- bDetectedAlive 
    eDieNear[i] ~ dbern(bDieNear)
    eDetectedDeadNear[i] <- bDetectedDeadNear 
    eDetectedDeadFar[i] <- bDetectedDeadFar 
    eDetectedDead[i] <- eDieNear[i] * eDetectedDeadNear[i] + 
      (1 - eDieNear[i]) * eDetectedDeadFar[i]
    
    logit(eMortality[i,PeriodCapture[i]]) <- bMortality + bMortalityAnnual[Annual[PeriodCapture[i]]] + bMortalitySize[Size[i, PeriodCapture[i]]] + bMortalityMonth[Month[PeriodCapture[i]]]

    eMovedH[i,PeriodCapture[i]] <- bMovedH
    eMovedV[i,PeriodCapture[i]] <- bMovedV
    eDetected[i,PeriodCapture[i]] <- Alive[i,PeriodCapture[i]] * eDetectedAlive[i] +
      (1-Alive[i,PeriodCapture[i]]) * eDetectedDead[i]
    eDetectedCensus[i,PeriodCapture[i]] <- bDetectedCensus
    logit(eRecaptured[i,PeriodCapture[i]]) <- bRecaptured + bRecapturedMonth[Month[PeriodCapture[i]]] + bRecapturedSize[Size[i,PeriodCapture[i]]] 
    eReported[i,PeriodCapture[i]] <- bReported
    eReleased[i,PeriodCapture[i]] <- bReleased

    InLake[i,PeriodCapture[i]] <- 1
    Alive[i,PeriodCapture[i]] <- 1
    TBarTag[i,PeriodCapture[i]] ~ dbern(1-bTagLoss)

    Detected[i,PeriodCapture[i]] ~ dbern(Monitored[i,PeriodCapture[i]] * eDetected[i,PeriodCapture[i]])
    DetectedCensus[i,PeriodCapture[i]] ~ dbern(InLake[i,PeriodCapture[i]] * Monitored[i,PeriodCapture[i]] * eDetectedCensus[i,PeriodCapture[i]] * Census[PeriodCapture[i]])
    MovedH[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * Detected[i,PeriodCapture[i]] * eMovedH[i,PeriodCapture[i]])
    MovedV[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * Sensor[i,PeriodCapture[i]] * Detected[i,PeriodCapture[i]] * eMovedV[i,PeriodCapture[i]])
    
    Recaptured[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * eRecaptured[i,PeriodCapture[i]])
    Reported[i,PeriodCapture[i]] ~ dbern(Recaptured[i,PeriodCapture[i]] * eReported[i,PeriodCapture[i]] * TBarTag[i,PeriodCapture[i]])
    Released[i,PeriodCapture[i]] ~ dbern(Recaptured[i,PeriodCapture[i]] * eReleased[i,PeriodCapture[i]])

    for(a in (AnnualCapture[i]):nAnnual){
      eSpawning[i,a] <- (((LengthSpawned[i,a]^bSp) / (bLs^bSp + LengthSpawned[i,a]^bSp)) * bEs) 
    }
  for(j in (PeriodCapture[i]+1):nPeriod) {

    eMortalityHandling[i,j] <- ilogit(bMortalityHandling) * step(bMonthsHandling - eMonthsSinceCapture[i,j-1] - 1)
    eMonthsSinceCapture[i,j] <- (1-Recaptured[i,j]) * (eMonthsSinceCapture[i,j-1] + 1)
    eMortalitySpawning[i,j] <- ilogit(bMortalitySpawning) * Spawned[i,j]
    logit(eMortality[i,j]) <- bMortality + bMortalityAnnual[Annual[j]] + bMortalitySize[Size[i, j]] * (1 - Spawned[i,j]) + bMortalityMonth[Month[j]] * (1 - Spawned[i,j])
    Spawned[i,j] ~ dbern(Monitored[i,j] * SpawningPeriod[j] * eSpawning[i,Annual[j]])
    
    eDetected[i,j] <- Alive[i,j] * eDetectedAlive[i] + (1-Alive[i,j]) * eDetectedDead[i] 
    eDetectedCensus[i,j] <- bDetectedCensus

    eMovedH[i,j] <- bMovedH
    eMovedV[i,j] <- bMovedV
    logit(eRecaptured[i,j]) <-  bRecaptured + bRecapturedMonth[Month[j]] + bRecapturedSize[Size[i,j]]
    eReported[i,j] <- bReported
    eReleased[i,j] <- bReleased

    InLake[i,j] ~ dbern(InLake[i,j-1] * ((1 - (Recaptured[i,j-1])) + Recaptured[i,j-1] * Released[i,j-1]))
    Alive[i,j] ~ dbern(InLake[i,j] * Alive[i,j-1] * (1-eMortality[i,j-1]) * (1-eMortalityHandling[i,j-1]) * (1-eMortalitySpawning[i,j-1])) 
    TBarTag[i,j] ~ dbern(TBarTag[i,j-1] * (1-Recaptured[i,j-1]) * (1-bTagLoss))

    Detected[i,j] ~ dbern(InLake[i,j] * Monitored[i,j] * eDetected[i,j])
    DetectedCensus[i,j] ~ dbern(InLake[i,j] * Monitored[i,j] * eDetectedCensus[i,j] * Census[j])
    MovedH[i,j] ~ dbern(Alive[i,j-1] * Monitored[i,j] * eMovedH[i,j])
    MovedV[i,j] ~ dbern(Alive[i,j] * Sensor[i,j] * Monitored[i,j] * eMovedV[i,j])
    
    Recaptured[i,j] ~ dbern(Alive[i,j] * eRecaptured[i,j])
    Reported[i,j] ~ dbern(Recaptured[i,j] * eReported[i,j] * TBarTag[i,j])
    Released[i,j] ~ dbern(Recaptured[i,j] * eReleased[i,j])
  }
  }

Block 1. Survival model description.

Results

Tables

Survival

Table 1. Parameter descriptions.

Parameter	Description
`bDetectedAlive`	Probability of being detected by a fixed receiver if alive
`bDetectedCensus`	Probability of being detected by a mobile receiver during a census period
`bDetectedDeadFar`	Probability of being detected if dead far from a fixed receiver
`bDetectedDeadNear`	Probability of being detected if dead near to a fixed receiver
`bDieNear`	Probability of dying near to (vs far from) a fixed receiver
`bEs`	The annual probability of a mature fish spawning
`bLs`	The length at which 50% of fish mature.
`bMonthsHandling`	The number of months for which handling affects mortality.
`bMortalityAnnual[i]`	Effect of `i`th study year on `bMortality`
`bMortalityHandling`	Log odds probability of dying after (re)capture in a monthly period.
`bMortalityMonth[i]`	Effect of `i`th month on `bMortality`
`bMortalitySize[i]`	Effect of `i`th size class on `bMortality`
`bMortalitySpawning`	Log odds probability of dying after spawning in a monthly period.
`bMortality`	Log odds probability of dying of natural causes in a monthly period in the 1st season of the 1st study year
`bMovedH`	Probability of moving horizontally if alive and detected in a monthly period
`bMovedV`	Probability of moving vertically if alive and detected in a monthly period
`bRecapturedMonth[i]`	Effect of `i`th month on `bRecaptured`
`bRecapturedSize[i]`	Effect of `i`th size class on `bRecaptured`
`bRecaptured`	Log odds probability of being recaptured if alive in a monthly period
`bReleased`	Probability of being released if recaptured
`bReported`	Probability of being reported if recaptured with a FLOY tag
`bSp`	The maturity (as a function of length) power
`sMortalityMonth`	Standard deviation of the random effect of `bMortalityMonth`
`sRecapturedMonth`	Standard deviation of the random effect of `bRecapturedMonth`

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
bDetectedAlive	0.8375755	0.8210177	0.8533005	10.551708
bDetectedCensus	0.2702202	0.2402739	0.3053491	10.551708
bDetectedDeadFar	0.0094250	0.0050473	0.0156601	10.551708
bDetectedDeadNear	0.9727015	0.9500951	0.9882297	10.551708
bDieNear	0.1737226	0.1156306	0.2454140	10.551708
bEs	0.2428071	0.2083370	0.2829326	10.551708
bLs	407.9968075	392.0481766	422.5025806	10.551708
bMortality	-2.7568576	-3.7142054	-2.1080086	10.551708
bMortalityAnnual[2]	-0.6255056	-1.3414411	0.0440292	3.922352
bMortalityAnnual[3]	-0.7645928	-1.3999920	-0.1520198	5.796821
bMortalityAnnual[4]	0.3050115	-0.2257240	0.8367744	1.820389
bMortalityHandling	-3.1992778	-4.8929420	-2.5421481	10.551708
bMortalitySize[2]	-0.2536786	-0.7361078	0.2431523	1.635829
bMortalitySize[3]	0.7534943	0.1446327	1.4163236	5.693727
bMortalitySpawning	-1.9172027	-2.6785626	-1.3887909	10.551708
bMovedH	0.5812060	0.5603906	0.6007258	10.551708
bMovedV	0.5006144	0.4492197	0.5534175	10.551708
bRecaptured	-4.9707351	-5.8107138	-4.2491255	10.551708
bRecapturedSize[2]	0.8362891	0.1445183	1.6479300	5.907852
bRecapturedSize[3]	1.4332053	0.5753343	2.3031767	8.966746
bReleased	0.4460089	0.3223654	0.5844766	10.551708
bReported	0.9885818	0.9317024	0.9996544	10.551708
bSp	16.8428520	12.0407090	22.9989434	10.551708
sMortalityMonth	0.7897345	0.2619868	1.7048814	10.551708
sRecapturedMonth	0.5277492	0.1083606	1.1209293	10.551708

Table 3. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
13824	25	3	500	30	189	1.038	TRUE

Table 4. The estimated annual interval probabilities (with 95% CIs). ‘Natural Mortality’ is the natural mortality averaged across study year and the three size classes for a non-spawning fish. ‘Natural Mortality Spawned’ is the natural mortality for a fish that spawns. ‘Spawning Mortality’ is the probability of mortality from spawning, assuming that the effect of spawning lasts for fourth months (i.e., the spawning window). ‘Handling Mortality’ is the probability of mortality from handling, assuming that the effect of handling lasts for two months and occurs once in a year. ‘Release’ is the probability of being released if recaptured. ‘Recapture’ is the probability of recapture as the average of the three size classes. ‘Harvest Mortality’ is the recapture probability multiplied by the probability of being harvested if recaptured, assuming that recapture can only occur once in a year.

derived quantity	estimate	lower	upper
Natural Mortality	0.5600	0.4890	0.628
Natural Mortality Spawned	0.7480	0.6540	0.823
Spawning Mortality	0.4220	0.2330	0.590
Handling Mortality	0.0768	0.0148	0.141
Release	0.4460	0.3220	0.584
Recapture	0.1970	0.1440	0.258
Harvest Mortality	0.1080	0.0715	0.152

Spawning

Table 5. The estimated parameters from the spawning by fork length sub-model. ‘Es’ is the probability of a mature fish spawning. ‘Ls’ is the fork length (mm) at which 50% of fish are mature. ‘Sp’ is the maturity power (as a function of fork length).

derived quantity	estimate	lower	upper
Es	0.243	0.208	0.283
Ls	408.000	392.000	423.000
Sp	16.800	12.000	23.000

Figures

Growth

Figure 1. Estimated fork length for each fish by date. Points represent recapture and harvest.

Survival

Figure 2. The estimated annual interval probabilities (with 95% CIs). ‘Natural Mortality’ is the natural mortality averaged across study year and the three size classes for a non-spawning fish. ‘Natural Mortality Spawned’ is the natural mortality for a fish that spawns. ‘Spawning Mortality’ is the probability of mortality from spawning, assuming that the effect of spawning lasts for fourth months (i.e., the spawning window). ‘Handling Mortality’ is the probability of mortality from handling, assuming that the effect of handling lasts for two months and occurs once in a year. ‘Release’ is the probability of being released if recaptured. ‘Recapture’ is the probability of recapture as the average of the three size classes. ‘Harvest Mortality’ is the recapture probability multiplied by the probability of being harvested if recaptured, assuming that recapture can only occur once in a year.

Figure 3. The estimated annual natural mortality by study year averaged across size class for a non-spawning fish (with 95% CIs).

Figure 4. The estimated monthly natural mortality averaged across study year and size class for a non-spawning fish (with 95% CIs).

Figure 5. The estimated annual natural mortality by size class, averaged across study year for a non-spawning fish (with 95% CIs). Size classes are defined as small (<= 350 mm), medium (351 - 500 mm) and large (> 500 mm).

Figure 6. The estimated monthly recapture probability averaged across size class (with 95% CIs).

Figure 7. The estimated annual recapture probability by size class (with 95% CIs). Size classes are defined as small (<= 350 mm), medium (351 - 500 mm) and large (> 500 mm).

Spawning

Figure 8. The probability of spawning by fork length (mm) (with 95% CIs). Points represent whether a fish spawned at each spawning opportunity (i.e., spawning period and monitored).

Recommendations

Data collection
- If feasible, leave receivers in spawning tributaries year-round to observe how often fish are entering outside of the assumed spawning window.
Analyses
- Add seasonal variation to growth model.
- Estimate effect of study year on spawning probability.
- Estimate effect of study year on recapture probability.
- Estimate effect of fork length on natural mortality using a mechanistic relationship.

Acknowledgements

The organisations and individuals whose contributions have made this analytic appendix possible include:

Kintama Research Services
- David Welch
- Aswea Porter
- Paul Winchell
DFO
- Erin Rechisky
MFLNRORD
- Brendan Andersen
- Mike McCulloch
- Jennifer Sibbald
- Scott Silvestri
- Jaro Szczot

References

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Boca Raton: Taylor & Francis.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Boca Raton: Taylor; Francis, CRC Press.

R Core Team. 2021. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Walters, Carl J., and Steven J. D. Martell. 2004. Fisheries Ecology and Management. Princeton, N.J: Princeton University Press.

How to find a wolverine: Factors affecting detection at wolverine (Gulo gulo) bait stations in western Canada

Tue, 19 Mar 2024 00:00:00 +0000

Guidelines for Winter Recreation near Wolverine Dens in Montane Western North America

Sun, 03 Mar 2024 00:00:00 +0000

Llgaay Gwii sdiihlda (Restoring Balance project) 2014-2019

Fri, 16 Feb 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Thorley, J.L. & Irvine, R.L. (2024) Llgaay Gwii sdiihlda (Restoring Balance project) 2014-2019. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/994043377.

An associated paper is available from publications.

Background

To restore balance, a deer removal program was implemented on various islands in Gwaii Haanas.

The primary goal of the current analyses is to answer the following questions:

What was the relative efficiency of the removal methods?

What would the cost of removing the remaining deer have been?

Data Preparation

The data were provided by Parks Canada in the form of Excel spreadsheets and prepared for analysis using R version 4.3.2 (R Core Team 2020).

Statistical Analysis

Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman et al. 2017). The posterior distributions were estimated from 4000 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 4 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor $\hat{R} \leq 1.01$ (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) $\textrm{ESS} \geq 1000$ for each of the monitored parameters (Kery and Schaub 2011, 61).

The parameters are summarised in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.3 bits, which is equivalent to a significant p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row.

Model selection was based on Leave-one-out cross-validation (LOO-CV) as implemented using the Pareto-smoothed importance sampling (PSIS) algorithm (Vehtari et al. 2017). LOO-CV is asymptotically equal to the Widely Applicable Information Criterion Watanabe (2013). Model weight ($w_i$) was based on $\exp(-0.5 \Delta_i)$ as proposed by Akaike (1978) for Akaike Information Criterion (AIC) and by Watanabe (2010) for WAIC where $\Delta_i$ is the absolute difference in the out-of-sample predictive density of the $i$th model relative to the best model (Burnham and Anderson 2002). Primary explanatory variables were evaluated based on their estimated effect sizes with 95% CLs (Bradford et al. 2005)

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the proportion of zeros in the data and the first four central moments (mean, variance, skewness and kurtosis) in the deviance residuals were compared to the expected values by simulating new data based on the posterior distribution and assumed sampling distribution and calculating the deviance residuals. In this context each s-value indicates how surprising it would be to discover that the actual data was generated using the same distribution as the simulated data.

The sensitivity of the parameters to the choice of prior distributions was evaluated by doubling the standard deviations of all the normal and half-normal priors and then using $\hat{R}$ to evaluate whether the samples were drawn from the same posterior distribution (Thorley and Andrusak 2017).

The analyses were implemented using R version 4.3.2 (R Core Team 2022) and the mbr family of packages.

Model Descriptions

Model

The data were analysed using a power function (Ward et al. 2013) for the relationship between efficiency and density

\[\text{Efficiency} = \alpha \cdot {\text{Density}}^{\beta} \]

Four models were considered which varied in whether they included variation in $\alpha$ and $\beta$ by method.

Key assumptions of all the models include:

There is no migration among the islands during the course of the study.
The relationship between efficiency and density is described by a power function.
The expected number of deer removed by island, method and day is the product of the efficiency and effort.
The residual variation in the number deer removed by island, method and day is described by an overdispersed Poisson distribution (negative binomial).

The full model (variation in $\alpha$ and $\beta$ by method = ambm) is defined algebraically as follows:

\[\text{Efficiency}_{m,i,d} = \alpha_{m} \cdot {\text{Density}_{i,d}}^{\beta_{m}} \] where $\text{Efficiency}_{m,i,d}$ is the efficiency of the $m$th method on the $i$th island during the $d$th day, $\alpha_{m}$ is the efficiency of the $m$th method at a density of 1 deer per km², $\text{Density}_{i,d}$ is the deer density (deer/km²) on the $i$th island at the start of the $d$th day and $\beta_{m}$ is the scaling constant.

The allometric coefficient and the scaling exponent in the power function are given by

\[\log(\alpha_{m}) = a_m \] and

\[\log(\beta_{m}) = b_m\]

, respectively, where $a_m$ and $b_m$ are the values by method.

The expected number of deer removed using the $m$th method on the $i$th island during the $d$th day ($\mu_{m,i,d}$) is given by

\[\lambda_{m,i,d} = \text{Efficiency}_{m,i,d} \cdot \text{Effort}_{m,i,d}\]

where $\text{Effort}_{m,i,d}$ is the effort (heli hours) of the $m$th method on the $i$th island during the $d$th day.

The actual number of deer removed using the $m$th method on the $i$th island during the $d$th day ($\mu_{m,i,d}$) is given by the relationship

\[\text{Removed}_{m,i,d} \sim \text{NBinomial}(\lambda_{m,i,d}, \phi)\]

The total population on each island at the start of the study was given by

\[\text{Population}_{i} = \text{TotalRemoved}_i + \Psi_i\]

where $\Psi_i$, which is the total number of remaining deer on the $i$th island, was based on the number of scent trails detected by dogs at the end of the surveys and the effective coverage of those dogs according to the relationship

\[\text{ScentTrails}_i \sim \text{Binomial}(\Psi_{i},\text{Coverage}_i)\]

The priors were as follows:

\[a_m \sim \text{Normal}(1, 2)\]

\[b_m \sim \text{Normal}(0, 2)\] \[\phi \sim \text{Normal}(0, 1)\ \text{T}(0,)\]

\[\Psi_i \sim \text{Normal}(0, \text{Area}_i \cdot 5) \text{T}(0,)\]

Model Templates

Removal

model{
  for(i in 1:nIsland) {
    bRemaining[i] ~ dnorm(0, (5 * Area[i])^-2) T(0,)
    Detections[i] ~ dbin(ObsEff[i], round(bRemaining[i]))
    bPopn[i,1] <- round(TotalRemoved[i] + bRemaining[i])
    bDensity[i,1] <- bPopn[i,1] / Area[i]
    for(j in 2:nDay) { 
      bPopn[i,j] <- bPopn[i,j-1] - DeerTotal[i,j-1]
      bDensity[i,j] <- bPopn[i,j] / Area[i]
    }
  }
  alpha0 <- 0

  for(i in 1:nMethod) {
    alpha_method[i] ~ dnorm(1, 2^-2)
  }

  beta0 <- 0

  for(i in 1:nMethod) {
    beta_method[i] ~ dnorm(0, 2^-2)
  }

  phi ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:nObs) {
    eDensity[i] <- bDensity[Island[i],Day[i]]
    eEffort[i] <- Hours[i] * HourlyRate[i]
    log(eAlpha[i]) <- alpha0 + alpha_method[Method[i]]
    log(eBeta[i]) <- beta0 + beta_method[Method[i]]
    log(eEfficiency[i]) <- eAlpha[i] +  log((eDensity[i] / 100)^eBeta[i])
    eDeer[i] <- eEffort[i] * eEfficiency[i] 
    eR[i] <- 1/phi
    eP[i] <- eR[i] / (eR[i] + eDeer[i])
    Deer[i] ~ dnegbin(eP[i], eR[i])
  }

Block 1. Model description.

Results

Tables

Costs

Table 1. The estimated hourly costs (in $) by crew member and equipment.

Type	HourlyRate
BaitHunter	180
BaitHunterLead	280
BaitHunterAvg	230
BoatPlusOperator	90
HunterAvg	135
Dog	40
DogHunter	90
HeliHunter	405
HeliPlusOperator	1560

Removal

Table 2. The number of scent trail detections by dogs and the effective coverage by island.

Island	ScentTrials	EffectiveCoverage
Ramsay Island	10	0.30
Murchison Island	2	0.75
House Island	0	0.90

Table 3. Model comparison using Pareto Smoothed Importance-Sampling Leave-One-Out Cross-Validation (PSIS) criterion. ‘ic’ is the information criterion value (IC) on the deviance scale; ‘se’ is the standard error of the IC; ‘npars’ is the number of effective parameters, ‘delta ic’ is the difference between the model’s IC and the minimum IC; ‘delta se’ is the standard error of the difference in IC; ‘weight’ summarizes the relative support for each model; and ‘k outliers’ is the proportion of data points with Pareto $\hat{k}$ values exceeding 0.7.

model	ic	se	npars	delta ic	delta se	weight
ambm	1019.026	41.54826	9.472834	0.00000	0.00000	9.995015e-01
amb0	1034.285	43.11210	7.304104	15.25862	8.83647	4.857535e-04
a0bm	1041.568	43.08811	7.276514	22.54128	10.85570	1.273524e-05
a0b0	1065.989	43.28138	3.383383	46.96279	15.62084	6.337856e-11

Table 4. Parameter descriptions.

Parameter	Description
`Area[i]`	Surface area of `i`^th island (km2)
`Day[i]`	Days since April 20th 2017 of `i`^th outing
`DeerTotal[i,j]`	Number of deer removed from `i`^th Island on `j`^th `Day` since April 20th 2017
`Deer[i]`	Number of deer removed during `i`^th outing
`Detections[i]`	Number of scent trail detections on `i`^th island at end of operations
`HourlyRate[i]`	Relative cost of `i`^th outing (helicopter crew hourly rate)
`Hours[i]`	Duration of `i`^th outing (hours)
`Island[i]`	Island of `i`^th outing
`ObsEff[i]`	Effective coverage of scent trail surveys on `i`^th island at end of operations
`TotalRemoved[i]`	Total number of deer removed from `i`^th island during operations
`alpha0`	Intercept for `log(eAlpha)`
`alpha_method[i]`	Effect of `i`^th method on `alpha0`
`bDensity[i,j]`	Expected density of deer per km2 on `i`^th island at start of `j`^th `Day` since April 20th 2017
`bPopn[i,j]`	Expected number of deer on `i`^th island at start of `j`^th `Day` since April 20th 2017
`bRemaining[i]`	Expected number of deer remaining on `i`^th island at end of operations
`beta0`	Intercept for `log(eBeta)`
`beta_method[i]`	Effect of `i`^th method on `beta0`
`eAlpha[i]`	`log(Efficiency)` at a density of 100 deer per km2
`eBeta[i]`	Effect of `log(bDensity * 100)` on `log(Efficiency)`
`phi`	Extra Poisson variation

Table 5. Model terms (with 95% CIs).

term	estimate	lower	upper	svalue
alpha_method[1]	1.6200	0.98100	2.030	12.000
alpha_method[2]	1.2700	1.01000	1.480	12.000
alpha_method[3]	1.1300	0.83300	1.370	12.000
alpha_method[4]	-0.9830	-3.28000	0.310	2.500
alpha_method[5]	-0.4640	-3.44000	0.985	0.624
bRemaining[1]	0.2730	0.01290	1.170	12.000
bRemaining[2]	2.8100	1.57000	5.870	12.000
bRemaining[3]	31.9000	19.20000	50.400	12.000
beta_method[1]	0.9550	0.00441	1.470	4.330
beta_method[2]	0.0176	-0.41600	0.334	0.124
beta_method[3]	0.1220	-0.25000	0.429	1.050
beta_method[4]	-2.0700	-4.64000	-0.662	12.000
beta_method[5]	-0.6440	-1.28000	0.155	3.040
phi	0.1330	0.02300	0.310	12.000

Table 6. Model convergence.

n	K	logLik	IC	nchains	niters	nthin	ess	rhat	converged
485	14	NA	NA	4	1000	200	1672	1.002	TRUE

Table 7. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.001	1.205

Table 8. Model sensitivity by parameter.

parameter	analysis	sensitivity	bound
alpha_method	1.000	1.000	1.193
bDensity	1.002	1.000	1.001
beta_method	1.000	1.001	1.205
bPopn	1.002	1.000	1.001
bRemaining	1.002	1.001	1.001

Table 9. Model sensitivity by fixed terms.

term	analysis	sensitivity	bound
alpha_method[1]	1.000	1.000	1.001
alpha_method[4]	1.000	1.000	1.193
alpha_method[5]	1.000	1.000	1.096
beta_method[1]	1.000	1.001	1.001
beta_method[4]	1.000	1.000	1.205
beta_method[5]	1.000	1.000	1.049
bRemaining[1]	1.001	1.001	1.001
bRemaining[2]	1.002	1.000	1.001

Table 10. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.5443299	0.5434298	0.4922049	0.5947105	0.0123122
mean	-0.2247702	-0.2483574	-0.3332791	-0.1558049	0.7757029
variance	0.9341431	0.8680227	0.7537828	0.9898375	1.8459070
skewness	0.7119498	0.7010032	0.4946737	0.9341840	0.1108871
kurtosis	0.1692013	-0.1078410	-0.5601321	0.6157612	1.4793099

Table 11. The total number of deer removed, the area (km2) and the removal density (deer/km2) by island.

Island	Removed	Area	RemovalDensity
House	10	0.3292	30.376671
Murchison	26	3.9987	6.502113
Ramsay	412	16.2276	25.388844

Table 12. The total number of deer removed and the estimated number of remaining deer by island (with CIs).

Island	Removed	lower98	lower95	estimate	upper95	upper98
House	10	0	0	0	1	1
Murchison	26	2	2	3	6	7
Ramsay	412	17	19	32	50	54

Table 13. The total number of deer removed and the estimated percent eradication completion by island (with CIs).

Island	Removed	lower98	lower95	estimate	upper95	upper98
House	10	0.9090909	0.9090909	1.0000000	1.0000000	1.0000000
Murchison	26	0.7878788	0.8125000	0.8965517	0.9285714	0.9285714
Ramsay	412	0.8841202	0.8917749	0.9279279	0.9559165	0.9603730

Table 14. The total costs spent in heli hours by island.

Island	Cost	Area	Cost/km2
House	10.44707	0.3292	31.734732
Murchison	33.69046	3.9987	8.425353
Ramsay	282.61980	16.2276	17.415995

Table 15. The estimated total completion cost by island and method (with CIs).

Island	Method	lower98	lower95	estimate	upper95	upper98
House	Indicator Dog	0.0000000	0.0000000	0.000000	1.376753	2.478155
House	Boat	0.0000000	0.0000000	0.000000	1.648855	2.748092
Murchison	Indicator Dog	0.2753505	0.5507011	3.854908	18.173136	24.781550
Murchison	Boat	1.0992366	2.1984733	18.137405	122.564886	169.837557
Ramsay	Indicator Dog	16.2429289	18.7169536	42.954686	129.696993	163.304904
Ramsay	Boat	35.7251908	46.7175573	194.015267	1124.093130	1777.075420

Table 16. The estimated proportional efficiency of indicator dog hunting by method and density.

Method	Density	estimate	lower	upper
Combined	2	2.5248852	0.6743702	6.582835
Boat	2	0.2672479	-0.4415746	1.954678

Table 17. The total number of deer removed and the proportion of the encounters removed by method.

Method	Removed	EncountersRemoved
Bait Station	90	0.90
Boat	101	0.57
Helicopter	138	0.49
Indicator Dog	75	0.71
Miscellaneous	15	0.75
NA	29	0.74

Table 18. The total number of deer removed on Ramsay Island and the proportion of the encounters removed by method.

Method	Removed	EncountersRemoved
Bait Station	90	0.90
Boat	97	0.56
Helicopter	136	0.49
Indicator Dog	64	0.68
Miscellaneous	5	0.71
NA	20	0.69

Figures

Removal

Figure 1. The removal efficiency by date, method and island with the estimated removal efficiency.

Figure 2. The estimated removal efficiency by density and method.

Figure 3. The estimated removal efficiency by method and density.

Figure 4. The posterior probability of the percent eradication completion by island.

Figure 5. The estimated eradication completion by cumulative effort by method for Ramsay with a starting population of 466 deer.

Acknowledgements

The organisations and individuals whose contributions have made this analytic appendix possible include:

Parks Canada
- Nadine Wilson
- Peter Dyment
- Charlotte Houston
- Patrick Bartier
- Christine Bentley
- Emily Gonzales
- Kent Prior
Coastal Conservation
- Chris Gill
- Greg Howald
Poisson Consulting
- Seb Dalgarno
Simon Fraser University
- Carl Schwarz
David Will
Pete McLelland
Norm MacDonald
Gerry Morigeau

References

Akaike, Hirotugu. 1978. “On the Likelihood of a Time Series Model.” Journal of the Royal Statistical Society: Series D (The Statistician) 27 (3-4): 217–35. https://doi.org/10.2307/2988185.

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Taylor & Francis.

Burnham, Kenneth P., and David R. Anderson. 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Second Edition. Springer.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Taylor; Francis, CRC Press.

R Core Team. 2020. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

R Core Team. 2022. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

Ward, Hillary G. M., Paul J. Askey, John R. Post, and Kenneth Rose. 2013. “A Mechanistic Understanding of Hyperstability in Catch Per Unit Effort and Density-Dependent Catchability in a Multistock Recreational Fishery.” Canadian Journal of Fisheries and Aquatic Sciences 70 (10): 1542–50. https://doi.org/10.1139/cjfas-2013-0264.

Watanabe, Sumio. 2010. “Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory.” Journal of Machine Learning Research 11 (Dec): 3571–94. http://www.jmlr.org/papers/v11/watanabe10a.html.

Watanabe, Sumio. 2013. “A Widely Applicable Bayesian Information Criterion.” Journal of Machine Learning Research 14 (Mar): 867–97. http://www.jmlr.org/papers/v14/watanabe13a.html.

Relative efficiency of hunting methods during an incomplete Sitka black-tailed deer eradication on Haida Gwaii, Canada

Wed, 07 Feb 2024 00:00:00 +0000

Impacts of a century of land‐use change on the eutrophication of large, shallow, prairie Lake Manitoba in relation to adjacent Lake Winnipeg (Manitoba, Canada)

Mon, 01 Jan 2024 00:00:00 +0000

Assessing diverse evidence to improve conservation decision‐making

Wed, 06 Sep 2023 00:00:00 +0000

Shiny Upload App User Guide

Thu, 27 Jul 2023 00:00:00 +0000

We have created a document to guide users through the steps for submitting their data to Poisson via the shiny upload app.

You can find the user guide at Shiny Upload App User Guide.

Lower Columbia River Fish Population Indexing 2022

Tue, 13 Jun 2023 00:00:00 +0000

The suggested citation for this analytic appendix is:

Thorley, J.L., Dalgarno, S., Lyons, S. (2023) Lower Columbia River Fish Population Indexing 2022. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1392245070.

Background

In the mid 1990s BC Hydro began operating Hugh L. Keenleyside (HLK) Dam to reduce dewatering of Mountain Whitefish and Rainbow Trout eggs.

The primary goal of the Lower Columbia River Fish Population Indexing program is to answer two key management questions:

What are the abundance, growth rate, survival rate, body condition, age distribution, and spatial distribution of subadult and adult Whitefish, Rainbow Trout, and Walleye in the Lower Columbia River?

What is the effect of inter-annual variability in the Whitefish and Rainbow Trout flow regimes on the abundance, growth rate, survival rate, body condition, and spatial distribution of subadult and adult Whitefish, Rainbow Trout, and Walleye in the Lower Columbia River?

The inter-annual variability in the Whitefish and Rainbow Trout flow regimes was quantified in terms of the percent egg dewatering as greater flow variability is associated with more egg stranding.

Methods

Data Preparation

The fish indexing data were provided by Okanagan Nation Alliance and Golder Associates in the form of an Access database. The discharge and temperature data were obtained from the Columbia Basin Hydrological Database maintained by Poisson Consulting. The Rainbow Trout egg dewatering estimates were provided by CLBMON-46 (Irvine, Baxter, and Thorley 2015) and the Mountain Whitefish egg stranding estimates by Golder Associates (2013).

The data were prepared for analysis using R version 4.3.0 (R Core Team 2018).

Discharge

Data Analysis

The one exception is the length-at-age estimates which were produced using the mixdist R package (P. Macdonald 2012) which implements Maximum Likelihood with Expectation Maximization.

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the number of zeros and the first four central moments (mean, variance, skewness and kurtosis) for the deviance residuals were compared to the expected values by simulating new residuals. In this context the s-value indicates how surprising each metric is given the estimated posterior probability distribution for the residual variation.

Where computationally practical, the sensitivity of the parameters to the choice of prior distributions was evaluated by increasing the standard deviations of all normal, half-normal and log-normal priors by an order of magnitude and then using $\hat{R}$ to test whether the samples where drawn from the same posterior distribution (Thorley and Andrusak 2017).

The analyses were implemented using R version 4.3.0 (R Core Team 2020) and the mbr family of packages.

Model Descriptions

Condition

The expected weight of fish of a given length were estimated from the data using an allometric mass-length model (He et al. 2008).

\[W = \alpha L^{\beta}\]

Key assumptions of the condition model include:

The expected weight is allowed to vary with length and date.
The expected weight is allowed to vary randomly with year.
The relationship between weight and length is allowed to vary with date.
The relationship between weight and length is allowed to vary randomly with year.
The residual variation in weight is log-normally distributed.

Growth

\[ \frac{\text{d}L}{\text{d}t} = k (L_{\infty} - L)\]

where $L$ is the length of the individual, $k$ is the growth coefficient and $L_{\infty}$ is the maximum length.

Integrating the above equation gives:

\[ L_t = L_{\infty} (1 - e^{-k(t - t_0)})\]

where $L_t$ is the length at time $t$ and $t_0$ is the time at which the individual would have had zero length.

The Fabens form allows

\[ L_r = L_c + (L_{\infty} - L_c) (1 - e^{-kT})\]

where $L_r$ is the length at recapture, $L_c$ is the length at capture and $T$ is the time between capture and recapture.

Key assumptions of the growth model include:

The mean maximum length $L_{\infty}$ is constant.
The growth coefficient $k$ is allowed to vary randomly with year.
The residual variation in growth is normally distributed.

The growth model was only fitted to Walleye with a fork length at release less than 450 mm.

Movement

The expected site fidelity is allowed to vary with fish length.
Observed site fidelity is Bernoulli distributed.

Length as a second-order polynomial was not found to be a significant predictor for site fidelity.

Length-At-Age

The expected length-at-age of Mountain Whitefish and Rainbow Trout were estimated from annual length-frequency distributions using a finite mixture distribution model (P. D. M. Macdonald and Pitcher 1979)

Rainbow Trout and Mountain Whitefish were categorized as Fry (Age-0), Juvenile (Age-1) and Adult (Age-2+) based on their length-based ages. All Walleye were considered to be Adults.

Survival

The annual adult survival rate was estimated by fitting a Cormack-Jolly-Seber model (Kery and Schaub 2011, 220–31) to inter-annual recaptures of adults.

Key assumptions of the survival model include:

Survival varies randomly with year.
The encounter probability for adults is allowed to vary with the total bank length sampled.

Preliminary analysis indicated that only including visits to index sites did not substantially change the results.

Observer Length Correction

The annual bias (inaccuracy) and error (imprecision) in observer’s fish length estimates were quantified from the divergence of the length distribution of their observed fish from the length distribution of the measured fish. More specifically, the length correction that minimised the Jensen-Shannon divergence (Lin 1991) between the two distributions provided a measure of the inaccuracy while the minimum divergence (the Jensen-Shannon divergence was calculated with log to base 2 which means it lies between 0 and 1) provided a measure of the imprecision.

Capture Efficiency

The probability of capture was estimated using a recapture-based binomial model (Kery and Schaub 2011, 134–36, 384–88).

Key assumptions of the capture efficiency model include:

The capture probability varies randomly by session within year.
The probability of a marked fish remaining at a site is the estimated site fidelity.
The number of recaptures is described by a binomial distribution.

Preliminary analyses indicated that the direction of effect of the frequency of the electrofishing current (30, 60 or 120 Hz) was uncertain.

Abundance

The abundance was estimated from the catch and bias-corrected observer count data using an overdispersed Poisson model (Kery and Schaub 2011, 55–56).

Key assumptions of the abundance model include:

The fish density varies randomly with site, year and site within year.
The change in fish density with overall abundance varies by site density.
The capture efficiency at a typical fish density is the point estimate for a typical session from the capture efficiency model.
The count efficiency varies from the capture efficiency.
The capture efficiency (but not the count efficiency) varies with density.
The overdispersion varies by visit type (count or catch).
The catches and counts are described by a gamma-Poisson distribution.

Distribution

\[ E = \frac{-\sum p_i \log(p_i)}{\log(S)}\]

Survival (Abundance-based)

The subadult ($S_t$) and adult ($A_t$) abundance estimates were used to calculate the subadult and adult survival ($\phi_t$) in year $t$ based on the relationship

\[\phi_t = \frac{A_t}{S_{t-1} + A_{t-1}}\]

Weight

The weight ($W_t$) in year $t$ was estimated from the expected adult length using the condition model.

Fecundity

Mountain Whitefish

\[F = \alpha W^{\beta}\]

Key assumptions of the fecundity model include:

The residual variation in fecundity is log-normally distributed.

Rainbow Trout

Following (Andrusak and Thorley 2019) the fecundity ($F_t$) in year $t$ of an adult female Rainbow Trout was calculated from the expected weight ($W_t$) in grams using the equation:

\[F_t = 3.8 \cdot W_t^{0.9}\]

Egg Deposition

The total egg deposition ($E_t$) in year $t$ was calculated according to the equation \[E_t = F_t * \frac{A_t}{2}\]

Stock-Recruitment

\[S_{t+1} = \frac{\alpha \cdot E_t}{1 + \beta \cdot E_t}\]

where $\alpha$ is the egg to age-1 survival at low density and $\beta$ is the density-dependence.

Key assumptions of the stock-recruitment model include:

The egg to recruit survival at low density ($\alpha$) was likely less than 1% (the prior distribution for $\alpha$ was a zero truncated normal with standard deviation of 0.005.
The expected log number of recruits varies with the proportional egg loss.
The residual variation in the number of recruits is log-normally distributed.

The expected egg survival for a given egg deposition is $S / E_t$ which is given by the equation

\[\phi_E = \frac{\alpha}{1 + \beta * E}\]

Age-Ratios

The proportion of Age-1 Mountain Whitefish $r^1_t$ from a given spawn year $t$ is calculated from the relative abundance of Age-1 & Age-2 fish $N^1_t$ & $N^2_t$ respectively, which were lead or lagged so that all values were with respect to the spawn year:

\[r^1_t = \frac{N^1_{t+2}}{N^1_{t+2} + N^2_{t+2}}\]

The relative abundances of Age-1 and Age-2 fish were taken from the proportions of each age-class in the length-at-age analysis.

As the number of Age-2 fish might be expected to be influenced by the percentage egg loss $Q_t$ three years prior, the predictor variable $\Pi_t$ used is:

\[\Pi_t = \textrm{log}(Q_t/Q_{t-1})\]

The ratio was logged to ensure it was symmetrical about zero (Tornqvist, Vartia, and Vartia 1985).

The relationship between $r^1_t$ and $\Pi_t$ was estimated using a Bayesian regression (Kery 2010) loss model.

Key assumptions of the final model include:

The log odds of the proportion of Age-1 fish varies with the log of the ratio of the percent egg losses.
The residual variation is normally distributed.

Adjusted Recruitment

The relationship(s) were estimated using a Generalized Linear Model (GLM).

Key assumptions of the final model include:

The abundance of Age-1 Rainbow Trout varies with proportion of the eggs dewatered and then number of Age-1 Mountain Whitefish caught in the same year.
The residual variation is log-normally distributed.

Model Templates

Condition

 data {
  int nYear;
  int nObs;

  vector[nObs] Length;
  vector[nObs] Weight;
  vector[nObs] Dayte;
  int Year[nObs];

parameters {
  real bWeight;
  real bWeightLength;
  real bWeightDayte;
  real bWeightLengthDayte;
  real<lower=0> sWeightYear;
  real<lower=0> sWeightLengthYear;

  vector[nYear] bWeightYear;
  vector[nYear] bWeightLengthYear;
  real<lower=0> sWeight;

model {

  vector[nObs] eWeight;

  bWeight ~ normal(5, 4);
  bWeightLength ~ normal(3, 1);

  bWeightDayte ~ normal(0, 1);
  bWeightLengthDayte ~ normal(0, 1);

  sWeightYear ~ normal(0, 1);
  sWeightLengthYear ~ normal(0, 1);

  for (i in 1:nYear) {
    bWeightYear[i] ~ normal(0, sWeightYear);
    bWeightLengthYear[i] ~ normal(0, sWeightLengthYear);
  }

  sWeight ~ normal(0, 5);
  for(i in 1:nObs) {
    eWeight[i] = bWeight + bWeightDayte * Dayte[i] + bWeightYear[Year[i]] + (bWeightLength + bWeightLengthDayte * Dayte[i] + bWeightLengthYear[Year[i]]) * Length[i];
    Weight[i] ~ lognormal(eWeight[i], sWeight);
  }

Block 1. Model description.

Growth

.model {
  bK ~ dnorm (0, 5^-2)
  sKYear ~ dnorm(0, 2^-2) T(0,)

  for (i in 1:nYear) {
    bKYear[i] ~ dnorm(0, sKYear^-2)
    log(eK[i]) <- bK + bKYear[i]
  }

  bLinf ~ dunif(200, 1000)
  sGrowth ~ dnorm(0, 25^-2) T(0,)
  for (i in 1:length(Year)) {
    eGrowth[i] <- max(0, (bLinf - LengthAtRelease[i]) * (1 - exp(-sum(eK[Year[i]:(Year[i] + dYears[i] - 1)]))))
    Growth[i] ~ dnorm(eGrowth[i], sGrowth^-2)
  }

Block 2.

Movement

.model {

  bFidelity ~ dnorm(0, 1^-2)
  bLength ~ dnorm(0, 1^-2)

  for (i in 1:length(Fidelity)) {
    logit(eFidelity[i]) <- bFidelity + bLength * Length[i]
    Fidelity[i] ~ dbern(eFidelity[i])
  }

Block 3.

Survival

.model{
  bEfficiency ~ dnorm(0, 4^-2)
  bEfficiencySampledLength ~ dnorm(0, 4^-2)

  bSurvival ~ dnorm(0, 4^-2)

  sSurvivalYear ~ dnorm(0, 4^-2) T(0,)
  for(i in 1:nYear) {
    bSurvivalYear[i] ~ dnorm(0, sSurvivalYear^-2)
  }

  for(i in 1:(nYear-1)) {
    logit(eEfficiency[i]) <- bEfficiency + bEfficiencySampledLength * SampledLength[i]
    logit(eSurvival[i]) <- bSurvival + bSurvivalYear[i]

    eProbability[i,i] <- eSurvival[i] * eEfficiency[i]
    for(j in (i+1):(nYear-1)) {
      eProbability[i,j] <- prod(eSurvival[i:j]) * prod(1-eEfficiency[i:(j-1)]) * eEfficiency[j]
    }
    for(j in 1:(i-1)) {
      eProbability[i,j] <- 0
    }
  }
  for(i in 1:(nYear-1)) {
    eProbability[i,nYear] <- 1 - sum(eProbability[i,1:(nYear-1)])
  }

  for(i in 1:(nYear - 1)) {
    Marray[i, 1:nYear] ~ dmulti(eProbability[i,], Released[i])
  }

Block 4.

Capture Efficiency

.model {

  bEfficiency ~ dnorm(-4, 2^-2)

  sEfficiencySessionAnnual ~ dnorm(0, 1^-2) T(0,)
  for (i in 1:nSession) {
    for (j in 1:nAnnual) {
      bEfficiencySessionAnnual[i, j] ~ dnorm(0, sEfficiencySessionAnnual^-2)
    }
  }

  for (i in 1:length(Recaptures)) {

    logit(eEfficiency[i]) <- bEfficiency + bEfficiencySessionAnnual[Session[i], Annual[i]]

    eFidelity[i] ~ dnorm(Fidelity[i], FidelitySD[i]^-2) T(FidelityLower[i], FidelityUpper[i])
    Recaptures[i] ~ dbin(eEfficiency[i] * eFidelity[i], Tagged[i])
  }

Block 5.

Abundance

.model {
  bDensity ~ dnorm(5, 4^-2)
  bDensitySiteAnnual2 ~ dnorm(0, 2^-2)

  sDensityAnnual ~ dnorm(0, 1^-2) T(0,)
  for (i in 1:nAnnual) {
    bDensityAnnual[i] ~ dnorm(0, sDensityAnnual^-2)
  }

  sDensitySite ~ dnorm(0, 1^-2) T(0,)
  sDensitySiteAnnual ~ dnorm(0, 1^-2) T(0,)
  for (i in 1:nSite) {
    bDensitySite[i] ~ dnorm(0, sDensitySite^-2)
    for (j in 1:nAnnual) {
      bDensitySiteAnnual[i, j] ~ dnorm(0, sDensitySiteAnnual^-2)
    }
  }

  bEfficiencyVisitType[1] <- 0
  bEfficiencyVisitTypeDensity[1] ~ dnorm(0, 2^-2)
  for (i in 2:nVisitType) {
    bEfficiencyVisitType[i] ~ dnorm(0, 2^-2)
    bEfficiencyVisitTypeDensity[i] <- 0
  }

  sDispersion ~ dnorm(0, 1^-2)
  sDispersionVisitType[1] <- 0
  for(i in 2:nVisitType) {
    sDispersionVisitType[i] ~ dnorm(0, 2^-2)
  }

  for (i in 1:length(Fish)) {
    log(eDensity[i]) <- bDensity + bDensitySite[Site[i]] + bDensityAnnual[Annual[i]] + bDensitySiteAnnual[Site[i],Annual[i]] + bDensitySiteAnnual2 * exp(bDensity + bDensitySite[Site[i]]) * bDensityAnnual[Annual[i]]

    eAbundance[i] <- eDensity[i] * SiteLength[i]

    logit(eEfficiency[i]) <- logit(Efficiency[i]) + bEfficiencyVisitType[VisitType[i]] + bEfficiencyVisitTypeDensity[VisitType[i]] * (eDensity[i] - exp(bDensity + sDensityAnnual^2/2 + sDensitySite^2/2 + sDensitySiteAnnual^2/2))

    log(esDispersion[i]) <- sDispersion + sDispersionVisitType[VisitType[i]]

    eDispersion[i] ~ dgamma(esDispersion[i]^-2 + 0.1, esDispersion[i]^-2 + 0.1)
    eFish[i] <- eAbundance[i] * ProportionSampled[i] * eEfficiency[i]
    Fish[i] ~ dpois(eFish[i] * eDispersion[i])
  }

Block 6.

Fecundity

model {
  bFecundity ~ dnorm(0, 5^-2)
  bFecundityWeight ~ dnorm(1, 1^-2) T(0,)

  sFecundity ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(Weight)) {
    eFecundity[i] = bFecundity + bFecundityWeight * log(Weight[i])
    Fecundity[i] ~ dlnorm(eFecundity[i], sFecundity^-2)
  }

Block 7.

Stock-Recruitment

.model {
  bAlpha ~ dnorm(0, 0.003^-2) T(0,)
  bBeta ~ dnorm(0, 0.007^-2) T(0, )
  bEggLoss ~ dnorm(0, 100^-2)

  sRecruits ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(Recruits)){
    log(eRecruits[i]) <- log(bAlpha * Eggs[i] / (1 + bBeta * Eggs[i])) + bEggLoss * EggLoss[i]
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

Block 8.

Age-Ratios

.model{
  bProbAge1 ~ dnorm(0, 1^-2)
  bProbAge1Loss ~ dnorm(0, 1^-2)

  sProbAge1 ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(Age1Prop)){
    eAge1Prop[i] <- bProbAge1 + bProbAge1Loss * LossLogRatio[i]
    Age1Prop[i] ~ dnorm(eAge1Prop[i], sProbAge1^-2)
  }

Block 9.

Adjusted Recruitment

.model{
  b0 ~ dnorm(0, 10^-2)
  bMw ~ dnorm(0, 2^-2)
  bEggLoss ~ dnorm(0, 2^-2)

  sRb ~ dnorm(0, 2^-2) T(0,)
  for (i in 1:nObs) {
    log(eRb[i]) <- b0 + bMw * Mw[i] + bEggLoss * EggLoss[i]
    Rb[i] ~ dlnorm(log(eRb[i]), sRb^-2)
  }

Block 10. Model description.

Results

Tables

Condition

Table 1. Parameter descriptions.

Parameter	Description
`Dayte[i]`	Standardised day of year `i`^th fish was captured
`Length[i]`	Log-transformed and centered fork length of `i`^th fish
`Weight[i]`	Recorded weight of `i`^th fish
`Year[i]`	Year `i`^th fish was captured
`bWeightDayte`	Effect of `Dayte` on `bWeight`
`bWeightLengthDayte`	Effect of `Dayte` on `bWeightLength`
`bWeightLengthYear[i]`	Effect of `i`^th `Year` on `bWeightLength`
`bWeightLength`	Intercept of effect of `Length` on `bWeight`
`bWeightYear[i]`	Effect of `i`^th `Year` on `bWeight`
`bWeight`	Intercept of `log(eWeight)`
`eWeight[i]`	Expected `Weight` of `i`^th fish
`sWeightLengthYear`	Log standard deviation of `bWeightLengthYear`
`sWeightYear`	Log standard deviation of `bWeightYear`
`sWeight`	Log standard deviation of residual variation in `log(Weight)`

Mountain Whitefish

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
bWeight	5.4732409	5.4547642	5.4935278	10.55171
bWeightDayte	-0.0203585	-0.0237843	-0.0171574	10.55171
bWeightLength	3.1698089	3.1228550	3.2100053	10.55171
bWeightLengthDayte	-0.0206998	-0.0296748	-0.0115355	10.55171
sWeight	0.1463908	0.1447608	0.1479743	10.55171
sWeightLengthYear	0.1094173	0.0800618	0.1553615	10.55171
sWeightYear	0.0485446	0.0365079	0.0672782	10.55171

Table 3. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
16748	7	3	500	2	330	1.013	TRUE

Table 4. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0001197	-0.0001267	-0.0151571	0.0149550	0.0409441
variance	0.9973298	1.0001656	0.9798047	1.0214428	0.3192873
skewness	-0.6017602	-0.0001625	-0.0374198	0.0364191	10.5517083
kurtosis	1.8037636	-0.0019891	-0.0724094	0.0731123	10.5517083

Table 5. Model sensitivity.

all	analysis	sensitivity	bound
all	1.013	1.016	1.01

Rainbow Trout

Table 6. Model coefficients.

term	estimate	lower	upper	svalue
bWeight	6.0370978	6.0270566	6.0475897	10.55171
bWeightDayte	-0.0039410	-0.0061160	-0.0017446	10.55171
bWeightLength	2.9264697	2.9052969	2.9479187	10.55171
bWeightLengthDayte	0.0380958	0.0312706	0.0443072	10.55171
sWeight	0.0999041	0.0988578	0.1009943	10.55171
sWeightLengthYear	0.0517524	0.0370216	0.0733665	10.55171
sWeightYear	0.0249434	0.0186728	0.0347163	10.55171

Table 7. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
18015	7	3	500	2	334	1.015	TRUE

Table 8. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0003917	0.0000477	-0.0144834	0.0149080	0.0830842
variance	0.9970646	1.0006205	0.9798205	1.0202371	0.4444912
skewness	-0.6767363	0.0003559	-0.0347633	0.0356899	10.5517083
kurtosis	2.2597359	-0.0012144	-0.0660660	0.0765119	10.5517083

Table 9. Model sensitivity.

all	analysis	sensitivity	bound
all	1.015	1.007	1.007

Walleye

Table 10. Model coefficients.

term	estimate	lower	upper	svalue
bWeight	6.2931412	6.2784639	6.3082132	10.551708
bWeightDayte	0.0154347	0.0130148	0.0178195	10.551708
bWeightLength	3.2299131	3.1996511	3.2615267	10.551708
bWeightLengthDayte	-0.0057003	-0.0203522	0.0090376	1.191959
sWeight	0.0918887	0.0906741	0.0931940	10.551708
sWeightLengthYear	0.0727259	0.0512623	0.1036775	10.551708
sWeightYear	0.0353325	0.0271249	0.0489660	10.551708

Table 11. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
10635	7	3	500	2	256	1.014	TRUE

Table 12. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0004620	0.0002671	-0.0171674	0.0199111	0.0330551
variance	0.9957787	1.0006582	0.9741117	1.0263279	0.4655720
skewness	-0.0187410	-0.0010659	-0.0458653	0.0452100	1.2365587
kurtosis	0.9703957	-0.0024500	-0.0918807	0.0940346	10.5517083

Table 13. Model sensitivity.

all	analysis	sensitivity	bound
all	1.014	1.005	1.01

Growth

Table 14. Parameter descriptions.

Parameter	Description
`Growth[i]`	Observed growth between release and recapture of `i`^th recapture
`LengthAtRelease[i]`	Length at previous release of `i`^th recapture
`Year[i]`	Release year of `i`^th recapture
`bKYear[i]`	Effect of `i`^th `Year` on `bK`
`bK`	Intercept of `log(eK)`
`bLinf`	Mean maximum length
`dYears[i]`	Years between release and recapture of `i`^th recapture
`eGrowth`	Expected `Growth` between release and recapture
`eK[i]`	Expected von Bertalanffy growth coefficient from `i-1`^th to `i`^th year
`sGrowth`	Log standard deviation of residual variation in `Growth`
`sKYear`	Log standard deviation of `bKYear`

Mountain Whitefish

Table 15. Model coefficients.

term	estimate	lower	upper	svalue
bK	-0.9384301	-1.1484168	-0.7404093	10.55171
bLinf	399.3945885	393.9249703	404.2050797	10.55171
sGrowth	11.5473791	10.6856803	12.6933727	10.55171
sKYear	0.3869568	0.2565911	0.5909488	10.55171

Table 16. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
298	4	3	500	50	938	1.002	TRUE

Table 17. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0268456	0.0000000	0.0000000	0.0000000	10.5517083
mean	0.0369528	-0.0012119	-0.1193993	0.1115377	0.9241744
variance	0.9355362	0.9927556	0.8435511	1.1775451	1.0538564
skewness	-0.2212812	-0.0064231	-0.2617183	0.2655630	3.4121569
kurtosis	0.4357409	-0.0645029	-0.4722527	0.5628374	3.0048138

Table 18. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.004	1.002

Rainbow Trout

Table 19. Model coefficients.

term	estimate	lower	upper	svalue
bK	-0.1091626	-0.2568478	0.0412570	2.816999
bLinf	480.7711096	476.0357000	485.3382333	10.551708
sGrowth	29.9495580	28.8233850	31.0386148	10.551708
sKYear	0.3014960	0.2208877	0.4419149	10.551708

Table 20. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1479	4	3	500	50	994	1.003	TRUE

Table 21. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0047329	0.0000000	0.0000000	0.0000000	10.5517083
mean	0.0113159	-0.0007548	-0.0489893	0.0500167	0.6493331
variance	0.9832970	0.9995337	0.9272834	1.0759454	0.6048020
skewness	0.2639783	-0.0017621	-0.1230891	0.1249517	10.5517083
kurtosis	0.6337016	-0.0127135	-0.2312639	0.2744559	10.5517083

Table 22. Model sensitivity.

all	analysis	sensitivity	bound
all	1.003	1.004	1.002

Walleye

Table 23. Model coefficients.

term	estimate	lower	upper	svalue
bK	-2.4702742	-2.9888898	-2.0531680	10.55171
bLinf	725.0025161	615.7184191	935.5042341	10.55171
sGrowth	17.6588686	16.2562102	19.2704080	10.55171
sKYear	0.3160298	0.1960807	0.4982604	10.55171

Table 24. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
293	4	3	500	50	285	1.017	TRUE

Table 25. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0068259	0.0000000	0.0000000	0.0000000	10.551708
mean	0.0458839	-0.0009061	-0.1241999	0.1136060	1.310917
variance	0.9357665	0.9965045	0.8463250	1.1696999	1.140197
skewness	0.1750667	-0.0049900	-0.2699366	0.2805990	2.155104
kurtosis	1.5559537	-0.0476233	-0.4748618	0.6507876	8.966746

Table 26. Model sensitivity.

all	analysis	sensitivity	bound
all	1.017	1.009	1.014

Movement

Table 27. Parameter descriptions.

Parameter	Description
`Fidelity[i]`	Whether the `i`^th recapture was encountered at the same site as the previous encounter
`Length[i]`	Length at previous encounter of `i`^th recapture
`bFidelity`	Intercept of `logit(eFidelity)`
`bLength`	Effect of length on `logit(eFidelity)`
`eFidelity[i]`	Expected site fidelity of `i`^th recapture

Mountain Whitefish

Table 28. Model coefficients.

term	estimate	lower	upper	svalue
bFidelity	-0.2657959	-0.6044753	0.0708722	3.0048138
bLength	-0.0453499	-0.4329178	0.2909701	0.3289134

Table 29. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
131	2	3	500	1	770	1.002	TRUE

Table 30. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.5648855	0.5648855	0.4427481	0.6793893	0.0193523
mean	-0.0427948	-0.0421908	-0.2553256	0.1598318	0.0019236
variance	1.3766540	1.3649081	1.2104312	1.4087225	0.4159990
skewness	0.2617577	0.2617691	-0.1991201	0.7783710	0.0019236
kurtosis	-1.9299736	-1.9011336	-1.9975464	-1.3654516	0.3097251

Table 31. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.001	1.002

Rainbow Trout

Table 32. Model coefficients.

term	estimate	lower	upper	svalue
bFidelity	0.6946167	0.5443672	0.8499499	10.55171
bLength	-0.3293639	-0.4721354	-0.1688172	10.55171

Table 33. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
856	2	3	500	1	861	1.006	TRUE

Table 34. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.3352804	0.3376168	0.2897196	0.3808411	0.1035920
mean	0.1058443	0.1020926	0.0312593	0.1760155	0.1433785
variance	1.2415013	1.2431609	1.1515570	1.3110939	0.0468893
skewness	-0.6874727	-0.6829440	-0.9075806	-0.4890093	0.0568527
kurtosis	-1.4645190	-1.4663365	-1.7021816	-1.0965910	0.0154610

Table 35. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.001	1.003

Walleye

Table 36. Model coefficients.

term	estimate	lower	upper	svalue
bFidelity	0.6571219	0.3775618	0.9239956	10.5517083
bLength	-0.0806242	-0.3202032	0.1809377	0.9837522

Table 37. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
237	2	3	500	1	978	1	TRUE

Table 38. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.3417722	0.3417722	0.2573840	0.4303797	0.0000000
mean	0.1019739	0.1015907	-0.0462431	0.2401701	0.0077098
variance	1.2775310	1.2765889	1.1139577	1.3768625	0.0135193
skewness	-0.6673407	-0.6644430	-1.0602884	-0.2712075	0.0468893
kurtosis	-1.5508735	-1.5503584	-1.9126616	-0.8350748	0.0019236

Table 39. Model sensitivity.

all	analysis	sensitivity	bound
all	1	1.002	1

Length-At-Age

Mountain Whitefish

Table 40. The estimated upper length cutoffs (mm) by age and year.

Year	Age0	Age1	Age2
1990	163	275	NA
1991	144	226	296
2001	141	257	343
2002	163	260	343
2003	159	263	353
2004	158	249	342
2005	168	263	362
2006	175	284	357
2007	171	279	337
2008	170	248	341
2009	169	265	355
2010	177	272	353
2011	163	269	349
2012	162	268	347
2013	185	282	350
2014	178	283	362
2015	167	278	366
2016	165	283	352
2017	158	269	354
2018	177	262	346
2019	188	281	363
2020	167	291	365
2021	165	275	350
2022	162	272	346

Rainbow Trout

Table 41. The estimated upper length cutoffs (mm) by age and year.

Year	Age0	Age1
1990	154	356
1991	129	348
2001	133	327
2002	154	352
2003	161	345
2004	142	335
2005	163	349
2006	170	367
2007	165	378
2008	145	342
2009	146	342
2010	143	340
2011	155	347
2012	151	347
2013	169	358
2014	154	341
2015	166	338
2016	154	340
2017	132	320
2018	138	313
2019	160	318
2020	154	350
2021	168	349
2022	140	354

Survival

Table 42. Parameter descriptions.

Parameter	Description
`SampledLength`	Total standardised length of river sampled
`bEfficiencySampledLength`	Effect of `SampledLength` on `bEfficiency`
`bEfficiency`	Intercept for `logit(eEfficiency)`
`bSurvivalYear[i]`	Effect of `Year` on `bSurvival`
`bSurvival`	Intercept for `logit(eSurvival)`
`eEfficiency[i]`	Expected recapture probability in `i`^th year
`eSurvival[i]`	Expected survival probability from `i-1`^th to `i`^th year
`sSurvivalYear`	Log SD of `bSurvivalYear`

Mountain Whitefish

Table 43. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-4.2707306	-4.4595929	-4.0944209	10.551708
bEfficiencySampledLength	0.4020345	0.1769052	0.6036249	10.551708
bSurvival	0.8816300	0.2801552	1.7311084	6.464245
sSurvivalYear	1.2104642	0.6351825	2.2528623	10.551708

Table 44. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
21	4	3	500	200	1272	1.005	TRUE

Rainbow Trout

Table 45. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-2.5300941	-2.6847303	-2.3822189	10.5517083
bEfficiencySampledLength	0.0344673	-0.1081122	0.1773570	0.6890709
bSurvival	-0.4514088	-0.6572149	-0.2228691	8.9667458
sSurvivalYear	0.3798247	0.2127659	0.6245149	10.5517083

Table 46. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
21	4	3	500	200	1300	1.003	TRUE

Walleye

Table 47. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-3.4860038	-3.6692440	-3.2909426	10.551708
bEfficiencySampledLength	0.1423139	-0.0109218	0.2961277	3.757292
bSurvival	0.1328372	-0.1631503	0.4713474	1.565866
sSurvivalYear	0.5107776	0.2233458	0.9176983	10.551708

Table 48. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
21	4	3	500	200	1233	1.003	TRUE

Capture Efficiency

Table 49. Parameter descriptions.

Parameter	Description
`Annual[i]`	Year of `i`^th visit
`FidelitySD[i]`	SD of site fidelity on `i`^th visit
`Fidelity[i]`	Mean site fidelity on `i`^th visit
`Recaptures[i]`	Number of marked fish recaught during `i`^th visit
`Session[i]`	Session of `i`^th visit
`Tagged[i]`	Number of marked fish tagged prior to `i`^th visit
`bEfficiencySessionAnnual`	Effect of `Session` within `Annual` on `logit(eEfficiency)`
`bEfficiency`	Intercept for `logit(eEfficiency)`
`eEfficiency[i]`	Expected efficiency on `i`^th visit
`eFidelity[i]`	Expected site fidelity on `i`^th visit
`sEfficiencySessionAnnual`	SD of `bEfficiencySessionAnnual`

Mountain Whitefish

Subadult

Table 50. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-4.3284960	-4.8490697	-3.934985	10.55171
sEfficiencySessionAnnual	0.5343559	0.0348258	1.216418	10.55171

Table 51. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1611	2	3	500	100	316	1.004	TRUE

Table 52. Model sensitivity.

all	analysis	sensitivity	bound
all	1.004	1.011	1.01

Adult

Table 53. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-4.4636275	-4.7575859	-4.1899651	10.55171
sEfficiencySessionAnnual	0.2198811	0.0058236	0.6827221	10.55171

Table 54. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1810	2	3	500	100	176	1.014	TRUE

Table 55. Model sensitivity.

all	analysis	sensitivity	bound
all	1.014	1.016	1.013

Rainbow Trout

Subadult

Table 56. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-3.0401384	-3.1648924	-2.9174620	10.55171
sEfficiencySessionAnnual	0.4017667	0.2982203	0.5523378	10.55171

Table 57. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1851	2	3	500	100	1172	1.004	TRUE

Table 58. Model sensitivity.

all	analysis	sensitivity	bound
all	1.004	1.006	1.003

Adult

Table 59. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-3.457437	-3.587065	-3.3351777	10.55171
sEfficiencySessionAnnual	0.160002	0.007375	0.3909227	10.55171

Table 60. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1929	2	3	500	100	292	1.019	TRUE

Table 61. Model sensitivity.

all	analysis	sensitivity	bound
all	1.019	1.007	1.012

Walleye

Table 62. Model coefficients.

term	estimate	lower	upper	svalue
bEfficiency	-3.9245961	-4.1742563	-3.716280	10.55171
sEfficiencySessionAnnual	0.5734736	0.3653922	0.829421	10.55171

Table 63. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1971	2	3	500	100	1214	1.005	TRUE

Table 64. Model sensitivity.

all	analysis	sensitivity	bound
all	1.005	1.005	1.003

Abundance

Table 65. Parameter descriptions.

Parameter	Description
`Annual`	Year
`Efficiency`	Capture efficiency
`Fish`	Number of fish captured or counted
`ProportionSampled`	Proportion of site surveyed
`SiteLength`	Length of site
`Site`	Site
`VisitType`	Survey type (catch versus count)
`bDensityAnnual`	Effect of `Annual` on `bDensity`
`bDensitySiteAnnual`	Effect of `Site` within `Annual` on `bDensity`
`bDensitySite`	Effect of `Site` on `bDensity`
`bDensity`	Intercept for `log(eDensity)`
`bEfficiencyVisitType`	Effect of `VisitType` on `Efficiency`
`eDensity`	Expected density
`esDispersion`	Overdispersion of `Fish`
`sDensityAnnual`	Log SD of effect of `Annual` on `bDensity`
`sDensitySiteAnnual`	Log SD of effect of `Site` within `Annual` on `bDensity`
`sDensitySite`	Log SD of effect of `Site` on `bDensity`
`sDispersionVisitType`	Effect of `VisitType` on `sDispersion`
`sDispersion`	Intercept for `log(esDispersion)`

Mountain Whitefish

Subadult

Table 66. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	4.7654135	4.4298072	5.1118994	10.551708
bDensitySiteAnnual2	0.0004267	-0.0002978	0.0014276	1.970508
bEfficiencyVisitType[2]	1.3070680	1.1847237	1.4435341	10.551708
bEfficiencyVisitTypeDensity[1]	0.0000286	-0.0001100	0.0003042	0.365594
sDensityAnnual	0.6077907	0.4450699	0.8612345	10.551708
sDensitySite	0.8038636	0.6594168	0.9823473	10.551708
sDensitySiteAnnual	0.4287876	0.3789356	0.4819790	10.551708
sDispersion	-0.8066521	-0.9006505	-0.7254798	10.551708
sDispersionVisitType[2]	0.6498328	0.4794663	0.8151093	10.551708

Table 67. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3176	9	3	500	300	380	1.009	TRUE

Table 68. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.2399244	0.2660579	0.2506297	0.2818010	8.229780
mean	-0.2198519	-0.2716164	-0.3068670	-0.2362125	10.551708
variance	0.6964590	0.9939585	0.9494096	1.0417335	10.551708
skewness	0.0372843	0.2530509	0.1839855	0.3311656	10.551708
kurtosis	-0.4753057	-0.3483064	-0.4843925	-0.1724373	3.837463

Adult

Table 69. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	5.6143437	5.3419341	5.8946215	10.55171
bDensitySiteAnnual2	-0.0012832	-0.0016188	-0.0009110	10.55171
bEfficiencyVisitType[2]	1.4437105	1.2671676	1.6023786	10.55171
bEfficiencyVisitTypeDensity[1]	0.0008742	0.0002957	0.0016557	8.22978
sDensityAnnual	0.4853793	0.3527698	0.6664810	10.55171
sDensitySite	0.8135312	0.6392286	1.0423186	10.55171
sDensitySiteAnnual	0.2929567	0.2348879	0.3615738	10.55171
sDispersion	-0.6759789	-0.7385486	-0.6178612	10.55171
sDispersionVisitType[2]	0.4821419	0.3476732	0.6119354	10.55171

Table 70. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3176	9	3	500	300	276	1.018	TRUE

Table 71. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1939547	0.2204030	0.2059194	0.2361618	10.5517083
mean	-0.2186165	-0.2749468	-0.3101131	-0.2394821	8.9667458
variance	0.6764664	1.0002512	0.9516328	1.0476540	10.5517083
skewness	0.0597644	0.2047577	0.1337467	0.2792528	10.5517083
kurtosis	-0.2699228	-0.2932358	-0.4262421	-0.1243632	0.4262948

Rainbow Trout

Subadult

Table 72. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	4.5247947	4.3090214	4.7330819	10.551708
bDensitySiteAnnual2	0.0030998	-0.0000194	0.0108526	4.211858
bEfficiencyVisitType[2]	1.4468067	1.3002071	1.6050459	10.551708
bEfficiencyVisitTypeDensity[1]	-0.0012638	-0.0015645	-0.0009376	10.551708
sDensityAnnual	0.2799273	0.1660076	0.4302089	10.551708
sDensitySite	0.8236282	0.6726549	1.0046244	10.551708
sDensitySiteAnnual	0.4594670	0.4171896	0.5067461	10.551708
sDispersion	-1.0040817	-1.0779921	-0.9286539	10.551708
sDispersionVisitType[2]	0.6729044	0.5084309	0.8178001	10.551708

Table 73. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3176	9	3	500	300	430	1.009	TRUE

Table 74. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1476700	0.1608942	0.1476700	0.1742837	4.266306
mean	-0.1883697	-0.2412219	-0.2761533	-0.2055251	10.551708
variance	0.6336320	1.0271523	0.9801174	1.0729555	10.551708
skewness	-0.1112058	0.1219910	0.0486218	0.1960913	10.551708
kurtosis	-0.1253851	-0.2649071	-0.3923112	-0.1047962	3.681344

Adult

Table 75. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	5.0857026	4.8899721	5.2665983	10.551708
bDensitySiteAnnual2	-0.0014601	-0.0024873	-0.0002079	5.125444
bEfficiencyVisitType[2]	1.2790530	1.1343707	1.4254093	10.551708
bEfficiencyVisitTypeDensity[1]	-0.0005252	-0.0009242	0.0005826	1.800164
sDensityAnnual	0.4065795	0.2962779	0.5934062	10.551708
sDensitySite	0.7381740	0.5693840	0.9367766	10.551708
sDensitySiteAnnual	0.2739269	0.2246509	0.3183488	10.551708
sDispersion	-1.0167195	-1.1001826	-0.9426470	10.551708
sDispersionVisitType[2]	0.5323403	0.3785799	0.6831950	10.551708

Table 76. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3176	9	3	500	300	821	1.014	TRUE

Table 77. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1120907	0.1319270	0.1199622	0.1448363	10.551708
mean	-0.1728934	-0.2351263	-0.2690604	-0.1992549	8.966746
variance	0.6351706	1.0359526	0.9884881	1.0855832	10.551708
skewness	-0.1411944	0.0816807	0.0082146	0.1575561	10.551708
kurtosis	-0.0135759	-0.2523869	-0.3709942	-0.0993513	7.744353

Walleye

Table 78. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	4.7583827	4.5379649	4.9874019	10.5517083
bDensitySiteAnnual2	-0.0030645	-0.0042116	-0.0010457	7.3817833
bEfficiencyVisitType[2]	0.9588480	0.8134857	1.0892770	10.5517083
bEfficiencyVisitTypeDensity[1]	0.0004414	-0.0008670	0.0031452	0.7524266
sDensityAnnual	0.6150421	0.4390139	0.9191938	10.5517083
sDensitySite	0.3259205	0.2201073	0.4573650	10.5517083
sDensitySiteAnnual	0.2649366	0.1844590	0.3463508	10.5517083
sDispersion	-0.8566934	-0.9304274	-0.7888412	10.5517083
sDispersionVisitType[2]	0.5134419	0.3574094	0.6724905	10.5517083

Table 79. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3176	9	3	500	300	802	1.008	TRUE

Table 80. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.1278338	0.1634131	0.1492443	0.1782116	10.551708
mean	-0.1873457	-0.2544441	-0.2923371	-0.2197764	10.551708
variance	0.6884585	1.0410894	0.9960040	1.0871595	10.551708
skewness	-0.2048957	0.1098760	0.0438272	0.1841277	10.551708
kurtosis	-0.2841428	-0.3482950	-0.4716007	-0.2035399	1.394361

Fecundity

Table 81. Parameter descriptions.

Parameter	Description
`Fecundity[i]`	Fecundity of `i`^th fish (eggs)
`Weight[i]`	Weight of `i`^th fish (g)
`bFecundityWeight`	Effect of `log(Weight)` on `log(bFecundity)`
`bFecundity`	Intercept of `eFecundity`
`eFecundity[i]`	Expected `Fecundity` of `i`^th fish
`sFecundity`	SD of residual variation in `log(Fecundity)`

Mountain Whitefish

Table 82. Model coefficients.

term	estimate	lower	upper	svalue
bFecundity	2.9161899	2.1241977	3.7162983	10.55171
bFecundityWeight	0.9976925	0.8754308	1.1191333	10.55171
sFecundity	0.1310897	0.1015794	0.1813106	10.55171

Table 83. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
28	3	3	500	500	768	1	TRUE

Table 84. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0059935	-0.0114609	-0.3694242	0.3722424	0.1118774
variance	0.9064797	0.9749290	0.5354749	1.5850613	0.2860932
skewness	0.0149416	0.0000306	-0.8470844	0.7914794	0.0548545
kurtosis	-0.7134985	-0.3808919	-1.1375460	1.5357451	0.9761690

Table 85. Model sensitivity.

all	analysis	sensitivity	bound
all	1	1.001	1.001

Stock-Recruitment

Table 86. Parameter descriptions.

Parameter	Description
`EggLoss`	Proportional egg loss
`Eggs`	Total egg deposition
`Recruits`	Number of Age-1 recruits
`bAlpha`	`eRecruits` per `Stock` at low `Stock` density
`bBeta`	Expected density-dependence
`bEggLoss`	Effect of `EggLoss` on `log(eRecruits)`
`eRecruits`	Expected `Recruits`
`sRecruits`	SD of residual variation in `log(Recruits)`

Mountain Whitefish

Table 87. Model coefficients.

term	estimate	lower	upper	svalue
bAlpha	0.0037057	0.0009752	0.0082626	10.5517083
bBeta	0.0000002	0.0000000	0.0000005	10.5517083
bEggLoss	-0.1500900	-2.2575553	2.1957646	0.1669245
sRecruits	0.5871984	0.4319101	0.8699800	10.5517083

Table 88. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
20	4	3	500	50	993	1.002	TRUE

Table 89. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	12.8804931	0.0077018	-0.4459022	0.4329071	10.5517083
variance	0.8871422	0.9581590	0.4607771	1.7371333	0.3049677
skewness	-0.1555486	0.0146468	-0.8950604	1.0137125	0.4735575
kurtosis	-1.2463233	-0.4169529	-1.2736863	1.8506637	3.9818527

Table 90. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.002	1.598

Rainbow Trout

Table 91. Model coefficients.

term	estimate	lower	upper	svalue
bAlpha	0.0039458	0.0012818	0.0083450	10.551708
bBeta	0.0000002	0.0000000	0.0000006	10.551708
bEggLoss	39.8063425	-9.1529902	86.3906578	3.391837
sRecruits	0.4280420	0.3231793	0.6096573	10.551708

Table 92. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
21	4	3	500	50	806	1.001	TRUE

Table 93. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	17.6244702	-0.0133078	-0.4319130	0.4139092	10.5517083
variance	0.9490031	0.9748977	0.4760945	1.7215223	0.1118774
skewness	0.1806145	-0.0060781	-0.9725603	0.9173140	0.6048020
kurtosis	0.5039790	-0.4041432	-1.2344735	1.7850030	1.9856542

Table 94. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.003	1.626

Age-Ratios

Table 95. Parameter descriptions.

Parameter	Description
`Age1[i]`	The number of Age-1 fish in the `i`^th year
`Age1and2[i]`	The number of Age-1 and Age-2 fish in the `i`^th year
`LossLogRatio[i]`	The `log` of the ratio of the percent egg losses
`bProbAge1Loss`	Effect of `LossLogRatio` on `bProbAge1`
`bProbAge1`	Intercept for `logit(eProbAge1)`
`eProbAge1[i]`	The expected proportion of Age-1 fish in the `i`^th year
`sDispersion`	SD of extra-binomial variation

Table 96. Model coefficients.

term	estimate	lower	upper	svalue
bProbAge1	0.2631247	-0.0638540	0.5845396	3.2207914
bProbAge1Loss	-0.1231114	-0.5857534	0.3175392	0.8426244
sProbAge1	0.7625138	0.5714295	1.0656212	10.5517083

Table 97. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
22	3	3	500	1	609	1.001	TRUE

Table 98. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0089939	0.0016096	-0.4331818	0.4377352	0.0449048
variance	0.9233599	0.9681931	0.4995536	1.6388864	0.2107455
skewness	-0.5146722	0.0060581	-0.8760666	0.9776663	2.0800330
kurtosis	-0.9497531	-0.4512111	-1.2150035	1.6382316	1.8687137

Table 99. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1	1.002

Adjusted Recruitment

Table 100. Parameter descriptions.

Parameter	Description
`Eggloss[i]`	Proportional RB egg loss for the `i`^th spawn year
`Mw[i]`	Abundance of age-1 MW caught in the same year as age-1 RB from the `i`^th spawn year
`Rb[i]`	Abundance of age-1 RB for the `i`^th spawn year
`b0`	Intercept for `eRb[i]`
`bEggLoss`	Effect of `EggLoss` on `eRb[i]`
`bMw`	Effect of `Mw` on `eRb[i]`
`eRb[i]`	Expected value of `Rb[i]`
`sRb`	SD of residual variation in `eRb[i]`

Table 101. Model coefficients.

term	estimate	lower	upper	svalue
b0	9.8695782	9.7273434	10.0179453	10.551708
bEggLoss	0.0896279	-0.0778609	0.2695086	1.754047
bMw	0.2910339	0.1242546	0.4577598	8.966746
sRb	0.3301308	0.2414721	0.4784315	10.551708

Table 102. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
22	4	3	500	50	1332	1.001	TRUE

Table 103. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0119982	0.0002479	-0.3887636	0.3970610	0.0790174
variance	0.8226930	0.9584356	0.4825587	1.7048865	0.7172372
skewness	0.5781854	-0.0130514	-0.9504658	0.8475869	2.4695592
kurtosis	1.5775049	-0.3813769	-1.1960595	1.4815960	4.4856191

Table 104. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1	1.001

Figures

Condition

Figure 1. Predicted length-mass relationship by species.

Subadult

Mountain Whitefish

Figure 2. Estimated change in condition relative to a typical year for a 200 mm Mountain Whitefish by year (with 95% CRIs).

Rainbow Trout

Figure 3. Estimated change in condition relative to a typical year for a 250 mm Rainbow Trout by year (with 95% CRIs).

Adult

Mountain Whitefish

Figure 4. Estimated change in condition relative to a typical year for a 350 mm Mountain Whitefish by year (with 95% CRIs).

Rainbow Trout

Figure 5. Estimated change in condition relative to a typical year for a 500 mm Rainbow Trout by year (with 95% CRIs).

Walleye

Figure 6. Estimated change in condition relative to a typical year for a 400 mm Walleye by year (with 95% CRIs).

Growth

Figure 7. Estimated von Bertalanffy growth curve by species. The growth curve for Walleye is not shown because the growth parameters were not representative for the younger age-classes.

Mountain Whitefish

Figure 8. Estimated change in von Bertalanffy growth coefficient (k) relative to a typical year by year (with 95% CIs).

Figure 9. Predicted maximum growth by year (with 95% CIs).

Rainbow Trout

Figure 10. Estimated change in von Bertalanffy growth coefficient (k) relative to a typical year by year (with 95% CIs).

Figure 11. Predicted maximum growth by year (with 95% CIs).

Walleye

Figure 12. Estimated change in von Bertalanffy growth coefficient (k) relative to a typical year by year (with 95% CIs).

Figure 13. Predicted maximum growth by year (with 95% CIs).

Movement

Figure 14. Probability of recapture at the same site versus a different site by fish length (with 95% CRIs).

Figure 15. Site fidelity by fish length (with 95% CRIs).

Length-At-Age

Mountain Whitefish

Figure 16. Length-frequency histogram with length-at-age predictions.

Figure 17. Average length of an age-0 individual by year.

Figure 18. Average length of an age-1 individual by year.

Figure 19. Average length of an age-2 individual by year.

Figure 20. Average length of an age-3+ individual by year.

Rainbow Trout

Figure 21. Length-frequency histogram with length-at-age predictions.

Figure 22. Average length of an age-0 individual by year.

Figure 23. Average length of an age-1 individual by year.

Figure 24. Average length of an age-2+ individual by year.

Survival

Adult

Mountain Whitefish

Figure 25. Predicted annual survival for an adult Mountain Whitefish.

Figure 26. Predicted annual efficiency for an adult Mountain Whitefish.

Rainbow Trout

Figure 27. Predicted annual survival for an adult Rainbow Trout.

Figure 28. Predicted annual efficiency for an adult Rainbow Trout.

Walleye

Figure 29. Predicted annual survival for an adult Walleye.

Figure 30. Predicted annual efficiency for an adult Walleye.

Observer Length Correction

Figure 31. Length inaccuracy and imprecision by observer, year and species.

Figure 32. Uncorrected length density plots by species, year and observer.

Figure 33. Corrected length density plots by species, year and observer.

Capture Efficiency

Figure 34. Predicted capture efficiency by species and life stage (with 95% CRIs).

Mountain Whitefish

Subadult

Figure 35. Predicted capture efficiency for a subadult Mountain Whitefish by session and year (with 95% CRIs).

Adult

Figure 36. Predicted capture efficiency for an adult Mountain Whitefish by session and year (with 95% CRIs).

Rainbow Trout

Subadult

Figure 37. Predicted capture efficiency for a subadult Rainbow Trout by session and year (with 95% CRIs).

Adult

Figure 38. Predicted capture efficiency for an adult Rainbow Trout by session and year (with 95% CRIs).

Walleye

Figure 39. Predicted capture efficiency for an adult Walleye by session and year (with 95% CRIs).

Abundance

Figure 40. Effect of density on capture efficiency by species and stage (with 95% CIs).

Figure 41. Estimated density in the lowest and highest abundance years for the lowest and highest abundance sites by species and stage (with 95% CIs).

Figure 42. Effect of counting (versus capture) on encounter efficiency at typical density by species and stage (with 95% CIs).

Figure 43. Effect of counting (versus capture) on overdispersion by species and stage (with 95% CIs).

Mountain Whitefish

Subadult

Figure 44. Estimated abundance of subadult Mountain Whitefish by year (with 95% CIs).

Figure 45. Estimated lineal river count density of subadult Mountain Whitefish by site in a typical year (with 95% CIs).

Figure 46. Estimated evenness of subadult Mountain Whitefish at index sites by year (with 95% CIs).

Figure 47. Estimated density of subadult Mountain Whitefish at non-index relative to index sites by year.

Adult

Figure 48. Estimated abundance of adult Mountain Whitefish by year (with 95% CIs).

Figure 49. Estimated lineal river count density of adult Mountain Whitefish by site in a typical year (with 95% CIs).

Figure 50. Estimated evenness of adult Mountain Whitefish at index sites by year (with 95% CIs).

Figure 51. Estimated density of adult Mountain Whitefish at non-index relative to index sites by year.

Rainbow Trout

Subadult

Figure 52. Estimated abundance of subadult Rainbow Trout by year (with 95% CIs).

Figure 53. Estimated lineal river count density of subadult Rainbow Trout by site in a typical year (with 95% CIs).

Figure 54. Estimated evenness of subadult Rainbow Trout at index sites by year (with 95% CIs).

Figure 55. Estimated density of subadult Rainbow Trout at non-index relative to index sites by year.

Adult

Figure 56. Estimated abundance of adult Rainbow Trout by year (with 95% CIs).

Figure 57. Estimated lineal river count density of adult Rainbow Trout by site in a typical year (with 95% CIs).

Figure 58. Estimated evenness of adult Rainbow Trout at index sites by year (with 95% CIs).

Figure 59. Estimated density of adult Rainbow Trout at non-index relative to index sites by year.

Walleye

Figure 60. Estimated abundance of adult Walleye by year (with 95% CIs).

Figure 61. Estimated lineal river count density of adult Walleye by site in a typical year (with 95% CIs).

Figure 62. Estimated evenness of adult Walleye at index sites by year (with 95% CIs).

Figure 63. Estimated density of adult Walleye at non-index relative to index sites by year.

Survival (Abundance-based)

Mountain Whitefish

Figure 64. Predicted annual survival for adult and subadult Mountain Whitefish.

Rainbow Trout

Figure 65. Predicted annual survival for adult and subadult Rainbow Trout.

Weight

Mountain Whitefish

Figure 66. Predicted weight of an adult Mountain Whitefish by year (with 95% CIs).

Rainbow Trout

Figure 67. Predicted weight of an adult Rainbow Trout by year (with 95% CIs).

Fecundity

Mountain Whitefish

Figure 68. The fecundity-weight relationship for Mountain Whitefish (with 95% CRIs). The data are from Boyer et al (2017).

Figure 69. Predicted fecundity of an adult female Mountain Whitefish by year (with 95% CIs).

Rainbow Trout

Figure 70. Predicted fecundity of an adult female Rainbow Trout by year (with 95% CIs).

Egg Deposition

Mountain Whitefish

Figure 71. Predicted total egg deposition by Mountain Whitefish by year (with 95% CIs).

Rainbow Trout

Figure 72. Predicted total egg deposition by Rainbow Trout by year (with 95% CIs).

Stock-Recruitment

Mountain Whitefish

Figure 73. Predicted stock-recruitment relationship by spawn year (with 95% CRIs).

Figure 74. Predicted egg to age-1 survival by total egg deposition (with 95% CRIs).

Figure 75. Predicted effect of egg loss on the number of age-1 recruits (with 95% CRIs).

Rainbow Trout

Figure 76. Predicted stock-recruitment relationship by spawn year (with 95% CRIs).

Figure 77. Predicted egg to age-1 survival by total egg deposition (with 95% CRIs).

Figure 78. Predicted effect of egg loss on the number of age-1 recruits (with 95% CRIs).

Age-Ratios

Figure 79. Proportion of Age-1 Mountain Whitefish by spawn year.

Figure 80. Percentage Mountain Whitefish egg loss by spawn year.

Figure 81. Proportion of Age-1 Mountain Whitefish by percentage egg loss ratio, labelled by spawn year. The predicted relationship is indicated by the solid black line (with 95% CRIs).

Figure 82. Predicted effect of egg loss on the number of age-1 Mountain Whitefish recruits by egg loss relative to 10% egg loss (with 95% CRIs).

Adjusted Recruitment

Figure 83. Relationship between age-1 Rainbow Trout and age-1 Mountain Whitefish in the same year of capture by Rainbow Trout spawn year (with 95% CIs).

Figure 84. Predicted effect of egg loss on the number of age-1 Rainbow Trout recruits by egg loss relative to 10% egg loss (with 95% CRIs).

Recommendations

Develop fecundity vs weight relationship for Mountain Whitefish and Rainbow Trout on the Lower Columbia River.

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

BC Hydro
- Matt Casselman
- Guy Martel
- Teri Neighbour
- Margo Sadler
- Gillian Kong
Okanagan Nation Alliance
- Evan Smith
Golder Associates
- David Roscoe
- Dustin Ford
- Sima Usvyatsov
- Dana Schmidt
- Demitria Burgoon
- Chris King

References

Andrusak, G. F., and J. L. Thorley. 2019. “Determination of Gerrard Rainbow Trout Stock Productivity at Low Abundance: Final Report.” A {Ministry} of {Forests}, {Lands} and {Natural} {Resource} {Operations} {Report} COL-F19-F-2703. Nelson, B.C.: Fish; Wildlife Compensation Program - Columbia Basin, Habitat Conservation Trust Foundation; Freshwater Fisheries Society of British Columbia.

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Boca Raton: Taylor & Francis.

Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.

Fabens, A J. 1965. “Properties and Fitting of the von Bertalanffy Growth Curve.” Growth 29 (3): 265–89.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Golder Associates Ltd. 2013. “Lower Columbia River Whitefish Spawning Ground Topography Survey: Year 3 Summary Report.” A {Golder} {Associates} {Ltd}. {Report} 10-1492-0142F. Castlegar, BC: BC Hydro.

Irvine, R. L., J. T. A. Baxter, and J. L. Thorley. 2015. “Lower Columbia River Rainbow Trout Spawning Assessment: Year 7 (2014 Study Period).” A {Mountain} {Water} {Research} and {Poisson} {Consulting} {Ltd}. {Report}. Castlegar, B.C.: BC Hydro. http://www.bchydro.com/content/dam/BCHydro/customer-portal/documents/corporate/environment-sustainability/water-use-planning/southern-interior/clbmon-46-yr7-2015-02-04.pdf.

Kery, Marc. 2010. Introduction to WinBUGS for Ecologists: A Bayesian Approach to Regression, ANOVA, Mixed Models and Related Analyses. Amsterdam; Boston: Elsevier. http://public.eblib.com/EBLPublic/PublicView.do?ptiID=629953.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

Lin, J. 1991. “Divergence Measures Based on the Shannon Entropy.” IEEE Transactions on Information Theory 37 (1): 145–51. https://doi.org/10.1109/18.61115.

Macdonald, Peter. 2012. “Mixdist: Finite Mixture Distribution Models.” https://CRAN.R-project.org/package=mixdist.

McElreath, Richard. 2016. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Chapman & Hall/CRC Texts in Statistical Science Series 122. Boca Raton: CRC Press/Taylor & Francis Group.

Plummer, Martyn. 2015. “JAGS Version 4.0.1 User Manual.” http://sourceforge.net/projects/mcmc-jags/files/Manuals/4.x/.

R Core Team. 2018. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

———. 2020. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Tornqvist, Leo, Pentti Vartia, and Yrjo O. Vartia. 1985. “How Should Relative Changes Be Measured?” The American Statistician 39 (1): 43. https://doi.org/10.2307/2683905.

von Bertalanffy, L. 1938. “A Quantitative Theory of Organic Growth (Inquiries on Growth Laws Ii).” Human Biology 10: 181–213.

Walters, Carl J., and Steven J. D. Martell. 2004. Fisheries Ecology and Management. Princeton, N.J: Princeton University Press.

Quesnel Exploitation Analysis 2022

Tue, 06 Jun 2023 00:00:00 +0000

The suggested citation for this analytic appendix is:

Dalgarno, S. & Thorley, J.L. (2023) Quesnel Exploitation Analysis 2022. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1230088647.

Background

Quesnel Lake supports a recreational fishery for Rainbow Trout. To provide information on the natural and fishing mortality, Rainbow Trout were caught by angling and tagged with acoustic transmitters and/or a $100 and $10 reward tag.

The objectives of this analysis are to:

Provide a complete history of tagged fish from capture to tag expiry and estimated fate.
Provide a summary of life history including variation in fork length at spawning under different lake conditions and any additional observations on change in population age structure and size.
Estimate annual in-lake abundance >= 50 cm from mark-recapture data in years with sufficient data.
Provide an assessment of conservation impacts and allowable exploitation if changing the lower slot limit to 60 cm or 70 cm.

Data Preparation

The outing, receiver deployment, detection, fish capture and recapture information were provided by the Ministry of Forests, Lands and Natural Resource Operations and added to a SQLite database.

The data were prepared for analysis using R version 4.3.0 (R Core Team 2022).

Receivers were assumed to have a detection range of 500 m. Detections were aggregated daily, where for each transmitter the receiver with the most number of detections was chosen. In the case of a tie, the receiver with the greatest coverage area was chosen. Only individuals with a fork length (FL) $\geq$ 450 mm, an acoustic tag life $\geq$ 365 days (if acoustically tagged) and a $100 and $10 reward tags were included in the survival analysis.

Statistical Analysis

Where possible, model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the number of zeros and the first four central moments (mean, variance, skewness and kurtosis) for the deviance residuals were compared to the expected values by simulating new residuals. In this context the s-value indicates how surprising each observed metric is given the estimated posterior probability distribution for the residual variation.

The analyses were implemented using R version 4.3.0 (R Core Team 2022) and the mbr family of packages.

Model Descriptions

Condition

The expected weight of fish of a given length were estimated from the data using a mass-length model (He et al. 2008).

More specifically the model was based on the allometric relationship

\[ W = \alpha L^{\beta}\]

where $W$ is the weight (mass), $\alpha$ is the coefficent, $\beta$ is the exponent and $L$ is the length.

To improve chain mixing the relation was log-transformed, i.e.,

\[ \log(W) = \log(\alpha) + \beta \cdot \log(L)\] Length and weights of fish were taken from two sources: captures from broodstock collection in the Horsefly River (2015-2022) and in-lake captures from tagging outings (2013-2022).

Key assumptions of the condition model include:

The expected weight is allowed to vary with length and date.
The expected weight is allowed to vary randomly with year and source within year.
The relationship between weight and length is allowed to vary with date.
The residual variation in weight is log-normally distributed.

Growth

Annual growth of fish were estimated from inter-annual recaptures (in-lake and broodstock) and aged broodstock captures using the Fabens method (Fabens 1965) for estimating the von Bertalanffy growth curve (von Bertalanffy 1938). This curve is based on the premise that:

\[ \frac{\text{d}L}{\text{d}t} = k (L_{\infty} - L)\]

where $L$ is the length of the individual, $k$ is the growth coefficient and $L_{\infty}$ is the maximum length.

Integrating the above equation gives:

\[ L_t = L_{\infty} (1 - e^{-k(t - t_0)})\]

where $L_t$ is the length at time $t$ and $t_0$ is the time at which the individual would have had zero length.

The Fabens form allows

\[ L_r = L_c + (L_{\infty} - L_c) (1 - e^{-kT})\]

where $L_r$ is the length at recapture, $L_c$ is the length at capture and $T$ is the time between capture and recapture.

For aged fish, length at capture is the length at Age-0 $L_0$, estimated with the modified form:
\[L_r = L_{0} - (L_{\infty} - L_0) (1 - e^{-kA})\] where $A$ is the age.

Key assumptions of the growth model include:

The mean maximum length $L_{\infty}$ is constant.
The growth coefficient $k$ varies randomly with growth year.
$L_{\infty}$ is likely between 800 and 1000.
The residual variation in growth is student-t distributed.
The student-t shape parameter is fixed at 2.

A growth year spans April 01 to March 31 the following year.

Preliminary analyses indicated lack of support for variation in $k$ by Horsefly River sockeye run size with a one or two year lag.

Survival

The natural mortality and recapture probability were estimated using a Bayesian individual state-space survival model (Thorley and Andrusak 2017) with three month intervals. The survival model incorporated natural and handling mortality, acoustic detections, inter-section movement, T-bar tag loss, spawning, recapture, reporting, fork length at spawning and sockeye run size in the previous year.

T-bar tag loss was estimated from the number of tags reported from recaught double-tagged individuals (Fabrizio et al. 1999). The model assumes that tag loss is not a time-dependent process and occurs following initial release.

Fork length at spawning was calculated from the measured length at capture ($L_{i,fi}$) plus the length increment expected under a Von Bertalanffy Growth Curve (Walters & Martell, 2004)

\[L_{i,t} = L_{i,fi} + (L_{∞} − L_{i,fi} )(1 − e^{(−0.25 \cdot k(t −fi))})\]

where the 0.25 in the exponent adjusts for the fact that there are four time periods per year. Values for $L_\infty$ and annual $k$ and were estimated in the growth model.

The probability of spawning at estimated length $L$ is determined by

\[S = \frac{L^{S_p}}{L_s^{S_p} + L^{S_p}} \cdot es\]

where $L_s$ is the length at which 50% of fish are mature, $es$ is the probability of a mature fish spawning and $S_p$ is the maturity (as a function of length) power.

Key assumptions of the spawning probability sub-model include:

$L_s$ varies randomly by year
$es$ varies randomly by year

A fish was considered to have spawned in a given year if one of the following conditions were met: detected within a spawning river during the spawning period with a detection hiatus (no detections in lake) of at least 7 days; or detected within a spawning gate during the spawning period with a detection hiatus of at least 14 days. Cutoff values for minimum hiatus periods were determined by inspecting the distribution of in-lake hiatus periods for fish detected in spawning rivers and the distribution of all hiatus periods for suspected spawners within the spawning period.

In addition to assumptions 1 to 2 and 4 to 10, in Thorley and Andrusak (2017), the model also assumes that:

The natural mortality varies by season, spawning and sockeye abundance for each size class.
The recapture probability varies randomly by year.
The recapture probability varies by season.
The probability of tag loss is independent between tags on a fish.
The probability of detection depends on whether a fish is alive or has died near or far from a receiver
All recaptured fish that are released have their FLOY tags removed.
Reporting of recaptured fish with one or more T-bar tags is likely greater than 90%.

Seasons are defined as winter (January to March), spring (April to June), summer (July to September), and fall (October to December).

Size classes are defined as small ($\le$ 600 mm), medium (601 $-$ 700 mm) and large ($>$ 700 mm).

Sockeye abundance is the Horsefly River run size with a two year lag (i.e., spawning in the spring of 2016 is associated with autumn 2014 sockeye abundance).

Preliminary analyses indicated lack of support for variation in mortality by year, sockeye year (October 01 to September 30 the following year) and size class within year.

Mark-Recapture

The in-lake and spawner abundance for individuals $\ge$ 500 mm were estimated using hierarchical Bayesian mark-recapture sub-models within the survival model, where the number of alive marked individuals in the lake at each time period was estimated from the ‘Alive’ latent state in the previous period. The number of spawning marked individuals was estimated from the spawning probability and the number of alive marked individuals in the previous period.

The ‘in-lake’ mark-recapture model estimated abundance from in-lake captures and recaptures from researcher tagging outings.

The ‘spawner’ mark-recapture model estimated spawner abundance from spawner captures and recaptures from broodstock collection in the Horsefly River.

An additional in-lake abundance $A_L$ estimate was derived from estimated spawner abundance $A_S$ and spawning probability $P_S$ given the following: \[A_L = \frac{A_S}{P_S} \]

Key assumptions of the mark-recapture models include:

Marked fish mix randomly with unmarked fish within the lake.
The probability of capturing a marked or unmarked fish is the same.
All recaptured fish are correctly identified as being marked.
The number of recaptures (of fish marked at that site in that year) and the total number of fish caught are each binomially distributed.

Additional assumptions for the lake model include:

Capture efficiency varies by total outing hours in a three month period.
Abundance varies randomly by year.

Yield-Per-Recruit

The optimal yield (defined as the harvested number of fish as percent of the number of recruits at carrying capacity) was calculated using a yield-per-recruit approach (Bison, O’Brien, and Martell 2003).

Scenarios explored include:

Lower slot limit (Llo) of 40, 50, 60, 70, 80 and 90, with other YPR parameter values held constant.
Rk (lifetime spawners per spawner at low density) values of 5, 7.5, and 10 and Ls (length at 50% mature) values of 55, 60, and 65, with Llo held at 60.
Rk values of 5, 7.5, and 10 and Ls values of 55, 60, and 65, with Llo held at 50.

Key assumptions include:

The population is at equilibrium.
All captured individuals above Llo are retained.
All captured individuals below Llo are released.
Handling mortality is 10%.
The life-history parameters are fixed.
There are no Allee effects.
There are no limitations on prey species.

Model Templates

Condition

.model{
  bWeight ~ dnorm(3, 2^-2)
  bWeightLength ~ dnorm(3, 1^-2)
  bWeightDayte ~ dnorm(0, 1^-2);
  bWeightLengthDayte ~ dnorm(0, 1^-2)
  sWeightAnnual ~ dexp(1)
  for(i in 1:nAnnual) {
    bWeightAnnual[i] ~ dnorm(0, sWeightAnnual^-2)
  }
  sWeightSourceAnnual ~ dexp(1)
  for(i in 1:nSource){
    for(j in 1:nAnnual){
      bWeightSourceAnnual[i,j] ~ dnorm(0, sWeightSourceAnnual^-2)
    }
  }
  sWeight ~ dexp(1)
  for(i in 1:nObs) {
    log(eWeight[i]) <- bWeight + bWeightDayte * Dayte[i] + bWeightAnnual[Annual[i]] + bWeightSourceAnnual[Source[i], Annual[i]] + (bWeightLength + bWeightLengthDayte * Dayte[i]) * (log(Length[i]) - log(100))
    Weight[i] ~ dlnorm(log(eWeight[i]), sWeight^-2)
  }

Block 1. Model description.

Growth

.model {
  bK ~ dnorm(0, 2^-2) 
  bL0 ~ dnorm(0, 100^-2)
  sKYear ~ dexp(1)
  theta <- 0.5

  bLinf ~ dnorm(900, 100^-2) T(0,)
  sGrowth ~ dexp(1)
  sLength ~ dexp(1)
  for (i in 1:nYear) {
    bKYear[i] ~ dnorm(0, sKYear^-2)
    log(eK[i]) <- bK + bKYear[i] 
  }
  for(i in 1:nAged){
    eLength[i] <- bLinf - (bLinf - bL0) * exp(-exp(bK) * Age[i])
    Length[i] ~ dnorm(eLength[i], sLength^-2)
  }
  for(i in 1:nFish){
    eL0[i] <- max(bL0, LengthAtRelease[i])
    for(j in 1:nYear){
      ekProp[i,j] <- eK[j] * pYear[i,j]
    }
    eSumEk[i] <- exp(-sum(ekProp[i,]))
    eGrowth[i] <- max(0, (bLinf - eL0[i]) * (1 -eSumEk[i]))
    Growth[i] ~ dt(eGrowth[i], sGrowth^-2, 1/theta)
  }

Block 2.

Tag Loss

model {
  bTagLoss ~ dbeta(1, 1)

  for (i in 1:nObs) {
    TagsRecap[i] ~ dbin(1 - bTagLoss, 2) T(1,)
  }

Block 3. Model description.

Survival

.model{
  bMortality ~ dnorm(-3, 3^-2)
  bMortalitySpawning ~ dnorm(0, 2^-2)
  bLs ~  dnorm(6, 2^-2)
  bSp ~ dexp(1/20)
  bEs ~ dnorm(0, 2^-2)

  bDetectedAlive ~ dbeta(1, 1)
  bDieNear ~ dbeta(1, 1)
  bDetectedDeadNear ~ dbeta(10, 1)
  bDetectedDeadFar ~ dbeta(1, 10)
  bMoved ~ dbeta(1, 1)
  bRecaptured ~ dnorm(0, 2^-2)
  bReported ~ dbeta(10, 1)
  bReleased ~ dbeta(1, 1)
  bReleasedResearcher ~ dbeta(1, 1)
  bRecapturedResearcher ~ dbeta(5, 1)
  bTagLoss <- 0.0545
  bTagLossT <- 0.0001

  bEfficiency ~ dnorm(-4, 2^-2)
  bAbundance ~ dnorm(7, 2^-2)
  bOutingHours ~ dnorm(0, 2^-2)
  bEfficiencySpawn ~ dnorm(-4, 2^-2)
  bAbundanceSpawn ~ dnorm(1000, 500^-2)

  bMortalitySeason[1] <- 0
  for (i in 2:nSeason) {
    bMortalitySeason[i] ~ dnorm(0, 2^-2)
  }
  for (i in 1:nSizeClass) {
    bMortalitySockeye[i] ~ dnorm(0, 2^-2)
  }
  sRecapturedAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bRecapturedAnnual[i] ~ dnorm(0, sRecapturedAnnual^-2)
  }
  bRecapturedSeason[1] <- 0
  for (i in 2:nSeason) {
    bRecapturedSeason[i] ~ dnorm(0, 2^-2)
  }
  sLsAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bLsAnnual[i] ~ dnorm(0, sLsAnnual^-2)
  }
  sEsAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bEsAnnual[i] ~ dnorm(0, sEsAnnual^-2)
  }

  for (i in 1:nCapture){
    eDetectedAlive[i] <- bDetectedAlive
    eDieNear[i] ~ dbern(bDieNear)
    eDetectedDeadNear[i] <- bDetectedDeadNear
    eDetectedDeadFar[i] <- bDetectedDeadFar
    eDetectedDead[i] <- eDieNear[i] * eDetectedDeadNear[i] + (1 - eDieNear[i]) * eDetectedDeadFar[i]
    
    logit(eMortality[i,PeriodCapture[i]]) <- bMortality + bMortalitySeason[Season[PeriodCapture[i]]] + bMortalitySockeye[SizeClass[i,PeriodCapture[i]]] * SockeyePrevious[PeriodCapture[i]]
    logit(eRecaptured[i,PeriodCapture[i]]) <- bRecaptured + bRecapturedSeason[Season[PeriodCapture[i]]] + bRecapturedAnnual[Annual[PeriodCapture[i]]] 

    eMoved[i,PeriodCapture[i]] <- bMoved 
    eReported[i,PeriodCapture[i]] <- bReported
    eReleased[i,PeriodCapture[i]] <- bReleased
    eReleasedResearcher[i,PeriodCapture[i]] <- bReleasedResearcher
    eRecapturedResearcher[i,PeriodCapture[i]] <- bRecapturedResearcher

    InLake[i,PeriodCapture[i]] <- 1
    Alive[i,PeriodCapture[i]] ~ dbern(1-eMortality[i,PeriodCapture[i]])
    Alive50[i,PeriodCapture[i]] <- Alive[i,PeriodCapture[i]] * (Length[i,PeriodCapture[i]] >= 500)

    TBarTag100[i,PeriodCapture[i]] ~ dbern(1-bTagLoss)
    TBarTag10[i,PeriodCapture[i]] ~ dbern(1-bTagLoss)
    
    eDetected[i,PeriodCapture[i]] <- Alive[i,PeriodCapture[i]] * eDetectedAlive[i] + (1-Alive[i,PeriodCapture[i]]) * eDetectedDead[i]
    Detected[i,PeriodCapture[i]] ~ dbern(Monitored[i,PeriodCapture[i]] * eDetected[i,PeriodCapture[i]])
    Moved[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * Detected[i,PeriodCapture[i]] * eMoved[i,PeriodCapture[i]])

    RecapturedResearcher[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * eRecapturedResearcher[i,PeriodCapture[i]])
    Recaptured[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * eRecaptured[i,PeriodCapture[i]])
    Reported[i,PeriodCapture[i]] ~ dbern(Recaptured[i,PeriodCapture[i]] * eReported[i,PeriodCapture[i]] * step(TBarTag100[i,PeriodCapture[i]] + TBarTag10[i,PeriodCapture[i]] - 1))
    Released[i,PeriodCapture[i]] ~ dbern(Recaptured[i,PeriodCapture[i]] * eReleased[i,PeriodCapture[i]])
    ReleasedResearcher[i,PeriodCapture[i]] ~ dbern(RecapturedResearcher[i,PeriodCapture[i]] * eReleasedResearcher[i,PeriodCapture[i]])

  for(j in (PeriodCapture[i]+1):nPeriod) {
    log(eLs[i,j]) <- bLs + bLsAnnual[Annual[j]]
    logit(eEs[i,j]) <- bEs + bEsAnnual[Annual[j]]
    eSpawning[i,j] <- (((Length[i,j]^bSp) / (eLs[i,j]^bSp + Length[i,j]^bSp)) * eEs[i,j]) 
    Spawned[i,j] ~ dbern(Alive[i,j-1] * Monitored[i,j] * SpawningPeriod[j] * eSpawning[i,j])
    logit(eMortality[i,j]) <- bMortality + bMortalitySpawning * Spawned[i,j-1] + bMortalitySeason[Season[j]] + bMortalitySockeye[SizeClass[i,j]] * SockeyePrevious[j]
    logit(eRecaptured[i,j]) <- bRecaptured + bRecapturedSeason[Season[j]] + bRecapturedAnnual[Annual[j]] 
    
    eMoved[i,j] <- bMoved 
    eRecapturedResearcher[i,j] <- bRecapturedResearcher
    eReported[i,j] <- bReported
    eReleased[i,j] <- bReleased
    eReleasedResearcher[i,j] <- bReleasedResearcher

    InLake[i,j] ~ dbern(InLake[i,j-1] * ((1 - (Recaptured[i,j-1])) + Recaptured[i,j-1] * Released[i,j-1]) * ((1 - (RecapturedResearcher[i,j-1])) + RecapturedResearcher[i,j-1] * ReleasedResearcher[i,j-1])) 
    Alive[i,j] ~ dbern(InLake[i,j] * Alive[i,j-1] * (1-eMortality[i,j]))
    Alive50[i,j] <- Alive[i,j] * (Length[i,j] >= 500)
    TBarTag100[i,j] ~ dbern(TBarTag100[i,j-1] * (1-Recaptured[i,j-1]) * (1-bTagLossT))
    TBarTag10[i,j] ~ dbern(TBarTag10[i,j-1] * (1-Recaptured[i,j-1]) * (1-bTagLossT))
    
    eDetected[i,j] <- Alive[i,j] * eDetectedAlive[i] + (1-Alive[i,j]) * eDetectedDead[i]
    Detected[i,j] ~ dbern(InLake[i,j] * Monitored[i,j] * eDetected[i,j])
    Moved[i,j] ~ dbern(Alive[i,j] * Detected[i,j] * eMoved[i,j])
    
    RecapturedResearcher[i,j] ~ dbern(Alive[i,j] * eRecapturedResearcher[i,j] * step(TBarTag100[i,j] + TBarTag10[i,j] - 1))
    Recaptured[i,j] ~ dbern(Alive[i,j] * eRecaptured[i,j])
    Reported[i,j] ~ dbern(Recaptured[i,j] * eReported[i,j] * step(TBarTag100[i,j] + TBarTag10[i,j] - 1))
    Released[i,j] ~ dbern(Recaptured[i,j] * eReleased[i,j])
    ReleasedResearcher[i,j] ~ dbern(RecapturedResearcher[i,j] * eReleasedResearcher[i,j])
  }
  }
  Marked[1] <- sum(Alive50[1:PeriodCaptures[1],1])
  MarkedSpawners[1] <- 1
  for(p in 2:nPeriod){
      Marked[p] <- sum(Alive50[1:PeriodCaptures[p],p])
      logit(eSpawningPeriod[p]) <- bEs + bEsAnnual[Annual[p]]
      MarkedSpawners[p] ~ dpois(Marked[p-1] * eSpawningPeriod[p])
  }
  sAbundanceAnnual ~ dexp(1)
  for (i in 1:nAnnualOuting) {
    bAbundanceAnnual[i] ~ dnorm(0, sAbundanceAnnual^-2)
  }
  eAbundanceSpawn ~ dpois(bAbundanceSpawn)
  for(l in 1:nPeriodCount){
      logit(eEfficiencySpawn[l]) <- bEfficiencySpawn
      RecaptureSpawners[l] ~ dbin(eEfficiencySpawn[l], MarkedSpawners[PeriodCount[l]])
      CaptureSpawners[l] ~ dbin(eEfficiencySpawn[l], eAbundanceSpawn)
  }

  for(k in 1:nPeriodOuting){
      log(eAbundance[k]) <- bAbundance + bAbundanceAnnual[AnnualOuting[k]]
      eAbundanceInt[k] ~ dpois(eAbundance[k])
      logit(eEfficiency[k]) <- bEfficiency + bOutingHours * OutingHours[PeriodOuting[k]]
      Recaptures[k] ~ dbin(eEfficiency[k], Marked[PeriodOuting[k]-1])
      Captures[k] ~ dbin(eEfficiency[k], eAbundanceInt[k])
  }

Block 4. Survival model description.

Results

Tables

Condition

Table 1. Parameter descriptions.

Parameter	Description
`Annual[i]`	Year the `i`^th fish was captured as a factor
`Dayte[i]`	Day of year that the `i`^th fish was captured
`Length[i]`	Fork length of the `i`^th fish
`Source[i]`	Data source for the `i`^th fish (lake or broodstock)
`Weight[i]`	Recorded weight of the `i`^th fish
`bWeightAnnual[i]`	Effect of the `i`^th `Annual` on `bWeight`
`bWeightDayte[i]`	Effect of the `i`^th `Dayte` on `bWeight`
`bWeightLengthDayte[i]`	Effect of the `i`^th `Dayte` on `bWeightLength`
`bWeightLength`	Effect of `log(Length[i])` on `bWeight`
`bWeightSourceAnnual[i]`	Effect of the `i`^th Source within the `j`^th `Annual` on `bWeight`
`bWeight`	Intercept of `log(eWeight[i])`
`eWeight[i]`	Expected `Weight[i]` of the `i`^th fish
`sWeightAnnual`	SD of `bWeightAnnual`
`sWeightSourceAnnual`	SD of `bSourceAnnual`
`sWeight`	SD of `log(Weight[i])`

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
bWeight	2.2722775	2.1647061	2.3739569	10.55171
bWeightDayte	-0.2371098	-0.3462110	-0.1279576	10.55171
bWeightLength	3.1081631	3.0577862	3.1634224	10.55171
bWeightLengthDayte	0.1527845	0.0930352	0.2145596	10.55171
sWeight	0.1086107	0.1042768	0.1132872	10.55171
sWeightAnnual	0.0455064	0.0034942	0.1121414	10.55171
sWeightSourceAnnual	0.0616008	0.0349995	0.1013398	10.55171

Table 3. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1124	7	3	500	100	178	1.008	TRUE

Table 4. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0000426	-0.0005523	-0.0619175	0.0582786	0.0271665
variance	0.9834447	1.0000853	0.9184694	1.0790217	0.5196625
skewness	-0.2243304	-0.0017212	-0.1427099	0.1503316	10.5517083
kurtosis	0.0653836	-0.0173751	-0.2628448	0.3158036	0.8186929

Table 5. Model sensitivity.

all	analysis	sensitivity	bound
all	1.008	1.015	1.013

Growth

Table 6. Parameter descriptions.

Parameter	Description
`Growth[i]`	Observed growth between release and recapture of `i`^th recapture
`LengthAtRelease[i]`	Length at previous release of `i`^th recapture
`Year[i]`	Release year of `i`^th recapture
`bKYear[i]`	Effect of `i`^th `Year` on `bK`
`bK`	Intercept of `log(eK)`
`bLinf`	Mean maximum length
`eGrowth`	Expected `Growth` between release and recapture
`eK[i]`	Expected von Bertalanffy growth coefficient for `i`^th `Year`
`eSumK[i]`	Expected sum of the proportion of annual von Bertalanffy growth coefficient between release and recapture
`pYear[i]`	Proortion of Years between release and recapture of `i`^th recapture
`sGrowth`	Log standard deviation of residual variation in `Growth`
`sKYear`	Log standard deviation of `bKYear`

Table 7. Model coefficients.

term	estimate	lower	upper	svalue
bK	-1.4320124	-1.5455592	-1.3142231	10.551708
bL0	-20.1326797	-65.1034093	13.9370296	1.918713
bLinf	854.8911464	818.6242759	894.1474971	10.551708
sGrowth	28.2133323	25.2070680	32.2250808	10.551708
sKYear	0.4060662	0.2429683	0.6943251	10.551708
sLength	49.8966766	46.1097545	54.2918162	10.551708

Table 8. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
5580	6	3	500	30	222	1.008	TRUE

Tag Loss

Table 9. Parameter descriptions.

Parameter	Description
`TagsRecap[i]`	Tags remaining on ith recapture
`bTagLossIntercept`	Intercept for logit(eTagLoss)
`eTagLoss[i]`	Predicted probability of single tag loss for the ith recapture

Table 10. Model coefficients.

term	estimate	lower	upper	svalue
bTagLoss	0.0537979	0.0312079	0.0862457	10.55171

Table 11. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
126	1	3	500	10	1500	1	TRUE

Table 12. The number of recaptured fish that were reported without their $100 versus $10 reward tag by species.

Species	Loss100	Loss10
Rainbow Trout	0	11

Table 13. Model sensitivity.

all	analysis	sensitivity	bound
all	1	1	1

Survival

Table 14. Parameter descriptions.

Parameter	Description
`TagLossT`	Seasonal probability of loss of a single tag in all periods subsequent to capture
`TagLoss`	Seasonal probability of loss of a single tag in the period of capture
`bAbundanceAnnual`	Effect of `i`^th `Year` on `bAbundance`
`bAbundanceSpawn`	Expected spawner abundance
`bAbundance`	Expected in-lake abundance
`bDetectedAlive`	Seasonal probability of detection if in-lake
`bDetectedDeadFar`	Seasonal probability of detection if in-lake and dead far from a receiver
`bDetectedDeadNear`	Seasonal probability of detection if in-lake and dead near a receiver
`bDieNear`	Lifetime probability of dying near versus far from a receiver
`bEfficiencySpawn`	Log odds spawner capture efficiency
`bEfficiency`	Log odds in-lake capture efficiency
`bEsAnnual`	Effect of `i`^th `Year` on `bEs`
`bEs`	The annual probability of a mature fish spawning
`bLsAnnual`	Effect of `i`^th `Year` on `bLs`
`bLs`	The length at which 50% of fish mature.
`bMortalitySeason`	Effect of `i`^th `Season` on `bMortality`
`bMortalitySockeye`	Effect of standardized sockeye abundance on `bMortality` for the `i`^th `Size Class`.
`bMortalitySpawning`	Effect of spawning on `bMortality`
`bMortality`	Log odds seasonal probability of dying of natural causes
`bMoved`	Seasonal probability of being detected moving between sections if alive
`bOutingHours`	Effect of standardized outing hours on `bEfficiency`
`bRecapturedAnnual`	Effect of `i`^th `Year` on `bRecaptured`
`bRecapturedResearcher`	Log odds Seasonal probability of being recaptured by a researcher if alive
`bRecapturedSeason`	Effect of `i`^th `Season` on `bRecaptured`
`bRecaptured`	Log odds Seasonal probability of being recaptured if alive
`bReleasedResearcher`	Probability of being released if recaptured by researcher
`bReleased`	Probability of being released if recaptured
`bReported`	Probability of being reported if recaptured with one or more T-bar tags
`bSp`	The maturity (as a function of length) power
`sAbundanceAnnual`	Standard deviation of residual variation in `bAbundanceAnnual`
`sEsAnnual`	Standard deviation of residual variation in `bEsAnnual`
`sLsAnnual`	Standard deviation of residual variation in `bLsAnnual`
`sRecapturedAnnual`	Standard deviation of residual variation in `bRecapturedAnnual`
`sSpawningAnnualSizeClass`	Standard deviation of residual variation in `bSpawningAnnualSizeClass`
`sSpawningAnnual`	Standard deviation of residual variation in `bSpawningAnnual`

Table 15. Model coefficients.

term	estimate	lower	upper	svalue
bAbundance	7.7024694	7.2883220	8.0580680	10.5517083
bAbundanceSpawn	1033.5691976	623.0275780	1632.7147157	10.5517083
bDetectedAlive	0.9575550	0.9500319	0.9649763	10.5517083
bDetectedDeadFar	0.0185010	0.0125331	0.0251879	10.5517083
bDetectedDeadNear	0.7156397	0.6715751	0.7586170	10.5517083
bDieNear	0.1646191	0.1280068	0.2038573	10.5517083
bEfficiency	-4.5292845	-4.8471735	-4.2563804	10.5517083
bEfficiencySpawn	-2.2603841	-2.7654726	-1.6840752	10.5517083
bEs	-0.0902006	-0.6365079	1.0105769	0.3121097
bLs	6.4078382	6.3434401	6.5121100	10.5517083
bMortality	-2.3912119	-2.6922047	-2.1132121	10.5517083
bMortalitySeason[2]	-0.6907013	-1.2561726	-0.2152811	7.7443533
bMortalitySeason[3]	0.7289708	0.3849013	1.0840406	10.5517083
bMortalitySeason[4]	0.5056264	0.1532712	0.9074002	8.2297802
bMortalitySockeye[1]	-0.1831197	-0.3702748	-0.0077123	4.6209709
bMortalitySockeye[2]	0.0395580	-0.1028514	0.1822475	0.7818704
bMortalitySockeye[3]	0.2830888	0.0433976	0.4997483	5.7968208
bMortalitySpawning	1.7486527	1.3874464	2.1252123	10.5517083
bMoved	0.8362576	0.8213738	0.8490445	10.5517083
bOutingHours	0.8287863	0.7432894	0.9152798	10.5517083
bRecaptured	-5.6091619	-6.8081781	-4.6506558	10.5517083
bRecapturedResearcher	0.0111486	0.0080106	0.0145835	10.5517083
bRecapturedSeason[2]	1.9441166	1.0207622	3.1353051	10.5517083
bRecapturedSeason[3]	2.4679565	1.4953470	3.6113419	10.5517083
bRecapturedSeason[4]	-1.3397190	-3.4156383	0.3297862	2.9742794
bReleased	0.7978703	0.6948177	0.8764043	10.5517083
bReleasedResearcher	0.9014618	0.7846933	0.9713065	10.5517083
bReported	0.9925239	0.9561869	0.9997260	10.5517083
bSp	16.2312302	7.3646727	30.6734894	10.5517083
sAbundanceAnnual	0.3888943	0.2214759	0.7549890	10.5517083
sEsAnnual	0.3867077	0.0190089	1.2427542	10.5517083
sLsAnnual	0.0564376	0.0075130	0.1442898	10.5517083
sRecapturedAnnual	0.6015764	0.1741662	1.3339886	10.5517083

Table 16. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
23870	33	3	500	150	63	1.079	FALSE

Table 17. The estimated annual interval probabilities (with 95% CIs). Natural mortalities are for a medium sized fish (600-700 mm). Harvest mortality is the recapture probability multiplied by the probability of being harvested if recaptured, assuming that recapture can only occur once in a year. Release is the probability of being released if recaptured.

derived quantity	estimate	lower	upper
Natural Mortality Spawner	0.6370	0.5760	0.6970
Natural Mortality Non-Spawner	0.3620	0.3320	0.3930
Recapture	0.0709	0.0422	0.1150
Release	0.7980	0.6950	0.8760
Harvest Mortality	0.0141	0.0070	0.0267

Mark-Recapture

Table 18. The number of captures, recaptures, estimated marked individuals alive and available for capture and total number of outing hours summarized by year.

Year	Marked	Captures	Recaptures	Outing Hours
2014	43	68	3	260
2015	62	96	6	297
2016	76	150	2	288
2017	134	71	3	226
2018	127	103	8	385
2019	138	98	3	385
2020	135	102	6	287
2021	117	85	7	163

Yield-per-Recruit

Slot Limit

Table 19. Rainbow Trout yield-per-recruit calculations including the capture probability (pi), exploitation rate (u), and average age, length and weight of fish harvested.

Type	pi	u	Yield	Age	Length	Weight	Effort
actual	0.0708714	0.0708714	0.0114817	6.951293	67.28609	3683.809	0.6976822
optimal	0.6843336	0.6843336	0.0438806	6.205631	64.49527	3193.747	10.9440364

Table 20. Rainbow Trout yield-per-recruit model parameters.

Parameter	Value	Description	Source
tmax	30.0000	The maximum age (yr).	Professional Judgement
k	0.2388	The VB growth coefficient (yr-1).	Current Study
Linf	85.4891	The VB mean maximum length (cm).	Current Study
t0	0.3000	The (theoretical) age at zero length (yr).	Professional Judgement
k2	0.1500	The VB growth coefficient after length L2 (yr-1).	Default
Linf2	100.0000	The VB mean maximum length after length L2 (cm).	Default
L2	1000.0000	The length (or age if negative) at which growth switches from the
first to second phase (cm or yr).	Default
Wb	3.1082	The weight (as a function of length) scaling exponent.	Current Study
Ls	60.6581	The length (or age if negative) at which 50 % mature (cm or yr).	Current Study
Sp	16.2312	The maturity (as a function of length) power.	Current Study
es	0.4775	The annual probability of a mature fish spawning.	Current Study
Sm	0.4299	The spawning mortality probability.	Current Study
fb	1.0000	The fecundity (as a function of weight) scaling exponent.	Default
tR	1.0000	The age from which survival is density-independent (yr).	Default
BH	1.0000	Recruitment follows a Beverton-Holt (1) or Ricker (0) relationship.	Default
Rk	7.5000	The lifetime spawners per spawner at low density (or the egg to tR survival if between 0 and 1).	(thorley_duncan_2018?)
n	0.3619	The \strong{annual interval	Current Study
nL	0.2000	The \strong{annual interval	Default
Ln	1000.0000	The length (or age if negative) at which the
annual interval natural mortality rate switches from n to nL (cm or yr).	Constant Mortality
Lv	30.0000	The length (or age if negative) at which 50 % vulnerable to harvest
(cm or yr).	Professional Judgement
Vp	20.0000	The vulnerability to harvest (as a function of length) power.	Professional Judgement
Llo	60.0000	The lower harvest slot length (cm).	Fishery Regulation
Lup	1000.0000	The upper harvest slot length (cm).	Fishery Regulation
Nc	0.0000	The slot limits non-compliance probability.	Default
pi	0.0709	The annual capture probability.	Current Study
rho	0.0000	The release probability.	Professional Judgement
Hm	0.1000	The hooking mortality probability.	Professional Judgement
Rmax	1.0000	The number of recruits at the carrying capacity (ind).	Default
Wa	0.0076	The (extrapolated) weight of a 1 cm individual (g).	Current Study
fa	1.0000	The (theoretical) fecundity of a 1 g female (eggs).	Default
q	0.1000	The catchability (annual probability of capture) for a unit of
effort.	Default
RPR	1.0000	The relative proportion of recruits that are of the ecotype.	Default

Figures

Spawners

Figure 51. Date of detections of Rainbow Trout spawners by year. Colour indicates whether fish was detected in a spawning river, at a spawning gate or at a receiver not associated with a spawning area. Spawning period (March 15th to May 15th) is shown by solid black vertical lines. A fish was deemed to be spawning in a given year if it satisfied one of the following criteria: 1. Detected within a spawning river during the spawning period and a detection hiatus period of at least 7 days; 2. Detected within a spawning gate during the spawning period and a detection hiatus period of at least 21 days. Detections have been aggregated daily.

Condition

Figure 52. The percent change in body weight of a 700 mm Rainbow Trout spawner relative to an in-lake capture, by year (with 95% CRIs).

Figure 53. The percent change in the body weight of a 100 mm Rainbow Trout relative to a typical year, by year (with 95% CRIs).

Growth

Figure 54. Estimated fork length for each fish by date.

Figure 55. Estimated von Bertalanffy growth curve for a typical growth year.

Figure 56. Estimated von Bertalanffy growth coefficient (k) by growth year (with 95% CIs).

Tag Loss

Figure 57. Estimated probability of the loss of a single T-bar tag (with 95% CIs).

Survival

Figure 58. The estimated annual interval probabilities (with 95% CIs). Natural mortalities are for a medium sized fish (600-700 mm). Harvest mortality is the recapture probability multiplied by the probability of being harvested if recaptured, assuming that recapture can only occur once in a year. Release is the probability of being released if recaptured.

Figure 59. The probability of spawning by fork length (mm) for a typical year (with 95% CIs). Points represent whether a fish spawned at each spawning opportunity (i.e. spawning period and monitored).

Figure 60. The estimated annual natural mortality by size class and sockeye abundance in the previous sockeye year (October 01 to September 30 the following year) (with 95% CIs).

Figure 61. The estimated annual recapture probability by year (with 95% CIs).

Figure 62. The estimated seasonal recapture probability for a typical year (with 95% CRIs).

Figure 63. The estimated probability of a mature fish spawning, by year (with 95% CRIs).

Figure 64. The estimated fork length (mm) at which 50% of fish are mature, by year (with 95% CRIs).

Mark-Recapture

Figure 65. The estimated abundance of fish > 50 cm from in-lake captures and recaptures in 2014 to 2021, by year (with 95% CIs).

Figure 66. The estimated in-lake abundance of fish > 50 cm from in-lake recaptures and captures, averaged across years (2014-2021), in-lake abundance from spawner captures and recaptures averaged across years (2019-2021) and spawner abundance (with 95% CIs).

Figure 67. The estimated number of alive marked fish in the lake and marked spawning fish by seasonal time period (with 95% CIs). The number of marked spawners is based on the alive marked fish in the previous period.

Yield-per-Recruit

Slot Limit

Figure 68. Calculated probability of spawning by length.

Figure 69. Calculated vulnerability to capture by length.

Figure 70. Calculated natural survivorship by length.

Figure 71. Calculated length by age.

Figure 72. Calculated weight by length.

Figure 73. Calculated harvested number of fish as percent of the number of recruits at carrying capacity by annual interval exploitation rate, by Llo (lower slot limit).

Figure 74. Calculated harvested number of fish as percent of the number of recruits at carrying capacity by annual interval exploitation rate, by Ls (length at 50% mature) and Rk (lifetime spawners per spawner at low density) with Llo (lower slot limit) held at 60 cm.

Figure 75. Calculated harvested number of fish as percent of the number of recruits at carrying capacity by annual interval exploitation rate, by Ls (length at 50% mature) and Rk (lifetime spawners per spawner at low density) with Llo (lower slot limit) held at 50 cm.

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

Quesnel Lake anglers for reporting their catches
Habitat Conservation Trust Foundation (HCTF)
- and the anglers, hunters, trappers and guides who contribute to the Trust
Ministry of Forests, Lands and Natural Resource Operations
- Lee Williston
- Mike Ramsay
- Greg Andrusak
Reel Adventures
- Kerry Reed
Vicky Lipinski

References

Bison, Robert, David O’Brien, and Steven J. D. Martell. 2003. “An Analysis of Sustainable Fishing Options for Adams Lake Bull Trout Using Life History and Telemetry Data.” Kamloops, B.C.: BC Ministry of Water Land; Air Protection.

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Boca Raton: Taylor & Francis.

Fabens, A J. 1965. “Properties and Fitting of the von Bertalanffy Growth Curve.” Growth 29 (3): 265–89.

Fabrizio, Mary C., James D. Nichols, James E. Hines, Bruce L. Swanson, and Stephen T. Schram. 1999. “Modeling Data from Double-Tagging Experiments to Estimate Heterogeneous Rates of Tag Shedding in Lake Trout (Salvelinus Namaycush).” Canadian Journal of Fisheries and Aquatic Sciences 56 (8): 1409–19. http://www.nrcresearchpress.com/doi/pdf/10.1139/f99-069.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Boca Raton: Taylor; Francis, CRC Press.

R Core Team. 2022. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

von Bertalanffy, L. 1938. “A Quantitative Theory of Organic Growth (Inquiries on Growth Laws Ii).” Human Biology 10: 181–213.

Kaslo Bull Trout Productivity 2022

Thu, 18 May 2023 00:00:00 +0000

The suggested citation for this analytic appendix is:

Amies-Galonski, E., Andrusak G.F. and Thorley, J.L. (2023) Kaslo Bull Trout Productivity 2022. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/798364147.

Background

The Kaslo River and Keen Creek, which is a tributary of the Kaslo River, are important Bull Trout spawning and rearing tributaries on Kootenay Lake. From 2012 to 2022, field crews have night-snorkeled these systems in the fall and recorded all Bull Trout less than 350 mm in length. Keen Creek was not snorkelled in 2020, 2021, or 2022 due to visibility issues. Snorkel and electrofishing marking crews have also captured and tagged juvenile Bull Trout for the snorkel crews to resight. Redd counts have been conducted in both systems since 2006, with the exception of 2020, when Keen Creek was not surveyed. The primary goal of the current analyses is to answer the following questions:

What is the observer efficiency when night-snorkeling for juvenile Bull Trout in the Kaslo River and Keen Creek?

What are the numbers of age-1 Bull Trout in the Kaslo River and Keen Creek?

What is the relationship between the stock (number of redds and eggs) and the resultant numbers of age-1 Bull Trout two years later?

Data Preparation

The data were cleaned, tidied and databased using R version 4.3.0 (R Core Team 2022).

Statistical Analysis

The parameters are summarised in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.3 bits, which is equivalent to a significant p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row.

The results are displayed graphically by plotting the modeled relationships between individual variables and the response with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their average values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% confidence/credible intervals (CIs, Bradford et al. 2005).

The analyses were implemented using R version 4.3.0 (R Core Team 2022) and the mbr family of packages.

Model Descriptions

Length Correction

The annual bias (inaccuracy) and error (imprecision) in observer’s fish length estimates when spotlighting (standing) and snorkeling were quantified from the divergence of their length distribution from the length distribution for JLT and SH (the two most experience snorkelers) in that year. More specifically, the length correction that minimised the Jensen-Shannon divergence (Lin 1991) provided a measure of the inaccuracy while the minimum divergence (the Jensen-Shannon divergence was calculated with log to base 2 which means it lies between 0 and 1) provided a measure of the imprecision.

Observer Efficiency

The observer efficiency varies by the fork length.

The preliminary analysis for observer efficiency indicated that system, observer, gradient, sinuosity, and river kilometre were not informative predictors.

Lineal Density

Both systems were broken into 100 m sites by bank. The lineal density at each site was estimated using an over-dispersed Poisson Generalized Linear Mixed Model.

Key assumptions of the Bayesian GLMM include:

The lineal density varies by system and stream distance.
The lineal density varies randomly by site and system within year.
The observer efficiency for each system is as estimated by the observer efficiency model.

The preliminary analysis for density indicated that site sinuosity and gradient were not informative predictors of lineal density.

Condition

The condition of adults was estimated using an allometric weight-length relationship. Key assumptions of the condition model include:

Weight varies by length.
Weight varies randomly by year.
The residual variation in weight is log normally distributed.

Fecundity

Female spawner fecundity was estimated using an allometric egg-weight relationship. The key assumptions of the fecundity model include:

Fecundity varies with weight.
The residual variation in fecundity is log normally distributed.

Spawner Length

The average length of spawners in each system and year was estimated using a linear regression. Key assumptions of the spawner length model include:

Length varies randomly with system, year, and system within year.
Length varies with sex and catch type.
The residual variation is normally distributed.

Egg Deposition

The total egg deposition in each year was estimated by

Converting the average spawner length to the average weight using the condition relationship for a typical year.
Adjusting the average weight by the annual condition effect (interpolating where unavailable)
Converting the average weight to the average fecundity using the fecundity relationship
Multiplying the average fecundity by the number of females (assuming 1 female per redd)

Eggs Stock-Recruitment

The prior uncertainty in the egg to age-1 survival is described by a Beta distribution with an alpha of 4 and beta of 49 which has a mean of 0.075 and a standard deviation of 0.036. This is based off the assumption that a recruit has a ~50% chance of surviving through each summer or winter and must pass through two winters and two summers to survive to age-1.
The carrying capacity varies between systems.
The residual variation in the recruits is log normally distributed.

The $E_{K/2}$ Limit Reference Point (Mace 1994) ($E_{0.5 R_{max}}$) was calculated, corresponding to the stock (number of eggs per 100 meters) that produce 50% of the maximum recruitment ($K$).

Model Templates

Observer Efficiency

model{
  bIntercept ~ dnorm(0, 2^-2)
  bLength ~ dnorm(0, 2^-2)

  for(i in 1:length(Observed)){
    logit(eObserved[i]) <-  bIntercept + bLength * Length[i]
    Observed[i] ~ dbern(eObserved[i])
  }

Block 1. Final model.

Lineal Density

model{
  bEfficiency ~ dnorm(Efficiency[1], EfficiencySD[1]^-2) T(0, 1)

  b0 ~ dnorm(0, 5^-2)
  bRkm ~ dnorm(0, 2^-2)
  bSystem[1] <- 0
  for(i in 2:nSystem) {
    bSystem[i] ~ dnorm(0, 2^2)
  }

  sSystemYear ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:nSystem) {
    for(j in 1:nYear) {
      bSystemYear[i,j] ~ dnorm(0, sSystemYear^-2)
    }
  }

  sSite ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:nSite) {
    bSite[i] ~ dnorm(0, sSite^-2)
  }

  sDispersion ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:length(Count)) {
    log(eDensity[i]) <- b0 + bSystem[System[i]] + bRkm * Rkm[i] + bSite[Site[i]] + bSystemYear[System[i],Year[i]]
    eDispersion[i] ~ dgamma(sDispersion^-2, sDispersion^-2)
    Count[i] ~ dpois(eDensity[i] * Length[i] * Coverage[i] * bEfficiency * eDispersion[i])
  }

Block 2. The final model.

Condition

model{
  b0 ~ dnorm(6, 2^-2)
  bLength ~ dnorm(3, 1^-2)
  sWeight ~ dnorm(0, 1^-2) T(0,)
  sYear ~ dnorm(0, 1^-2) T(0,)

  for (i in 1:nYear) {
    bYear[i] ~ dnorm(0, sYear^-2)
  }

  for (i in 1:nObs) {
    log(eWeight[i]) <- b0 + (log(Length[i]) - log(500)) * bLength + bYear[Year[i]]
    Weight[i] ~ dlnorm(log(eWeight[i]), sWeight^-2)
  }

Block 3. Model description.

Fecundity

model{
  b0 ~ dnorm(0, 2^-2)
  bWeight ~ dnorm(1, 2^-2)
  sFecundity ~ dnorm(0, 2^-2) T(0,)

  for (i in 1:nObs) {
    log(eFecundity[i]) <- b0 + (log(Weight[i]) - log(2300)) * bWeight
    Fecundity[i] ~ dlnorm(log(eFecundity[i]), sFecundity^-2)
  }

Block 4. Model description.

Spawner Length

model{
  bLength ~ dnorm(650, 100^-2)
  sLength ~ dnorm(0, 100^-2) T(0,)
  sYear ~ dnorm(0, 100^-2) T(0,)
  sSystem ~ dnorm(0, 100^-2) T(0,)
  sYearSystem ~ dnorm(0, 100^-2) T(0,)
  bSex ~ dbeta(1, 1)
  bFemale ~ dnorm(0, 100^-2)
  bBias ~ dnorm(0, 100^-2)

  for (i in 1:nYear) {
    bYear[i] ~ dnorm(0, sYear^-2)
  }

  for (i in 1:nSystem) {
    bSystem[i] ~ dnorm(0, sSystem^-2)
  }

  for (i in 1:nYear) {
    for (j in 1:nSystem) {
        bYearSystem[i, j] ~ dnorm(0, sYearSystem^-2)
    }
  }

  bCatchType[1] <- 0
  for(i in 2:nCatchType) {
    bCatchType[i] ~ dnorm(0, 100^-2)
  }

  for (i in 1:nObs) {
    Female[i] ~ dbern(bSex)
    eLength[i] <- bLength + bSystem[System[i]] + bFemale * Female[i] + bCatchType[CatchType[i]] + bYear[Year[i]] + bYearSystem[Year[i], System[i]] + bBias * Bias[i]
    Length[i] ~ dnorm(eLength[i], sLength^-2)
  }

Block 5. Model description.

Stock-Recruiment

model {
  bAlpha ~ dbeta(4, 49)

  bK ~ dnorm(3, 1^-2)
  bKPopulation ~ dnorm(0, 1^-2)
  sRecruits ~ dexp(1)

  for(i in 1:nObs){
    log(eK[i]) <- bK + (Kaslo[i] * bKPopulation - (1-Kaslo[i]) * bKPopulation)
    eBeta[i] <-  bAlpha/eK[i]
    eRecruits[i] <- bAlpha * Stock[i] / (1 + eBeta[i] * Stock[i])
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

Block 6. Final model.

Results

Tables

Coverage

Table 1. Total length of river bank counted (including replicates) by system and year.

System	Year	Length (km)
Kaslo River	2012	10.57 [km]
Kaslo River	2013	13.43 [km]
Kaslo River	2014	11.45 [km]
Kaslo River	2015	7.32 [km]
Kaslo River	2016	11.59 [km]
Kaslo River	2017	9.79 [km]
Kaslo River	2018	8.44 [km]
Kaslo River	2019	7.19 [km]
Kaslo River	2020	10.42 [km]
Kaslo River	2021	9.56 [km]
Kaslo River	2022	8.05 [km]
Keen Creek	2012	1.44 [km]
Keen Creek	2013	0.95 [km]
Keen Creek	2014	0.67 [km]
Keen Creek	2015	0.72 [km]
Keen Creek	2016	0.85 [km]
Keen Creek	2017	1.68 [km]
Keen Creek	2018	3.37 [km]
Keen Creek	2019	1.48 [km]

Observer Efficiency

Table 2. Parameter descriptions.

Parameter	Description
`Length`	The standardized fork length
`Observed`	The number of individuals observed (`0` or `1`)
`Tagged`	The number of tagged individuals (`1`)
`bIntercept`	The intercept for `logit(eObserved)`
`bLength2`	The effect of `Length` on the effect of `Length` on `bIntercept`
`bLength`	The effect of `Length` on `bIntercept`
`eObserved`	The expected probability of observing an individual

Table 3. Final parameter estimates.

term	estimate	lower	upper	svalue
bIntercept	-1.6658088	-2.0536553	-1.311882	10.55171
bLength	0.7290748	0.3666931	1.119661	10.55171

Table 4. Observer Efficiency estimates for a 123 mm Bull Trout.

System	Efficiency	EfficiencyLower	EfficiencyUpper	EfficiencySD
Kaslo River	0.1248111	0.0796234	0.1774935	0.0249669

Table 5. Final model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
256	2	3	500	1	554	1.003	TRUE

Table 6. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.8164062	0.8164062	0.7460938	0.8789062	0.0409441
mean	-0.1648526	-0.1641625	-0.2750668	-0.0430340	0.0057785
variance	0.8619843	0.8557727	0.6207338	1.0613685	0.0608574
skewness	1.5615010	1.5597321	1.0895702	2.2111012	0.0135193
kurtosis	0.7988234	0.8219999	-0.5282195	3.5177447	0.0213019

Table 7. Model sensitivity.

all	analysis	sensitivity	bound
all	1.003	1.005	1.003

Lineal Density

Table 8. Parameter descriptions.

Parameter	Description
`Count`	Number of fish counted
`Coverage`	Proportion of site surveyed
`Dispersion`	Factor for random effect of overdispersion
`Efficiency`	The observer efficiency from the observer efficiency model
`Length`	Length of site (m)
`Site`	The site
`System`	The system
`Year`	The year
`b0`	Intercept of log(eDensity)
`bDispersion`	The random effect of overdispersion
`bEfficiency`
`bSite`	The random effect of `Site` on `bSystemYear`
`bSystemYear`	The effect of `System` and `Year` on `log(eDensity)`
`bSystem`	The effect of `System` on `log(eDensity)`
`eCount`	The expected `Count`
`eDensity`	The expected lineal density
`log_sDispersion`	`log(sDispersion)`
`log_sSite`	`log(sSite)`
`sDispersion`	The SD of `bDispersion`
`sSite`	The SD of `bSite`
`sSystemYear`	The SD of `bSystemYear`

Table 9. Parameter estimates.

term	estimate	lower	upper	svalue
b0	-2.4738446	-2.9648525	-1.7767605	10.551708
bEfficiency	0.1240374	0.0703511	0.1748241	10.551708
bRkm	0.4456673	0.3508408	0.5397803	10.551708
bSystem[1]	0.0000000	0.0000000	0.0000000	0.000000
bSystem[2]	1.0692784	0.5273687	1.5287126	8.966746
sDispersion	0.5417775	0.4441352	0.6387831	10.551708
sSystemYear	0.4487018	0.2846204	0.7200574	10.551708

Table 10. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1295	6	3	500	50	216	1.018	TRUE

Table 11. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.4239382	0.4548263	0.4223938	0.4857143	4.092277
mean	-0.2613550	-0.2959537	-0.3486514	-0.2445796	2.299043
variance	0.7388195	0.9107643	0.8484822	0.9776519	10.551708
skewness	0.3276331	0.5445753	0.4236996	0.6679827	10.551708
kurtosis	-0.7229660	-0.3879293	-0.6289873	-0.0472131	8.229780

Table 12. Model sensitivity.

all	analysis	sensitivity	bound
all	1.018	1.009	1.013

Condition

Table 13. Parameter descriptions.

Parameter	Description
`Length[i]`	The `i`^th Length value
`Weight[i]`	The `i`^th Weight value
`Year[i]`	The `i`^th Year value
`b0`	Intercept for `eLength`
`bLength`	The effect of Length on `eWeight`
`bYear[i]`	The effect of year on `eWeight`
`eWeight[i]`	Expected value of `Weight[i]`
`sWeight`	SD of residual variation in `Weight`
`sYear`	SD of Year

Table 14. Model coefficients.

term	estimate	lower	upper	svalue
b0	7.0908427	6.9843972	7.1702566	10.55171
bLength	2.9169114	2.8592542	2.9753611	10.55171
sWeight	0.1287263	0.1226121	0.1352580	10.55171
sYear	0.1075035	0.0633133	0.2307799	10.55171

Table 15. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
795	4	3	500	50	236	1.009	TRUE

Table 16. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0004476	0.0002064	-0.0705615	0.0672957	0.0350233
variance	0.9883079	1.0013822	0.9061050	1.1032157	0.3483603
skewness	-0.4436651	-0.0000260	-0.1722547	0.1665849	10.5517083
kurtosis	1.7350490	-0.0175825	-0.3051569	0.3830626	10.5517083

Table 17. Model sensitivity.

all	analysis	sensitivity	bound
all	1.009	1.028	1.017

Fecundity

Table 18. Parameter descriptions.

Parameter	Description
`Fecundity[i]`	The `i`^th Fecundity value
`b0`	Intercept for `eFecundity`
`bWeight`	Effect of `Weight` on `b0`
`eFecundity[i]`	Expected value of `Fecundity[i]`
`sFecundity`	SD of residual variation in `Fecundity`

Table 19. Model coefficients.

term	estimate	lower	upper	svalue
b0	8.4865471	8.4361069	8.540152	10.55171
bWeight	1.0353648	0.9154408	1.155619	10.55171
sFecundity	0.1222765	0.0919066	0.171978	10.55171

Table 20. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
28	3	3	500	1	718	1.006	TRUE

Table 21. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0109091	-0.0008000	-0.3508327	0.3817987	0.0588536
variance	0.9014474	0.9753021	0.5335099	1.6020362	0.3581829
skewness	0.3395831	0.0024970	-0.7783873	0.8028030	1.3398200
kurtosis	-0.5677115	-0.3245532	-1.1359166	1.6072526	0.6106607

Table 22. Model sensitivity.

all	analysis	sensitivity	bound
all	1.006	1.001	1.004

Spawner Length

Table 23. Parameter descriptions.

Parameter	Description
`Bias[i]`	The `i`^th Bias Value
`CatchType[i]`	The `i`^th CatchType value
`Length[i]`	The `i`^th Length value
`Sex[i]`	The `i`^th Sex value
`System[i]`	The `i`^th System value
`Year[i]`	The `i`^th Year value
`bBias`	The effect of Bias on `bLength`
`bCatchType`	Effect of `CatchType` on `bLength`
`bLength`	Intercept for `eLength`
`bSex`	Effect of `Sex` on `bLength`
`bSystem`	Effect of `System` on `bLength`
`bYearSystem`	The effect of `Year` and `System` on `bLength`
`bYear`	Effect of `Year` on `bLength`
`eLength[i]`	Expected value of `Length[i]`
`sLength`	SD of residual variation in `Length`
`sYearSystem`	SD of `bYearSystem`

Table 24. Model coefficients.

term	estimate	lower	upper	svalue
bBias	85.1474062	28.0404411	131.5031725	6.851268
bCatchType[2]	-32.4961719	-56.8785455	-7.9729875	6.644818
bFemale	-82.5287978	-92.7655264	-72.9287353	10.551708
bLength	644.7371241	591.0138818	692.2094625	10.551708
bSex	0.3688462	0.3429386	0.3955583	10.551708
sLength	83.7530927	80.7409272	87.0345789	10.551708
sSystem	60.8708995	36.3204960	94.5656038	10.551708
sYear	54.5167351	32.9082313	87.7340891	10.551708
sYearSystem	17.9002560	2.0329852	41.9829273	10.551708

Table 25. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
1423	9	3	500	50	369	1.007	TRUE

Table 26. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.000000
mean	0.0079779	-0.0011206	-0.0498701	0.0505297	0.345915
variance	0.9485976	1.0005454	0.9251700	1.0724468	2.546084
skewness	-0.3025628	0.0046228	-0.1250822	0.1296527	10.551708
kurtosis	2.5073545	-0.0147488	-0.2430001	0.2711679	10.551708

Table 27. Model sensitivity.

all	analysis	sensitivity	bound
all	1.007	1.007	1.004

Egg Deposition

Table 28. The estimated total egg deposition by system and year.

Year	System	Length	Condition	Weight	Fecundity	Redds	Eggs
2012	Kaslo River	627.8450	1.1050547	2575.078	5453.752	433	2361474.79
2012	Keen Creek	661.9578	1.1050547	3004.671	6399.570	80	511965.63
2013	Kaslo River	670.6432	1.0819725	3055.623	6512.430	305	1986291.11
2013	Keen Creek	720.6144	1.0819725	3770.185	8091.041	50	404552.06
2014	Kaslo River	579.7287	1.0588902	1956.015	4101.159	113	463430.96
2014	Keen Creek	625.3021	1.0588902	2438.532	5153.111	17	87602.89
2015	Kaslo River	546.0136	1.0358079	1607.037	3350.012	135	452251.68
2015	Keen Creek	563.0800	1.0358079	1757.923	3673.386	80	293870.86
2016	Kaslo River	549.8609	1.0159445	1608.864	3353.936	340	1140338.10
2016	Keen Creek	588.9884	1.0159445	1965.087	4120.446	34	140095.17
2017	Kaslo River	561.9084	1.0149568	1712.189	3574.516	360	1286825.60
2017	Keen Creek	594.7727	1.0149568	2020.177	4240.097	113	479130.96
2018	Kaslo River	534.6373	0.9586071	1398.646	2900.493	267	774431.74
2018	Keen Creek	574.9807	0.9586071	1728.693	3610.353	51	184127.99
2019	Kaslo River	569.4663	0.8503298	1491.077	3099.131	131	405986.20
2019	Keen Creek	605.7864	0.8503298	1784.946	3731.536	33	123140.69
2020	Kaslo River	542.4144	0.9346675	1422.369	2952.108	111	327683.99
2020	Keen Creek	580.8238	0.9346675	1736.034	3625.971	NA	NA
2021	Kaslo River	490.5934	0.9346675	1061.906	2180.495	180	392489.05
2021	Keen Creek	534.7120	0.9346675	1364.274	2827.593	23	65034.65
2022	Kaslo River	507.3020	0.9346675	1170.917	2411.344	290	699289.84
2022	Keen Creek	549.7956	0.9346675	1479.643	3074.458	44	135276.15

Stock-Recruiment

Table 29. Parameter descriptions.

Parameter	Description
`Recruits`	Age-1 Bull Trout density
`Stock`	Bull Trout egg density
`bK`	Density Carrying Capacity
`eBeta[i]`	Density-dependence for the `i`^th population
`eRecruits`	Expected density of `Recruits`
`bKPopulation`	Population effect on `bK`
`bAlpha`	Recruits per stock per 100 meters at low stock density
`sRecruits`	SD of residual variation in `Recruits`

Table 30. Density Carrying Capacity (K).

System	estimate	upper	lower	svalue
Kaslo River	22.09028	34.95172	15.84668	10.55171
Keen Creek	40.38173	70.47400	26.21022	10.55171

Table 31. Model coefficients.

term	estimate	lower	upper	svalue
bAlpha	0.0456208	0.0176069	0.1243621	10.551708
bK	3.3885488	3.0988822	3.8125180	10.551708
bKPopulation	-0.3056275	-0.5562366	-0.0446364	5.125444
sRecruits	0.3822650	0.2667180	0.5897838	10.551708

Table 32. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
15	4	3	500	100	1292	1.001	TRUE

Table 33. Ek/2 Limit Reference Point estimates.

Kaslo	Recruits	Stock	estimate	lower	upper	svalue
TRUE	1	1	474.0307	145.9616	1775.006	10.55171
FALSE	1	1	870.4489	257.5997	3571.140	10.55171

Table 34. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0222671	-0.0007621	-0.5017827	0.5116441	0.0851219
variance	0.7990110	0.9581722	0.4059042	1.8775531	0.6048020
skewness	0.2337462	0.0044709	-1.0808559	1.0521947	0.6283808
kurtosis	-0.6650047	-0.5111128	-1.3849358	1.9097934	0.3361753

Table 35. Model sensitivity.

all	analysis	sensitivity	bound
all	1.001	1.001	1.002

Figures

Systems

Figure 1. Spatial distribution of fish-bearing channel.

Figure 2. Channel elevation by river kilometre and system.

Figure 3. Catchment area by river kilometre and system.

Coverage

Figure 4. Snorkel count coverage by year and bank.

Sites

Figure 5. Site gradient by river kilometre and system.

Figure 6. Site sinuosity by river kilometre and system.

Length Correction

Figure 7. Corrected length-frequency histogram by year and observation type.

Fish

Figure 8. Length-frequency plot of marked Bull Trout by year and system, coloured by tag colour.

Figure 9. Corrected length-frequency plot of observed Bull Trout by year and age class.

Observer Efficiency

Figure 10. Distribution of marked juvenile Bull Trout by year and tag color.

Figure 11. Estimated observer efficiency by fork length and system (with 95% CIs).

Lineal Density

Figure 12. Percent coverage by year and system.

Figure 13. Estimated density by year and system (with 95% CIs).

Figure 14. The estimated abundance by year and system (with 95% CIs).

Figure 15. The estimated density by site and system (with 95% CIs).

Redds

Figure 16. Complete redds by system and spawn year.

Condition

Figure 17. The weight-length relationship (with 95% CIs).

Figure 18. The percent change in the body condition for an average length fish relative to a typical year by year (with 95% CIs).

Fecundity

Figure 19. The predicted relationship between Fecundity and Weight for Bull Trout (with 95% CIs), data from Brunson (1952).

Spawner Length

Figure 20. Length frequency of Bull Trout spawners by sex.

Figure 21. Length frequency of Bull Trout spawners by catch type in 2018 and 2019.

Figure 22. Average Length of Bull Trout spawners in Keen Creek compared to all other surveyed tributaries.

Figure 23. The expected length of female spawners by year for Kaslo River and Keen Creek (with 95% CIs).

Egg Deposition

Figure 24. The estimated spawner fecundity by year uncorrected for condition (with 95% CIs).

Figure 25. The estimated spawner fecundity by year corrected for condition.

Figure 26. The estimated total egg deposition by system and year.

Stock-Recruiment

Figure 27. Estimated stock-recruitment relationship (with 95% CIs). Additional modeled relationships (grey lines) derived from randomly sampled parameter values are also displayed. The points are labelled by spawn year.

Figure 28. Predicted egg survival to age-1 by egg deposition (with 95% CRIs).

Conclusions

Observer efficiency is approximately 17% for age-1 Bull Trout.
Age-1 Bull Trout are relatively evenly distributed with respect to mesohabitat.
Lineal densities of age-1 Bull Trout increase with river kilometre in both systems.
The age-1 carrying capacity is estimated to be around 170 fish per km in Kaslo River and around 310 fish per km in Keen Creek.

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

Habitat Conservation Trust Foundation
Ministry of Forest, Lands and Natural Resource Operations
- Greg Andrusak
- Emmanuel Abecia
Ministry of Environment
- Jen Sarchuk
BC Fish and Wildlife
- Trina Radford
Stephan Himmer
Gillian Sanders
Jeff Berdusco
Vicky Lipinski
Jimmy Robbins
Jason Bowers
Seb Dalgarno

References

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Taylor & Francis.

Gelman, Andrew, Daniel Simpson, and Michael Betancourt. 2017. “The Prior Can Often Only Be Understood in the Context of the Likelihood.” Entropy 19 (10): 555. https://doi.org/10.3390/e19100555.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Academic Press. http://www.vogelwarte.ch/bpa.html.

Lin, J. 1991. “Divergence Measures Based on the Shannon Entropy.” IEEE Transactions on Information Theory 37 (1): 145–51. https://doi.org/10.1109/18.61115.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. 2nd ed. CRC Texts in Statistical Science. Taylor; Francis, CRC Press.

R Core Team. 2022. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.

Duncan Lardeau Juvenile Rainbow Trout Abundance 2023

Mon, 17 Apr 2023 00:00:00 +0000

The suggested citation for this analytic report is:

Thorley, J.L., Andrusak, G.F. and Amies-Galonski, E. (2023) Duncan Lardeau Juvenile Rainbow Trout Abundance 2023. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1491296951.

Background

The primary aims of the current analyses were to:

Estimate the spring abundance of age-1 fish by year.
Estimate the egg deposition.
Estimate the stock-recruitment relationship between the egg deposition and the abundance of age-1 recruits the following spring.
Estimate the survival from age-1 to age-2.
Estimate the expected number of spawners without in river and/or in lake variation.

Methods

Data Preparation

The data were provided by the Ministry of Forests, Lands and Natural Resource Operations (MFLNRO). The historical and current snorkel count data were manipulated using R version 4.2.2 (R Core Team 2022) and organised in an SQLite database.

Statistical Analysis

The parameters are summarised in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of $>$ 4.3 bits, which is equivalent to a significant p-value $<$ 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row.

The condition that parameters describing the effects of secondary (nuisance) explanatory variable(s) have significant p-values was used as a model selection heuristic (Kery and Schaub 2011). Based on a similar argument, the condition that random effects have a standard deviation with a lower 95% compatibility interval (CL) $>$ 5% of the estimate was used as an additional model selection heuristic. Primary explanatory variables are evaluated based on their estimated effect sizes with 95% CLs (Bradford et al. 2005)

The analyses were implemented using R version 4.2.2 (R Core Team 2022) and the mbr family of packages.

Model Descriptions

Length Correction

After correcting the fish lengths, age-1 individuals were assumed to be those with a fork length $\leq$ 100 mm.

Abundance

Key assumptions of the abundance model include:

The lineal fish density varies with year, useable width and river kilometer as a polynomial, and randomly with site.
The observer efficiency at marking sites varies by study design (GPS versus Index).
The observer efficiency also varies by visit type (marking versus count) within study design and randomly by snorkeller.
The expected count at a site is the expected lineal density multiplied by the site length, the observer efficiency and the proportion of the site surveyed.
The residual variation in the actual count is gamma-Poisson distributed.

Condition

The condition of fish with a fork length $\geq$ 500 mm was estimated via an analysis of mass-length relations (He et al. 2008).

More specifically the model was based on the allometric relationship

\[ W = \alpha_c L^{\beta_c}\]

where $W$ is the weight (mass), $\alpha_c$ is the coefficent, $\beta_c$ is the exponent and $L$ is the length.

To improve chain mixing the relation was log-transformed, i.e.,

\[ \log(W) = \log(\alpha_c) + \beta_c \log(L).\]

Key assumptions of the condition model include:

$\alpha_c$ can vary randomly by year.
The residual variation in weight is log-normally distributed.

Fecundity

The fecundity of females with a fork length $\geq$ 500 mm was estimated via an analysis of fecundity-mass relations.

More specifically the model was based on the allometric relationship

\[ F = \alpha_f W^{\beta_f}\]

where $F$ is the fecundity, $\alpha_f$ is the coefficent, $\beta_f$ is the exponent and $W$ is the weight.

To improve chain mixing the relation was log-transformed.

Key assumptions of the fecundity model include:

The residual variation in fecundity is log-normally distributed.

Spawner Size

Egg Deposition

The egg deposition in each year was estimated by

converting the average length of spawners to the average weight using the condition relationship for a typical year
adjusting the average weight by the annual condition effect (interpolating where unavailable)
converting the average weight to the average fecundity using the fecundity relationship
multiplying the average fecundity by the AUC based estimate of the number of females (assuming a sex ratio of 1:1)

Stock-Recruitment

\[ R = \frac{\alpha_s \cdot E}{1 + \beta_s \cdot E} \quad,\]

where $\alpha_s$ is the maximum number of recruits per egg (egg survival), and $\beta_s$ is the density dependence.

Key assumptions of the stock-recruitment model include:

The residual variation in the number of recruits is log-normally distributed with the standard deviation scaling with the uncertainty in the number of recruits.

The age-1 carrying capacity ($K$) is given by:

\[ K = \frac{\alpha_s}{\beta_s} \quad.\]

The LRP was also converted into a number of spawners in a typical year (assuming 6,000 eggs per spawner and a sex ratio of 1:1).

Age-1 to Age-2 Survival

Key assumptions of the survival rate model include:

The residual variation in the number of age-2 individuals is log-normally distributed.

Reproductive Rate

The maximum reproductive rate (the number of spawners per spawner at low density) not accounting for fishing mortality was calculated by multiplying $\beta_s$ (number of recruits per egg at low density) from the stock-recruitment relationship by the inlake survival by the estimated eggs per spawner in each year (assuming a sex ratio of 1:1). The inlake survival from age-1 to spawning was calculated by dividing the subsequent number of spawners by the number of recruits assuming that equal numbers of fish spawn at age 5, 6 and 7.

Expected Spawners

The expected spawners without in river and/or in lake variation was calculated from the stock-recruitment relationship and assuming an age-1 to spawner survival of 0.67%.

Results

Model Templates

Abundance

  data {

    int<lower=0> nMarked;
    int<lower=0> Marked[nMarked];
    int<lower=0> Resighted[nMarked];
    int<lower=0> IndexMarked[nMarked];

    int<lower=0> nObs;
    int Marking[nObs];
    int Index[nObs];
    int<lower=0> nSwimmer;
    int<lower=0> Swimmer[nObs];
    int<lower=0> nYear;
    int<lower=0> Year[nObs];
    real Rkm[nObs];
    int<lower=0> nSite;
    int<lower=0> Site[nObs];

    real SiteLength[nObs];
    real SurveyProportion[nObs];

    int Count[nObs];
  }
  parameters {
    real bEfficiency;
    real bEfficiencyIndex;

    real bDensity;
    real<lower=0> sDensityYear;
    vector[nYear] bDensityYear;

    vector[4] bDensityRkm;
    real<lower=0> sDensitySite;
    vector[nSite] bDensitySite;

    real bEfficiencyMarking;
    real bEfficiencyMarkingIndex;

    real<lower=0,upper=5> sEfficiencySwimmer;
    vector[nSwimmer] bEfficiencySwimmer;

    real<lower=0> sDispersion;
  }
  model {

    vector[nObs] eDensity;
    vector[nObs] eEfficiency;
    vector[nObs] eAbundance;
    vector[nObs] eCount;

    sDispersion ~ gamma(0.01, 0.01);

    bDensity ~ normal(0, 2);
    bDensityRkm ~ normal(0, 2);
    sDensitySite ~ uniform(0, 5);
    bDensitySite ~ normal(0, sDensitySite);
    sDensityYear ~ uniform(0, 5);
    bDensityYear ~ normal(0, sDensityYear);

    bEfficiency ~ normal(0, 5);
    bEfficiencyIndex ~ normal(0, 5);
    bEfficiencyMarking ~ normal(0, 5);
    bEfficiencyMarkingIndex ~ normal(0, 5);
    sEfficiencySwimmer ~ uniform(0, 5);
    bEfficiencySwimmer ~ normal(0, sEfficiencySwimmer);

    for (i in 1:nMarked) {
      target += binomial_lpmf(Resighted[i] | Marked[i],
        inv_logit(
          bEfficiency +
          bEfficiencyIndex * IndexMarked[i] +
          bEfficiencyMarking +
          bEfficiencyMarkingIndex * IndexMarked[i]
        ));
    }

    for (i in 1:nObs) {
      eDensity[i] = exp(bDensity +
                        bDensityRkm[1] * Rkm[i] +
                        bDensityRkm[2] * pow(Rkm[i], 2.0) +
                        bDensityRkm[3] * pow(Rkm[i], 3.0) +
                        bDensityRkm[4] * pow(Rkm[i], 4.0) +
                        bDensitySite[Site[i]] +
                        bDensityYear[Year[i]]);

      eEfficiency[i] = inv_logit(
        bEfficiency +
        bEfficiencyIndex * Index[i] +
        bEfficiencyMarking * Marking[i] +
        bEfficiencyMarkingIndex * Index[i] * Marking[i] +
        bEfficiencySwimmer[Swimmer[i]]);

      eAbundance[i] = eDensity[i] * SiteLength[i];

      eCount[i] = eAbundance[i] * eEfficiency[i] * SurveyProportion[i];
    }

    target += neg_binomial_2_lpmf(Count | eCount, sDispersion);
  }

Block 1. Abundance model description.

Condition

 data {
  int nYear;
  int nObs;

  vector[nObs] Length;
  vector[nObs] Weight;
  int Year[nObs];

parameters {
  real bWeight;
  real bWeightLength;
  real sWeightYear;

  vector[nYear] bWeightYear;
  real sWeight;

model {

  vector[nObs] eWeight;

  bWeight ~ normal(-10, 5);
  bWeightLength ~ normal(3, 2);

  sWeightYear ~ normal(-2, 5);

  for (i in 1:nYear) {
    bWeightYear[i] ~ normal(0, exp(sWeightYear));
  }

  sWeight ~ normal(-2, 5);
  for(i in 1:nObs) {
    eWeight[i] = bWeight + bWeightLength * log(Length[i]) + bWeightYear[Year[i]];
    Weight[i] ~ lognormal(eWeight[i], exp(sWeight));
  }

Block 2.

Fecundity

 data {
  int nObs;

  vector[nObs] Weight;
  vector[nObs] Fecundity;

parameters {
  real bFecundity;
  real bFecundityWeight;
  real sFecundity;

model {

  vector[nObs] eFecundity;

  bFecundity ~ uniform(0, 5);
  bFecundityWeight ~ uniform(0, 2);
  sFecundity ~ uniform(0, 1);

  for(i in 1:nObs) {
    eFecundity[i] = log(bFecundity) + bFecundityWeight * log(Weight[i]);
    Fecundity[i] ~ lognormal(eFecundity[i], sFecundity);
  }

Block 3.

Stock-Recruitment

model {
  a ~ dunif(0, 1)
  b ~ dunif(0, 0.1)
  sScaling ~ dunif(0, 5)

  eRecruits <- a * Stock / (1 + Stock * b)

  for(i in 1:nObs) {
    esRecruits[i] <- SDLogRecruits[i] * sScaling
    Recruits[i] ~ dlnorm(log(eRecruits[i]), esRecruits[i]^-2)
  }

Block 4. Stock-Recruitment model description.

Age-1 to Age-2 Survival

model {
  bSurvival ~ dnorm(0, 2^-2)
  sRecruits ~ dnorm(0, 2^-2) T(0,)

  for(i in 1:nObs) {
    logit(eSurvival[i]) <- bSurvival
    eRecruits[i] <- Stock[i] * eSurvival[i]
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }

Block 5. In-river survival model description.

Tables

Abundance

Table 1. Parameter descriptions.

Parameter	Description
`Index`	Whether the `i`^th visit was to an index site
`Marking[i]`	Whether the `i`^th visit was to a site with marked fish
`Rkm[i]`	River kilometer of `i`^th visit
`SiteLength[i]`	Length of site of `i`^th visit
`Site[i]`	Site of `i`^th visit
`SurveyProportion[i]`	Proportion of site surveyed on `i`^th visit
`Swimmer[i]`	Snorkeler on `i`^th site visit
`Year[i]`	Year of `i`^th site visit
`bDensityRkm[i]`	`i`^th-order polynomial coefficients of effect of river kilometer on `bDensity`
`bDensitySite[i]`	Effect of `i`^th `Site` on `bDensity`
`bDensityYear[i]`	Effect of `i`^th `Year` on `bDensity`
`bDensity`	Intercept for `log(eDensity)`
`bEfficiencyIndex`	Effect of `Index` on `bEfficiency`
`bEfficiencyMarkingIndex`	Effect of `Marking` and `Index` on `bEfficiency`
`bEfficiencyMarking`	Effect of `Marking` on `bEfficiency`
`bEfficiencySwimmer[i]`	Effect of `i`^th `Swimmer` on `bEfficiency`
`bEfficiency`	Intercept of `logit(eEfficiency)`
`eAbundance[i]`	Expected abundance of fish at site of `i`^th visit
`eCount[i]`	Expected total number of fish at site of `i`^th visit
`eDensity[i]`	Expected lineal density of fish at site of `i`^th visit
`eEfficiency[i]`	Expected observer efficiency on `i`^th visit
`sDensitySite`	SD of `bDensitySite`
`sDispersion`	Overdispersion of `Count[i]`
`sEfficiencySwimmer`	SD of `bEfficiencySwimmer`

Age-1

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	-1.2992426	-1.7582160	-0.8349936	11.551228
bDensityRkm[1]	-0.1705547	-0.3205906	-0.0188394	5.211378
bDensityRkm[2]	0.5741459	0.3731147	0.7746487	11.551228
bDensityRkm[3]	-0.1244389	-0.1970181	-0.0533908	11.551228
bDensityRkm[4]	-0.2618053	-0.3327087	-0.1959403	11.551228
bEfficiency	-1.7997011	-2.0588426	-1.5291016	11.551228
bEfficiencyIndex	0.3902163	-0.1970283	1.0653021	2.263515
bEfficiencyMarking	0.5387898	0.3322680	0.7394791	11.551228
bEfficiencyMarkingIndex	0.8219006	0.1752764	1.4436940	6.421945
sDensitySite	0.6871717	0.6198630	0.7551973	11.551228
sDensityYear	0.6898622	0.4809528	1.0513731	11.551228
sDispersion	1.3123239	1.1933859	1.4448227	11.551228
sEfficiencySwimmer	0.3795079	0.2268709	0.6855667	11.551228

Table 3. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3913	13	3	1000	5	1419	1.004	TRUE

Table 4. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.3677485	0.4219269	0.3999425	0.4439049	9.966265
mean	-0.2987924	-0.3675765	-0.3981296	-0.3372166	11.551228
variance	0.6032294	0.8452765	0.8040410	0.8864911	11.551228
skewness	0.2345063	0.5620084	0.4834390	0.6412612	11.551228
kurtosis	-0.4722507	-0.1689507	-0.3399505	0.0368888	9.229299

Table 5. Model sensitivity.

all	analysis	sensitivity	bound
all	1.004	1.004	1.002

Age-2

Table 6. Model coefficients.

term	estimate	lower	upper	svalue
bDensity	-2.2225123	-2.6706912	-1.7718504	11.5512276
bDensityRkm[1]	-0.2249711	-0.4090499	-0.0442557	5.8233071
bDensityRkm[2]	0.0839440	-0.1737145	0.3388965	0.9747433
bDensityRkm[3]	0.0449129	-0.0450734	0.1335851	1.5868867
bDensityRkm[4]	-0.1032100	-0.1876697	-0.0163603	5.7698679
bEfficiency	-1.9634576	-2.3298731	-1.5893538	11.5512276
bEfficiencyIndex	0.5461119	-0.0913767	1.2325885	3.4584705
bEfficiencyMarking	0.8766722	0.5712271	1.1853509	11.5512276
bEfficiencyMarkingIndex	1.9609922	0.9199813	3.0653768	11.5512276
sDensitySite	0.8326922	0.7428837	0.9194001	11.5512276
sDensityYear	0.5354524	0.3692774	0.8382351	11.5512276
sDispersion	1.1967609	1.0442236	1.3892824	11.5512276
sEfficiencySwimmer	0.3121140	0.1882937	0.5415842	11.5512276

Table 7. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
3913	13	3	1000	5	1608	1.003	TRUE

Table 8. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.5803731	0.6023511	0.5808778	0.6243292	4.432287
mean	-0.3053955	-0.3354219	-0.3639946	-0.3067131	4.731049
variance	0.5642587	0.7107472	0.6655310	0.7553889	11.551228
skewness	0.7478776	0.9123587	0.8242427	1.0006221	11.551228
kurtosis	-0.1121314	0.4160019	0.1574572	0.7294242	11.551228

Table 9. Model sensitivity.

all	analysis	sensitivity	bound
all	1.003	1.003	1.002

Condition

Table 10. Parameter descriptions.

Parameter	Description
`Length[i]`	Fork length of `i`^th fish
`Weight[i]`	Recorded weight of `i`^th fish
`Year[i]`	Year `i`^th fish was captured
`bWeightLength`	Intercept of effect of `log(Length)` on `bWeight`
`bWeightYear[i]`	Effect of `i`^th `Year` on `bWeight`
`bWeight`	Intercept of `log(eWeight)`
`eWeight[i]`	Expected `Weight` of `i`^th fish
`sWeightYear`	Log standard deviation of `bWeightYear`
`sWeight`	Log standard deviation of residual variation in `log(Weight)`

Table 11. Model coefficients.

term	estimate	lower	upper	svalue
bWeight	-12.742314	-13.161984	-12.329061	11.55123
bWeightLength	3.212932	3.151046	3.276499	11.55123
sWeight	-1.925831	-1.966975	-1.885132	11.55123
sWeightYear	-1.993317	-2.259050	-1.678059	11.55123

Table 12. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
1201	4	3	1000	10	2085	1.002	TRUE

Table 13. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0000833	0.0010569	-0.0574337	0.0572287	0.0429376
variance	0.9775820	0.9982287	0.9217832	1.0798540	0.7334445
skewness	-0.4264927	0.0021968	-0.1394569	0.1418448	11.5512276
kurtosis	2.0129975	-0.0129126	-0.2500804	0.2940716	11.5512276

Table 14. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.003	1.001

Fecundity

Table 15. Parameter descriptions.

Parameter	Description
`Fecundity[i]`	Fecundity of `i`^th fish (eggs)
`Weight[i]`	Weight of `i`^th fish (mm)
`bFecundityWeight`	Effect of `log(Weight)` on `log(bFecundity)`
`bFecundity`	Intercept of `eFecundity`
`eFecundity[i]`	Expected `Fecundity` of `i`^th fish
`sFecundity`	SD of residual variation in `log(Fecundity)`

Table 16. Model coefficients.

term	estimate	lower	upper	svalue
bFecundity	3.8126217	1.1964373	4.9461786	11.55123
bFecundityWeight	0.8627896	0.8317399	0.9917802	11.55123
sFecundity	0.1282124	0.0952329	0.1854032	11.55123

Table 17. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
22	3	3	1000	10	1320	1.022	TRUE

Table 18. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	0.0085082	0.0015395	-0.4134554	0.4072162	0.0399691
variance	0.9072617	0.9739389	0.4897938	1.6969432	0.2652475
skewness	-2.1071656	0.0053045	-0.9293090	0.8902141	11.5512276
kurtosis	6.4702377	-0.4161188	-1.2178146	1.6509297	11.5512276

Table 19. Model sensitivity.

all	analysis	sensitivity	bound
all	1.022	1.001	1.013

Stock-Recruitment

Table 20. Parameter descriptions.

Parameter	Description
`Recruits[i]`	Number of recruits from `i`^th spawn year
`SDLogRecruits[i]`	Standard deviation of uncertainty in `log(Recruits[i])`
`Stock[i]`	Number of eggs in `i`^th spawn year
`a`	`Recruits` per `Stock` at low density
`b`	Density-dependence
`eRecruits[i]`	Expected number of recruits from `i`^th spawn year
`esRecruits[i]`	Expected SD of residual variation in `Recruits`
`sScaling`	Scaling term for SD of residual variation in `log(eRecruits)`

Age-1

Table 21. Model coefficients.

term	estimate	lower	upper	svalue
a	0.4028589	0.2081573	0.9183933	11.55123
b	0.0000037	0.0000014	0.0000111	11.55123
sScaling	2.6046477	1.8366964	4.0935494	11.55123

Table 22. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
17	3	3	1000	100	2739	1	TRUE

Table 23. Estimated carry capacity (with 95% CRIs).

estimate	lower	upper
109000	71700	168000

Table 24. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0434895	-0.0023107	-0.4784338	0.4810838	0.2119343
variance	0.8256236	0.9564922	0.4364768	1.8285862	0.5020597
skewness	0.3421157	0.0040927	-1.0533288	1.0324028	1.0989864
kurtosis	-0.7801808	-0.4874005	-1.3113585	1.7070623	0.7576243

Table 25. Estimated reference points (with 80% CRIs).

Metric	estimate	lower	upper
eggs	268000.00000	117000	528000
spawners	89.33333	39	176

Table 26. Model sensitivity.

all	analysis	sensitivity	bound
all	1	1	1

Age-1 to Age-2 Survival

Table 27. Parameter descriptions.

Parameter	Description
`Recruits[i]`	Number of age-2 juveniles from `i`^th spawn year
`Stock[i]`	Number of age-1 juveniles from `i`^th spawn year
`bSurvival`	`logit(eSurvival)`
`eSurvival[i]`	Expected annual survival for `i`^th spawn year
`sRecruits`	SD of residual variation in `Recruits`

Age-2

Table 28. Model coefficients.

term	estimate	lower	upper	svalue
bSurvival	-0.8345667	-1.1665037	-0.432871	8.091796
sRecruits	0.4658847	0.3337759	0.743539	11.551228

Table 29. Model summary.

n	K	nchains	niters	nthin	ess	rhat	converged
15	2	3	1000	1	1110	1.002	TRUE

Table 30. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
mean	-0.0197959	0.0039662	-0.4798786	0.5011234	0.1171207
variance	0.8883267	0.9536296	0.3942445	1.8768558	0.2163957
skewness	-0.2878148	0.0297094	-1.0106613	1.0837573	0.9100790
kurtosis	-0.9725506	-0.5338619	-1.4081306	1.6738521	1.2080419

Table 31. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.004	1.002

Figures

Length Correction

Figure 1. Measured length-frequency histogram by year.

Figure 2. Corrected length-frequency histogram by year and observation type.

Abundance

Age-1

Figure 3. Predicted abundance of age-1 Rainbow Trout in the Duncan and Lardeau Rivers by year (with 95% CRIs).

Figure 4. Predicted lineal density of age-1 Rainbow Trout in 2010 by river kilometre (with 95% CRIs).

Figure 5. Predicted observer efficiency for age-1 Rainbow Trout by visit type and study design (with 95% CRIs).

Age-2

Figure 6. Predicted abundance of age-2 Rainbow Trout in the Duncan and Lardeau Rivers by year (with 95% CRIs).

Figure 7. Predicted lineal density of age-2 Rainbow Trout in 2010 by river kilometre (with 95% CRIs).

Figure 8. Predicted observer efficiency for age-2 Rainbow Trout by visit type and study design (with 95% CRIs).

Condition

Figure 9. The percent change in the body condition for an average length fish relative to a typical year by year (with 95% CRIs).

Fecundity

Figure 10. The fecundity-weight relationship (with 95% CRIs).

Spawner Size

Figure 11. The mean length of spawning Rainbow Trout by the mean weight of Rainbow Trout in the KLRT.

Figure 12. The mean weight of Rainbow Trout in the KLRT by year.

Egg Deposition

Figure 13. The spawner abundance by year.

Figure 14. The estimated spawner fecundity by year.

Figure 15. The egg deposition by year.

Stock-Recruitment

Age-1

Figure 16. Predicted stock-recruitment relationship between spawners and age-1 recruits (with 95% CRIs).

Figure 17. Predicted egg survival to age-1 by egg deposition (with 95% CRIs).

Figure 18. Predicted percent of age-1 recruits carry capacity by egg deposition (with 80% CRIs).

Age-1 to Age-2 Survival

Age-2

Figure 19. Predicted relationship between age-1 and age-2 abundance (with 95% CRIs).

Figure 20. Estimated age-1 and age-2 survival by spawn year.

Inlake

Figure 21. The number of spawners by return year.

Figure 22. The estimated egg deposition by return year.

Figure 23. The number of expected age-1 recruits by spawn year based on the estimated egg deposition.

Figure 24. Survival from expected age-1 to spawners by return year.

Reproductive Rate (age-1)

Figure 25. Survival from age-1 to spawning by return year.

Reproductive Rate (age-2)

Figure 26. Survival from age-2 to spawning by return year.

Expected Spawners

Figure 27. Expected spawners by return year and in river and in lake variation based on age-1 recruitment.

Figure 28. Expected spawners by return year and in river and in lake variation based on age-2 recruitment.

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

Habitat Conservation Trust Foundation (HCTF) and the anglers, hunters, trappers and guides who contribute to the Trust.
Fish and Wildlife Compensation Program (FWCP) and its program partners BC Hydro, the Province of BC and Fisheries and Oceans Canada.
Ministry of Forests, Lands and Natural Resource Operations (MFLNRO)
- Greg Andrusak
- Matt Neufeld
- Jeff Burrows
- Tyler Weir
- Rob Bison
Poisson Consulting
- Evan Amies-Galonski
- Seb Dalgarno
- Robyn Irvine
Stefan Himmer
Vicky Lipinski
John Hagen
Scott Decker
Jody Schick
Gillian Sanders
Jeremy Baxter
Jeff Berdusco
Dave Derosa