Duncan Lardeau Juvenile Rainbow Trout Abundance 2017

The suggested citation for this analytic report is:

Thorley, J.L., Muir, C.D., Dalgarno, S. and Hogan P.M. (2017) Duncan Lardeau Juvenile Rainbow Trout Abundance 2017. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/492093238.

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 site. 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 are to:

  1. Estimate the spring abundance of age-1 fish by year.
  2. Estimate the stock-recruitment relationship between the number of spawners and the abundance of age-1 recruits the following spring.

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 manipulate using R version 3.4.2 (R Core Team 2017) and organised in an SQLite database. Due to the loss of the primary data manager, the estimates should be considered preliminary until additional verification of the restructured data has been conducted. Age-1 individuals were assumed to be those with a fork length \(\geq\) 100 mm.

Data Analysis

Hierarchical Bayesian models were fitted to the data using R version 3.4.2 (R Core Team 2017), Stan 2.16.0 (Carpenter et al. 2017) and JAGS 4.2.0 (Plummer 2015) and the mbr family of packages.

Unless indicated otherwise, the models used prior distributions that were vague in the sense that they did not affect the posterior distributions (Kery and Schaub 2011, 36). The posterior distributions were estimated from 2,000 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of three chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that \(\hat{R} < 1.1\) (Kery and Schaub 2011, 40) for each of the monitored parameters in the model (Kery and Schaub 2011, 61). Where relevant, model adequacy was confirmed by examination of residual plots.

The posterior distributions of the fixed (Kery and Schaub 2011, 75) parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% confidence/credible limits (CLs) and the p-value (Kery and Schaub 2011, 37, 42). The estimate is the median (50th percentile) of the MCMC samples, the z-score is \(\mathrm{sd}/\mathrm{mean}\) and the 95% CLs are the 2.5th and 97.5th percentiles. A p-value of 0.05 indicates that the lower or upper 95% CL is 0.

Variable selection was achieved by dropping fixed (Kery and Schaub 2011, 77–82) variables with two-sided p-values \(\geq\) 0.05 (Kery and Schaub 2011, 37, 42) and random variables with percent relative errors \(\geq\) 80%.

The results are displayed graphically by plotting the modeled relationships between particular variables and the response with 95% credible intervals (CIs) 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). Where informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% CIs (Bradford, Korman, and Higgins 2005).

Model Descriptions

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:

Stock-Recruitment

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

\[ R = \frac{a \cdot S}{1 + b \cdot S} \quad,\]

where \(a\) is the maximum reproductive performance per spawner, and \(b\) is the density dependence.

Key assumptions of the stock-recruitment model include:

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

\[ K = \frac{a}{b} \quad.\]

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 Width[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;
    vector[nYear] bDensityYear;
    real bDensityWidth;
    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);
    bDensityWidth ~ normal(0, 2);
    bDensityYear ~ normal(0, 5);

    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]] +
                        bDensityWidth * log(Width[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);
  }

Template 1. Abundance model description.

Stock-Recruitment

model {
  a ~ dnorm(8000 * 0.5^4, 250^-2) T(0, )
  b ~ dunif(0, 0.1)
  sScaling ~ dunif(0, 5)

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

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

Template 2. Stock-Recruitment model description.

Results

Tables

Abundance

Table 1. Parameter descriptions.

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

Table 2. Model coefficients.

term estimate sd zscore lower upper pvalue
bDensity -0.5609736 1.2717405 -0.4748684 -3.0208802 1.7686930 0.6730
bDensityRkm[1] -0.2245419 0.0970234 -2.3050645 -0.4188671 -0.0320095 0.0240
bDensityRkm[2] 0.6614224 0.1137607 5.8202288 0.4212796 0.8853642 0.0005
bDensityRkm[3] 0.0309741 0.0382207 0.8185385 -0.0419120 0.1073823 0.4310
bDensityRkm[4] -0.3002494 0.0383004 -7.8520963 -0.3760327 -0.2241509 0.0005
bDensityWidth 0.0816910 0.0283956 2.8775942 0.0254561 0.1377320 0.0010
bDensityYear[1] 0.4807932 1.2962360 0.4227568 -1.8357639 3.0299739 0.7250
bDensityYear[2] -0.3464799 1.2935789 -0.2387211 -2.6933792 2.1272273 0.7980
bDensityYear[3] -0.7047454 1.2807716 -0.4904629 -3.0051360 1.8556464 0.6430
bDensityYear[4] -0.7545865 1.2846435 -0.5454590 -3.0895398 1.7895746 0.6050
bDensityYear[5] -0.3468520 1.2966775 -0.2186230 -2.6968581 2.2429440 0.8190
bDensityYear[6] 0.0395578 1.2717297 0.0627167 -2.2843352 2.5126749 0.9710
bDensityYear[7] 0.0103889 1.2666607 0.0462209 -2.3013677 2.4891637 0.9930
bDensityYear[8] -0.3197939 1.2688035 -0.2281621 -2.6648160 2.1053966 0.8310
bDensityYear[9] -0.7803547 1.2670294 -0.5927232 -3.1533316 1.6692387 0.5830
bDensityYear[10] -1.6874992 1.2756542 -1.2863060 -4.0218208 0.7823581 0.2100
bDensityYear[11] -1.2588629 1.2687599 -0.9555919 -3.6158763 1.1808410 0.3660
bEfficiency -1.7819382 0.1703980 -10.4469313 -2.1061139 -1.4321656 0.0005
bEfficiencyIndex 0.1934140 0.4060052 0.4667747 -0.5683309 0.9881197 0.6640
bEfficiencyMarking 0.1022605 0.1238488 0.7975761 -0.1474622 0.3439068 0.4510
bEfficiencyMarkingIndex 1.9852189 0.3887763 5.0685101 1.2059535 2.7109611 0.0005
sDensitySite 0.6598949 0.0429870 15.3610139 0.5757140 0.7456404 0.0005
sDispersion 1.1754351 0.0673789 17.4748628 1.0549137 1.3210813 0.0005
sEfficiencySwimmer 0.3535370 0.1081607 3.4335531 0.2132199 0.6352534 0.0005

Table 3. Model summary.

n K nsamples nchains nsims rhat converged
2916 24 2000 4 4000 1.01 TRUE

Table 4. Summary survey information by year and river.

Year River Sites SurveyLength SurveyPercent Fish Marked Resighted
2006 Duncan 0 0.0 0 0 0 0
2006 Lardeau 35 2.0 1 657 122 79
2007 Duncan 0 0.0 0 0 0 0
2007 Lardeau 47 2.7 2 260 0 0
2008 Duncan 0 0.0 0 0 0 0
2008 Lardeau 97 7.5 5 618 94 54
2009 Duncan 0 0.0 0 0 0 0
2009 Lardeau 83 5.0 4 384 0 0
2010 Duncan 0 0.0 0 0 0 0
2010 Lardeau 48 2.5 2 307 0 0
2011 Duncan 32 1.8 4 149 0 0
2011 Lardeau 264 15.8 11 1947 0 0
2012 Duncan 71 3.8 7 481 0 0
2012 Lardeau 257 15.2 11 1691 43 6
2013 Duncan 117 7.0 14 828 56 10
2013 Lardeau 308 18.0 13 1236 145 27
2014 Duncan 77 4.9 9 242 70 14
2014 Lardeau 273 16.2 12 928 96 13
2016 Duncan 70 3.6 7 137 0 0
2016 Lardeau 257 14.7 10 324 14 2
2017 Duncan 99 6.2 12 250 30 5
2017 Lardeau 604 38.4 27 1326 45 1

Table 5. Summary survey information by year.

Year Sites SurveyLength SurveyPercent Fish Marked Resighted
2006 35 2.0 1 657 122 79
2007 47 2.7 1 260 0 0
2008 97 7.5 4 618 94 54
2009 83 5.0 3 384 0 0
2010 48 2.5 1 307 0 0
2011 296 17.6 9 2096 0 0
2012 328 19.0 10 2172 43 6
2013 425 25.0 13 2064 201 37
2014 350 21.0 11 1170 166 27
2016 327 18.3 10 461 14 2
2017 703 44.6 23 1576 75 6

Stock-Recruitment

Table 6. Parameter descriptions.

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

Table 7. Model coefficients.

term estimate sd zscore lower upper pvalue
a 481.4567932 200.8357326 2.520724 203.6611240 944.6992620 5e-04
b 0.0040898 0.0026784 1.705611 0.0009107 0.0107306 5e-04
sScaling 2.3845734 0.6332814 3.909790 1.5157516 4.0364412 5e-04

Table 8. Model summary.

n K nsamples nchains nsims rhat converged
11 3 2000 4 4e+05 1 TRUE

Table 9. Estimated carry capacity.

estimate sd zscore lower upper
120849.6 48569.61 2.660642 70856.94 234359.9

Figures

Abundance

figures/abundance/efficiency-type.png
Figure 1. Predicted observer efficiency for age-1 Rainbow Trout by visit type and study design (with 95% CRIs).
figures/abundance/abundance-year.png
Figure 2. Predicted abundance of age-1 Rainbow Trout in the Duncan and Lardeau Rivers by year (with 95% CRIs).
figures/abundance/density-site.png
Figure 3. Predicted lineal density of age-1 Rainbow Trout in 2010 by river kilometre (with 95% CRIs).
figures/abundance/density-width.png
Figure 4. Predicted lineal density of age-1 Rainbow Trout in 2010 by width (with 95% CRIs).

Stock-Recruitment

figures/sr/stock-recruitment.png
Figure 5. Predicted stock-recruitment relationship between spawners and age-1 recruits (with 95% CRIs).
figures/sr/recruits-per-spawner.png
Figure 6. Predicted reproductive performance per spawner by spawners (with 95% CRIs).
figures/sr/percent-carry-capacity.png
Figure 7. Predicted percent of age-1 recruits carry capacity by spawners (with 95% CRIs).

Recommendations

Acknowledgements

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

References

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.

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.

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

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

R Core Team. 2017. “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.