Duncan Lardeau Juvenile Rainbow Trout Abundance 2024

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

1. converting the average length of spawners to the average weight using the condition relationship for a typical year
2. adjusting the average weight by the annual condition effect (interpolating where unavailable)
3. converting the average weight to the average fecundity using the fecundity relationship
4. 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.

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.

Abundance

Table 1. Parameter descriptions.

Condition

Table 10. Parameter descriptions.

Table 11. Model coefficients.

Table 12. Model summary.

Table 13. Model posterior predictive checks.

Table 14. Model sensitivity.

Fecundity

Table 15. Parameter descriptions.

Table 16. Model coefficients.

Table 17. Model summary.

Table 18. Model posterior predictive checks.

Table 19. Model sensitivity.

Stock-Recruitment

Table 20. Parameter descriptions.

Age-1 to Age-2 Survival

Table 28. Parameter descriptions.

Modelled In-lake Survival

Table 33. Parameter descriptions.

Length Correction

Abundance

Age-1

Age-2

Condition

Fecundity

Spawner Size

Egg Deposition

Stock-Recruitment

Age-1

Age-1 to Age-2 Survival

Age-2

Calculated In-lake Survival

Expected Spawners

Modelled In-lake Survival

Stock-Recruitment

Returning Spawners

All 5

All 6

All 7

All Equal

Pre 7 Post 6

Pre 7 Post Half 6

Active Spawners Only

All 6

Reproductive Rate (age-1)

Reproductive Rate (age-2)