Snorkel Efficiency

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 logistic regression model. Key assumptions of the full Maximum Likelihood logistic regression include:

• The observer efficiency varies by the fork length as a second order polynomial.
• The observer efficiency varies between systems.
• The observer efficiency varies by the gradient, sinuosity and river kilometre.

Preliminary analysis indicated that river kilometre received similar support to watershed area.

The final Bayesian logistic regression assumed that:

• The observer efficiency varies by the fork length.

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 full Maximum Likelihood GLMM include:

• The lineal density varies between systems and years.
• The lineal density varies randomly by site.
• The lineal density varies by site sinuosity, gradient and river kilometre.
• The observer efficiency for each system is as estimated by the observer efficiency model.

Preliminary analysis indicated that river kilometre was better supported as a predictor of lineal density than watershed area.

The final Bayesian GLMM dropped sinuosity and gradient and took into account the uncertainty in the observer efficiencies as estimated by the observer efficiency model.

Stock-Recruitment

The stock-recruitment relationship was estimated using a Bayesian Beverton-Holt stock-recruitment curve. Key assumptions of the final BH SR model include:

• The strength of density-dependence varies between systems.

Maximum Likelihood

#include <TMB.hpp>

template<class Type>
Type objective_function<Type>::operator() () {
DATA_VECTOR(Observed);
DATA_FACTOR(System);
DATA_VECTOR(Tagged);
DATA_VECTOR(Length);
DATA_VECTOR(Gradient);
DATA_VECTOR(Sinuosity);
DATA_VECTOR(Rkm);

PARAMETER(bIntercept);
PARAMETER(bSystem);
PARAMETER(bLength);
PARAMETER(bLength2);
PARAMETER(bGradient);
PARAMETER(bSinuosity);
PARAMETER(bRkm);

vector<Type> eObserved = Observed;

int n = Observed.size();

Type nll = 0.0;

for(int i = 0; i < n; i++){
eObserved(i) = invlogit(bIntercept + bSystem * System(i) + bLength * Length(i) + bLength2 * pow(Length(i), 2) + bGradient * Gradient(i) + bSinuosity * Sinuosity(i) + bRkm * Rkm(i));
nll -= dbinom(Observed(i), Tagged(i), eObserved(i), true);
return nll;

Template 1. Full model.

Bayesian

model{
  bIntercept ~ dnorm(0, 5^-2)
  bLength ~ dnorm(0, 5^-2)

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

Template 2. Final model.

Maximum Likelihood

#include <TMB.hpp>

template<class Type>
Type objective_function<Type>::operator() () {
DATA_VECTOR(Count);
DATA_VECTOR(Length);
DATA_VECTOR(Coverage);
DATA_VECTOR(Efficiency);
DATA_VECTOR(Dispersion);

DATA_VECTOR(Gradient);
DATA_VECTOR(Sinuosity);
DATA_VECTOR(Rkm);

DATA_FACTOR(System);

DATA_FACTOR(Site);
DATA_INTEGER(nSite);
DATA_FACTOR(Year);

PARAMETER_MATRIX(bSystemYear);
PARAMETER_VECTOR(bSite);

PARAMETER(bGradient);
PARAMETER(bSinuosity);
PARAMETER(bRkm);

PARAMETER_VECTOR(bDispersion);
DATA_INTEGER(nDispersion);
PARAMETER(log_sDispersion);
PARAMETER(log_sSite);

Type sSite = exp(log_sSite);
Type sDispersion = exp(log_sDispersion);

ADREPORT(sSite);
ADREPORT(sDispersion);

vector<Type> eDensity = Count;
vector<Type> eCount = Count;
vector<Type> eNorm = pnorm(bDispersion, Type(0), Type(1));
vector<Type> eDispersion = qgamma(eNorm, pow(sDispersion, -2), pow(sDispersion, 2));

Type nll = 0.0;

for(int i = 0; i < nSite; i++){
nll -= dnorm(bSite(i), Type(0), sSite, true);

for(int i = 0; i < nDispersion; i++){
nll -= dnorm(bDispersion(i), Type(0), Type(1), true);

eDensity(i) = exp(bSystemYear(System(i), Year(i)) + bGradient * Gradient(i) + bSinuosity * Sinuosity(i) +  bRkm * Rkm(i) + bSite(Site(i)));
eCount(i) = eDensity(i) * Length(i) * Coverage(i);

nll -= dpois(Count(i), eCount(i) * Efficiency(i) * eDispersion(i), true);

return nll;

Template 3. The full model.

Bayesian

model{

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

  for(i in 1:nSystem) {
    for(j in 1:nYear) {
      bSystemYear[i,j] ~ dnorm(0, 5^-2)
    }
  }

  log_sSite ~ dnorm(0, 5^-2)
  log(sSite) <- log_sSite
  for(i in 1:nSite) {
    bSite[i] ~ dnorm(0, sSite^-2)
  }

  bRkm ~ dnorm(0, 5^-2)

  log_sDispersion ~ dnorm(0, 5^-2)
  log(sDispersion) <- log_sDispersion


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

Template 4. The final model.

model {
  log_bAlpha ~ dnorm(5, 5^-2)
  log(bAlpha) <- log_bAlpha

  for(i in 1:nSystem) {
    log_bBeta[i] ~ dnorm(0, 5^-2)
    log(bBeta[i]) <- log_bBeta[i]
  }

  log_sRecruits ~ dnorm(0, 5^-2)
  log(sRecruits) <- log_sRecruits

  for(i in 1:length(Recruits)){
    eRecruits[i] <- bAlpha * Stock[i] / (1 + bBeta[System[i]] * Stock[i])
    Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
  }
  bCarryingCapacity <- bAlpha / bBeta

Template 5. Final model.