Quesnel Exploitation Analysis 2021

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.,

\[ \log(W) = \log(\alpha) + \beta \cdot \log(L)\]

Key assumptions of the condition model include:

• \(\alpha\) can vary randomly by year.
• The residual variation in weight is log-normally distributed.

Growth

The expected length of fish at a given age were estimated from broodstock data collected in the Horsefly River from 2015-2021 using a Von Bertalanffy growth curve model (von Bertalanffy 1938).

\[ L = L_\infty \cdot (1 - \exp(-k \cdot (A - t_0) ))\]

where \(A\) is the scale age in years and \(t_0\) is the extrapolated age at zero length.

Key assumptions of the growth model include:

• The residual variation in length is normally distributed.

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 and reporting.

T-bar tag loss was estimated from the number of tags reported from recaught double-tagged individuals (Fabrizio et al. 1999).

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. Parameters k and LInf were estimated from the growth model.

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 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 effect of handling mortality is restricted to the first three month period after initial capture.
• The natural mortality varies by season, year and spawning.
• The spawning probability varies by year and fork length.
• 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 have their tags removed.
• Reporting of recaptured fish with one or more T-bar tags is likely greater than 90%.

Yield-Per-Recruit

The optimal recapture rate (to maximize number of individuals caught) was calculated using a yield-per-recruit approach (Bison, O’Brien, and Martell 2003). Key assumptions include:

• The population is at equilibrium.
• All captured individuals < 500 mm are retained.
• All captured individuals \(\geq\) 500 mm are released.
• Handling mortality is 10%.
• The life-history parameters are fixed.
• There are no Allee effects.
• There are no limitations on prey species.

The optimal yield was also calculated with no slot limit.

Growth

.model {
  bLInf ~ dnorm(900, 200^-2) T(0, )
  bK ~ dnorm(0.2, 1^-2) T(0, )
  bT0 ~ dnorm(0, 2^-1)

  sLength ~ dnorm(0, 100^-2) T(0, )
  for (i in 1:length(Length)) {
    eLength[i] <- bLInf * (1 - exp(-bK * (Age[i] - bT0))) 
    Length[i] ~ dnorm(eLength[i], sLength^-2) T(0, )
  }

Block 1. Growth model description.

Condition

  data {
    int nannual;
    int nObs;

    vector[nObs] length;
    vector[nObs] weight;
    int annual[nObs];
  }

  parameters {
    real bWeight;
    real bLength;
    real<lower=0> sWeightAnnual;

    vector[nannual] bWeightAnnual;
    real<lower=0> sWeight;
  }

  model {

    vector[nObs] eWeight;

    bWeight ~ normal(-11, 2);
    bLength ~ normal(3, 1);

    sWeightAnnual ~ normal(0, 1);

    for (i in 1:nannual) {
        bWeightAnnual[i] ~ normal(0, sWeightAnnual);
    }

    sWeight ~ normal(0, 1);
    for(i in 1:nObs) {
      eWeight[i] = exp(bWeight + bLength * log(length[i]) + bWeightAnnual[annual[i]]);
      weight[i] ~ lognormal(log(eWeight[i]), sWeight);
    }
  }

Block 2. Condition model description.

Survival

.model{
  bMortality ~ dnorm(-3, 3^-2)
  bMortalitySpawning ~ dnorm(0, 2^-2)
  bSpawning ~ dnorm(0, 2^-2)
  bSpawningLength ~ dnorm(0, 2^-2)
  bTagLoss ~ dbeta(101, 1001)

  bDetectedAlive ~ dbeta(1, 1)
  bDieNear ~ dbeta(1, 1)
  bDetectedDeadNear ~ dbeta(10, 1)
  bDetectedDeadFar ~ dbeta(1, 10)
  bMoved ~ dbeta(1, 1)
  bRecaptured ~ dbeta(1, 1)
  bReported ~ dbeta(10, 1)
  sSpawningAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bSpawningAnnual[i] ~ dnorm(0, sSpawningAnnual^-2)
  }
  bMortalitySeason[1] <- 0
  for (i in 2:nSeason) {
    bMortalitySeason[i] ~ dnorm(0, 2^-2)
  }
  sMortalityAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bMortalityAnnual[i] ~ dnorm(0, sMortalityAnnual^-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[i,PeriodCapture[i]]] + bMortalityAnnual[Annual[i,PeriodCapture[i]]] 
    
    eTagLoss[i,PeriodCapture[i]] <- bTagLoss
    eMoved[i,PeriodCapture[i]] <- bMoved 
    eRecaptured[i,PeriodCapture[i]] <- bRecaptured
    eReported[i,PeriodCapture[i]] <- bReported

    InLake[i,PeriodCapture[i]] <- 1
    Alive[i,PeriodCapture[i]] ~ dbern(1-eMortality[i,PeriodCapture[i]])
    TBarTag100[i,PeriodCapture[i]] ~ dbern(1-eTagLoss[i,PeriodCapture[i]])
    TBarTag10[i,PeriodCapture[i]] ~ dbern(1-eTagLoss[i,PeriodCapture[i]])
    
    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]])

    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))

  for(j in (PeriodCapture[i]+1):nPeriod) {
    logit(eSpawning[i,j]) <- bSpawning + bSpawningLength * Length[i,j] + bSpawningAnnual[Annual[i,j]] 
    Spawned[i,j] ~ dbern(Alive[i,j-1] * Monitored[i,j] * SpawningPeriod[i,j] * eSpawning[i,j])
    
    logit(eMortality[i,j]) <- bMortality + bMortalitySpawning * Spawned[i,j] + bMortalitySeason[Season[i,j]] + bMortalityAnnual[Annual[i,j]] 
    eDetected[i,j] <- Alive[i,j] * eDetectedAlive[i] + (1-Alive[i,j]) * eDetectedDead[i]

    eTagLoss[i,j] <- bTagLoss
    eMoved[i,j] <- bMoved 
    eRecaptured[i,j] <- bRecaptured
    eReported[i,j] <- bReported

    InLake[i,j] ~ dbern(InLake[i,j-1] * (1 - Recaptured[i,j-1]))
    Alive[i,j] ~ dbern(Alive[i,j-1] * (1 - Recaptured[i,j-1]) * (1-eMortality[i,j]))
    TBarTag100[i,j] ~ dbern(TBarTag100[i,j-1] * (1-Recaptured[i,j-1]) * (1-eTagLoss[i,j]))
    TBarTag10[i,j] ~ dbern(TBarTag10[i,j-1] * (1-Recaptured[i,j-1]) * (1-eTagLoss[i,j]))
    
    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])
    
    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))
  }
  }

Block 3. Survival model description.

Growth

Table 1. Model coefficients.

Table 2. Model summary.

Condition

Table 3. Parameter descriptions.

Table 4. Model coefficients.

Table 5. Model summary.

Survival

Table 6. Parameter descriptions.

Table 7. Model coefficients.

Table 8. Model summary.

Yield-per-Recruit

Captures

Sections

Coverage

Detections

Spawners

Growth

Condition

Survival

Yield-per-Recruit