Horse Lake Exploitation Analysis 2021

Condition

The expected weight, \(W\), of a fish of a given length, \(L\), was 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 prior distribution on the power term (\(\beta\)) is a normal distribution with a mean of 3.18 and a standard deviation of 0.19 (McDermid, Shuter, and Lester 2010).
• The weight varies by day of the year.
• The residual variation in weight is log-normally distributed.

Preliminary analysis indicated year as a random effect and capture method (angling vs SPIN) were not important predictors of condition.

Growth and Spawning

The growth curve and length at spawning for each ecotype were estimated from the length, scale age, and spawning data using an integrated model. The model consists of a biphasic Von Bertalanffy (von Bertalanffy 1938) growth curve and length at spawning model.

The Von Bertalanffy growth curve (von Bertalanffy 1938) for the small ecotype is

\[ L = L_{\infty_{1}} \cdot (1 - \exp(-k_1 \cdot A ))\]

where \(A\) is the scale age in years.

The large ecotype transitions to the second growth phase at the transitions age (\(A_T\)), where \(L_T\) is the transition length (mm)

\[ L_T = L_{\infty_{1}} \cdot (1 - \exp(-k_1 \cdot A_T )) \]

The Von Bertalanffy growth curve (von Bertalanffy 1938) in the second growth phase the large ecotype

\[ L = L_T + (L_{\infty_{2}} - L_T) \cdot (1 - \exp(-k_2 \cdot (A - A_T)))\]

The probability of spawning at length \(L\) is determined by

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

Key assumptions of the model include:

• \(Ls\) varies by ecotype
• \(Es\) varies by ecotype
• Transition age, \(A_T\), varies randomly by fish
• The residual variation in length is log normally distributed.

The sensitivity analysis indicated that the estimate of bBigEs is sensitive to the choice of prior. Increasing the standard deviation of the prior by an order of magnitude increased the estimate from 1.71 (95% CI 0.06 - 4.15) to 8.97 (95% CI 0.153 - 42.0).

Mobile

A fish was classified as being in the lake at the time of the survey if it was detected 1 or more times. A fish was classified as having moved if the weighted centriod had shifted more then 800 m since the previous survey.

Survival

The natural mortality and capture probability for individuals \(\geq\) 300 mm was estimated using a Bayesian individual state-space survival model (Thorley and Andrusak 2017) with three month intervals. The survival model incorporated natural mortality, fixed and mobile acoustic detection and movement, T-bar tag loss, recapture and reporting. Key assumptions of the survival model include:

• The natural mortality varies by season.
• The fishing mortality varies by year.

The survival model treats all recaptured fish as if they had been retained. Only individuals with a fork length (FL) \(\geq\) 300 mm that were tagged with an acoustic tag and/or $100 and $10 reward tags were included in the survival analysis.

Preliminary analysis indicated that there was not a clear difference between the natural mortality and fishing mortality rates for individuals \(\geq\) 500 mm.

Yield-Per-Recruit

The optimal capture rate (to maximize number of individuals harvested assuming 100% retention) was calculated using yield-per-recruit analysis (Walters and Martell 2004). A yield-per-recruit analysis was performed for the two different Lake Trout ecotypes separately and in combination. The \(Rk\) of 25 from Myers et al. (Myers, Bowen, and Barrowman 1999) was converted into the equivalent egg to age-1 survival of 0.025 based on the large ecotype life history

Condition

.model {
  bWeight ~ dnorm(-12, 2^-2) 
  bWeightLength ~ dnorm(3.18, 0.19^-2)
  bDayte ~ dnorm(0, 2^-2)
  bDayte2 ~ dnorm(0, 2^-2)
  sWeight ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:length(LogLength)) {
    eWeightLength[i] <- bWeightLength + bDayte * Dayte[i] + bDayte2 * Dayte[i]^2 
    eWeight[i] <- bWeight + eWeightLength[i] * LogLength[i] 
    Weight[i] ~ dlnorm(eWeight[i], sWeight^-2)
  }

Block 1. Condition model description.

Growth and Spawning

.model {
  bT0 <- 0
               
  bLInf1 ~ dnorm(6, 2^-2)
  bLInf2  ~ dnorm(7, 2^-2)
  bK1 ~ dnorm(0, 2^-2)
  bK2 ~ dnorm(0, 2^-2)             
  bTransitionAge ~ dnorm(2, 1^-2)
  sTransitionAgeFish ~ dnorm(0, 2^-2) T(0,)
  bSmall ~ dnorm(0, 2^-2) 
  sLength ~ dnorm(0, 2^-2) T(0, )
   
  Ls ~ dnorm(6, 2^-2)
  Sp ~ dnorm(20, 20^-2) T(0,)
  Es ~ dnorm(0, 2^-2)
  bBigLs ~ dnorm(0, 2^-2)
  bBigEs ~ dnorm(0, 2^-2)
  for (i in 1:nFishID) {
    bTransitionAgeFish[i] ~ dnorm(-sTransitionAgeFish^2/2, sTransitionAgeFish^-2)
  }
  for (i in 1:nObs) {
    logit(eSmall[i]) <- bSmall
    log(eLInf1[i]) <- bLInf1 
    log(eLInf2[i]) <- bLInf2
    log(eK1[i]) <- bK1
    log(eK2[i]) <- bK2  
    log(eLs[i]) <- Ls  + bBigLs * (1 - SmallEcotype[i])
    logit(eEs[i]) <- Es + bBigEs * (1 - SmallEcotype[i])
    eSpawning[i] <- ( (Length[i] ^ Sp) / (eLs[i]^Sp + Length[i]^Sp) ) * eEs[i]
    eSmallLength[i] <- eLInf1[i] * (1 - exp(-eK1[i] * (Age[i] - bT0))) 
    
    log(eTransitionAge[i]) <- bTransitionAge + bTransitionAgeFish[FishID[i]]
    eTransitionLength[i] <- eLInf1[i] * (1 - exp(-eK1[i] * (eTransitionAge[i] - bT0))) 
    eBigLength[i] <- eSmallLength[i] * step(eTransitionAge[i] - Age[i]) + step(Age[i] - eTransitionAge[i] - 0.01) * (eTransitionLength[i] + (eLInf2[i] - eTransitionLength[i]) * (1 - exp(-eK2[i] * (Age[i] - eTransitionAge[i]))))
    eLength[i] <-  eSmallLength[i] * SmallEcotype[i] + eBigLength[i] * (1 - SmallEcotype[i])
    Spawning[i] ~ dbin(eSpawning[i], 1)
    SmallEcotype[i] ~ dbin(eSmall[i], 1)
    Length[i] ~ dlnorm(log(eLength[i]), sLength^-2)
  }

Block 2. Growth model description.

Survival

.model{
  bMortality ~ dnorm(-3, 3^-2)
  bTagLoss ~ dbeta(1, 1)
  bMoved ~ dbeta(1, 1)
  bMobileDetected ~ dbeta(1, 1)
  bMobileMoved ~ dbeta(1, 1)
  bRecaptured ~ dnorm(0, 2^-2)
  bReported ~ dbeta(1, 1)
  bDetectedAlive ~ dbeta(1, 1)
  bDieNear ~ dbeta(1, 1)
  bDetectedDeadNear ~ dbeta(10, 1)
  bDetectedDeadFar ~ dbeta(1, 10)
  bMortalitySeason[1] <- 0
  for (i in 2:nSeason) {
    bMortalitySeason[i] ~ dnorm(0, 2^-2)
  }
  sRecapturedAnnual ~ dnorm(0, 2^-2) T(0,)
  for(i in 1:nAnnual) {
    bRecapturedAnnual[i] ~ dnorm(0, sRecapturedAnnual^-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]]]
    eTagLoss[i,PeriodCapture[i]] <- bTagLoss
    eDetected[i,PeriodCapture[i]] <- Alive[i,PeriodCapture[i]] * eDetectedAlive[i] + (1-Alive[i,PeriodCapture[i]]) * eDetectedDead[i]
    eMoved[i,PeriodCapture[i]] <- bMoved
    eMobileDetected[i,PeriodCapture[i]] <- bMobileDetected
    eMobileMoved[i,PeriodCapture[i]] <- bMobileMoved
    
    logit(eRecaptured[i,PeriodCapture[i]]) <- bRecaptured + bRecapturedAnnual[Annual[PeriodCapture[i]]]
    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]])
    Detected[i,PeriodCapture[i]] ~ dbern(Monitored[i,PeriodCapture[i]] * eDetected[i,PeriodCapture[i]])
    Moved[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * Monitored[i,PeriodCapture[i]] * eMoved[i,PeriodCapture[i]])
    MobileDetected[i,PeriodCapture[i]] ~ dbern(Monitored[i,PeriodCapture[i]] * eMobileDetected[i,PeriodCapture[i]] * MobileSurvey[i,PeriodCapture[i]])
    MobileMoved[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * MobileDetected[i,PeriodCapture[i]] * eMobileMoved[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(eMortality[i,j]) <- bMortality + bMortalitySeason[Season[j]]
    eTagLoss[i,j] <- bTagLoss

    eDetected[i,j] <- Alive[i,j] * eDetectedAlive[i] + (1-Alive[i,j]) * eDetectedDead[i]
    eMoved[i,j] <- bMoved
    eMobileDetected[i,j] <- bMobileDetected
    eMobileMoved[i,j] <- bMobileMoved
    logit(eRecaptured[i,j]) <- bRecaptured + bRecapturedAnnual[Annual[j]]
    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-1]))
    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] * Monitored[i,j] * eMoved[i,j])
    MobileDetected[i,j] ~ dbern(InLake[i,j] * Monitored[i,j] * eMobileDetected[i,j] * MobileSurvey[i,j])
    MobileMoved[i,j] ~ dbern(Alive[i,j] * MobileDetected[i,j] * eMobileMoved[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.

Condition

Table 1. Parameter descriptions.

Table 2. Model coefficients.

Table 3. Model convergence.

Table 4. Model posterior predictive checks.

Table 5. Model sensitivity.

Growth and Spawning

Table 6. Parameter descriptions.

Table 7. Model coefficients.

Table 8. Model convergence.

Table 9. Model posterior predictive checks.

Table 10. Model sensitivity.

Survival

Table 11. Parameter descriptions.

Table 12. Model coefficients.

Table 13. Model convergence.

Yield-per-Recruit

Table 14. Summary of parameters in yield-per-recruit model.

Table 15. Lake Trout small ecotype yield-per-recruit calculations including the capture probability (pi), exploitation rate (u) and average age, length and weight of fish harvested.

Table 16. Lake Trout large ecotype yield-per-recruit calculations including the capture probability (pi), exploitation rate (u) and average age, length and weight of fish harvested.

Table 17. Lake Trout (both ecotypes) yield-per-recruit calculations including the capture probability (pi), exploitation rate (u) and average age, length and weight of fish harvested.

Table 18. Lake Trout (both ecotypes) stock-recruitment calculations including the capture probability (pi), exploitation rate (u) the relative number of eggs, recruits and spawners, and the average fecundity.

Captures

Length Distribution

Coverage

Detections

Mobile Detection Survey

Condition

Growth and Spawning

Survival

Yield-per-Recruit