Horse Lake Exploitation Analysis 2019

Condition

The expected weight of a fish of a given length was estimated from the data using a allometric mass-length model (He et al. 2008)

\[ L = \alpha W^\beta\]

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 residual variation in weight is log-normally distributed.

Growth

Growth was estimated from length and scale age data for Horse Lake 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.

1 fish with a fork length \(\geq\) 500 mm was excluded from the analysis.

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 monthly intervals. The survival model incorporated natural and handling mortality, acoustic detection, movement, T-bar tag loss, recapture and reporting. Key assumptions of the survival model include:

• The tagged individuals are a random sample from the population.
• The only effects of tagging are change in the natural mortality for three months following acoustic tagging.
• The prior probability on the annual interval individual external tag loss rate is a uniform distribution between 1 and 30% (Fabrizio et al. 1996).
• All individuals are correctly identified.
• There is no emigration out of the lake.
• There are no unmodelled individual differences in the probability of recapture in each time interval.
• There are no unmodelled individual differences in the probability of survival in each time interval.
• There are no unmodelled individual differences in the probability of a living individual with an active acoustic transmitter being detected moving among sections in each time period.
• The fate of each individual is independent of the fate of any other individual.
• The sampling periods are instantaneous and all individuals are immediately released.

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 the difference between the natural mortality and fishing mortality rates for individuals \(\geq\) 500 mm were not sigificant.

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 separate yield-per-recruit analysis was performed for the two different Lake Trout ecotypes in Horse Lake.

Condition

.model {
  bWeight ~ dnorm(0, 1^-2) 
  bWeightLength ~ dnorm(3.18, 0.19^-2)

  sWeight ~ dnorm(0, 1^-2) T(0,) 
  for(i in 1:length(LogLength)) {
    eWeight[i] <- bWeight + bWeightLength * LogLength[i]
    Weight[i] ~ dlnorm(eWeight[i], exp(sWeight)^-2)
  }
..

Block 1. Condition model description.

Growth

.model {
  bLInf ~ dnorm(500, 100^-2) T(0, )
  bK ~ dnorm(0, 1^-2) T(0, )
  bT0 ~ dnorm(0, 1^-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 2. Growth model description.

Survival

.model{
  bMortality ~ dnorm(-3, 3^-2)
  bMortalityHandling ~ dnorm(0, 2^-2)
  bTagLoss ~ dunif(1 - (1 - 0.01)^(1/12), 1 - (1 - 0.30)^(1/12))
  bDetected ~ dunif(0, 1)
  bMoved ~ dunif(0, 1)
  bRecaptured ~ dunif(0, 1 - (1 - 0.50)^(1/12))
  bReported ~ dunif(0.9, 1.00)

  for (i in 1:nCapture){
    logit(eMortality[i,PeriodCapture[i]]) <- bMortality + bMortalityHandling
    eTagLoss[i,PeriodCapture[i]] <- bTagLoss
    eDetected[i,PeriodCapture[i]] <- bDetected
    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]])
    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]])
    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 + bMortalityHandling * step(PeriodCapture[i] - j + 2)
    eTagLoss[i,j] <- bTagLoss
    eDetected[i,j] <- bDetected
    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-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])
    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.

Captures

Table 1. Number of fish captured by year, species and size class. Eight Rainbow Trout were transplanted in 2019.

Table 2. Number of Lake Trout captured by size class, and tag class. Any fish > 500mm is classified as ‘large’.

Recaptures

Table 3. All Lake Trout recaptures to date. Recaptures with natural mortality were tags found in the stomach of another fish!.

Condition

Table 4. Parameter descriptions.

Table 5. Model coefficients.

Table 6. Model summary.

Growth

Table 7. Parameter descriptions.

Table 8. Model coefficients.

Table 9. Model summary.

Survival

Table 10. Parameter descriptions.

Table 11. Model coefficients.

Table 12. Model summary.

Yield-per-Recruit

Table 13. Summary of parameters in yield-per-recruit model.

Table 14. 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 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.

Captures

Coverage

Detections

Condition

Growth

Survival

Yield-per-Recruit