Quesnel Exploitation Analysis 2022

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.,

\[ \log(W) = \log(\alpha) + \beta \cdot \log(L)\] Length and weights of fish were taken from two sources: captures from broodstock collection in the Horsefly River (2015-2022) and in-lake captures from tagging outings (2013-2022).

Key assumptions of the condition model include:

• The expected weight is allowed to vary with length and date.
• The expected weight is allowed to vary randomly with year and source within year.
• The relationship between weight and length is allowed to vary with date.
• The residual variation in weight is log-normally distributed.

Growth

Annual growth of fish were estimated from inter-annual recaptures (in-lake and broodstock) and aged broodstock captures using the Fabens method (Fabens 1965) for estimating the von Bertalanffy growth curve (von Bertalanffy 1938). This curve is based on the premise that:

\[ \frac{\text{d}L}{\text{d}t} = k (L_{\infty} - L)\]

where \(L\) is the length of the individual, \(k\) is the growth coefficient and \(L_{\infty}\) is the maximum length.

Integrating the above equation gives:

\[ L_t = L_{\infty} (1 - e^{-k(t - t_0)})\]

where \(L_t\) is the length at time \(t\) and \(t_0\) is the time at which the individual would have had zero length.

The Fabens form allows

\[ L_r = L_c + (L_{\infty} - L_c) (1 - e^{-kT})\]

where \(L_r\) is the length at recapture, \(L_c\) is the length at capture and \(T\) is the time between capture and recapture.

For aged fish, length at capture is the length at Age-0 \(L_0\), estimated with the modified form:
\[L_r = L_{0} - (L_{\infty} - L_0) (1 - e^{-kA})\] where \(A\) is the age.

Key assumptions of the growth model include:

• The mean maximum length \(L_{\infty}\) is constant.
• The growth coefficient \(k\) varies randomly with growth year.
  
• \(L_{\infty}\) is likely between 800 and 1000.
• The residual variation in growth is student-t distributed.
  
• The student-t shape parameter is fixed at 2.

A growth year spans April 01 to March 31 the following year.

Preliminary analyses indicated lack of support for variation in \(k\) by Horsefly River sockeye run size with a one or two year lag.

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, reporting, fork length at spawning and sockeye run size in the previous year.

T-bar tag loss was estimated from the number of tags reported from recaught double-tagged individuals (Fabrizio et al. 1999). The model assumes that tag loss is not a time-dependent process and occurs following initial release.

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. Values for \(L_\infty\) and annual \(k\) and were estimated in the growth model.

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

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

where \(L_s\) is the length at which 50% of fish are mature, \(es\) is the probability of a mature fish spawning and \(S_p\) is the maturity (as a function of length) power.

Key assumptions of the spawning probability sub-model include:

• \(L_s\) varies randomly by year
• \(es\) varies randomly by year

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 (no detections in lake) 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 natural mortality varies by season, spawning and sockeye abundance for each size class.
  
• The recapture probability varies randomly by year.
  
• The recapture probability varies by season.
  
• 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 that are released have their FLOY tags removed.
• Reporting of recaptured fish with one or more T-bar tags is likely greater than 90%.

Seasons are defined as winter (January to March), spring (April to June), summer (July to September), and fall (October to December).

Size classes are defined as small (\(\le\) 600 mm), medium (601 \(-\) 700 mm) and large (\(>\) 700 mm).

Sockeye abundance is the Horsefly River run size with a two year lag (i.e., spawning in the spring of 2016 is associated with autumn 2014 sockeye abundance).

Preliminary analyses indicated lack of support for variation in mortality by year, sockeye year (October 01 to September 30 the following year) and size class within year.

Mark-Recapture

The in-lake and spawner abundance for individuals \(\ge\) 500 mm were estimated using hierarchical Bayesian mark-recapture sub-models within the survival model, where the number of alive marked individuals in the lake at each time period was estimated from the ‘Alive’ latent state in the previous period. The number of spawning marked individuals was estimated from the spawning probability and the number of alive marked individuals in the previous period.

The ‘in-lake’ mark-recapture model estimated abundance from in-lake captures and recaptures from researcher tagging outings.

The ‘spawner’ mark-recapture model estimated spawner abundance from spawner captures and recaptures from broodstock collection in the Horsefly River.

An additional in-lake abundance \(A_L\) estimate was derived from estimated spawner abundance \(A_S\) and spawning probability \(P_S\) given the following: \[A_L = \frac{A_S}{P_S} \]

Key assumptions of the mark-recapture models include:

• Marked fish mix randomly with unmarked fish within the lake.
  
• The probability of capturing a marked or unmarked fish is the same.
• All recaptured fish are correctly identified as being marked.
  
• The number of recaptures (of fish marked at that site in that year) and the total number of fish caught are each binomially distributed.

Additional assumptions for the lake model include:

• Capture efficiency varies by total outing hours in a three month period.
  
• Abundance varies randomly by year.

Yield-Per-Recruit

The optimal yield (defined as the harvested number of fish as percent of the number of recruits at carrying capacity) was calculated using a yield-per-recruit approach (Bison, O’Brien, and Martell 2003).

Scenarios explored include:

• Lower slot limit (Llo) of 40, 50, 60, 70, 80 and 90, with other YPR parameter values held constant.
  
• Rk (lifetime spawners per spawner at low density) values of 5, 7.5, and 10 and Ls (length at 50% mature) values of 55, 60, and 65, with Llo held at 60.
  
• Rk values of 5, 7.5, and 10 and Ls values of 55, 60, and 65, with Llo held at 50.

Key assumptions include:

• The population is at equilibrium.
• All captured individuals above Llo are retained.
• All captured individuals below Llo are released.
• Handling mortality is 10%.
• The life-history parameters are fixed.
• There are no Allee effects.
• There are no limitations on prey species.

Condition

.model{
  bWeight ~ dnorm(3, 2^-2)
  bWeightLength ~ dnorm(3, 1^-2)
  bWeightDayte ~ dnorm(0, 1^-2);
  bWeightLengthDayte ~ dnorm(0, 1^-2)
  sWeightAnnual ~ dexp(1)
  for(i in 1:nAnnual) {
    bWeightAnnual[i] ~ dnorm(0, sWeightAnnual^-2)
  }
  sWeightSourceAnnual ~ dexp(1)
  for(i in 1:nSource){
    for(j in 1:nAnnual){
      bWeightSourceAnnual[i,j] ~ dnorm(0, sWeightSourceAnnual^-2)
    }
  }
  sWeight ~ dexp(1)
  for(i in 1:nObs) {
    log(eWeight[i]) <- bWeight + bWeightDayte * Dayte[i] + bWeightAnnual[Annual[i]] + bWeightSourceAnnual[Source[i], Annual[i]] + (bWeightLength + bWeightLengthDayte * Dayte[i]) * (log(Length[i]) - log(100))
    Weight[i] ~ dlnorm(log(eWeight[i]), sWeight^-2)
  }

Block 1. Model description.

Growth

.model {
  bK ~ dnorm(0, 2^-2) 
  bL0 ~ dnorm(0, 100^-2)
  sKYear ~ dexp(1)
  theta <- 0.5

  bLinf ~ dnorm(900, 100^-2) T(0,)
  sGrowth ~ dexp(1)
  sLength ~ dexp(1)
  for (i in 1:nYear) {
    bKYear[i] ~ dnorm(0, sKYear^-2)
    log(eK[i]) <- bK + bKYear[i] 
  }
  for(i in 1:nAged){
    eLength[i] <- bLinf - (bLinf - bL0) * exp(-exp(bK) * Age[i])
    Length[i] ~ dnorm(eLength[i], sLength^-2)
  }
  for(i in 1:nFish){
    eL0[i] <- max(bL0, LengthAtRelease[i])
    for(j in 1:nYear){
      ekProp[i,j] <- eK[j] * pYear[i,j]
    }
    eSumEk[i] <- exp(-sum(ekProp[i,]))
    eGrowth[i] <- max(0, (bLinf - eL0[i]) * (1 -eSumEk[i]))
    Growth[i] ~ dt(eGrowth[i], sGrowth^-2, 1/theta)
  }

Block 2.

Tag Loss

model {
  bTagLoss ~ dbeta(1, 1)

  for (i in 1:nObs) {
    TagsRecap[i] ~ dbin(1 - bTagLoss, 2) T(1,)
  }

Block 3. Model description.

Survival

.model{
  bMortality ~ dnorm(-3, 3^-2)
  bMortalitySpawning ~ dnorm(0, 2^-2)
  bLs ~  dnorm(6, 2^-2)
  bSp ~ dexp(1/20)
  bEs ~ dnorm(0, 2^-2)

  bDetectedAlive ~ dbeta(1, 1)
  bDieNear ~ dbeta(1, 1)
  bDetectedDeadNear ~ dbeta(10, 1)
  bDetectedDeadFar ~ dbeta(1, 10)
  bMoved ~ dbeta(1, 1)
  bRecaptured ~ dnorm(0, 2^-2)
  bReported ~ dbeta(10, 1)
  bReleased ~ dbeta(1, 1)
  bReleasedResearcher ~ dbeta(1, 1)
  bRecapturedResearcher ~ dbeta(5, 1)
  bTagLoss <- 0.0545
  bTagLossT <- 0.0001

  bEfficiency ~ dnorm(-4, 2^-2)
  bAbundance ~ dnorm(7, 2^-2)
  bOutingHours ~ dnorm(0, 2^-2)
  bEfficiencySpawn ~ dnorm(-4, 2^-2)
  bAbundanceSpawn ~ dnorm(1000, 500^-2)

  bMortalitySeason[1] <- 0
  for (i in 2:nSeason) {
    bMortalitySeason[i] ~ dnorm(0, 2^-2)
  }
  for (i in 1:nSizeClass) {
    bMortalitySockeye[i] ~ dnorm(0, 2^-2)
  }
  sRecapturedAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bRecapturedAnnual[i] ~ dnorm(0, sRecapturedAnnual^-2)
  }
  bRecapturedSeason[1] <- 0
  for (i in 2:nSeason) {
    bRecapturedSeason[i] ~ dnorm(0, 2^-2)
  }
  sLsAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bLsAnnual[i] ~ dnorm(0, sLsAnnual^-2)
  }
  sEsAnnual ~ dexp(1)
  for (i in 1:nAnnual) {
    bEsAnnual[i] ~ dnorm(0, sEsAnnual^-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]]] + bMortalitySockeye[SizeClass[i,PeriodCapture[i]]] * SockeyePrevious[PeriodCapture[i]]
    logit(eRecaptured[i,PeriodCapture[i]]) <- bRecaptured + bRecapturedSeason[Season[PeriodCapture[i]]] + bRecapturedAnnual[Annual[PeriodCapture[i]]] 

    eMoved[i,PeriodCapture[i]] <- bMoved 
    eReported[i,PeriodCapture[i]] <- bReported
    eReleased[i,PeriodCapture[i]] <- bReleased
    eReleasedResearcher[i,PeriodCapture[i]] <- bReleasedResearcher
    eRecapturedResearcher[i,PeriodCapture[i]] <- bRecapturedResearcher

    InLake[i,PeriodCapture[i]] <- 1
    Alive[i,PeriodCapture[i]] ~ dbern(1-eMortality[i,PeriodCapture[i]])
    Alive50[i,PeriodCapture[i]] <- Alive[i,PeriodCapture[i]] * (Length[i,PeriodCapture[i]] >= 500)

    TBarTag100[i,PeriodCapture[i]] ~ dbern(1-bTagLoss)
    TBarTag10[i,PeriodCapture[i]] ~ dbern(1-bTagLoss)
    
    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]])

    RecapturedResearcher[i,PeriodCapture[i]] ~ dbern(Alive[i,PeriodCapture[i]] * eRecapturedResearcher[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))
    Released[i,PeriodCapture[i]] ~ dbern(Recaptured[i,PeriodCapture[i]] * eReleased[i,PeriodCapture[i]])
    ReleasedResearcher[i,PeriodCapture[i]] ~ dbern(RecapturedResearcher[i,PeriodCapture[i]] * eReleasedResearcher[i,PeriodCapture[i]])

  for(j in (PeriodCapture[i]+1):nPeriod) {
    log(eLs[i,j]) <- bLs + bLsAnnual[Annual[j]]
    logit(eEs[i,j]) <- bEs + bEsAnnual[Annual[j]]
    eSpawning[i,j] <- (((Length[i,j]^bSp) / (eLs[i,j]^bSp + Length[i,j]^bSp)) * eEs[i,j]) 
    Spawned[i,j] ~ dbern(Alive[i,j-1] * Monitored[i,j] * SpawningPeriod[j] * eSpawning[i,j])
    logit(eMortality[i,j]) <- bMortality + bMortalitySpawning * Spawned[i,j-1] + bMortalitySeason[Season[j]] + bMortalitySockeye[SizeClass[i,j]] * SockeyePrevious[j]
    logit(eRecaptured[i,j]) <- bRecaptured + bRecapturedSeason[Season[j]] + bRecapturedAnnual[Annual[j]] 
    
    eMoved[i,j] <- bMoved 
    eRecapturedResearcher[i,j] <- bRecapturedResearcher
    eReported[i,j] <- bReported
    eReleased[i,j] <- bReleased
    eReleasedResearcher[i,j] <- bReleasedResearcher

    InLake[i,j] ~ dbern(InLake[i,j-1] * ((1 - (Recaptured[i,j-1])) + Recaptured[i,j-1] * Released[i,j-1]) * ((1 - (RecapturedResearcher[i,j-1])) + RecapturedResearcher[i,j-1] * ReleasedResearcher[i,j-1])) 
    Alive[i,j] ~ dbern(InLake[i,j] * Alive[i,j-1] * (1-eMortality[i,j]))
    Alive50[i,j] <- Alive[i,j] * (Length[i,j] >= 500)
    TBarTag100[i,j] ~ dbern(TBarTag100[i,j-1] * (1-Recaptured[i,j-1]) * (1-bTagLossT))
    TBarTag10[i,j] ~ dbern(TBarTag10[i,j-1] * (1-Recaptured[i,j-1]) * (1-bTagLossT))
    
    eDetected[i,j] <- Alive[i,j] * eDetectedAlive[i] + (1-Alive[i,j]) * eDetectedDead[i]
    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])
    
    RecapturedResearcher[i,j] ~ dbern(Alive[i,j] * eRecapturedResearcher[i,j] * step(TBarTag100[i,j] + TBarTag10[i,j] - 1))
    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))
    Released[i,j] ~ dbern(Recaptured[i,j] * eReleased[i,j])
    ReleasedResearcher[i,j] ~ dbern(RecapturedResearcher[i,j] * eReleasedResearcher[i,j])
  }
  }
  Marked[1] <- sum(Alive50[1:PeriodCaptures[1],1])
  MarkedSpawners[1] <- 1
  for(p in 2:nPeriod){
      Marked[p] <- sum(Alive50[1:PeriodCaptures[p],p])
      logit(eSpawningPeriod[p]) <- bEs + bEsAnnual[Annual[p]]
      MarkedSpawners[p] ~ dpois(Marked[p-1] * eSpawningPeriod[p])
  }
  sAbundanceAnnual ~ dexp(1)
  for (i in 1:nAnnualOuting) {
    bAbundanceAnnual[i] ~ dnorm(0, sAbundanceAnnual^-2)
  }
  eAbundanceSpawn ~ dpois(bAbundanceSpawn)
  for(l in 1:nPeriodCount){
      logit(eEfficiencySpawn[l]) <- bEfficiencySpawn
      RecaptureSpawners[l] ~ dbin(eEfficiencySpawn[l], MarkedSpawners[PeriodCount[l]])
      CaptureSpawners[l] ~ dbin(eEfficiencySpawn[l], eAbundanceSpawn)
  }

  for(k in 1:nPeriodOuting){
      log(eAbundance[k]) <- bAbundance + bAbundanceAnnual[AnnualOuting[k]]
      eAbundanceInt[k] ~ dpois(eAbundance[k])
      logit(eEfficiency[k]) <- bEfficiency + bOutingHours * OutingHours[PeriodOuting[k]]
      Recaptures[k] ~ dbin(eEfficiency[k], Marked[PeriodOuting[k]-1])
      Captures[k] ~ dbin(eEfficiency[k], eAbundanceInt[k])
  }

Block 4. Survival model description.

Condition

Table 1. Parameter descriptions.

Table 2. Model coefficients.

Table 3. Model summary.

Table 4. Model posterior predictive checks.

Table 5. Model sensitivity.

Growth

Table 6. Parameter descriptions.

Table 7. Model coefficients.

Table 8. Model summary.

Tag Loss

Table 9. Parameter descriptions.

Table 10. Model coefficients.

Table 11. Model summary.

Table 12. The number of recaptured fish that were reported without their $100 versus $10 reward tag by species.

Table 13. Model sensitivity.

Survival

Table 14. Parameter descriptions.

Table 15. Model coefficients.

Table 16. Model summary.

Table 17. The estimated annual interval probabilities (with 95% CIs). Natural mortalities are for a medium sized fish (600-700 mm). Harvest mortality is the recapture probability multiplied by the probability of being harvested if recaptured, assuming that recapture can only occur once in a year. Release is the probability of being released if recaptured.

Mark-Recapture

Table 18. The number of captures, recaptures, estimated marked individuals alive and available for capture and total number of outing hours summarized by year.

Yield-per-Recruit

Spawners

Condition

Growth

Tag Loss

Survival

Mark-Recapture

Yield-per-Recruit