Quesnel Lake Exploitation Analysis 2017

The suggested citation for this analytic report is:

Thorley, J.L. and Dalgarno, S. (2018) Quesnel Lake Exploitation Analysis 2017. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/942646731.

Background

Quesnel Lake supports a recreational fishery for large Bull Trout, Lake Trout and Rainbow Trout. To provide information on the natural and fishing mortality, trout were caught by angling and tagged with acoustic transmitters and/or reward tags.

Methods

Data Preparation

The outing, receiver deployment and fish capture and recapture information were provided by the Ministry of Forests, Lands and Natural Resource Operations and entered by Vicky Lipinski and databased by Gary Pavan.

The data were prepared for analysis using R version 3.4.4 (R Core Team 2018). Receivers were assumed to have a detection range of 500 m. Only individuals with a fork length (FL) \(\geq\) 450 mm, an acoustic tag life \(\geq\) 365 days (if acoustically tagged) and a $100 and $10 reward tags were included in the survival analysis.

Data Analysis

Hierarchical Bayesian models were fitted to the data using R version 3.4.4 (R Core Team 2018) and JAGS 4.2.0 (Plummer 2015) which interfaced with each other via the jmbr package. For additional information on hierarchical Bayesian modelling in the BUGS language, of which JAGS uses a dialect, the reader is referred to Kery and Schaub (2011, 41–44).

Unless indicated otherwise, the Bayesian analyses used normal and uniform prior distributions that were vague in the sense that they did not constrain the posteriors (Kery and Schaub 2011, 36). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61). Where \(\hat{R}\) is the potential scale reduction factor and \(\textrm{ESS}\) is the effective sample size (Brooks et al. 2011).

The posterior distributions of the fixed (Kery and Schaub 2011, 75) parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% confidence/credible limits (CLs) and the p-value (Kery and Schaub 2011, 37, 42). The estimate is the median (50th percentile) of the MCMC samples, the z-score is \(\mathrm{sd}/\mathrm{mean}\) and the 95% CLs are the 2.5th and 97.5th percentiles. A p-value of 0.05 indicates that the lower or upper 95% CL is 0.

The results are displayed graphically by plotting the modeled relationships between particular variables and the response with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values respectively while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). Where informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% CIs (Bradford, Korman, and Higgins 2005).

Model Descriptions

Survival

The natural mortality and exploitation were 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, inter-section movement, T-bar tag loss, recapture and reporting. In addition to assumptions 1 to 10, in Thorley and Andrusak (2017), the model also assumes that:

Condition

The expected weight of fish of a given length were estimated from the data using a mass-length model (He et al. 2008). Key assumptions of the condition model include:

Yield-Per-Recruit

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

Results

Templates

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))
    }
  }
..

Template 1.

Condition

model {
  bWeight ~ dnorm(0, 2^-2)
  bWeightLength ~ dnorm(3, 2^-2)
  bWeightDayte ~ dnorm(0, 2^-2)

  sWeight ~ dnorm(0, 5^-2)
  for(i in 1:length(Length)) {
    eWeight[i] <- bWeight + bWeightDayte * Dayte[i] + bWeightLength * Length[i]
    Weight[i] ~ dlnorm(eWeight[i], exp(sWeight)^-2)
  }
  W_p <- bWeightLength
  W_500 <- exp(bWeight)
..

Template 2.

Tables

Survival

Table 1. Parameter descriptions.

Parameter Description
bDetected Logit monthly probability of detection if in-lake
bMortality Logit monthly probability of dying of natural causes
bMortalityHandling Effect of capture and handling on bMortality
bMoved Logit monthly probability of being detected moving between sections if alive
bRecaptured Logit monthly probability of being recaptured
bReported Monthly probability of being reported if recaptured with one or more T-bar tags
bTagLoss Monthly probability of loss for a single T-bar tag

Bull Trout

Table 2. Model coefficients.

term estimate sd zscore lower upper pvalue
bDetected 0.4169320 0.0107617 38.750084 0.3959670 0.4382811 0.0007
bMortality -3.4923604 0.2457833 -14.271143 -4.0301724 -3.0534896 0.0007
bMortalityHandling 1.0727630 0.4133397 2.552433 0.2286481 1.8572029 0.0147
bMoved 0.1773531 0.0129811 13.699811 0.1540948 0.2041313 0.0007
bRecaptured 0.0048835 0.0019410 2.650326 0.0020804 0.0097071 0.0007
bReported 0.9572536 0.0289644 32.956158 0.9027288 0.9981692 0.0007
bTagLoss 0.0099161 0.0062226 1.754576 0.0021213 0.0259348 0.0007

Table 3. Model summary.

n K nchains niters nthin ess rhat converged
4675 7 3 500 500 53 1.085 FALSE

Lake Trout

Table 4. Model coefficients.

term estimate sd zscore lower upper pvalue
bDetected 0.7952307 0.0064009 124.219474 0.7818868 0.8076625 0.0007
bMortality -4.2166810 0.1889221 -22.314892 -4.5937970 -3.8623131 0.0007
bMortalityHandling 0.9918517 0.3149927 3.130487 0.3507571 1.5607728 0.0013
bMoved 0.4540679 0.0094711 47.914867 0.4355635 0.4720405 0.0007
bRecaptured 0.0030575 0.0005579 5.508281 0.0020813 0.0042957 0.0007
bReported 0.9799874 0.0235677 41.294378 0.9128165 0.9992992 0.0007
bTagLoss 0.0013730 0.0008288 1.965480 0.0008621 0.0039681 0.0007

Table 5. Model summary.

n K nchains niters nthin ess rhat converged
24860 7 3 500 500 58 1.041 FALSE

Rainbow Trout

Table 6. Model coefficients.

term estimate sd zscore lower upper pvalue
bDetected 0.5603467 0.0068584 81.683821 0.5462199 0.5738301 0.0007
bMortality -2.8386594 0.0989192 -28.732253 -3.0410289 -2.6652109 0.0007
bMortalityHandling -0.4992109 0.2245297 -2.239697 -0.9644613 -0.0874545 0.0253
bMoved 0.7066050 0.0085314 82.812592 0.6899037 0.7224901 0.0007
bRecaptured 0.0125388 0.0017724 7.164344 0.0096511 0.0163317 0.0007
bReported 0.9871643 0.0165066 59.497222 0.9382342 0.9996264 0.0007
bTagLoss 0.0033280 0.0016213 2.206780 0.0011467 0.0076593 0.0007

Table 7. Model summary.

n K nchains niters nthin ess rhat converged
20185 7 3 500 500 99 1.06 FALSE

Condition

Table 8. Parameter descriptions.

Parameter Description
bWeight Intercept of eWeight
bWeightDayte Effect of Dayte on bWeight
bWeightLength Intercept of effect of Length on bWeight
Dayte Standardised day of year of capture
eWeight Log expected Weight
Length Log-transformed and centered on 500mm fork length
sWeight Log standard deviation of residual variation in log Weight
W500 Weight at 500 mm (kg)
Weight Mass (kg)
Wp Weight power term

Bull Trout

Table 9. Model coefficients.

term estimate sd zscore lower upper pvalue
bWeight 0.2470391 0.0463567 5.369646 0.1586015 0.3390328 0.0007
bWeightDayte 0.0294158 0.0275804 1.062235 -0.0265500 0.0806554 0.2733
bWeightLength 3.2939603 0.2086140 15.778112 2.8908608 3.7156016 0.0007
sWeight -1.6326989 0.0995449 -16.353686 -1.8212933 -1.4198944 0.0007

Table 10. Derived coefficients.

term estimate sd zscore lower upper pvalue
W_500 1.280229 0.0596763 21.51637 1.171871 1.403590 7e-04
W_p 3.293960 0.2086140 15.77811 2.890861 3.715602 7e-04

Table 11. Model summary.

n K nchains niters nthin ess rhat converged
56 4 3 500 1 186 1.007 TRUE

Lake Trout

Table 12. Model coefficients.

term estimate sd zscore lower upper pvalue
bWeight 0.2956832 0.0173038 17.112181 0.2613740 0.3298025 0.0007
bWeightDayte 0.0284693 0.0085946 3.319793 0.0111546 0.0460031 0.0013
bWeightLength 3.1414125 0.0773853 40.564267 2.9889160 3.2848537 0.0007
sWeight -1.9593239 0.0438079 -44.716025 -2.0389539 -1.8703952 0.0007

Table 13. Derived coefficients.

term estimate sd zscore lower upper pvalue
W_500 1.344044 0.0232833 57.75877 1.298713 1.390693 7e-04
W_p 3.141412 0.0773853 40.56427 2.988916 3.284854 7e-04

Table 14. Model summary.

n K nchains niters nthin ess rhat converged
262 4 3 500 1 189 1.012 TRUE

Rainbow Trout

Table 15. Model coefficients.

term estimate sd zscore lower upper pvalue
bWeight 0.4093824 0.0162670 25.175583 0.3768944 0.4425569 7e-04
bWeightDayte 0.0578334 0.0108988 5.294111 0.0355493 0.0797535 7e-04
bWeightLength 3.0596462 0.0772934 39.581225 2.9060432 3.2092630 7e-04
sWeight -1.7163188 0.0433542 -39.538284 -1.7961453 -1.6295262 7e-04

Table 16. Derived coefficients.

term estimate sd zscore lower upper pvalue
W_500 1.505888 0.0245204 61.43093 1.457750 1.556682 7e-04
W_p 3.059646 0.0772934 39.58122 2.906043 3.209263 7e-04

Table 17. Model summary.

n K nchains niters nthin ess rhat converged
277 4 3 500 1 246 1.017 TRUE

Yield-Per-Recruit

Table 18. Assumed life-history parameters values in yield-per-recruit calculation by trout species.

Parameter Symbol Bull Lake Rainbow
Asymptotic length (mm) \(L_{\infty}\) 1000.00 1000.00 1000.0
Growth constant (y-1) \(k\) 0.13 0.15 0.2
Weight at 500 mm (kg) \(W_{500}\) 1.30 1.40 1.6
Weight power \(W_{p}\) 3.40 3.10 3.0
Length at maturity (mm) \(L_{m}\) 500.00 500.00 500.0
Length at which 50% vulnerable to angling (mm) \(L_{v}\) 400.00 400.00 400.0
Vulnerability power \(V_{p}\) 10.00 10.00 10.0
Maximum annual reproductive rate \(R_{k}\) 3.80 2.50 6.0

Table 19. Yield-per-recruit calculations.

Parameter Symbol Relationship
Annual interval fishing mortality rate \(U\) \(\text{Optimized by yield-per-recruit analysis}\)
Annual interval natural mortality rate \(V\) \(\text{From survival analysis}\)
Annual instantaneous fishing mortality rate \(F\) \(-\text{log}(1-U)\)
Annual instantaneous natural mortality rate \(M\) \(-\text{log}(1-V)\)
Length at age a (mm) \(L_{a}\) \(L_{\infty}(1-\exp(-k \cdot a))\)
Weight at age a (kg) \(W_{a}\) \((W_{500} / 500^{W_p}) \cdot L_{a}^{W_p}\)
Relative fecundity at age a \(F_{a}\) \(W_{a} \: \text{if} \: (L_{a} \geq L_{m}) \: \mathrm{otherwise}\: 0\)
Vulnerability to fishing at age a \(V_{a}\) \(L_{a}^{V_p}/(L_{v}^{V_p} + L_{a}^{V_p})\)
Unfished survivorship to age a \(\lambda_{a}\) \(\exp(-M)^{a-1}\)
Fished survivorship to age a \(\lambda'_{a}\) \(\exp(-M+F \cdot V_a)^{a-1}\)
Unfished relative fecundity per age-1 recruit \(\phi_{0}\) \(\sum(\lambda_{a} \cdot F_{a})\)
Fished relative fecundity per age-1 recruit \(\phi'_{0}\) \(\sum(\lambda'_{a} \cdot F_{a})\)
Unfished age-1 recruits as percent of total unfished fish population (TUFP) (%) \(R_{0}\) \(100/\sum\lambda_{a}\)
Stock productivity at low density \(\alpha\) \(R_{k} \cdot \phi^{-1}\)
Stock density dependence \(\beta\) \((R_{k} - 1)/(R_{0} \cdot \phi)\)
Fished age-1 recruits as percent of TUFP (%) \(R'_{0}\) \((\alpha \cdot \phi'_{0} - 1)/(\beta \cdot \phi'_{0})\)
Equilibrium yield as percent of TUFP (%) \(Y_{e}\) \(U \cdot R'_{0} \cdot \sum(\lambda'_{a} \cdot V_{a})\)

Recapture

Table 20.

Species Year Captured Reported
Bull Trout 2013 23 0
Bull Trout 2014 16 0
Bull Trout 2015 13 2
Bull Trout 2016 24 1
Bull Trout 2017 9 3
Lake Trout 2013 101 1
Lake Trout 2014 182 9
Lake Trout 2015 74 2
Lake Trout 2016 59 9
Lake Trout 2017 36 11
Rainbow Trout 2013 55 1
Rainbow Trout 2014 53 13
Rainbow Trout 2015 55 6
Rainbow Trout 2016 140 18
Rainbow Trout 2017 64 13

Figures

Capture

figures/capture/CaptureHistogram.png
Figure 1. Captures by fork length, year, species and tag type.

Sections

figures/sections/section-map.png
Figure 2. Quesnel Lake sections. Color code is used to identify sections throughout analysis.

Coverage

figures/coverage/ReceiverSpatialCoverage.png
Figure 3. Receiver coverage of section area by date.

Detections

figures/detection/detection-overview.png
Figure 4. Location (color) and date of detections by species for each acoustic tagged fish. Grey segments indicate estimated tag life from capture date. Black shapes indicate recapture date (square indicates that fish was not released; triangle indicates release).

Section Use

figures/movement/DetectionMap2BT.png
Figure 5. Percent of Bull Trout detections by section and season, weighted by receiver coverage.
figures/movement/DetectionMap2LT.png
Figure 6. Percent of Lake Trout detections by section and season, weighted by receiver coverage.
figures/movement/DetectionMap2RB.png
Figure 7. Percent of Rainbow Trout detections by section and season, weighted by receiver coverage.

Survival

figures/survival/mortality.png
Figure 8. The annual interval natural mortality by species.
figures/survival/recapture.png
Figure 9. The annual interval probability of recapture by species.
figures/survival/tagloss.png
Figure 10. The annual probability of T-bar tag loss by species.

Yield-Per-Recruit

figures/yield/length_age.png
Figure 11. Calculated length by age.
figures/yield/weight_length.png
Figure 12. Calculated weight by length.
figures/yield/fecundity_length.png
Figure 13. Calculated relative fecundity by length.
figures/yield/vulnerability_length.png
Figure 14. Calculated vulnerability to capture by length.
figures/yield/survivorship_length.png
Figure 15. Calculated natural survivorship by length.
figures/yield/stock_recruit.png
Figure 16. Calculated age-1 recruits as a percentage of the total unfished fish population by percent of the total annual fecundity when unfished.
figures/yield/yield.png
Figure 17. Calculated yield as percent of total unfished population by annual interval fishing mortality rate and species.
figures/yield/exploitation.png
Figure 18. Angling mortality by type and species. The modeled mortalities assume all captured fish are retained.

Conclusions

Recommendations

Recommendations include:

Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

References

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He, Ji X., James R. Bence, James E. Johnson, David F. Clapp, and Mark P. Ebener. 2008. “Modeling Variation in Mass-Length Relations and Condition Indices of Lake Trout and Chinook Salmon in Lake Huron: A Hierarchical Bayesian Approach.” Transactions of the American Fisheries Society 137 (3): 801–17. https://doi.org/10.1577/T07-012.1.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

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R Core Team. 2018. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Thorley, Joseph L., and Greg F. Andrusak. 2017. “The Fishing and Natural Mortality of Large, Piscivorous Bull Trout and Rainbow Trout in Kootenay Lake, British Columbia (2008–2013).” PeerJ 5 (January): e2874. https://doi.org/10.7717/peerj.2874.