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Hierarchical Bayesian Analysis of the Spiny Lobster Fishery in California

Hierarchical Bayesian Analysis of the Spiny Lobster Fishery in California. Brian Kinlan, Steve Gaines, Deborah McArdle, Katherine Emery UCSB. The Original Data – An Exceptionally Long Catch-Effort Time Series. Goals.

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Hierarchical Bayesian Analysis of the Spiny Lobster Fishery in California

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  1. Hierarchical Bayesian Analysis of the Spiny Lobster Fishery in California Brian Kinlan, Steve Gaines, Deborah McArdle, Katherine Emery UCSB

  2. The Original Data – An Exceptionally Long Catch-Effort Time Series

  3. Goals • Estimate dynamics of lobster population (including recruits and sublegals) over history of fishery • Evaluate alternative models for population replenishment • Examine interactive effects of variation in effort, climate, and population dynamics over long time series • NEED: Model linking catch-effort time series to population dynamics

  4. Methods Overview • Size-Structured State-Space Model • Length-Weight relationships used to link biomass catch data to abundance in underlying model • Development of priors on growth, mortality, and size structure • Implementation in WinBUGS, analysis in Matlab

  5. Components of an Hierarchical Bayesian Model • Data • Likelihood model for data (Observation Error – assumed lognormal) • Process Model • Prior distributions for parameters • Logical links specifying functional form of deterministic relationships among parameters

  6. Process Equations Abundance Ns,y=[Ns−1,y−1 * exp(−M) − Ca−1,y−1* exp(−0.5M)] * Gs,s-1,y N1,y=Ry Ry ~ LogNormal(μrecruits,σ2recruits) Catch log(CPUEy) = log(q) + log(Ny) Growth Ly=Ly-1+B0 exp(B1Ly-1)

  7. Results • Posterior distributions of parameters summarized by their mean • Evaluation of Model Fit • Patterns emerging from model – stock-recruitment relationships, climate correlations

  8. 6 legal stock x 10 3 total including catch escapement 2.5 2 1.5 No. of Lobsters 1 0.5 0 1900 1920 1940 1960 1980 2000 Year

  9. 7 x 10 2 total stock(1-8) sublegal stock(1-3) legal stock(4-8) 1.8 reproductive stock(3-8) 1.6 1.4 1.2 No. of Lobsters 1 0.8 0.6 0.4 0.2 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Year

  10. exploitation rate exploitation rate 1 0.8 0.6 Fraction Exploited 0.4 0.2 0 1900 1920 1940 1960 1980 2000 Year

  11. weight of avg legal lobster (lbs) 3.2 3 2.8 2.6 2.4 2.2 Weight (Lbs) 2 1.8 1.6 1.4 1900 1920 1940 1960 1980 2000 Year

  12. 6 recruitment x 10 11 10 9 8 7 6 No. of Lobsters 5 4 3 2 1 1900 1920 1940 1960 1980 2000 Year

  13. growth parameter 35 34 ) 1 - y 1 - 33 32 31 30 Asymptotic Growth Rate (mm indiv 29 28 27 1900 1920 1940 1960 1980 2000 Year

  14. Comparison with simpler standard fisheries models • DeLury depletion model (abundance) • Shaeffer surplus production model (biomass) • Both assume constant r, K, q and fit unknown No; model estimated by least-squares or MLE

  15. Y Bayesian total biomass Schaeffer biomass (7 outliers) Schaefer biomass (12 outliers) DeLury biomass Comparison with Standard Fisheries Models Biomass Comparison: Schaeffer, De Lury and Bayesian (1888-2005) 9000000 8000000 7000000 6000000 Y 5000000 4000000 3000000 2000000 1000000 0 1880 1900 1920 1940 1960 1980 2000 Year

  16. Model Fit and Residuals • Model vs. Predicted Total Catch • Model vs. Predicted CPUE • Residuals – Effect of Constant Catchability Assumptions

  17. 5 Actual by Predicted Catch in Lbs x 10 12 Data Regression 1:1 Line 10 8 6 Catch(Lbs) Observed 4 2 0 0 2 4 6 8 10 12 Catch(Lbs) Model 5 x 10

  18. Catch-Effort Model Fit (Test Constant q Assumption) 4 10 Data 2005 Regression 1:1 Line 3 10 1950 CPUE(Lbs/Trap) Observed Data Points Color Coded by Year 2 10 1895 1 10 1 2 3 10 10 10 Expected CPUE = q*N[y]*P(g|s)*exp(-M/2)

  19. ‘Empirical’ Stock-Recruitment Relationships • No assumptions or priors specifying a relationship between stock and recruitment were included in model • Recruitment was fit based on Catch, Effort, and the dynamic state equations • Does an ‘empirical’ relationship arise in the model fit?

  20. Stock-Recruitment Relationship 10000000 9000000 8000000 7000000 6000000 5000000 4000000 Recruitment in Year Y (No. of lobsters) 3000000 2000000 500000 1000000 5000000 Reproductive Stock in Year Y-1 (No. of lobsters)

  21. Recruits per Adult vs. No. of Adults 5.5 5 4.5 Recruits in Year Y per Adult in Year Y-1 4 3.5 3 2.5 2 1.5 0 1 2 3 4 5 6 6 x 10 No. of Adults in Year Y-1

  22. Future Model Directions • allow time-dependency of catchability, time+size dependent mortality • additional growth, mortality, size info via priors • age-structured version with explicit modeling of cohort growth-in-length • ocean climate covariates • Spatial Model

  23. Future Model Directions • Spatial Model • use regional (port-based) catch-effort data • compare alternative models of connectivity via larval movement and/or juvenile migration • will help clarify the population dynamic mechanism underlying the compensatory recruits-per-spawner relationship (pre- or post-dispersal density dependence)

  24. Applications in Context of the Sustainable Fisheries Group • Evaluate forecast and hindcast scenarios of changing temporal (and spatial) patterns of effort • Incorporate process and observation uncertainty explicitly using bayesian posteriors • Assess value of information in this fishery

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