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 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

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

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

N s,y =[N s−1,y−1 * exp(−M) − C a−1,y−1 * exp(−0.5M)] * G s,s-1,y Process Equations log(CPUE y ) = log(q) + log(N y ) Abundance Catch N 1,y =R y R y ~ LogNormal(μ recruits,σ 2 recruits ) L y =L y-1 +B 0 exp(B 1 L y-1 ) Growth

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

x 10 6 legal stock Year No. of Lobsters total including catch escapement

x 10 6 recruitment Year No. of Lobsters

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

Comparison with Standard Fisheries Models Biomass Comparison: Schaeffer, De Lury and Bayesian ( ) Y Bayesian total biomass Schaeffer biomass (7 outliers) Schaefer biomass (12 outliers) DeLury biomass Y Year

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

‘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?

Stock-Recruitment Relationship Reproductive Stock in Year Y-1 (No. of lobsters) Recruitment in Year Y (No. of lobsters)

x Recruits per Adult vs. No. of Adults No. of Adults in Year Y-1 Recruits in Year Y per Adult in Year Y-1

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

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)

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