Modelling stochastic fish stock dynamics using Markov Chain Monte Carlo

1. Modelling stochastic fish stock dynamics using Markov Chain Monte Carlo Reporter: Hsu Hsiang-Jung ENRIC CORTÉS ICES Journal of Marine Science, 60: 743-752. 2003

2. Introduction • The precautionary approach has become a basis concept in fish stock management. • The quantification of there uncertainties has emphasised the need for developing stochastic assessment approaches. • Estimates of parameters including process variances and predicted stock number have been obtained using likelihood Markov Chain Monte Carlo.

3. Population dynamics models • Catch-at-age in numbers and effort data for commercial fleets. • Catch-at-age in numbers without effort data for the remaining part of total international catches. • CPUE by age for research surveys.

4. Population dynamics models

5. Population dynamics models

6. Population dynamics models

7. Population dynamics models “N” denotes the stock number. “F” the fishing mortality . “Z” the total mortality. “ “ the standard deviations for the survival and fishing processes.

8. Population dynamics models “q” the catchability. “e” the effort. “T” the day of year when the survey takes place. “ “ the standardised normal distribution.

9. Estimation methods • For complex models with strutural relationships between variables and parameters, such as the stochastic survival model considered, the so-called single component Metropolis-Hastings or Gibbs sampling is an MCMC method especially suitable for simulating the likelihood function. • The difference between the MLE and this estimator lies in the MLE being the maximum of the likelihood function while the new estimator being the mean.

10. Simulation experiments The catch observations were simulated in the following way: 1.The model used was the same as described by Equations (1)-(7). 2.The parameters , Θ, used were the values estimated applying data described in the next section. 3.Fres,a,y and Ff,a,y were calculated, the latter using effort data and catchabilities, qf,a. 4.NminA,1 was predicted by randomly darwing from the lognormal distribution,(Equation(6)) 5.Na,1 a=2,…,A were predicted by randomly drawing from the lognormal distribution ,(Equation(7))

11. Simulation experiments 6. SSB1 was calculated. 7. For y = 2 recruitment NminA,ywas randomly drewn from Equation(5). 8. For a = 2,…,A Na,y was randomly drawn from Equation(1a) and SSBy calculated. 9. Steps 7and 8 were repeated as long as y < Y. 10. The catch observations, Cres,a,y, Cf,a,y and Is,a,y, were generated from the lognormal distributions(Equations(2)-(4)).

12. Materials and software used • Catch-at and effort data for the Dutch and English commercial beam trawl fleets. • Catch-at-age data for the combined fleet without effort data. • Survey indices for the Dutch beam trawl and the Sole Net Survey.

13. Materials and software used • The software package, WinBUGS1.4 was used to simulate the posterior distributions of the parameters.

14. Results

15. Results

16. Results

17. Results

18. Results

19. Results

20. Results of simulation experiments

21. Discussion • Our model of fish stock assessment which includes stochastic survival and recruitment. • Errors associated with the catch-at-age by fleet used in stock assessment consist of sampling errpr and other errors denoted as process errors.

22. Discussion • The MCMC methodology, in particular the single component Metropolis-Hastings and graphical models. • It is easy to implement such complex model in the WinBUGS program.

23. THANK