1. J. Fernando Márquez  Farías, Ph. D Raúl E. Lara  Mendoza Suitability of the use of the Bayesian approach for the estimation of growth parameters for viviparous Chondrichthyans. Facultad de Ciencias del Mar. Universidad Autónoma de Sinaloa. Center for the Advancement of Population Assessment Methodology Mazatlán, Sinaloa, México Growth workshop. November 3-7, La Jolla, California. USA.

2. Problems with the rational explotation Low productivity…. Time series of catch/effort and key biological parameters still lacking. Uncertainty on the status of most species In absence of formal stock asssessment, alternative quantitative and qualitative techiques (demographic analysis, PSA). International Plan of Actions induce to take action for research, monitoring and management. Chondrichthyan fisheries worldwide

3. Importance of age and growth studies in chondrichthyans Determine growth rates of individuals and populations Determine age at sexual maturity Estimate mortality rates Determine productivity Determine demographic parameters and fishery model parameters (Ro, r,, MSY, Yield/Recruit) Develop fisheries management strategies (Size/Age Limits, Quotas, Area and Season Closures, etc.)

4. Useful structures for ageing sharks Vertebral centra Spines Neural Arches Courtesy of Prof. Gregor Cailliet, MLML

5. Estimation of growth parameters Raw age-length data mean length–age keys

6. Estimation of growth parameters Raw age-length data mean length–age keys

7. Fabens (1965) von Bertalanffy (1938) Ricker (1975) Define the model to describe growth and the likelihood function Estimation of growth parameters Likelihood (Normal/log-normal) Least squares Goodness of fit Growth models…

8. Carcharhinus leucas (northern Gulf of Mexico)

9. Rhinobatos rhinobatos (Mediterranean Sea) Carcharhinus limbatus (southeastern Gulf of Mexico)

10. Carcharhinus falciformis (Pacific Ocean) Mustelus canis (Northwest Atlantic Ocean)

11. ….questionable results??

12. 120 cm

13. Opportunity to improve the estimation of parameters

14. The Bayesian approach offers a robust, elegant and transparent procedure to estimate growth parameters. Previous knowledge of parameters can be utilized. Parameter estimation do not depend only upon age/length keys which may be biased by selectivity, migration, or misrepresentation of large size individuals that once were abundant. Derived parameters can be assessed for biological consistence. Goodness of the approach

15. Prior distributions of parameters The implementation of the Bayesian approach in parameter estimation is subject to the challenge that prior distributions for each parameter must be specified. If information exists about the growth parameters why ignore it?. Ideally, this should be reflected in the priors. Prior distribution for the parameter can be informative or non- informative and can be elaborated from different sources including data, expert opinion and biology.

16. Posterior probability of the parameter(s) Prior probability of the parameter(s) Components of Bayes formula Likelihood of the data given the parameter(s)

17. L  = Asintotic length k = Growth coeficient Lo = Length at age-zero (lenght at first year of life? or birth length?) For growth parameters…

18. Example with scalloped hammerhead (Sphyrna lewini) from the Gulf of California, Mexico

19. Growth of hammerhead (females) Loo = 353 k = 0.156 to =-0.633

20. Growth of hammerhead

22. Changing stage from newborn to young of the year

23. Growth during first year of life

24. 46-67 cm 46-97 cm Lo= 51 cm L o = 67 cm

25. Different criteria for Lo

26. 32 cm 51 cm 67cm If Lo is birth size, then should represent final size of embryonic growth

27. k high Noninformative prior for L oo Informative prior for L oo If Lo is birth size, then should represent final size of embryonic growth

30. Some final thoughts It is difficult to accept a complete ignorance to make a prior distribution for growth parameters. In biological systems, the definition of limits can demonstrate knowledge. The elaboration of prior distributions of growth parameters can be intuitive and plausible. There is no justification for the value of the parameter be fixed (Lo), that represents a perfect knowledge of the parameter.

31. Little changes in growth parameters would result in significant changes in biomass and demographic parameters. Some final thoughts

32. Gracias!!!

33. Acknowledgements Consejo Nacional de Ciencia y Tecnología Universidad Autónoma de Sinaloa