On the stochastic approach to marine population dynamics


  • Eduardo Ferrandis Departamento de Ciencias del Mar y Biología Aplicada, Universidad de Alicante, Alicante




marine population dynamics, stock assessment, survival analysis, mortality models, stochastic processes


The purpose of this article is to deepen and structure the statistical basis of marine population dynamics. The starting point is the correspondence between the concepts of mortality, survival and lifetime distribution. This is the kernel of the possibilities that survival analysis techniques offer to marine population dynamics. A rigorous definition of survival and mortality based on their properties and their probabilistic versions is briefly presented. Some well established models for lifetime distribution, which generalise the usual simple exponential distribution, might be used with their corresponding survivals and mortalities. A critical review of some published models is also made, including original models proposed in the way opened by Caddy (1991) and Sparholt (1990), which allow for a continuously decreasing natural mortality. Considering these elements, the pure death process dealt with in the literature is used as a theoretical basis for the evolution of a marine cohort. The elaboration of this process is based on Chiang´s study of the probability distribution of the life table (Chiang, 1960) and provides specific structured models for stock evolution as a Markovian process. These models may introduce new ideas in the line of thinking developed by Gudmundsson (1987) and Sampson (1990) in order to model the evolution of a marine cohort by stochastic processes. The suitable approximation of these processes by means of Gaussian processes may allow theoretical and computational multivariate Gaussian analysis to be applied to the probabilistic treatment of fisheries issues. As a consequence, the necessary catch equation appears as a stochastic integral with respect to the mentioned Markovian process of the stock. The solution of this equation is available when the mortalities are proportional, hence the use of the proportional hazards model (Cox, 1959). The assumption of these proportional mortalities leads naturally to the construction of a survival model based on the Weibull distribution for the population lifetime. Finally, the Weibull survival model is elaborated in order to obtain some reference parameters that are useful for management purposes. This section does not deal exhaustively with the biological and fishery reference parameters covered in the specialised monographs (Caddy and Mahon, 1996; Cadima, 2000). We focused our work in two directions. Firstly, the principal tools generating the usual reference parameters were adapted to the proposed Weibull model. This is the case of biomass per recruit and yield per recruit, which generate some of the important reference points used for management purposes, such as the FMSY, F0.1, Fmed. They also provide important and useful concepts such as virgin biomass and overexploitation growth. For this adaptation, it was necessary to previously adapt the critical age as well as the overall natural, fishing and total mortality rates. Secondly, we analysed some indices broadly used in all population dynamics (including human populations) but only marginally dealt with in fishery science, such as life expectancy, mean residual lifetime and median survival time. These parameters are redundant with mortality rates in the classical exponential model, but are not so trivial in a more general framework.


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Abella, R., J.F. Caddy and F. Serena. – 1997. Do natural mortality and availability decline with age? An alternative yield paradigm for juvenile fisheries, illustrated by the hake, Merluccius merluccius fishery in the Mediterranean. Aquat. Living Resour., 10: 257-269. doi:10.1051/alr:1997029

Altman, D.G. – 1999. Practical statistics for medical research. Chapman & Hall / Crc., New York.

Beverton, R.H.J. and S.J. Holt. – 1957. On the dynamics of exploited fish populations. U.K. Minist. Agric. Fish. Invest., Ser. II (19), London.

Blaxter, J.H.S. (ed.) – 1974. The early life history of fish. Springer- Verlag, Berlin.

Buckland, S.T., I.B.J. Goudie and D.L. Borchers. – 2000. Wildlife population assesment: past developments and future directions. Biometrics, 56(1): 1-12. doi:10.1111/j.0006-341X.2000.00001.x

Cadima, E.L. – 2000. Manual de avaliaçao de recursos pesqueiros. F.A.O. Doc Téc. Pescas, 393, Rome.

Caddy, J.F. – 1991. Death rates and time intervals: Is there an alternative to the constant natural mortality axiom? Rev. Fish Biol. Fish., 1: 109-138. doi:10.1007/BF00157581

Caddy, J.F. and R. Mahon. – 1996. Puntos de Referencia para la ordenación pesquera. F.A.O. Doc. Téc. Pesca, 347, Rome.

Cox, D.R. – 1959. The analysis of exponentially distributed lifetimes with two types of failure. J. R. Stat. Soc. Ser. B (Methodological), 21(2): 411-421.

Curtis, F.G. and O.W. Patrick. – 1994. Applied Numerical Analysis. Addison-Wesley, New York.

Chen, S. and S. Watanabe. – 1989. Age dependence of natural mortality coefficient in fish population dynamics. Nippon Suisan Gakkaishi, 55(2): 205-208.

Chiang, C. L. – 1960. A stochastic study of the life table and its applications: I. Probability distributions of the biometric functions. Biometrics, 16(4): 618-635. doi:10.2307/2527766

Chiang, C.L. – 1970. Competing risks and conditional probabilities. Biometrics, 26(4): 767-776. doi:10.2307/2528722

Chiang, C.L. – 1991. Competing risks in mortality analysis. Annu. Rev. Publ. Health, 12: 281-307. doi:10.1146/annurev.pu.12.050191.001433

David, H.A. – 1970. On Chiang’s proportionality assumption in the theory of competing risks. Biometrics, 26(2): 336-339. doi:10.2307/2529082

Dieudonné, J. – 1963. Foundations of Modern Analysis. Academic Press inc., New York.

Doob, J.L. – 1953. Stochastic Processes. John Wiley and sons, New York.

Dobson, A.J.– 2002. An introduction to generalized linear models. Chapman & Hall / Crc., New York.

Einstein, A.- 1905. Investigation on the Theory of the Brownian Movement. Methuen, London.

Ferrandis, E and P. Hernández. – 2007. Direct Survival Analysis: a new stock assessment method. Sci. Mar., 71(1): 175-185.

Gail, M. – 1975. A review and critique of some models used in competing risk analysis. Biometrics, 31(1): 209-222. doi:10.2307/2529721

Gudmundsson, G. – 1987. Time series models of fishing mortality rates. ICES. Doc. C.M. 1988/D: 6.

Gulland, J.A. – 1971. The Fish Resources of the Ocean. Fishing News Book, West Byfleet, UK.

Gulland, J.A. – 1974. The Management of Marine Fisheries. Scientica, Bristol.

Gulland, J.A. – 1977. Fish Population Dynamics. John Wiley and sons, New York.

Gulland, J.A. – 1983. Fish Stock Assessment: A Manual of Basic Methods. John Wiley and sons, New York.

Hildebrandt, F.B. – 1974. Introduction to Numerical Analysis. McGraw-Hill, New York.

McShane, E.J. – 1969. A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals. An. Math. Soc. Memoirs., vol. 88.

McShane, E.J. – 1974. Stochastic Calculus and Stochastic Models. Academic Press, London.

Munroe, M.E. – 1953. Measure and Integration. Addison-Wesley, Massachusetts.

Oksendal, B. – 1985. Stochastic Differential Equations. An Introduction with Applications. Springer-Verlag, Berlin.

Ricker, W.E. – 1975. Computations and Interpretation of Biological Statistics of Fish Populations. Fish Res. Board Can. Bull., 191.

Sampson, D.B. – 1988. Fish capture as stochastic process. J. Cons. Int. Explor. Mer., 45: 39-60.

Shepherd, J.G. – 1983. Two measures of overall fishing mortality. J. Cons. Int. Explor. Mer., 41: 76-80.

Smith, P.J. – 2002. Analysis of Failure and Survival Data. Chapman & Hall / Crc., New York.

Sparholt, H. – 1990. Improved estimates of the natural mortality rates of nine commercially important fish species included in the North Sea multispecies VPA model. J. Cons. Int. Explor. Mer., 46(2): 211-223.

Wiener, N. – 1923. Differential space. J. Math. Physics, 2.

Yeh, J. – 1973. Stochastic Processes and the Wiener Integral. Marcel and Dekker, New York.

Young, L.C. – 1970. Some new stochastic integrals and Stieltjes integrals. Part I: Analogues to Hardy and Littlewood classes. In: P. Ney (ed.): Advances in probability vol. 2, pp. 161-239. Marcel and Dekker, New York.

Young, L.C. – 1974. Some new stochastic integrals and Stieltjes integrals. Part II: Nigh-martingales. In: P. Ney and S. Port (eds.), Advances in probability vol. 3, pp. 101-177. Marcel and Dekker, New York.




How to Cite

Ferrandis E. On the stochastic approach to marine population dynamics. scimar [Internet]. 2007Mar.30 [cited 2022Nov.30];71(1):153-74. Available from: https://scientiamarina.revistas.csic.es/index.php/scientiamarina/article/view/41




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