## Scientia Marina, Vol 71, No 1 (2007)

### On the stochastic approach to marine population dynamics

https://doi.org/10.3989/scimar.2007.71n1153

*Eduardo Ferrandis*

Departamento de Ciencias del Mar y Biología Aplicada, Universidad de Alicante, Alicante, Spain

#### Abstract

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 F_{MSY}, F_{0.1}, F_{med}. 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.

#### Keywords

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