Seminário de Biomatemática

10 de Março 2009, 16h00

Growth and Extinction of Populations and Individuals in Randomly Varying Environments
Complexo Interdisciplinar, Sala B3-01

Carlos A. Braumann (Centro de Investigação em Matemática e Aplicações, Universidade de Évora)

Abstract:

In a randomly varying environment, the per capita growth rate (abbreviately growth rate) of a population can been described by an "average" rate g(N) (usually dependent on population size N) perturbed by a white noise (as a reasonable approximation to a noise with low correlations). So, with N=N(t) being the population size at time t, we consider the general stochastic differential equation (SDE) model dN=g(N)Ndt+sNdW(t), where W(t) is a standard Wiener process and s is a (constant) noise intensity.

These models have been studied in the literature for specific functional forms of the "average" growth rate" g (like, for example, the logistic model g(N)=r(1-N/K)). Since it is hard to determine the "true" functional form of g, one wonders whether the qualitative results (concerning population extinction or existence of a stationary density) are model robust. We have managed to prove the usual qualitative results for a general function g satisfying only some basic assumptions dictated by biological considerations (and some mild technical assumptions).

From the applied point of view, it was embarrassing that the two main stochastic calculus, Itô and Stratonovich, lead to apparently different qualitative results regarding important issues like population extinction and that led to a controversy in the literature on which calculus is more appropriate to model population growth. We have resolved the controversy by showing that g means different types of "average" growth rate according to the calculus used and the apparent difference was due to the wrong implicit assumption that g represented the same "average". Taking into account the different meaning of g, there is no difference (qualitative or quantitative) between the two calculi.

We have also recently considered (with Patrícia A. Filipe) SDE models for the individual growth from birth to maturity of the size (weight, volume, length,.) of individual animals (or plants) and we briefly report some results with applications to cattle breeding.

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