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Bragg-Williams approximation for the dynamics of prey-predator biological associations

The dynamics of an association of interactive biological species is studied theoretically. We explore a mean field approximation to describe the temporal evolution of an ecological system with the basic prey-predator interspecies relation, as well as an approximation to introduce time correlations in the dynamics. We start by discussing the solution of the Volterra-Lotka model in a mean field approximation based in an analogy with the Weiss solution to the Ising model for ferromagnetic materials. In order to explore the effects of long-range time correlations, we describe the time evolution of the system within a kind of Bragg-Williams approximation. This approach allows us to evaluate a characteristic life-time of the ecosystem. This quantity could be very useful to discuss the time evolution of the system under a wide diversity of environmental conditions of the ecosystem which is not usually considered. We discuss the general trends of the temporal evolution of the association with some data from real ecosystems.