Competitive or weak cooperative stochastic Lotka–V

Authors: Patrick Cattiaux Sylvie Méléard

Publish Date: 2009/08/01

Volume: 60, Issue: 6, Pages: 797-829

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Abstract

We are interested in the long time behavior of a twotype densitydependent biological population conditioned on nonextinction in both cases of competition or weak cooperation between the two species This population is described by a stochastic Lotka–Volterra system obtained as limit of renormalized interacting birth and death processes The weak cooperation assumption allows the system not to blow up We study the existence and uniqueness of a quasistationary distribution that is convergence to equilibrium conditioned on nonextinction To this aim we generalize in twodimensions spectral tools developed for onedimensional generalized Feller diffusion processes The existence proof of a quasistationary distribution is reduced to the one for a ddimensional Kolmogorov diffusion process under a symmetry assumption The symmetry we need is satisfied under a local balance condition relying the ecological rates A novelty is the outlined relation between the uniqueness of the quasistationary distribution and the ultracontractivity of the killed semigroup By a comparison between the killing rates for the populations of each type and the one of the global population we show that the quasistationary distribution can be either supported by individuals of one the strongest one type or supported by individuals of the two types We thus highlight two different long time behaviors depending on the parameters of the model either the model exhibits an intermediary time scale for which only one type the dominant trait is surviving or there is a positive probability to have coexistence of the two species

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