Population dynamical behavior of Lotka–Volterra system under regime switching

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In this paper, we investigate a Lotka–Volterra system under regime switching dx(t)=diag(x1(t),…,xn(t))[(b(r(t))+A(r(t))x(t))dt+σ(r(t))dB(t)], where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.

论文关键词:60H10,34F05,92B05,Brownian motion,Stochastic differential equation,Generalized Itô’s formula,Markov chain,Stochastic permanence

论文评审过程:Received 7 August 2008, Revised 2 May 2009, Available online 1 July 2009.

论文官网地址:https://doi.org/10.1016/j.cam.2009.06.021