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Population dynamical behavior of Lotka-Volterra system under regime switching

Li, X. and Jiang, D. and Mao, X. and , National Natural Science Foundation of China (Funder) and , Royal Society of Edinburgh (Funder) (2009) Population dynamical behavior of Lotka-Volterra system under regime switching. Journal of Computational and Applied Mathematics, 232 (2). pp. 427-448. ISSN 0377-0427

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Abstract

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 + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to nd out what happens under regime switching. We rst obtain the sucient conditions for the existence of global positive solutions, stochastic permanence, extinction. We nd out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Morkov 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 coecients. Finally, the main results are illustrated by several examples.