Population dynamical behavior of Lotka-Volterra system under regime switching

Xiaoyue Li, Daqing Jiang, Xuerong Mao, National Natural Science Foundation of China (Funder), Royal Society of Edinburgh (Funder)

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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 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.
Original languageEnglish
Pages (from-to)427-448
Number of pages21
JournalJournal of Computational and Applied Mathematics
Volume232
Issue number2
DOIs
Publication statusPublished - 2009

Keywords

  • brownian motion
  • stochastic differential equation
  • generalized It^o's formula
  • markov chain
  • stochastic permanence.

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