American Institute of Mathematical Sciences

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June  2009, 24(2): 523-545. doi: 10.3934/dcds.2009.24.523

Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation

Xiaoyue Li 1, and  Xuerong Mao 2,

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China
2. 

Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland

Received  July 2008 Revised  October 2008 Published  March 2009

In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system $ dx_i (t) = x_i(t)$[($b_i(t)$-$\sum_{j=1}^{n} a_{ij}(t)x_j(t))$$dt$$+ \sigma_i(t) d B_i(t)]$, where $B_i(t)$($i=1 ,\ 2,\cdots,\ n$) are independent standard Brownian motions. Some dynamical properties are discussed and the sufficient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated.
Keywords: Brownian motion, Itô's formula, stochastic differential equation, stochastic permanence, global attractivity..
Mathematics Subject Classification: Primary: 62F10, 34F05; Secondary: 92B0.
Citation: Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523
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