Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations

Meng Liu; Ke Wang

doi:10.3934/dcds.2013.33.2495

Article Contents

2013, Volume 33, Issue 6: 2495-2522. Doi: 10.3934/dcds.2013.33.2495

This issue Previous Article Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term Next Article Ergodicity of certain cocycles over certain interval exchanges

Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations

Meng Liu^1, and
Ke Wang^2,

1.
School of Mathematical Science, Huaiyin Normal University, Huaian 223300
2.
Department of Mathematics, Harbin Institute of Technology, Weihai 264209

Received: February 28, 2011

Revised: September 30, 2012

Published: June 2013

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Abstract

This paper is concerned with two $n$-species stochastic cooperative systems. One is autonomous, the other is non-autonomous. For the first system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant is negative, then the corresponding species will go to extinction with probability 1; If the constant is positive, then the corresponding species will be persistent with probability 1. For the second system, sufficient conditions for stochastic permanence and global attractivity are established. In addition, the upper- and lower-growth rates of the positive solution are investigated. Our results reveal that, firstly, the stochastic noise of one population is unfavorable for the persistence of all species; secondly, a population could be persistent even the growth rate of this population is less than the half of the intensity of the white noise.

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Mathematics Subject Classification: Primary: 34F05, 60H10; Secondary: 92B05, 60J27.

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