On the effects of migration and inter-specific competitions  in steady state of some Lotka-Volterra  model

Fang Li; Liping Wang; Yang Wang

doi:10.3934/dcdsb.2011.15.669

Article Contents

2011, Volume 15, Issue 3: 669-686. Doi: 10.3934/dcdsb.2011.15.669

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On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model

1.
Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907
2.
Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
3.
Institute of Mathematics, Hangzhou Dianzi Universitye, Xiasha Hangzhou Zhejiang 310018

Received: December 31, 2009

Revised: February 28, 2010

Published: September 2011

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Abstract

Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we investigate in a shadow system the effects of migration and interspecific competition coefficients on the existence of positive solutions. Our study shows that for any given migration, if the interspecific competition coefficient of the invader is small, then the shadow system has coexistence state; otherwise we can always find some migration such that it has no coexistence state. Moreover, these findings can be applied to steady state of a two-species Lotka-Volterra competition-diffusion model. Particularly, we show that if the interspecific competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to coexistence state.

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Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J55, 92D10.

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