On the steady state of a shadow system to the SKT competition model

Qi Wang

doi:10.3934/dcdsb.2014.19.2941

Article Contents

2014, Volume 19, Issue 9: 2941-2961. Doi: 10.3934/dcdsb.2014.19.2941

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On the steady state of a shadow system to the SKT competition model

Qi Wang^1,

1.
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130

Received: November 30, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

We study a boundary value problem with an integral constraint that arises from the modelings of species competition proposed by Lou and Ni in [10]. Through local and global bifurcation theories, we obtain the existence of non-constant positive solutions to this problem, which are small perturbations from its positive constant solution, over a one-dimensional domain. Moreover, we investigate the stability of these bifurcating solutions. Finally, for the diffusion rate being sufficiently small, we construct infinitely many positive solutions with single transition layer, which is represented as an approximation of a step function. The transition-layer solution can be used to model the segregation phenomenon through interspecific competition.

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Mathematics Subject Classification: Primary: 35B40, 35J57; Secondary: 92D25, 35B32, 35B35.

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