Qualitative analysis of a Lotka-Volterra competition system with advection

Qi Wang; Chunyi Gai; Jingda Yan

doi:10.3934/dcds.2015.35.1239

Article Contents

2015, Volume 35, Issue 3: 1239-1284. Doi: 10.3934/dcds.2015.35.1239

This issue Previous Article Minimal sets and $\omega$-chaos in expansive systems with weak specification property Next Article Equilibrium states and invariant measures for random dynamical systems

Qualitative analysis of a Lotka-Volterra competition system with advection

1.
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130
2.
Department of Mathematics and Statistics, Dalhousie University, 6316 Coburg Road, Halifax, Nova Scotia, Canada B3H 4R2
3.
Hanqing Advanced Institute of Economics and Finance, Renmin University of China, No. 59 Zhongguancun Street, Haidian District, Beijing 100872

Received: December 31, 2013

Revised: July 31, 2014

Published: February 2015

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Abstract

We study a diffusive Lotka-Volterra competition system with advection under Neumann boundary conditions. Our system models a competition relationship that one species escape from the region of high population density of their competitors in order to avoid competition. We establish the global existence of bounded classical solutions to the system over one-dimensional finite domains. For multi-dimensional domains, globally bounded classical solutions are obtained for a parabolic-elliptic system under proper assumptions on the system parameters. These global existence results make it possible to study bounded steady states in order to model species segregation phenomenon. We then investigate the one-dimensional stationary problem. Through bifurcation theory, we obtain the existence of nonconstant positive steady states, which are small perturbations from the positive equilibrium; we also rigourously study the stability of these bifurcating solutions when diffusion coefficients of the escaper and its competitor are large and small respectively. In the limit of large advection rate, we show that the reaction-advection-diffusion system converges to a shadow system involving the competitor population density and an unknown positive constant. Existence and stability of positive nonconstant solutions to the shadow system have also been obtained through bifurcation theories. Finally, we construct infinitely many single interior transition layers to the shadow system when crowding rate of the escapers and diffusion rate of their interspecific competitors are sufficiently small. The transition-layer solutions can be used to model the interspecific segregation phenomenon.

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Mathematics Subject Classification: Primary: 92C17, 35B32, 35B35, 35B36, 35B40, 35J47.

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