Spatial heterogeneity of resources versus Lotka-Volterra dynamics
type: Item
instance of: scholarly article
title: Spatial heterogeneity of resources versus Lotka-Volterra dynamics
zbMATH Open document ID: 1023.35047
author: Yuan Lou
author: Vivian Hutson
author: K. Mischaikow
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: This paper considers the Lotka-Volterra dynamical system \(\dot{u}=u(1-u-bv)\), \(\dot{v} = v(1-cu-v)\). To study this, the authors consider the following system of reaction-diffusion equations \[ \mu \Delta \widetilde{u}+\widetilde{u}[1+\gamma \beta(x) - \widetilde{u}] =0,\quad \nu \Delta \widetilde{v}+\widetilde{v}[1+\gamma \beta(x) - \widetilde{v}] =0, \] with zero Neumann boundary conditions. The authors investigate the stability of the equilibria and the existence of coexistence solutions with emphasis on cases where the spatial variation of the environment becomes large. The authors discuss the stability of \((\widetilde{u}, 0)\) and \( ( 0, \widetilde{v})\) respectively. The authors also discuss the implications of their results for the principle of competitive exclusions and for the equation of the evolution of diffusion discussed in [\textit{J. Dockery}, \textit{V. Hutson}, \textit{K. Mischaikow} and \textit{M. Peruarowski}, J. Math. Biol. 37, 61-83 (1998; Zbl 0921.92021)].
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1884196
zbMATH Keywords: zero Neumann boundary conditions
zbMATH Keywords: stability of equilibria
zbMATH Keywords: coexistence solutions
zbMATH Keywords: competitive exclusion
reviewed by: Zhang Wanguo
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4157
OpenAlex ID: W1990636052
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DOI: 10.1006/JDEQ.2001.4157
description: scientific article; zbMATH DE number 1884196