Global positive coexistence of a nonlinear elliptic biological interacting model
type: Item
instance of: scholarly article
title: Global positive coexistence of a nonlinear elliptic biological interacting model
zbMATH Open document ID: 0698.92022
DOI: 10.1016/0025-5564(89)90040-0
author: Lige Li
published in: Mathematical Biosciences
publication date: 1989-01-01T00:00:00Z
review text: The author studies an elliptic predator-prey system of the Dirichlet problem of the type \[ (1) \Delta u+u M(u,v)=0,\;d\Delta v+v(g(u)- m(v))=0,\;(u,v)|_{\partial \Omega}=(0,0). \] Under some assumptions on \(M\), \(g\) and \(m\), and providing that the domain \(\Omega\) is large, he gives a necessary and sufficient condition for the coexistence of positive solutions to (1). This necessary and sufficient condition is equivalent to the condition that the corresponding o.d.e. system \[ du/dt=u M(u,v),\;dv/dt=v(g(u)-m(v)) \] has positive equilibrium \({\tilde u}>0\), \({\tilde v}>0\). So, when \(\Omega\) is large, this condition does not depend on the shape of \(\Omega\) and is algebraically computable. Also, some stability properties of the positive solutions to (1) are studied.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B99
zbMATH DE Number: 4145731
zbMATH Keywords: interaction model
zbMATH Keywords: elliptic predator-prey system
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: necessary and sufficient condition for the coexistence of positive solutions
zbMATH Keywords: positive equilibrium
zbMATH Keywords: stability properties
Wikidata QID: Q52425099
MaRDI profile type: Publication
full work available at URL: 0025-5564(89)90040-0
OpenAlex ID: W2064779003
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description: scientific article; zbMATH DE number 4145731