Existence and uniqueness of coexistence states for a predator-prey model with diffusion
type: Item
instance of: scholarly article
title: Existence and uniqueness of coexistence states for a predator-prey model with diffusion
zbMATH Open document ID: 0823.35050
author: V. Pereyra
published in: Differential and Integral Equations
publication date: 1995-11-01T00:00:00Z
review text: The authors investigate the problem \[ -\Delta u= \lambda u- u^ 2- b{\textstyle {{uv} \over {\gamma+u}}},\;\;-\Delta v= \mu v- v^ 2+ c{\textstyle {{uv} \over {\gamma+u}}} \text{ in } \Omega \subset \mathbb{R}^ n, \quad u,v=0 \text{ on } \partial \Omega. \] \(\lambda\), \(\mu\), \(b\) and \(c\) are nonnegative constants and \(\gamma\) is a positive constant. This is a predator-prey problem modelling the interaction between a predator, with population density \(v(x)\) and a prey, with population density \(u(x)\) living in the region \(\Omega\). One is interested in positive solutions \((u,v)\). First the authors present numerical results for coexistence states for sets of parameter values. Then, some necessary and some sufficient conditions for coexistence are given. The problem of uniqueness and of stability is also discussed. Simple conditions insuring coexistence in terms of the principal eigenvalues of some semilinear elliptic equation are given.
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 527357
zbMATH Keywords: uniqueness
zbMATH Keywords: stability
zbMATH Keywords: principal eigenvalues
reviewed by: Giovanni Porru
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 527357