Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case
type: Item
instance of: scholarly article
title: Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case
zbMATH Open document ID: 0813.34022
author: Julián López-Gómez
author: Rosa Pardo
published in: Differential and Integral Equations
publication date: 1993-12-14T00:00:00Z
review text: The authors consider the boundary value problem (1) \(u'' = - \lambda u + u^ 2 + bu \nu\), \(\nu'' = - \mu \nu - cu \nu+\nu^ 2\), \(x \in (0,1)\), \(u(0) = u(1) = \nu (0) = \nu (1) = 0\). This problem usually arises in biology and chemistry in modeling the behaviour of two interacting species on (0,1). The authors show that the problem (1) has a coexistence state if and only if the relations \(\lambda > \lambda_ 1 (b \theta_ \mu)\) and \(\mu > \lambda_ 1 (-c \theta_ \lambda)\) hold. Moreover if these relations are satisfied then (1) has exactly one coexistence state, and this state is linearly stable, provided \(b\) (or \(c)\) is small enough.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 459187
zbMATH Keywords: boundary value problem
zbMATH Keywords: behaviour of two interacting species
zbMATH Keywords: coexistence state
reviewed by: Jibin Li
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 459187