Effects of a degeneracy in the competition model. I: Classical and generalized steady-state solutions
type: Item
instance of: scholarly article
title: Effects of a degeneracy in the competition model. I: Classical and generalized steady-state solutions
zbMATH Open document ID: 1042.35016
author: Yihong Du
published in: Journal of Differential Equations
publication date: 2002-07-08T00:00:00Z
review text: This paper is devoted to how heterogeneous environmental effects can greatly change the dynamical behaviour of a population model. The author considers the competition model given by \[ \begin{cases} \frac{\partial u}{\partial t}-d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv\\ \frac{\partial v}{\partial t}-d_2(x)\Delta v=\mu a_2(x)v-e(x)v^2-d(x)uv\end{cases}\tag{1} \] where \(x\in\Omega\) and \(t\geq 0\), \(\Omega\) denotes a smooth bounded domain in \(\mathbb{R}^N\) \((N\geq 2)\), \(d_1,d_2,a_1,a_2,b,c,d,e\) are nonnegative functions over \(\Omega\), and \(\lambda,\mu\) are positive constants. Throughout this paper the author supposes that \(u\) and \(v\) satisfy homogeneous Dirichlet boundary conditions. The author shows that there exists a critical number \(\lambda_*\) such that (1) behaves as if \(b(x)\) is a positive constant if \(\lambda<\lambda_*\), but if \(\lambda>\lambda_*\), then interesting new phenomena occur. This shows that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1764710
zbMATH Keywords: heterogeneous environmental effects
zbMATH Keywords: population model
zbMATH Keywords: competition model
zbMATH Keywords: homogeneous Dirichlet boundary conditions
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4074
OpenAlex ID: W2766024708
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DOI: 10.1006/JDEQ.2001.4074
description: scientific article; zbMATH DE number 1764710