On diffusive population models with toxicants and time delays
type: Item
instance of: scholarly article
title: On diffusive population models with toxicants and time delays
zbMATH Open document ID: 0927.35049
author: W. Feng
author: X. Lu
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-12-13T00:00:00Z
review text: The authors study two generalized reaction-diffusion systems, as logistic systems in mathematical ecology. The first of these is the following: \[ \partial u(t,x)/\partial t- Au(t,x)= r(x) u(t,x)(K(x)- u(t,x))/(K(x)+ c(x)u(t,x))\text{ in }[0,\infty)\times \Omega, \] \[ B[u](t,x)= 0\quad\text{on }(0,\infty)\times \partial\Omega,\quad u(0,x)= u_0(x)\quad\text{on }\overline\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\); the functions \(r\), \(c\) and \(K\) are positive in \(\Omega\) and Hölder continuous on \(\overline\Omega\); \(B[u]= u\) or \(B[u]= \partial u/\partial\nu+ \gamma(x)u\), with \(\gamma\in C^{1+\alpha}(\partial\Omega)\) and \(\gamma(x)\geq 0\) on \(\partial\Omega\); the initial function \(u_0\) is a Hölder continuous function on \(\overline\Omega\), and the differential operator \(A\) is a uniformly strongly elliptic operator. The main goal of this paper is to show the existence of a unique positive steady-state solution in these models and investigate the asymptotic behavior of the time-dependent solutions, in both models, in relation to such steady solutions.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 92D40
zbMATH DE Number: 1289968
zbMATH Keywords: unique positive steady-state solution
reviewed by: Ion Onciulescu
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6332
OpenAlex ID: W2064413724
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DOI: 10.1006/JMAA.1999.6332
description: scientific article; zbMATH DE number 1289968