Global periodicity in a class of reaction-diffusion systems with time delays
type: Item
instance of: scholarly article
title: Global periodicity in a class of reaction-diffusion systems with time delays
zbMATH Open document ID: 1078.35126
author: W. Feng
author: X. Lu
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2003-01-01T00:00:00Z
review text: The authors study a class of reaction-diffusion systems modelling the dynamics of food-limited populations with periodic environmental data and time delays. The following two generalized periodic-parabolic systems are considered: \[ {\partial u(x,t)\over \partial t}-Au(t,x)=r(t,x)u(t,x){K(t,x)-u(t,x)\over K(t,x)+c(t,x)u(t,x)}\tag{1} \] in \([0,\infty )\times \Omega \) subject to the boundary condition \(B[u](t,x)=0\) on \((0,\infty )\times \partial \Omega \) and initial data \(u(0,x)=u_0(x)\) on \(\overline\Omega \), \[ {\partial w(x,t)\over \partial t}-Aw(t,x)=r(t,x)w(t,x){K(t,x)-a(t,x)w(t,x)-b(t,x)w(t-mT,x)\over K(t,x)+c(t,x)[a(t,x)w(t,x)+b(t,x)w(t-mT,x)]}\tag{2} \] in \([0,\infty )\times \Omega \) subject to the boundary condition \(B[w](t,x)=0\) on \((0,\infty )\times \partial \Omega \) and initial data \(w(s,x)=\eta (s,x)\) on \([-mT,0]\times \overline\Omega \) (\(T>0\)), \(\Omega \) is a bounded domain with smooth boundary \(\partial\Omega \). Here the boundary operator is either \(B[u]=u\) or \(B[u]={\partial u\over \partial \nu }+\gamma (x)u\) with \(\gamma\in C^{1+\alpha }(\partial \Omega )\), \(\gamma (x)\geq 0\) on \(\partial \Omega \), \(u_0\) and \(\eta \) are Hölder continuous functions, the functions \(K,r,c\) are positive in \(\Omega\times (0,\infty )\) and \(T\)-periodic in \(t\) and Hölder continuous on \([0,\infty )\times \overline\Omega \), \(a+b=1\) (\(a,b\) are nonnegative numbers), \(A\) is a linear differential operator with coefficients depending on \(t\) and \(x\), which are Hölder continuous in \(x\) and \(T\)-periodic in \(t\). The main result is that there exists a unique positive \(T\)-periodic solution \(U(t,x)\) for the problem (1). It depends on the growth rate \(r(t,x)\). It turns out that \(U(t,x)\) is the global attractor for the considered problem with any positive initial density function \(u_0(x)\). For the model (2) with time delays, it is shown that the \(T\)-periodic function \(U(t,x)\) is still a positive global attractor, …
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 35B41
zbMATH DE Number: 1894897
zbMATH Keywords: dynamics of food-limited populations
zbMATH Keywords: unique positive \(T\)-periodic solution
zbMATH Keywords: global attractor
zbMATH Keywords: finite-difference approximations
zbMATH Keywords: monotone iteration method
reviewed by: Drumi Bainov
MaRDI profile type: Publication
full work available at URL: dcdsb.2003.3.69
OpenAlex ID: W2067374047
DOI: 10.3934/DCDSB.2003.3.69
description: scientific article; zbMATH DE number 1894897