Uniform persistence for nonautonomous and random parabolic Kolmogorov systems
type: Item
instance of: scholarly article
title: Uniform persistence for nonautonomous and random parabolic Kolmogorov systems
zbMATH Open document ID: 1092.35015
author: Janusz Mierczyński
author: Wenxian Shen
author: Xiao-Qiang Zhao
published in: Journal of Differential Equations
publication date: 2004-11-23T00:00:00Z
review text: The authors considered the following nonautonomous parabolic Kolmogorov system: \[ \frac{\partial u_{i}}{\partial t}= \Delta u_{i}+f_{i}(t,x,u)u_{i},\quad x\in D,\;t>0, \] \[ B_{i}u_{i}=0,\quad x\in \partial D,\;t>0,\;i=1,2,3,\dots,n, \] where \(u=(u_{1},u_{2},u_{3},\dots, u_{n}),D\) is a bounded domain in \(\mathbb R^{N}\) with sufficiently smooth boundary \(\partial D,\) \(f=(f_{1},f_{2},f_{3},\dots,f_{n}):\mathbb R \times cl(D)\times [ 0,\infty )^{n}\rightarrow \mathbb R^{n}\) is a function bounded on sets of the form \(\mathbb R\times cl(D)\times B\) with bounded \(B\subset [ 0,\infty )^{n},\) uniformly Hölder continuous in \((t,x)\) and \( C^{2}\) in \(u,\) and \(B_{i}\) denotes a boundary operator either of the Dirichlet type or of the Robin type. Embedding the above system into system \(\Sigma \) of skew-product semiflow, they presented a sufficient condition for the above system to be uniformly persistent. This is proved using the notions of global attractor of \(\Sigma \) and a Morse decomposition of maximal invariant set of \(\Sigma \) in \(\partial X_{+}\times Y.\) That is, the main theorem is Theorem 3.1. Assume that \(\Sigma \) has a global attractor and that there is a Morse decomposition \(\left\{ M_{1},M_{2},\dots,M_{k}\right\} \) of the maximal compact invariant set of \(\Sigma \) in \(\partial X_{+}\times Y\) such that for any \(1\leq j\leq k,\) there is an \(1\leq i\leq n\) with the property \(\lambda _{\min }^{i}(M_{j})>0.\) Then the above system is uniformly persistent. Two sufficient conditions of another types are also given. Furthermore the strong uniform persistence of random Kolmogorov system also is investigated. As the applications strong uniform persistence in \(n\)-species random competitive system is discussed.
reviewed by: Takeshi Taniguchi
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 37B55
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 37H15
Mathematics Subject Classification ID: 37L55
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 2117822
zbMATH Keywords: Skew-product semiflows
zbMATH Keywords: Principal spectrum
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: Unsaturated invariant set
zbMATH Keywords: Random equilibrium
MaRDI profile type: Publication
full work available at URL: j.jde.2004.02.014
OpenAlex ID: W2035362826
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DOI: 10.1016/J.JDE.2004.02.014
description: scientific article; zbMATH DE number 2117822