The long-run behavior of periodic competitive Kolmogorov systems
type: Item
instance of: scholarly article
title: The long-run behavior of periodic competitive Kolmogorov systems
zbMATH Open document ID: 1020.92026
DOI: 10.1016/S1468-1218(01)00034-7
author: Yi Wang
author: Jifa Jiang
published in: Nonlinear Analysis. Real World Applications
publication date: 2003-03-11T00:00:00Z
review text: The paper is concerned with the non-autonomous periodic competitive system \[ \dot{x}_i=x_iN_i(t,x_1,\ldots,x_n). \tag{1} \] Assuming dissipation and irreducibility, the authors prove the existence of a canonically defined countable family of unordered, disjoint invariant sets which attract all persistent trajectories whose \(\omega\)-limit sets are not cycles. Then a more precise characterization of this family is obtained: it is shown that each member of the family (a) lies on an invariant \((n-1)\)-dimensional Lipschitz submanifold; (b) is the lower boundary of the basin of lower repulsion of a certain cycle in the interior of \(C\); (c) consists of finite connected components which are open \((n-1)\)-cells. Applying these results to the case studied by \textit{H. L. Smith} [J. Differ. Equations 64, 165-194 (1986; Zbl 0596.34013)], the authors obtain more information on the long-time behavior of trajectories of the system (1). Finally, they also study the topological entropy of the Poincaré map \(T\) generated by the system (1) and give a more explicit formula compared to that one obtained by Liang and Jiang in a preprint from 2000.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 37N25
zbMATH DE Number: 1880130
zbMATH Keywords: non-autonomous periodic competitive systems
zbMATH Keywords: omega-limit sets
zbMATH Keywords: invariant sets
zbMATH Keywords: limit cycles
zbMATH Keywords: topological entropy
reviewed by: Svitlana P. Rogovchenko
MaRDI profile type: Publication
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full work available at URL: s1468-1218(01)00034-7
OpenAlex ID: W2089316312
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description: scientific article; zbMATH DE number 1880130