Forwards attractors for non-autonomous Lotka-Volterra cooperative systems: a detailed geometrical description
type: Item
instance of: scholarly article
title: Forwards attractors for non-autonomous Lotka-Volterra cooperative systems: a detailed geometrical description
author: Juan Garcia-Fuentes
author: Piotr Kalita
author: J. A. Langa
author: Antonio Suárez
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2024-10-08T00:00:00Z
review text: The authors provide a detailed analysis on the asymptotic behavior of cooperative \(n\)-dimensional nonautonomous Lotka-Volterra systems \N\[\N\dot u_i=u_i\Big(a_i(t)-b_{ii}(t)u_i-\sum_{\substack{j=1\\ j\neq i}}^nb_{ij}(t)u_j\Big),\qquad 1\leq i\leq n,\N\]\Nfrom population dynamics. Assuming that the coefficient functions are continuous, sufficient conditions for the existence of a globally (forward) attractive solution existing on the entire time axis are given. The solution found is such that one species becomes extinct, or every species except one becomes extinct. In \(n=1\), \(n=2\) and \(n=3\) dimensions the precise geometrical structure of the forward attractor is obtained by establishing heteroclinic connections between the globally attractive solution and semi-stable solutions in case of the species permanence and extinction. \N\NThese results are particularly interesting, because time-variant equations can exhibit highly intricate dynamics, and in general it is rare to obtain corresponding invariant compact forward attracting sets. Moreover, only few papers in the literature explore the geometric structure of such sets.
reviewed by: Christian Pötzsche
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 37N25
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 7923046
zbMATH Keywords: nonautonomous systems
zbMATH Keywords: theory of attractors
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: population systems
zbMATH Keywords: dynamical systems
MaRDI profile type: Publication
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DOI: 10.3934/DCDSB.2024085
description: scientific article; zbMATH DE number 7923046