Balancing survival and extinction in nonautonomous competitive Lotka- Volterra systems
type: Item
instance of: scholarly article
title: Balancing survival and extinction in nonautonomous competitive Lotka- Volterra systems
zbMATH Open document ID: 0830.34039
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-01-28T00:00:00Z
review text: The non-autonomous Lotka-Volterra system \(\dot x_i= x_i\left(b_i(t)- \sum^n_{j= 1} a_{ij}(t) x_j\right)\), \(i= 1,\dots, n\) was studied in the late nineteen eighties under the assumption that the functions \(b_i\) and \(a_{ij}\) are positive and periodic by Lazer, Alvarez, Tineo, Gopalsamy and others (for a survey see the referee's book ``Periodic Motions'', New York: Springer-Verlag (1994; Zbl 0805.34037)). Conditions were given for the existence, uniqueness and global asymptotic stability of a periodic solution. In the nineteen nineties the same system was studied under the condition that \(b_i\) and \(a_{ij}\) are positive and bounded by Lazer, Ahmad, Coleman, Tineo and the second author of the present paper (see the references in the paper). In this paper, the authors prove conditions under which \(n- r\) of the species die out and the system has a unique bounded positive solution \((x_1(t),\dots, x_r(t), 0,\dots, 0)\) which is globally asymptotically stable.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 785118
zbMATH Keywords: non-autonomous Lotka-Volterra system
zbMATH Keywords: unique bounded positive solution
author: Francisco Montes de Oca
author: Mary-Lou Zeeman
reviewed by: M. Farkas
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1177
OpenAlex ID: W2033430649
DOI: 10.1006/JMAA.1995.1177
description: scientific article; zbMATH DE number 785118