Global stability of Lotka-Volterra systems with delay
type: Item
instance of: scholarly article
title: Global stability of Lotka-Volterra systems with delay
zbMATH Open document ID: 0880.34074
author: Lian Wang
author: Zhang Yi
published in: Differential Equations and Dynamical Systems
publication date: 1998-02-12T00:00:00Z
review text: The authors investigate the system with infinite delay \[ x_i'(t)= x_i(t)\Biggl[r_i(t)+ \sum^n_{j=1} a_{ij}(t) x_j(t-\tau_{ij}(t))+ \sum^n_{j=1} b_{ij}(t) \int^t_{-\infty} k_{ij}(t- s)x_j(s)ds\Biggr], \] where \(i= 1,\dots,n\), and \(r_i\), \(a_{ij}\), \(\tau_{ij}\geq 0\), \(k_{ij}\geq 0\) are continuous functions with \(\tau_{ii}\equiv 0\), \(t-\tau_{ij}(t)\to\infty\) as \(t\to\infty\), and \(\int^\infty_0 k_{ij}(s)ds= 1\). Under certain conditions, involving in particular \(a_{ii}(t)<0\) and \(\int^\infty_0|a_{ii}(t)|dt=\infty\), it is shown that the positive equilibrium is globally asymptotically stable. Bounded delays are also considered.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 1014367
zbMATH Keywords: functional-differential equations
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Lotka-Volterra systems
zbMATH Keywords: infinite delay
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1014367