Persistence and global stability for discrete models of nonautonomous Lotka-Volterra type.
type: Item
instance of: scholarly article
title: Persistence and global stability for discrete models of nonautonomous Lotka-Volterra type.
zbMATH Open document ID: 1033.39013
DOI: 10.1016/S0022-247X(02)00261-5
author: Yoshiaki Muroya
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The paper is concerned with the following discrete model of Lotka-Volterra type \[ \begin{gathered} N_{i}(p+1)=N_{i}(p)\exp\left\{ c_{i}(p)-a_{i}(p)N_{i}(p)-\sum_{j=1}^{n} \sum_{l=0}^{m}a^{l}_{ij}(p)N_{i}(p-k_{l})\right\}, \;p\geq 0,\\ N_{i}(p)=N_{ip}\geq0\;\text{ for} \;p\leq0,\;\text{and} \;N_{i0}>0, \;\;\;\;1\leq i\leq n, \end{gathered} \] where each \(c_{i}(p)\), \(a_{i}(p)\), \(a^{l}_{ij}(p)\) are bounded for \(p\geq0\); \(\inf_{p\geq0}a_{i}(p)>0\), \(a^{0}_{ii}(p)\equiv0\) for \(1\leq i\leq n\); \(a^{l}_{ij}(p)\geq0\) for \(1\leq i\leq j\leq n\), \(0\leq l\leq m\); \(k_{0}=0\), integers \(k_{l}\geq0\) for \(1\leq l\leq m\). By using the technique developed recently by \textit{S. Ahmad} and \textit{A. C. Lazer} [Nonlinear Anal., Theory Methods Appl. 40 A, 37--49 (2000; Zbl 0955.34041)] for competitive Lotka-Volterra differential system the author establishes similar averaged conditions for the persistence and global asymptotic stability of the above discrete system. These conditions depend on the infimum and supremum bounds of the coefficients \(a_{i}(p)\), \(a^{l}_{ij}(p)\), and the lower and upper averages \(m[c_{i}]=\liminf_{q\to\infty}\{A(c_{i},p_{1},p_{2}): p_{2}-p_{1}\geq q\}\), \(M[c_{i}]=\limsup_{q\to\infty}\{A(c_{i},p_{1},p_{2}): p_{2}-p_{1}\geq q\}\), where \(A(c_{i},p_{1},p_{2})=(p_{2}-p_{1})^{-1} \sum_{p=p_{1}}^{p_{2}-1}c_{i}(p)\).
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1866580
zbMATH Keywords: persistence
zbMATH Keywords: global stability
zbMATH Keywords: discrete nonautonomous Lotka-Volterra model
zbMATH Keywords: asymptotic stability
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866580