Global stability and oscillations in classical Lotka-Volterra loops
type: Item
instance of: scholarly article
title: Global stability and oscillations in classical Lotka-Volterra loops
zbMATH Open document ID: 0609.92032
DOI: 10.1007/BF00275505
author: A. B. Roy
author: Fortunata Solimano
published in: Journal of Mathematical Biology
publication date: 1987-01-01T00:00:00Z
review text: The classical Lotka-Volterra model \[ \dot N_ i=N_ i(b_ i+\sum^{n}_{j=1}a_{ij}N_ j);\quad N_ i(0)>0;\quad i,j=1,...,n;\quad n\geq 3, \] is investigated under the assumptions \(a_{ii}<0\), \(a_{ij}a_{ji}<0\), \(i\neq j\), and \(a_{ij}=0\) if the remainder classes of i and j mod n are not neighbouring. This last assumption expresses that the system is an n-loop. It is called cyclic, if sign \(a_{i,i+1}=sign a_{n1}\), for all \(i=1,2,...,n-1\); otherwise it is called acyclic. Sufficient conditions are provided for the existence of a nonnegative and stable equilibrium. Boundedness of the solutions is proved for acyclic loops and sufficient conditions are given for the case of cyclic loops. The entire parameter space is studied for a 3-species cyclic loop.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 80A30
zbMATH DE Number: 3987110
zbMATH Keywords: global stability
zbMATH Keywords: sufficient conditions for global asymptotic stability
zbMATH Keywords: global boundedness
zbMATH Keywords: predator-prey system
zbMATH Keywords: stability
zbMATH Keywords: Lotka-Volterra model
zbMATH Keywords: nonnegative and stable equilibrium
zbMATH Keywords: acyclic loops
zbMATH Keywords: 3-species cyclic loop
Wikidata QID: Q113909311
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00275505
OpenAlex ID: W2052985345
description: scientific article; zbMATH DE number 3987110