Permanence in Lotka-Volterra equations: Linked prey-predator systems
type: Item
instance of: scholarly article
title: Permanence in Lotka-Volterra equations: Linked prey-predator systems
zbMATH Open document ID: 0607.92022
DOI: 10.1016/0025-5564(86)90136-7
author: Gabriela Kirlinger
published in: Mathematical Biosciences
publication date: 1986-01-01T00:00:00Z
review text: \textit{V. Hutson} and \textit{G. T. Vickers}, Math. Biosci. 63, 253-269 (1983; Zbl 0524.92023), have shown that a bistable Lotka-Volterra system of two competing species can not be stabilized by the introduction of one predator. In the present paper, two predators are introduced one feeding on one prey, the other one on the second prey. An intuitively meaningful inequality condition is established implying that the four-dimensional system is permanent provided that a fixed point exists in the interior of the positive orthant of the four-dimensional space. Interesting numerical examples show that the sufficient condition for permanence may hold even when the interior equilibrium is not stable, and conversely, a system may not be permanent in spite of the fact that the interior equilibrium is (locally) asymptotically stable. The results are extended to six-dimensional systems (three predators feeding on three preys), and to two predators two prey systems with interspecific competition between the predators. The fairly sophisticated proof of the first theorem is lucidly presented.
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 3982986
zbMATH Keywords: linked prey-predator systems
zbMATH Keywords: bistable Lotka-Volterra system
zbMATH Keywords: two predators
zbMATH Keywords: inequality condition
zbMATH Keywords: four-dimensional system
zbMATH Keywords: fixed point
zbMATH Keywords: numerical examples
zbMATH Keywords: permanence
zbMATH Keywords: interior equilibrium
zbMATH Keywords: six-dimensional systems
zbMATH Keywords: three predators feeding on three preys
zbMATH Keywords: two predators two prey systems
zbMATH Keywords: interspecific competition
Wikidata QID: Q115599892
reviewed by: M. Farkas
MaRDI profile type: Publication
full work available at URL: 0025-5564(86)90136-7
OpenAlex ID: W2037408821
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description: scientific article; zbMATH DE number 3982986