Smoothly intertwined basins of attraction in a prey-predator model
type: Item
instance of: scholarly article
title: Smoothly intertwined basins of attraction in a prey-predator model
zbMATH Open document ID: 0831.58050
DOI: 10.1007/BF00992843
published in: Acta Applicandae Mathematicae
publication date: 1996-02-13T00:00:00Z
review text: The relative alignment of basins of attraction in a prey-predator model which is of interest for the biological control of parasites in agriculture is considered. The model is assumed to be given by the two- dimensional system of Lotka-Volterra type \[ \dot x= (h_1- ky)x, \quad \dot y= (-h_2+ kx- f(x) y)y, \qquad (x,y\geq 0), \tag{1} \] where \(k, h_1, h_2>0\), for \(x\geq 0\), \(f(x)\) behaves likewise \(bx^3/ (x^5 +c)\). By suitable choices of parameters, one can achieve the system (1) to have three equilibria in \(\{ x>0\), \(y>0\}\): two sinks \(L\), \(R\) and a saddle \(S\). It is shown that if the system (1) has no periodic orbits then the common boundary of the basins of attraction of \(L\) and \(R\) generically intersects the line \(\{y= h_1/ k\}\) and the curve \(\{y= (kx- h_2)/ f(x)\}\) in countably many points, accumulating at \((0, h_1/ k)\) and \((h_2/ k, 0)\), respectively. Such behavior of the boundary leads to the intertwining of the basins of attraction of \(L\), \(R\) and makes difficult the prediction of which final equilibrium (\(L\) or \(R\)) will be attained by the system. Refining of the model by introducing a friction between preys is discussed.
Mathematics Subject Classification ID: 37N99
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 65J99
zbMATH DE Number: 790355
zbMATH Keywords: basin of attraction
zbMATH Keywords: prey-predator model
Wikidata QID: Q115610366
author: Laura Tedeschini-Lalli
reviewed by: Eugene Ershov
MaRDI profile type: Publication
cites work: Q5582924
cites work: Fractal basin boundaries
cites work: Basin boundary metamorphoses: Changes in accessible boundary orbits
cites work: Bifurcations of the forced van der Pol oscillator
description: scientific article; zbMATH DE number 790355