Dynamical properties of discrete Lotka-Volterra equations
type: Item
instance of: scholarly article
title: Dynamical properties of discrete Lotka-Volterra equations
zbMATH Open document ID: 0999.39017
DOI: 10.1016/S0960-0779(00)00214-9
author: Jerry Chen
author: John Perez
author: Michelle Savescu
author: Denis Blackmore
published in: Chaos, Solitons and Fractals
publication date: 2002-06-13T00:00:00Z
review text: The authors consider the Lotka-Volterra system of differential equations in the plane of the form \[ \frac{dx}{dt}=\lambda x(1-ax-by),\quad \frac{dy}{dt}=\mu y(1-cx-d y) \] where \(\lambda>0\), \(\mu>0\), \(a\geq 0\), \(b\geq 0\), \(c\geq 0\), \(d\geq 0\), \(a+b+c+d>0\). Define the Lotka-Volterra map \(F:\mathbb{R}^2\to\mathbb{R}^2\) by \[ F(x,y)=(\lambda x(1-ax-by),\;\mu y(1-cx-d y)),\quad (x,y)\in\mathbb{R}^2. \] Denote by \(F^n\) the iterates of \(F\) under composition of functions. Consider the smooth analytic semigroup action \(\mathbb{Z}_+\times \mathbb{R}^2\to\mathbb{R}^2\) defined by \((n,X)\mapsto F^n(X)\) constituting a smooth semidynamical system on \(\mathbb{R}^n\), where \(\mathbb{Z}_+\) is the additive semigroup of nonnegative integers and \(X=(x,y)\in\mathbb{R}^2\). The authors consider the special uncoupled case of Lotka-Volterra map \[ F(x,y)=(\lambda x(1-x),\mu y(1-y)),\quad (x,y)\in\mathbb{R}^2 \] and point out a few of the features of the iterates of this map. Then the dynamics of the simple quadratic coupled special case of the Lotka-Volterra map \[ F(x,y)=(\lambda x(1-y),\mu y(1-x)),\quad (x,y)\in\mathbb{R}^2 \] is investigated. In this case for \(0<\lambda\), \(\mu <1\) the unit square is invariant with respect to the map \(F\), for \(\lambda=\mu\) we have \(F(L_\sigma)\subset L_{\lambda\sigma}\) where \[ L_\sigma= \{(x,y)\in\mathbb{R}^2;\;y=x+\sigma,\quad \sigma\in\mathbb{R}\}, \] for \(\lambda=\mu>1\) the authors give some properties of the dynamical system of iterates of \(F\) and also if \(\lambda>1\), \(\mu>1\) and \((\lambda-1)(\mu-1)<4\). Finally the authors consider \[ F(x,y)=(\lambda x(1-x-\alpha y),\mu y(1-\beta x-y)),\quad (x,y)\in\mathbb{R}^2 \] with \(\alpha, \beta, \lambda, \mu\) positive numbers, summarize some basic features of this map and its dynamic behaviour by proving that there is a chaotic regime generated by an orientation-reversing horseshoe map that the authors call a twisted horseshoe. Moreover the authors show that …
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1755351
zbMATH Keywords: discrete Lotka-Volterra equations
zbMATH Keywords: population dynamics
zbMATH Keywords: chaotic regime
zbMATH Keywords: Lotka-Volterra system
zbMATH Keywords: semidynamical system
zbMATH Keywords: twisted horseshoe
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
cites work: Q3893024
cites work: On the abundance of aperiodic behaviour for maps on the interval
cites work: Q3696187
cites work: Q3993502
cites work: Quantitative universality for a class of nonlinear transformations
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: The dynamics of density dependent population models
cites work: Q3995254
cites work: A computer-assisted proof of the Feigenbaum conjectures
cites work: Q3238801
cites work: Simple mathematical models with very complicated dynamics
cites work: Chaos and the dynamics of biological populations
cites work: Q3814426
cites work: Q3744918
cites work: Q3946835
cites work: Q3279257
full work available at URL: s0960-0779(00)00214-9
OpenAlex ID: W2028429404
description: scientific article; zbMATH DE number 1755351