Bifurcations and chaos in a time-discrete integral model of population dynamics
type: Item
instance of: scholarly article
title: Bifurcations and chaos in a time-discrete integral model of population dynamics
zbMATH Open document ID: 0761.92046
DOI: 10.1016/0025-5564(92)90041-T
author: A. V. Tuzinkevich
published in: Mathematical Biosciences
publication date: 1992-09-27T00:00:00Z
review text: The purpose of this paper is to study the dynamic properties of integral models, mainly one-species models. The study is made using the model \[ u_{n+1}(x)=a\int_ M\alpha(x,y)u_ n(x,y)dy\;\exp\left[-\int_ M\beta(x,y)u_ n(x,y)dy\right], \] where \(M\) is a one-dimensional uniform space and the kernels \(\alpha\) and \(\beta\) are of the form \[ \alpha(x,y)=\begin{cases} a_ 0 & \| x-y\|\leq r_ \alpha\\ 0 & \| x-y\|>r_ \alpha\end{cases},\qquad\beta(x,y)=\begin{cases} b_ 0 & \| x- y\|\leq r_ \beta\\ 0 & \| x-y\|>r_ \beta\end{cases}. \] The following normalization condition is assumed to be satisfied: \[ \int_ M\alpha(x,y)dy=\int_ M\beta(x,y)dy=1, \] that is \(2 r_ \alpha a_ 0=2 r_ \beta b_ 0=1\). \(L_ 2(M)\) will be used as a phase space. The presence of dissipator caused by spatial interaction is detected. Several bifurcations of codimensions 1 and 2 are investigated. The Lyapunov exponents and dimension of attractors is shown to be a common feature of chaotic behaviour.
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 39A99
zbMATH DE Number: 64799
zbMATH Keywords: chaos
zbMATH Keywords: L2-spaces
zbMATH Keywords: time-discrete integral model
zbMATH Keywords: cone of nonnegative functions
zbMATH Keywords: period-doubling bifurcations
zbMATH Keywords: one-species models
zbMATH Keywords: one- dimensional uniform space
zbMATH Keywords: normalization condition
zbMATH Keywords: dissipator
zbMATH Keywords: spatial interaction
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: dimension of attractors
Wikidata QID: Q52421466
reviewed by: Ion Onciulescu
MaRDI profile type: Publication
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full work available at URL: 0025-5564(92)90041-t
OpenAlex ID: W1975663064
description: scientific article; zbMATH DE number 64799