Dynamics of a predator-prey model with non-monotonic response function

H. W. Broer; K. Saleh; V. Naudot; R. Roussarie

doi:10.3934/dcds.2007.18.221

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2007, Volume 18, Issue 2&3: 221-251. Doi: 10.3934/dcds.2007.18.221

This issue Previous Article Preface Next Article Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory

Dynamics of a predator-prey model with non-monotonic response function

1.
Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen
2.
Mathematics Institute, University of Warwick, Coventry CV4 7AL
3.
Institut Mathématiques de Bourgogne, CNRS, 9, avenue Alain Savary, B.P. 47 870, 21078 Dijon cedex

Received: February 28, 2006

Revised: December 31, 2006

Published: May 2007

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Abstract

A five-parameter family of planar vector fields, which models the dynamics of certain populations of predators and their prey, is discussed. The family is a variation of the classical Volterra-Lotka system by taking into account group defense strategy, competition between prey and competition between predators. Also we initiate computer-assisted research on time-periodic perturbations, which model seasonal dependence. We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multi-stability occurs. Further, the bifurcations between the various domains of structural stability are investigated. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. Here we find several codimension 3 bifurcations that form organizing centres for the global bifurcation set. Studying the time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors.

Keywords:

Predator-prey dynamics,
bifurcation,
organizing centre and nonmonotonic response function.

Mathematics Subject Classification: 58K45, 34C23, 34C60,37D15, 37D45.

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