Mathematics Faculty Articles

Title

Multiple Bifurcations in a Predator-Prey System of Holling and Lesile Type with Constant-Yield Prey Harvesting

Document Type

Article

Publication Date

2013

Publication Title

International Journal of Bifurcation and Chaos

Keywords

Predator–prey system of Holling and Leslie type, constant-yield harvesting, cusp of codimension at least 4, Hopf bifurcation, Bogdanov–Takens bifurcations of codimensions 2 and 3

ISSN

0218-1274

Volume

23

Issue/No.

10

First Page

1350164

Abstract

The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

DOI

10.1142/S0218127413501642

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