Mathematics Faculty Articles

Document Type

Article

Publication Date

2013

Publication Title

Mathematical Modelling of Natural Phenomena

Keywords

Predator-prey model, Constant-yield harvesting, Seasonal harvesting, Bogdanov-Takens bifurcation, Degenerate Hopf bifurcation, Periodic solution, Invariant torus

ISSN

0973-5348

Volume

8

Issue/No.

5

First Page

95

Last Page

118

Abstract

In this paper we study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant-yield for prey, it is shown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established by numerical simulations. When the harvesting is seasonal for both species, sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and transition from invariant tori to periodic solutions, respectively, as the amplitude of seasonal harvesting increases.

Comments

© EDP Sciences, 2013

DOI

10.1051/mmnp/20138507

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