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September  2007, 8(2): 417-433. doi: 10.3934/dcdsb.2007.8.417

Multiple bifurcations of a predator-prey system

Dongmei Xiao 1, and  Kate Fang Zhang 2,

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, China
2. 

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada

Received  October 2006 Revised  March 2007 Published  June 2007

The bifurcation analysis of a generalized predator-prey model depending on all parameters is carried out in this paper. The model, which was first proposed by Hanski et al. [6], has a degenerate saddle of codimension 2 for some parameter values, and a Bogdanov-Takens singularity (focus case) of codimension 3 for some other parameter values. By using normal form theory, we also show that saddle bifurcation of codimension 2 and Bogdanov-Takens bifurcation of codimension 3 (focus case) occur as the parameter values change in a small neighborhood of the appropriate parameter values, respectively. Moreover, we provide some numerical simulations using XPPAUT to show that the model has two limit cycles for some parameter values, has one limit cycle which contains three positive equilibria inside for some other parameter values, and has three positive equilibria but no limit cycles for other parameter values.
Keywords: Predator-prey system, Bogdanov-Takens bifurcation of codimension 3., limit cycles, saddle bifurcation of codimension 2.
Mathematics Subject Classification: Primary: 34C25, 34C23; Secondary: 92D2.
Citation: Dongmei Xiao, Kate Fang Zhang. Multiple bifurcations of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 417-433. doi: 10.3934/dcdsb.2007.8.417
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