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July  2010, 14(1): 159-176. doi: 10.3934/dcdsb.2010.14.159

Bifurcations of a discrete prey-predator model with Holling type II functional response

Shuping Li 1, and  Weinian Zhang 2,

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2. 

Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received  August 2009 Revised  November 2009 Published  April 2010

Recently a discrete-time prey-predator model with Holling type II was discussed for its bifurcations so as to show its complicated dynamical properties. Simulation illustrated the occurrence of invariant cycles. In this paper we first clarify the parametric conditions of non-hyperbolicity, correcting a known result. Then we apply the center manifold reduction and the method of normal forms to completely discuss bifurcations of codimension 1. We give bifurcation curves analytically for transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation separately, showing bifurcation phenomena not indicated in the previous work for the system.
Keywords: discrete prey-predator model; invariant cycle; transcritical bifurcation; flip bifurcation; Neimark-Sacker bifurcation..
Mathematics Subject Classification: 37G10, 39A05, 34C23, 92D3.
Citation: Shuping Li, Weinian Zhang. Bifurcations of a discrete prey-predator model with Holling type II functional response. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 159-176. doi: 10.3934/dcdsb.2010.14.159
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