American Institute of Mathematical Sciences

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May  2006, 6(3): 559-572. doi: 10.3934/dcdsb.2006.6.559

Bifurcations in a discrete time Lotka-Volterra predator-prey system

Xiaoli Liu 1, and  Dongmei Xiao 1,

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China, China

Received  July 2005 Revised  December 2005 Published  February 2006

In this paper, a discrete-time system, derived from a predator-prey system by Euler's method with step one, is investigated in the closed first quadrant $R_+^2$. It is shown that the discrete-time system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and the discrete-time system has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. Numerical simulations are provided to verify the theoretical analysis and show the complicated dynamical behavior. These results reveal far richer dynamics of the discrete model compared with the same type continuous model.
Keywords: fixed point, numerical simulations., Neimark-Sacker bifurcations, Discrete time, fold and flip bifurcation, predator prey system.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
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