Dynamics in a Discrete-time Predator-prey System with Allee Effect  

作　　者：Xian-wei Chen Xiang-ling Fu Zhu-jun Jing 

机构地区：[1]Department of Mathematics, Hunan Normal University [2]School of Mathematics and Computational Science, Hunan University of Science and Technology [3]Academy of Mathematics and Systems Science, Chinese Academy of Sciences

出　　处：《Acta Mathematicae Applicatae Sinica》2013年第1期143-164,共22页应用数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China (No. 11071066)

摘　　要：In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.

关 键 词：Predator-prey System Allee effect flip bifurcation Hopf bifurcation Marotto＇s chaos transientchaos invariant circle periodic window 

分 类 号：O175[理学—数学]