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Acta Mathematicae Applicatae Sinica, English Series 2013, Vol. 29 Issue (1) :143-164    DOI: 10.1007/s10255-013-0207-5
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Dynamics in a Discrete-time Predator-prey System with Allee Effect
Xian-wei Chen1,2, Xiang-ling Fu2, Zhu-jun Jing1,3
1 Department of Mathematics, Hunan Normal University, Changsha 410081, China;
2 School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China;
3 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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Abstract 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.
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Xian-wei Chen
Xiang-ling Fu
Zhu-jun Jing
KeywordsPredator-prey System   Allee effect   flip bifurcation   Hopf bifurcation   Marotto's chaos   transient chaos   invariant circle   periodic window     
Abstract: 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.
KeywordsPredator-prey System,   Allee effect,   flip bifurcation,   Hopf bifurcation,   Marotto's chaos,   transient chaos,   invariant circle,   periodic window     
Received: 2011-01-19;
Fund:Supported by the National Natural Science Foundation of China (No. 11071066).
Cite this article:   
Xian-wei Chen, Xiang-ling Fu, Zhu-jun Jing .Dynamics in a Discrete-time Predator-prey System with Allee Effect[J]  Acta Mathematicae Applicatae Sinica, English Serie, 2013,V29(1): 143-164
URL:  
http://www.applmath.com.cn/jweb_yysxxb_en/EN/10.1007/s10255-013-0207-5      或     http://www.applmath.com.cn/jweb_yysxxb_en/EN/Y2013/V29/I1/143
 
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