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Complex dynamics in a discrete-time predator-prey system without Allee effect

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Abstract

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse perioddoubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, China

    Xian-wei Chen & Xiang-ling Fu

  2. Department of Mathematics, Hunan Normal University, Changsha, 410081, China

    Xian-wei Chen & Zhu-Jun Jing

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhu-Jun Jing

Authors
  1. Xian-wei Chen
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  2. Xiang-ling Fu
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  3. Zhu-Jun Jing
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Correspondence to Zhu-Jun Jing.

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

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Chen, Xw., Fu, Xl. & Jing, ZJ. Complex dynamics in a discrete-time predator-prey system without Allee effect. Acta Math. Appl. Sin. Engl. Ser. 29, 355–376 (2013). https://doi.org/10.1007/s10255-013-0221-7

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  • DOI: https://doi.org/10.1007/s10255-013-0221-7

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