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Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response

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Abstract

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.

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Acknowledgements

This work is supported by the Plan for Scientific Innovation Talent of Henan Province (164200510011), Innovative Research Team of Science and Technology in Henan Province (17IRTSTHN007), Opening fund of State Key Laboratory of Nonlinear Mechanics (LNM201710) and the NSFC (11271339) project and the National Key Research and Development Program of China (2017YFB0702504).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Jingli Ren & Liping Yu

  2. Department of Mathematics, TU Dresden, 01062, Dresden, Germany

    Stefan Siegmund

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  1. Jingli Ren
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  2. Liping Yu
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  3. Stefan Siegmund
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Correspondence to Jingli Ren.

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Ren, J., Yu, L. & Siegmund, S. Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response. Nonlinear Dyn 90, 19–41 (2017). https://doi.org/10.1007/s11071-017-3643-6

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  • DOI: https://doi.org/10.1007/s11071-017-3643-6

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