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Homoclinic bifurcation of a ratio-dependent predator–prey system with impulsive harvesting

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Abstract

In this paper, we investigate a ratio-dependent prey–predator model with state-dependent impulsive harvesting where the prey growth rate is subject to a strong Allee effect. The existence of order-1 homoclinic cycle is obtained, and choosing \(\alpha \) as a control parameter, the existence, uniqueness and stability of order-1 periodic solution of the system are discussed by means of the geometry theory of semi-continuous dynamic system. We also investigate that system exhibits the phenomenon of homoclinic bifurcation about parameter \(\alpha \). Moreover, the numerical simulations are provided to show the main results. The used methods are intuitive to prove the existence of order-1 periodic solution and homoclinic bifurcation.

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Acknowledgements

This work is supported by the Natural Science Foundation of China (11301216, 11371306) and Fujian Provincial Natural Science Foundation of China (2016J01667).

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Authors and Affiliations

  1. Science College, Jimei University, Xiamen, 361021, Fujian, People’s Republic of China

    Chunjin Wei & Junnan Liu

  2. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Lansun Chen

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  1. Chunjin Wei
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  2. Junnan Liu
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  3. Lansun Chen
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Correspondence to Chunjin Wei.

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Wei, C., Liu, J. & Chen, L. Homoclinic bifurcation of a ratio-dependent predator–prey system with impulsive harvesting. Nonlinear Dyn 89, 2001–2012 (2017). https://doi.org/10.1007/s11071-017-3567-1

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