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Cyclicity of the Limit Periodic Sets for a Singularly Perturbed Leslie–Gower Predator–Prey Model with Prey Harvesting

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Abstract

In this paper, we study the Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting. Our main focus is on the cyclicity of diverse limit periodic sets, including a generic contact point, canard slow–fast cycles, transitory canards, slow–fast cycles with two canard mechanisms, singular slow–fast cycle, etc. We develop new techniques for finding the maximum number of limit cycles produced by slow–fast cycles containing both the generic and degenerate contact point away from the origin (such slow–fast cycles are the transitory canards and cycles with two canard mechanisms). It can be applied not only to the Leslie–Gower predator–prey model, but more general systems as well. The main tool is geometric singular perturbation theory including cylindrical blow-up and the notion of slow divergence integral. We also study dynamics near the origin using non-standard techniques (constructing generalized normal sectors). The uniqueness and stability of a relaxation oscillation is shown using the notion of entry–exit function. Some interesting dynamical phenomena, such as relaxation oscillation and canard explosion, are simulated to illustrate the theoretical results.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

The author Jinhui Yao thanks Prof. Guihua Li for her many helpful comments on this work, and thanks Gang Guo for his simulations in Sect. 6. The author Jinhui Yao would also like to thank Prof. Jicai Huang for his constructive comments.

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Authors and Affiliations

  1. Department of Mathematics, North University of China, Taiyuan, 030051, Shan’xi, People’s Republic of China

    Jinhui Yao

  2. Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590, Diepenbeek, Belgium

    Renato Huzak

  3. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, Hubei, People’s Republic of China

    Jinhui Yao

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  1. Jinhui Yao
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Correspondence to Renato Huzak.

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Yao, J., Huzak, R. Cyclicity of the Limit Periodic Sets for a Singularly Perturbed Leslie–Gower Predator–Prey Model with Prey Harvesting. J Dyn Diff Equat (2022). https://doi.org/10.1007/s10884-022-10242-2

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