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The Application of Liénard Transformations to Predator–Prey Systems

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Abstract

This paper discusses the different ways a planar autonomous predator–prey system can be transformed to a Liénard system. The procedure is applied to several families like Gause and Leslie–Gower systems with functional response functions of different types ranging from simple prey-dependent functions to a Beddington–DeAngelis function, Hassell–Varley function, Crowley–Martin function, all of which depend in a nonlinear way on the predator or prey density.To illustrate the methodology we prove for three specific predator–prey systems -two Gause systems and one Leslie–Gower system—that at most one limit cycle exists using Liénard theorems. In two cases adjustments of the classical Zhang Zhifen theorem are applied, while the third case is solved by using a more powerful theorem, which is a generalization of Coppel’s theorem.

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Acknowledgements

This research was partially supported by the National Natural Science Foundation of China (12061016).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, Guangxi, China

    Yuan Liu, André Zegeling & Wentao Huang

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  1. Yuan Liu
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  2. André Zegeling
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  3. Wentao Huang
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Correspondence to André Zegeling.

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Liu, Y., Zegeling, A. & Huang, W. The Application of Liénard Transformations to Predator–Prey Systems. Qual. Theory Dyn. Syst. 23, 91 (2024). https://doi.org/10.1007/s12346-023-00947-0

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