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Qualitative Structures Near a Degenerate Fixed Point of a Discrete Ratio-Dependent Predator–Prey System

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Abstract

This paper investigates the qualitative structures of a discrete ratio-dependent predator–prey model near a degenerate fixed point whose eigenvalues are \(\pm 1\). By the normal form theory, Picard iteration and Takens’s theorem, this model is transformed into an ordinary differential system. Then the qualitative structures of this differential system near the highly degenerate equilibrium are analyzed with the blowing-up method, which yields the ones of the discrete model near the fixed point by the conjugacy between the discrete model and the time-one mapping of the vector field.

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Data Availibility Statement

The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. This research is supported by the National Natural Science Foundation of China (12171171), the Natural Science Foundation of Fujian Province of China (2022J01303 and 2023J01121) and the Scientific Research Funds of Huaqiao University.

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  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, Fujian, China

    Jinling Yang & Shengfu Deng

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  1. Jinling Yang
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S.D. proposed the idea of this paper and performed all the steps of the proofs. J. Y. wrote the whole paper. All authors read and approved the final manuscript.

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Correspondence to Shengfu Deng.

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Yang, J., Deng, S. Qualitative Structures Near a Degenerate Fixed Point of a Discrete Ratio-Dependent Predator–Prey System. Qual. Theory Dyn. Syst. 23, 191 (2024). https://doi.org/10.1007/s12346-024-01052-6

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