Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model

Yajie Sun; Ming Zhao; Yunfei Du; Yajie Sun; Ming Zhao; Yunfei Du

doi:10.3934/mbe.2023904

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 12: 20437-20467. doi: 10.3934/mbe.2023904

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Research article

Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model

1.
School of Science, China University of Geosciences (Beijing), Beijing 100083, China
2.
School of Basic Education, Beijing Institute of Graphic Communication, Beijing 102600, China

Received: 12 August 2023 Revised: 24 October 2023 Accepted: 06 November 2023 Published: 13 November 2023

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.
- discrete predator-prey model,
- fold bifurcation,
- 1:1 strong resonance bifurcation,
- fold-flip bifurcation,
- 1:2 strong resonance bifurcation
Citation: Yajie Sun, Ming Zhao, Yunfei Du. Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20437-20467. doi: 10.3934/mbe.2023904

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Abstract

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.

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