Flip bifurcation of a discrete predator-prey model with modified Leslie-Gower and Holling-type III schemes

Yangyang Li; Fengxue Zhang; Xianglai Zhuo; Yangyang Li; Fengxue Zhang; Xianglai Zhuo

doi:10.3934/mbe.2020106

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2003-2015. doi: 10.3934/mbe.2020106

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Research article

Flip bifurcation of a discrete predator-prey model with modified Leslie-Gower and Holling-type III schemes

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Received: 10 October 2019 Accepted: 12 December 2019 Published: 23 December 2019

The continuous predator-prey model is one of the main models studied in recent years. The dynamical properties of these models are so complex that it is an urgent topic to be studied. In this paper, we transformed a continuous predator-prey model with modified Leslie-Gower and Hollingtype Ⅲ schemes into a discrete mode by using Euler approximation method. The existence and stability of fixed points for this discrete model were investigated. Flip bifurcation analyses of this discrete model was carried out and corresponding bifurcation conditions were obtained. Provided with these bifurcation conditions, an example was given to carry out numerical simulations, which shows that the discrete model undergoes flip bifurcation around the stable fixed point. In addition, compared with previous studies on the continuous predator-prey model, our discrete model shows more irregular and complex dynamic characteristics. The present research can be regarded as the continuation and development of the former studies.
- flip bifurcation,
- Euler approximation method,
- discrete predator-prey model,
- stability,
- center manifold theorem
Citation: Yangyang Li, Fengxue Zhang, Xianglai Zhuo. Flip bifurcation of a discrete predator-prey model with modified Leslie-Gower and Holling-type III schemes[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2003-2015. doi: 10.3934/mbe.2020106

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Abstract

The continuous predator-prey model is one of the main models studied in recent years. The dynamical properties of these models are so complex that it is an urgent topic to be studied. In this paper, we transformed a continuous predator-prey model with modified Leslie-Gower and Hollingtype Ⅲ schemes into a discrete mode by using Euler approximation method. The existence and stability of fixed points for this discrete model were investigated. Flip bifurcation analyses of this discrete model was carried out and corresponding bifurcation conditions were obtained. Provided with these bifurcation conditions, an example was given to carry out numerical simulations, which shows that the discrete model undergoes flip bifurcation around the stable fixed point. In addition, compared with previous studies on the continuous predator-prey model, our discrete model shows more irregular and complex dynamic characteristics. The present research can be regarded as the continuation and development of the former studies.

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