Complicate dynamics of a discrete predator-prey model with double Allee effect

Shaosan Xia; Xianyi Li; Shaosan Xia; Xianyi Li

doi:10.3934/mmc.2022027

Mathematical Modelling and Control

2022, Volume 2, Issue 4: 282-295. doi: 10.3934/mmc.2022027

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Complicate dynamics of a discrete predator-prey model with double Allee effect

Shaosan Xia ,
Xianyi Li ^,

Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received: 02 November 2022 Revised: 11 December 2022 Accepted: 18 December 2022 Published: 27 December 2022

In this paper, a discrete predator-prey model with double Allee effect is discussed. We first simplify the corresponding continuous predator-prey model, and use the semidiscretization method to obtain a new discrete model. Next, the existence and local stability of nonnegative fixed points of the new discrete model are studied by using a key lemma. Then, by using the center manifold theorem and bifurcation theory, the sufficient conditions for the occurrences of transcritical bifurcation and Neimark-Sacker bifurcation and the stability of closed orbit bifurcated are obtained. Finally, the numerical simulations are presented, which not only verify the existence of Neimark-Sacker bifurcation but also reveal some new dynamic phenomena of this model.
- discrete predator-prey system,
- double Allee effect,
- transcritical bifurcation,
- Neimark-Sacker bifurcation
Citation: Shaosan Xia, Xianyi Li. Complicate dynamics of a discrete predator-prey model with double Allee effect[J]. Mathematical Modelling and Control, 2022, 2(4): 282-295. doi: 10.3934/mmc.2022027

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Abstract

In this paper, a discrete predator-prey model with double Allee effect is discussed. We first simplify the corresponding continuous predator-prey model, and use the semidiscretization method to obtain a new discrete model. Next, the existence and local stability of nonnegative fixed points of the new discrete model are studied by using a key lemma. Then, by using the center manifold theorem and bifurcation theory, the sufficient conditions for the occurrences of transcritical bifurcation and Neimark-Sacker bifurcation and the stability of closed orbit bifurcated are obtained. Finally, the numerical simulations are presented, which not only verify the existence of Neimark-Sacker bifurcation but also reveal some new dynamic phenomena of this model.

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