Period-doubling bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with Allee effect and cannibalism

Zhuo Ba; Xianyi Li; Zhuo Ba; Xianyi Li

doi:10.3934/era.2023072

Electronic Research Archive

2023, Volume 31, Issue 3: 1405-1438. doi: 10.3934/era.2023072

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Period-doubling bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with Allee effect and cannibalism

Zhuo Ba ,
Xianyi Li ^,

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

Academic Editor: Fengqi Yi

Received: 10 October 2022 Revised: 27 November 2022 Accepted: 07 December 2022 Published: 11 January 2023

In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation.
- predator-prey system with Allee effect,
- cannibalism,
- semidiscretization method,
- period-doubling bifurcation,
- Neimark-Sacker bifurcation
Citation: Zhuo Ba, Xianyi Li. Period-doubling bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with Allee effect and cannibalism[J]. Electronic Research Archive, 2023, 31(3): 1405-1438. doi: 10.3934/era.2023072

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Abstract

In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation.

References

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