Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect

Yingzi Liu; Zhong Li; Mengxin He; Yingzi Liu; Zhong Li; Mengxin He

doi:10.3934/mbe.2023379

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8632-8665. doi: 10.3934/mbe.2023379

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Research article

Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect

1.
School of Mathematics and Statistics, Fuzhou University, Fuzhou, Fujian 350108, PR China
2.
College of Mathematics and Data Science, Minjiang University, Fuzhou, Fujian 350108, PR China

Received: 30 December 2022 Revised: 10 February 2023 Accepted: 23 February 2023 Published: 06 March 2023

In this paper, we analyze the bifurcation of a Holling-Tanner predator-prey model with strong Allee effect. We confirm that the degenerate equilibrium of system can be a cusp of codimension 2 or 3. As the values of parameters vary, we show that some bifurcations will appear in system. By calculating the Lyapunov number, the system undergoes a subcritical Hopf bifurcation, supercritical Hopf bifurcation or degenerate Hopf bifurcation. We show that there exists bistable phenomena and two limit cycles. By verifying the transversality condition, we also prove that the system undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. The main conclusions of this paper complement and improve the previous paper ^[30]. Moreover, numerical simulations are given to verify the theoretical results.
- Allee effect,
- Holling-Tanner,
- predator-prey,
- Bogdanov-Takens bifurcation,
- Hopf bifurcation
Citation: Yingzi Liu, Zhong Li, Mengxin He. Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8632-8665. doi: 10.3934/mbe.2023379

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Abstract

In this paper, we analyze the bifurcation of a Holling-Tanner predator-prey model with strong Allee effect. We confirm that the degenerate equilibrium of system can be a cusp of codimension 2 or 3. As the values of parameters vary, we show that some bifurcations will appear in system. By calculating the Lyapunov number, the system undergoes a subcritical Hopf bifurcation, supercritical Hopf bifurcation or degenerate Hopf bifurcation. We show that there exists bistable phenomena and two limit cycles. By verifying the transversality condition, we also prove that the system undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. The main conclusions of this paper complement and improve the previous paper ^[30]. Moreover, numerical simulations are given to verify the theoretical results.

References

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