More complex dynamics in a discrete prey-predator model with the Allee effect in prey

Mianjian Ruan; Xianyi Li; Bo Sun; Mianjian Ruan; Xianyi Li; Bo Sun

doi:10.3934/mbe.2023868

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 11: 19584-19616. doi: 10.3934/mbe.2023868

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More complex dynamics in a discrete prey-predator model with the Allee effect in prey

1.
Institute of Applied Mathematics, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China
2.
Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
3.
School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 102206, China

Academic Editor: Carlo Cattani

Received: 26 July 2023 Revised: 16 October 2023 Accepted: 16 October 2023 Published: 25 October 2023

In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point $ E_*, $ unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point $ E_*, $ we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point $ E_1, $ the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point $ E_2, $ the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point $ E_* $. The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly.
- prey-predator model,
- Allee effect,
- transcritical bifurcation,
- period-doubling bifurcation,
- 1:2 resonance bifurcation
Citation: Mianjian Ruan, Xianyi Li, Bo Sun. More complex dynamics in a discrete prey-predator model with the Allee effect in prey[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19584-19616. doi: 10.3934/mbe.2023868

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Abstract

In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point $ E_*, $ unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point $ E_*, $ we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point $ E_1, $ the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point $ E_2, $ the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point $ E_* $. The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly.

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