Dynamics study of nonlinear discrete predator-prey system with Michaelis-Menten type harvesting

Xiaoling Han; Xiongxiong Du; Xiaoling Han; Xiongxiong Du

doi:10.3934/mbe.2023755

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 16939-16961. doi: 10.3934/mbe.2023755

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Research article

Dynamics study of nonlinear discrete predator-prey system with Michaelis-Menten type harvesting

Xiaoling Han ^,,
Xiongxiong Du

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received: 09 May 2023 Revised: 16 July 2023 Accepted: 07 August 2023 Published: 25 August 2023

In this paper, we study a discrete predator-prey system with Michaelis-Menten type harvesting. First, the equilibrium points number, local stability and boundedness of the system are discussed. Second, using the bifurcation theory and the center manifold theorem, the bifurcation conditions for the system to go through flip bifurcation and Neimark-Sacker bifurcation at the interior equilibrium point are obtained. A feedback control strategy is used to control chaos in the system, and an optimal harvesting strategy is introduced to obtain the optimal value of the harvesting coefficient. Finally, the numerical simulation not only shows the complex dynamic behavior, but also verifies the correctness of our theoretical analysis. In addition, the results show that the system causes nonlinear behaviors such as periodic orbits, invariant rings, chaotic attractors, and periodic windows by bifurcation.
- discrete system,
- stability,
- flip bifurcation,
- Neimark-Sacker bifurcation,
- optimal harvesting
Citation: Xiaoling Han, Xiongxiong Du. Dynamics study of nonlinear discrete predator-prey system with Michaelis-Menten type harvesting[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 16939-16961. doi: 10.3934/mbe.2023755

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Abstract

In this paper, we study a discrete predator-prey system with Michaelis-Menten type harvesting. First, the equilibrium points number, local stability and boundedness of the system are discussed. Second, using the bifurcation theory and the center manifold theorem, the bifurcation conditions for the system to go through flip bifurcation and Neimark-Sacker bifurcation at the interior equilibrium point are obtained. A feedback control strategy is used to control chaos in the system, and an optimal harvesting strategy is introduced to obtain the optimal value of the harvesting coefficient. Finally, the numerical simulation not only shows the complex dynamic behavior, but also verifies the correctness of our theoretical analysis. In addition, the results show that the system causes nonlinear behaviors such as periodic orbits, invariant rings, chaotic attractors, and periodic windows by bifurcation.

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