Controlling the chaos and bifurcations of a discrete prey-predator model

A. Q. Khan; Ibraheem M. Alsulami; S. K. A. Hamdani; A. Q. Khan; Ibraheem M. Alsulami; S. K. A. Hamdani

doi:10.3934/math.2024087

AIMS Mathematics

2024, Volume 9, Issue 1: 1783-1818. doi: 10.3934/math.2024087

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Research article

Controlling the chaos and bifurcations of a discrete prey-predator model

1.
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan
2.
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia

Received: 18 August 2023 Revised: 27 November 2023 Accepted: 04 December 2023 Published: 14 December 2023
MSC : 40A05, 70K50, 92D25

In this paper, we explore the existence of fixed points, local dynamics at fixed points, bifurcations and chaos of a discrete prey-predator fishery model with harvesting. More specifically, it is proved that, for all involved parameters, the model has trivial fixed point, but it has semitrivial and interior fixed points under definite parametric condition(s). We study the local behavior at fixed points by applying the theory of linear stability. Furthermore, it is shown that flip bifurcation does not occur at semitrivial and trivial fixed points, but that the model undergoes Neimark-Sacker bifurcation at interior fixed point. It is also proved that, at interior fixed point, the model undergoes the flip bifurcation. By using a feedback control strategy, the chaos control is also examined. Finally, to illustrate the theoretical findings, detailed numerical simulations are provided.
- chaos,
- discrete fishery model,
- flip bifurcation,
- fish harvesting,
- numerical simulation,
- Neimark-Sacker bifurcation
Citation: A. Q. Khan, Ibraheem M. Alsulami, S. K. A. Hamdani. Controlling the chaos and bifurcations of a discrete prey-predator model[J]. AIMS Mathematics, 2024, 9(1): 1783-1818. doi: 10.3934/math.2024087

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Abstract

In this paper, we explore the existence of fixed points, local dynamics at fixed points, bifurcations and chaos of a discrete prey-predator fishery model with harvesting. More specifically, it is proved that, for all involved parameters, the model has trivial fixed point, but it has semitrivial and interior fixed points under definite parametric condition(s). We study the local behavior at fixed points by applying the theory of linear stability. Furthermore, it is shown that flip bifurcation does not occur at semitrivial and trivial fixed points, but that the model undergoes Neimark-Sacker bifurcation at interior fixed point. It is also proved that, at interior fixed point, the model undergoes the flip bifurcation. By using a feedback control strategy, the chaos control is also examined. Finally, to illustrate the theoretical findings, detailed numerical simulations are provided.

References

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