Discrete Leslie's model with bifurcations and control

A. Q. Khan; Ibraheem M. Alsulami; A. Q. Khan; Ibraheem M. Alsulami

doi:10.3934/math.20231146

AIMS Mathematics

2023, Volume 8, Issue 10: 22483-22506. doi: 10.3934/math.20231146

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Discrete Leslie's model with bifurcations and control

A. Q. Khan ^{1
,
,},
Ibraheem M. Alsulami ²

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

Received: 02 May 2023 Revised: 24 June 2023 Accepted: 29 June 2023 Published: 14 July 2023
MSC : 70K50, 92D25, 39A10

We explored a local stability analysis at fixed points, bifurcations, and a control in a discrete Leslie's prey-predator model in the interior of $ \mathbb{R}_+^2 $. More specially, it is examined that for all parameters, Leslie's model has boundary and interior equilibria, and the local stability is studied by the linear stability theory at equilibrium. Additionally, the model does not undergo a flip bifurcation at the boundary fixed point, though a Neimark-Sacker bifurcation exists at the interior fixed point, and no other bifurcation exists at this point. Furthermore, the Neimark-Sacker bifurcation is controlled by a hybrid control strategy. Finally, numerical simulations that validate the obtained results are given.
- discrete Leslie's model,
- stability,
- hybrid control,
- bifurcations
Citation: A. Q. Khan, Ibraheem M. Alsulami. Discrete Leslie's model with bifurcations and control[J]. AIMS Mathematics, 2023, 8(10): 22483-22506. doi: 10.3934/math.20231146

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Abstract

We explored a local stability analysis at fixed points, bifurcations, and a control in a discrete Leslie's prey-predator model in the interior of $ \mathbb{R}_+^2 $. More specially, it is examined that for all parameters, Leslie's model has boundary and interior equilibria, and the local stability is studied by the linear stability theory at equilibrium. Additionally, the model does not undergo a flip bifurcation at the boundary fixed point, though a Neimark-Sacker bifurcation exists at the interior fixed point, and no other bifurcation exists at this point. Furthermore, the Neimark-Sacker bifurcation is controlled by a hybrid control strategy. Finally, numerical simulations that validate the obtained results are given.

References

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