Discrete-time predator-prey model with flip bifurcation and chaos control

A. Q. Khan; I. Ahmad; H. S. Alayachi; M. S. M. Noorani; A. Khaliq; A. Q. Khan; I. Ahmad; H. S. Alayachi; M. S. M. Noorani; A. Khaliq

doi:10.3934/mbe.2020317

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5944-5960. doi: 10.3934/mbe.2020317

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Research article

Discrete-time predator-prey model with flip bifurcation and chaos control

1.
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan
2.
Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur-10250 (AJK), Pakistan
3.
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia
4.
Department of Mathematics, Riphah International University, Lahore Campus, Lahore, Pakistan

Received: 24 June 2020 Accepted: 17 August 2020 Published: 09 September 2020

We explore the local dynamics, flip bifurcation, chaos control and existence of periodic point of the predator-prey model with Allee effect on the prey population in the interior of $\mathbb{R}^*{_+^2}$. Nu-merical simulations not only exhibit our results with the theoretical analysis but also show the complex dynamical behaviors, such as the period-2, 8, 11, 17, 20 and 22 orbits. Further, maximum Lyapunov exponents as well as fractal dimensions are also computed numerically to show the presence of chaotic behavior in the model under consideration.
- chaos control,
- Discrete-time model,
- flip bifurcation
Citation: A. Q. Khan, I. Ahmad, H. S. Alayachi, M. S. M. Noorani, A. Khaliq. Discrete-time predator-prey model with flip bifurcation and chaos control[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5944-5960. doi: 10.3934/mbe.2020317

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Abstract

We explore the local dynamics, flip bifurcation, chaos control and existence of periodic point of the predator-prey model with Allee effect on the prey population in the interior of $\mathbb{R}^*{_+^2}$. Nu-merical simulations not only exhibit our results with the theoretical analysis but also show the complex dynamical behaviors, such as the period-2, 8, 11, 17, 20 and 22 orbits. Further, maximum Lyapunov exponents as well as fractal dimensions are also computed numerically to show the presence of chaotic behavior in the model under consideration.

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