Bifurcations and hybrid control in a 3×3 discrete-time predator-prey model

Abdul Qadeer Khan; Azhar Zafar Kiyani; Imtiaz Ahmad; Abdul Qadeer Khan; Azhar Zafar Kiyani; Imtiaz Ahmad

doi:10.3934/mbe.2020360

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6963-6992. doi: 10.3934/mbe.2020360

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Research article

Bifurcations and hybrid control in a 3×3 discrete-time predator-prey model

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

Received: 12 June 2020 Accepted: 28 September 2020 Published: 16 October 2020

In this paper, we explore the bifurcations and hybrid control in a $3\times3$ discrete-time predator-prey model in the interior of $\mathbb{R}_+^3$. It is proved that $3\times3$ model has four boundary fixed points: $P_{000}(0, 0, 0)$, $P_{0y0}\left(0, \frac{r-1}{r}, 0\right)$, $P_{0yz}\left(0, \frac{d}{f}, \frac{rf-f-dr}{cf}\right)$, $P_{x0z}\left(\frac{d}{e}, 0, \frac{a}{b}\right)$, and the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber}, \frac{br-b-ac}{br}, \frac{a}{b}\right)$ under certain restrictions to the involved parameters. By utilizing method of Linearization, local dynamics along with topological classifications about fixed points have been investigated. Existence of prime period and periodic points of the model are also investigated. Further for $3\times3$ model, we have explored the occurrence of possible bifurcations about each fixed point, that gives more insight about the under consideration model. It is proved that the model cannot undergo any bifurcation about $P_{000}(0, 0, 0)$ and $P_{x0z}\left(\frac{d}{e}, 0, \frac{a}{b}\right)$, but the model undergo P-D and N-S bifurcations respectively about $P_{0y0}\left(0, \frac{r-1}{r}, 0\right)$ and $P_{0yz}\left(0, \frac{d}{f}, \frac{rf-f-dr}{cf}\right)$. For the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber}, \frac{br-b-ac}{br}, \frac{a}{b}\right)$, we have proved the N-S as well as P-D bifurcations by explicit criterion. Further, theoretical results are verified by numerical simulations. We have also presented the bifurcation diagrams and corresponding maximum Lyapunov exponents for the $3\times3$ model. The computation of the maximum Lyapunov exponents ratify the appearance of chaotic behavior in the under consideration model. Finally, the hybrid control strategy is applied to control N-S as well as P-D bifurcations in the discrete-time model.
- predator-prey model,
- bifurcation and hybrid control,
- center manifold theorem,
- numerical simulation
Citation: Abdul Qadeer Khan, Azhar Zafar Kiyani, Imtiaz Ahmad. Bifurcations and hybrid control in a 3×3 discrete-time predator-prey model[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6963-6992. doi: 10.3934/mbe.2020360

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Abstract

In this paper, we explore the bifurcations and hybrid control in a $3\times3$ discrete-time predator-prey model in the interior of $\mathbb{R}_+^3$. It is proved that $3\times3$ model has four boundary fixed points: $P_{000}(0, 0, 0)$, $P_{0y0}\left(0, \frac{r-1}{r}, 0\right)$, $P_{0yz}\left(0, \frac{d}{f}, \frac{rf-f-dr}{cf}\right)$, $P_{x0z}\left(\frac{d}{e}, 0, \frac{a}{b}\right)$, and the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber}, \frac{br-b-ac}{br}, \frac{a}{b}\right)$ under certain restrictions to the involved parameters. By utilizing method of Linearization, local dynamics along with topological classifications about fixed points have been investigated. Existence of prime period and periodic points of the model are also investigated. Further for $3\times3$ model, we have explored the occurrence of possible bifurcations about each fixed point, that gives more insight about the under consideration model. It is proved that the model cannot undergo any bifurcation about $P_{000}(0, 0, 0)$ and $P_{x0z}\left(\frac{d}{e}, 0, \frac{a}{b}\right)$, but the model undergo P-D and N-S bifurcations respectively about $P_{0y0}\left(0, \frac{r-1}{r}, 0\right)$ and $P_{0yz}\left(0, \frac{d}{f}, \frac{rf-f-dr}{cf}\right)$. For the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber}, \frac{br-b-ac}{br}, \frac{a}{b}\right)$, we have proved the N-S as well as P-D bifurcations by explicit criterion. Further, theoretical results are verified by numerical simulations. We have also presented the bifurcation diagrams and corresponding maximum Lyapunov exponents for the $3\times3$ model. The computation of the maximum Lyapunov exponents ratify the appearance of chaotic behavior in the under consideration model. Finally, the hybrid control strategy is applied to control N-S as well as P-D bifurcations in the discrete-time model.

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