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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

Special Issues: Numerical Linear Algebra for Large-Scale Dynamical Systems

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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1. J. Beddington, C. Free, J. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 225 (1975), 58-60.

2. X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solit. Fract., 32 (2007), 80-94.

3. M. Zhao, L. Zhang, Permanence and chaos in a host-parasitoid model with prolonged diapause for the host, Comm. Nonlinear. Sci. Numer. Simulat., 14 (2009), 4197-4203.

4. J. Yan, C. Li, X. Chen, L. Ren, Dynamic complexities in 2− dimensional discrete-time predatorprey systems with Allee effect in the prey, Discrete Dyn. Nat. Soc., 2106 (2016), 1-14.

5. J. Zhao, Y. Yan, Stability and bifurcation analysis of a discrete predator-prey system with modified Holling-Tanner functional response, Adv. Differ. Equ., 2108 (2018), 402.

6. Q. Fang, X. Li, Complex dynamics of a discrete predator-prey system with a strong Allee effect on the prey and a ratio-dependent functional response, Adv. Differ. Equ., 2108 (2018), 320.

7. F. Kangalgi, S. Kartal, Stability and bifurcation analysis in a host-parasitoid model with Hassell growth function, Adv. Differ. Equ., 2108 (2018), 240.

8. L. Li, J. Shen, Bifurcations and dynamics of a predator-prey model with double Allee effects and time delays, Int. J. Bifurcat. Chaos., 28 (2018), 1-14.

9. G. Stápán, Great delay in a predator-prey model, Nonli. Analy. Theory Meth. Appl., 10 (1986), 913-929.

10. L. Cheng, H. Cao, Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee effect, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 288-302.

11. W. Liu, D. Cai, J. Shi, Dynamic behaviors of a discrete-time predator-prey bioeconomic system, Adv. Differ. Equ., 2018 (2018), 133.

12. X. Liu, Y. Chu, Y. Liu, Bifurcation and chaos in a host-parasitoid model with a lower bound for the host, Adv. Differ. Equ., 2018 (2018), 31.

13. S. M. Sohel Rana, Chaotic dynamics and control of discrete ratio-dependent predator-prey system, Discrete Dyn. Nat. Soc., 2017 (2017), 1-13.

14. P. K. Santra, G. S. Mahapatra, G. R. Phaijoo, Bifurcation and chaos of a discrete predator-prey model with crowley-martin functional response incorporating proportional prey refuge, Discrete Dyn. Nat. Soc., 2020 (2020), 1-18.

15. A. Mareno, L. Q. English, Flip and Neimark-Sacker bifurcations in a coupled logistic map system, Discrete Dyn. Nat. Soc., 2020 (2020), 1-14.

16. A. Q. Khan, Bifurcation analysis of a discrete-time two-species model, Discrete Dyn. Nat. Soc, 2020 (2020), 1-12.

17. W. Znegui, H. Gritli, S. Belghith, Design of an explicit expression of the Poincaré map for the pas-sive dynamic walking of the compass-gait biped model, Chaos Solit. Fract., 130 (2020),109436.

18. C. Çelik, O. Duman, Allee effect in a discrete-time predatorprey system, Chaos Solit. Fract., 40 (2009), 1956-1962.

19. A. Q. Khan, M. Alesemi, M. A. El-Moneam, E. S. A. Elgarib, Bifurcation analysis of a discretetime predator-prey model, Wulfenia, 26 (2019), 23-39.

20. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, New York, Springer-Verlag, . 1983.

21. Y. A. Kuznetsov, Elements of applied bifurcation theorey, 3rd edition, Springer-Verlag, New York, 2004.

22. J. H. E. Cartwright, Nonlinear stiffness Lyapunov exponents and attractor dimension, Phys. Lett. A, 264 (1999), 298-304.

23. J. L. Kaplan, J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Comm. Math. Phys., 67 (1979), 93-108.

24. S. N. Elaydi, An introduction to difference equations, Springer-Verlag, New York, USA, 1996.

25. S. Lynch, Dynamical Systems with Applications Using Mathematica, Birkhäuser, Boston, Mass, USA, 2007.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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