Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

Abstract

This article is about a discrete-time predator-prey model obtained by the forward Euler method. The stability of the fixed point of the model and the existence conditions of the Neimark-Sacker bifurcation are investigated. In addition, the direction of the Neimark-Sacker bifurcation is given. Moreover, OGY control method is to implement to control chaos caused by the Neimark-Sacker bifurcation. Finally, Neimark-Sacker bifurcation, chaos control strategy, and asymptotic stability of the only positive fixed point are verified with the help of numerical simulations. The existence of chaotic behavior in the model is confirmed by computing of the maximum Lyapunov exponents.

Keywords

• Predator-Prey Model
• Fixed Point
• Stability
• Neimark-Sacker Bifurcation
• Chaotic Behavior
• OGY feedback control method

References

1. [1] A. Atabaigi, Multiple bifurcations and dynamics of a discrete-time predator-prey sys- tem with group defense and non-monotonic functional response, Differ. Equ. Dyn. Syst. 28, 107-132, 2020.
2. [2] C. Celik, The stability and Hopf bifurcation for a predator-prey system with time delay, Chaos Solitons Fractals, 37, 87–99, 2008.
3. [3] L. Cheng and H. Cao, Bifurcation analysis of a discrete-time ratio-dependent predator- prey model with Allee Effect, Commun. Nonlinear Sci. Numer. Simul. 38, 288–302, 2016.
4. [4] M. Danca, S. Condreanu and B. Bako, Detailed analysis of a nonlinear prey-predator model, J. Biol. Phys. 23 (1), 11–20, 1997.
5. [5] Q. Din, A Novel chaos control strategy for discrete-time brusselator models, J. Math. Chem. 56, 3045–3075, 2018.
6. [6] Q. Din, Bifurcation analysis and chaos control in discrete-time glycolysis Models, J. Math. Chem. 56 (3), 904–931, 2018.
7. [7] Q. Din, Bifurcation analysis and chaos control in a second-order rational difference equation, Int. J. Nonlinear Sci. Numer. Simul. 19 (1), 53–68, 2018.
8. [8] Q. Din, Stability, bifurcation analysis and chaos control for a predator-prey system, J. Vib. Control 25 (3), 612–626, 2019.

9. [9] Q. Din and M. Hussain, Controlling chaos and Neimark-Sacker bifurcation in a host- parasitoid model, Asian J. Control, 21 (4), 1–14, 2019.
10. [10] W. Du, J. Zhang, S. Qin and J.Yu, Bifurcation analysis in a discrete SIR epidemic model with the saturated contact rate and vertical transmission, J. Nonlinear Sci. Appl. 9, 4976–4989, 2016.
11. [11] E.M. Elabbasy, A.A. Elsadany and Y.Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput. 228, 184–194, 2014.
12. [12] S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, NY, USA, 1996.
13. [13] A. Gkana and L. Zachilas, Incorporating prey refuge in a prey-predator model with a Holling Type I functional response: random dynamics and population outbreaks, J. Biol. Phys. 39 (4), 587–606, 2013.
14. [14] A. Gkana and L. Zachilas, Non-overlapping generation species: Complex Prey- Predator Interactions, Int. J. Nonlinear Sci. Numer. Simul. 16 (5), 207–219, 2015.
15. [15] Z. He and X. Lai, Bifurcation and Chaotic Behaviour of a Discete Time Predator-Prey System, Nonlinear Anal. Real World Appl. 12 (1), 403–417, 2011.
16. [16] Z.M. He and B.O. Li, Complex dynamic behavior of a discrete-time predator-prey system of Holling-III Type, Adv. Difference Equ. 2014, Art. No. 180, 2014.
17. [17] Z.Hu, Z.Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator- prey model with nonmonotonic functional response, Nonlinear Anal. RealWorld Appl. 12, 2356–2377, 2011.
18. [18] Z. Jing and Y. Jianping, Bifurcation and chaos in discrete-time predatorprey system, Chaos Solitons Fractals 27 (1), 259–277, 2006.
19. [19] S. Kartal, Dynamics of a plant-herbivore model with differential-difference equations, Cogent Math. 3 (1), 1136198, 2016.
20. [20] S. Kartal, Flip and Neimark–Sacker bifurcation in a differential equation with piece- wise constant arguments model, J. Difference Equ. Appl. 23, 763–778, 2017.
21. [21] S.Kartal and F. Gurcan, Global behaviour of a predator–prey like model with piecewise constant arguments, J. Biol. Dyn. 9 (1), 157–171, 2015.
22. [22] A.Q. Khan, Neimark-Sacker bifurcation of a two-dimensional discrete-time predator- prey model, SpringerPlus 5 (1), Art. No. 126, 2016.
23. [23] A.Q. Khan, Stability and Neimark-Sacker bifurcation of a ratio-dependence predator- prey model, Math. Methods Appl. Sci. 40, 4109–4119, 2017.
24. [24] Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, NY, USA, 2nd edition, 1998.
25. [25] P.H. Leslie and J.C Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47, 219–234, 1960.
26. [26] S. Li and W. Zhang, Bifurcations of a discrete prey-predator model with Holling type II functional response, Discrete Contin. Dyn. Syst. Ser. B. 14, 159–176, 2010.
27. [27] J. Liu, P. Baoyang and Z. .Tailei, Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence, Appl. Math. Lett. 39, 60–66, 2015.
28. [28] X. Liu and X. Dongmei, Complex dynamic behaviors of a discrete-time predatorprey system, Chaos Solitons Fractals, 32 (1), 80–94, 2007.
29. [29] D. Lv, W. Zhang and Y. Tang, Bifurcation analysis of a ratio-dependent predator-prey system with multipla delays, J. Nonlinear Sci. Appl. 9, 3479–3490, 2016.
30. [30] R.M. May, Simple mathematical models with very complicated dynamics, Nature, 261, 459–467, 1976.
31. [31] E. Ott, C. Grebogi and J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (11), 1196–1199, 1990.
32. [32] P.J. Pal and P.K. Mandal, Bifurcation Analysis of a Modified Leslie-Gower Predator- Prey Model with Beddington-De Angelis Functional Response and Strong Allee Effect, Math. Comput. Simulation 97, 123–146, 2014.
33. [33] S.M Rana and U. Kulsum, Bifurcation analysis and chaos control in a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response, Discrete Dyn. Nat. Soc. 2017, Art. No. 9705985, 2017.
34. [34] H. Singh, J. Dhar and H.S. Bhatti, Discrete-time bifurcation behavior of a prey- predator system with generalized predator, Adv. Difference Equ. 2015, Art. No. 206, 2015.
35. [35] G. Sucu, Bir Ayrık Zamanlı Av-Avcı Modelinin Kararlılık ve Çatallanma, TOBB Ekonomi ve Teknoloji Üniversitesi Fen Bilimler Enstitüsü, Yüksek lisans Tezi, Ankara, 2016.
36. [36] S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, Springer-Verlag, New York, NY, USA, 2003.
37. [37] L. Zhang and L. Zou, Bifurcations and control in a discrete predator-prey model with strong Allee effect, Int. J. Bifur. Chaos, 28 (5), 1850062, 2018.
38. [38] J. Zhang, T. Deng, Y. Chu, S. Qin, W. Du, and H. Luo, Stability and bifurcation analysis of a discrete predator-prey model with Holling type III functional response, J. Nonlinear Sci. Appl. 9, 6228–6243, 2016.
39. [39] S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theoret. Population Biol. 67, 23–31, 2005.