Dynamics of a Prey-Predator System with Harvesting Effect on Prey

Abstract

This article is about the dynamic behavior of a prey-predator model exposed to the harvesting effect on prey. Firstly, the existence and stability of the fixed points of the model are obtained, and then the presence and direction of Neimark-Sacker bifurcation is examined. By using the bifurcation theory, we show that the system undergoes Neimark-Sacker bifurcation. The hybrid control strategy is applied to control the chaos caused by the Neimark-Sacker bifurcation. In addition, some numerical simulations are given to verify the theoretical results obtained.

Keywords

• Prey-Predator Model
• Fixed point
• Stability
• Harvesting
• Bifurcation
• Chaos control

References

1. Clark, CW., 1985 Bioeconomic modelling and fisheries management. Wiley: New York.
2. Clark CW., 1990 Mathematical bioeconomics: The optimal management of renewable resource. 2nd edition. John Wiley and Sons: New York.
3. Danca, M.F., M. Feckan, N. Kuznetsov and G. Chen, 2019 Rich dynamics and anticontrol of extinction in a prey–predator system. Nonlinear Dynamics 98:1421–1445.
4. Din Q., Ö.A. Gümüş, H. Khalil, 2017 Neimark–Sacker bifurcation and chaotic behaviour of a modified Host–Parasitoid model. Zeitschrift für Naturforschung A. 72: 25–37.
5. Elaydi S.N., 1996 An Introduction to Difference Equations Springer-Verlag, New York, NY, USA.
6. Elsadany, A.A., H.A. El-Metwally, E. M. Elabbasy and H.N. Agiza, 2012 Chaos and bifurcation of a nonlinear discrete prey–predator system. Computational Ecology and Software 2: 169–180.
7. Gümüş Ö.A., 2015 Dynamical consequences and stability analysis of a new host-parasitoid model, General Mathematics Notes 27: 9-16.
8. Gümüş, Ö.A., 2014 Global and local stability analysis in a nonlinear discrete time population model, Advances in Difference Equations 299: 1687–1847.

9. Gümüş Ö.A., 2020 Neimark-Sacker bifurcation and stability a prey-predator system. Miskolc Mathematical Notes 21: 873-875.
10. [Gümüş Ö.A., Q. Cui, A.G.M Selvam and A. Vianny, 2022 Global stability and bifurcation analysis of a discrete time SIR epidemic model. Miskolc Mathematical Notes 23: 193-210.
11. Gümüş, Ö.A. and M. Feckan, 2021 Stability, Neimark-Sacker bifurcation and chaos control for a prey-predator system with harvesting effect on predator. Miskolc Mathematical Notes 22: 663-679.
12. Gümüş, Ö.A. and F. Kangalgil, 2015 Allee effect and stability in a discrete-time host-parasitoid model. Journal of Advanced research in Applied Mathematics 7: 94-99.
13. Gümüş Ö.A. and H. Köse, 2012 Allee effect on a new delay population model and stability analysis. Journal of Pure and Applied Mathematics: Advances and Applications, 7: 21-31.
14. Gümüş, Ö.A., A.G.M. Selvam R. and Dhineshbabu, 2022 Bifurcation analysis and chaos control of the population model with harvest. International Journal of Nonlinear Analysis and Applications 13: 115-125.
15. Gümüş, Ö.A., A.G.M. Selvam and R. Janagaraj, 2020 Stability of Modified Host-Parasitoid Model with Allee Effect, Applications and Applied Mathematics: An International Journal 15: 1032-1045.
16. Gümüş, Ö.A., A.G.M. Selvam and D. A. Vianny, 2019 Bifurcation and stability analysis of a discrete time SIR epidemic model with vaccination, International Journal of Analysis and Applications 17: 809-820.
17. Gümüş Ö.A., A.G.M. Selvam, and D. Vighnes, 2022 The effect of Allee factor on a nonlinear delayed population model with harvesting. Journal of Science and Arts 22: 159-176.
18. Kapçak S., 2018 Discrete Dynamical Systems with SageMath. The Electronic Journal of Mathematics & Technology 12: 292-308.
19. Kuznetsov, Y.A., 1998 Elements of Applied Bifurcation Theory. Springer-Verlag, New York.
20. Liu X. and D. Xiao, 2007 Complex dynamic behaviors of a discrete-time predator-prey system. Chaos Solitons and Fractals, 32: 80-94.
21. Liu, C., Q. Zhang, Y. Zhang and X. Duan, 2008 Bifurcation and control in a differential-algebraic harvested prey–predator model with stage structure for predator, International Journal of Bifurcation and Chaos, 18, 3159–3168.
22. Lotka A.J., 1925 Elements of physical biology,Williams and Wilkins Company, Baltimore.
23. Madhusudanan V, K. Anitha, S. Vijaya, M. Gunasekaran, 2014 Complex effects in discrete time prey predator model with harvesting on prey, The International Journal of Engineering and Science (IJES), 3:1-5.
24. Merdan H. and Ö.A. Gümüş, 2012 Stability analysis of a general discrete-time population model involving delay and Allee effects. Applied Mathematics and Computation 219:1821–1832.
25. Merdan H., Ö.A. Gümüş and G. Karahisarlı 2018 Global stability analysis of a general scalar difference equation. Journal of Discontinuity, Nonlinearity and Complexity 7: 225–232.
26. Murray, JD., 1993 Mathematical Biology, Springer, New York.
27. Paula P., T. K. Kar., E. Das., 2021 Reactivity in prey–predator models at equilibrium under selective harvesting efforts. The European Physical Journal Plus 136-510.
28. Peng, G., Y. Jiang and C. Li, 2009 Bifurcations of a Hollingtype II predator–prey system with constant rate harvesting. International Journal of Bifurcation and Chaos, 19: 2499-2514.
29. Rana S.M., 2015 Bifurcation and Complex Dynamics of a Discrete-Time Predator-Prey System. Computational Ecology and Software, 5: 187-200.
30. Robinson, C., 1999 Dynamical Systems, Stability, Symbolic Dynamics and Chaos. 2nd edn. CRC Press, Boca Raton.
31. Selvam A.G.M, R. Dhineshbabu, Ö.A. Gümüş, 2020 Complex Dynamics Behaviors of a Discrete Prey - Predator - Scavenger Model with Fractional Order. 17: 2136-2146.
32. Selvam A.G.M., R. Dhineshbabu, Ö.A. Gümüş, 2020 Stability and Neimark-.Sacker bifurcation for a discrete system of one - scroll chaotic attractor with fractional order. Journal of Physics: Conference Series, IOP Publishing, 1597: 012009.
33. Singh, A., A. A. Elsadany and A. Elsonbaty 2019 Complex dynamics of a discrete fractional-order Leslie–Gower predator– prey model. Mathematical Methods in the Applied Sciences 42: 3992–4007.
34. Volterra V., 1927 Variations and fluctuations in the numbers of coexisting animal species. Lecture notes in Biomathematics 65-273.
35. Wiggins S., 2003 Introduction to Applied Nonlinear Dynamical System and Chaos, 2, Springer-Verlag, New York, USA.
36. Yuan L.G, Q.G. Yang 2015 Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system. Applied Mathematical Modelling 39: 2345–2362.