Qualitative Analysis of a Nicholson-Bailey Model in Patchy Environment

Year 2023, Volume: 6 Issue: 1, 30 - 42, 28.03.2023

Rizwan Ahmed , Shehraz Akhtar

https://doi.org/10.32323/ujma.1167907

Abstract

We studied a host-parasite model qualitatively. The host-parasitoid model is obtained by modifying the Nicholson-Bailey model so that the number of hosts that parasitoids can't attack is fixed. Topological classification of equilibria is achieved with the implementation of linearization. Furthermore, Neimark-Sacker bifurcation is explored using the bifurcation theory of normal forms at interior steady-state. The bifurcation in the model is controlled by implementing two control strategies. The theoretical studies are backed up by numerical simulations, which show the conclusions and their importance.

Keywords

Bifurcation, Chaos control, Nicholson-Bailey model, Stability

References

• [1] V. A. Bailey, A. J. Nicholson, The balance of animal populations, I Proc. Zool. Soc. Lond., 3 (1935), 551-598.
• [2] M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Princeton (NJ) Princeton University Press, 1978.
• [3] R. Wongsathan, Numerical simulation of coexist between host and parasitoid for improved modification of Nicholson-Bailey model, IEEE, (2009), 1002-1006.
• [4] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, London, 1993.
• [5] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic, Dordrecht, 2003.
• [6] L. Edelstein-Keshet, Mathematical models in Biology, McGraw-Hill, British Columbia, 1988.
• [7] S. N. Elaydi, (2nd edn) Discrete chaos with applications in science and engineering, Champan and Hall/CRC, Texas, 2008.
• [8] R. Ahmed, M. S. Yazdani, Complex dynamics of a discrete-time model with prey refuge and Holling type-II functional response, J. Math. Comput. Sci., 12 (2022), Article ID 113.
• [9] S. Akhtar, R. Ahmed, M. Batool, N. A. Shah, J. D. Chung, Stability, bifurcation and chaos control of a discretized Leslie prey-predator model, Chaos, Solitons & Fractals, 152 (2021), 111345.
• [10] A. Q. Khan, E. Abdullah, T. F. ˙Ibrahim, Supercritical Neimark-Sacker bifurcation and hybrid control in a discrete-time glycolytic oscillator model, Mathematical Problems in Engineering, 2020(3) (2020), 1-15.
• [11] M. R. S. Kulenovic, D. T. McArdle, Global dynamics of Leslie-Gower competitive systems in the plane, Mathematics, 7(1) (2019), 76.
• [12] P. A. Naik, Z. Eskandari, H. E. Shahraki, Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model, Mathematical Modelling and Numerical Simulation with Applications, 1(2) (2021), 95-101.
• [13] P. A. Naik, Z. Eskandari, M. Yavuz, J. Zu, Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect. Journal of Computational and Applied Mathematics, 413 (2022), 114401.
• [14] M. N. Qureshi, A. Q. Khan, Q. Din, Asymptotic behavior of a Nicholson-Bailey model, Adv. Differ. Equ., 2014 (2014), Article number: 62.
• [15] U¨ . Ufuktepe, S. Kapc¸ak, Stability analysis of a host parasite model, Adv. Differ. Equ., 2013 (2013), Article number: 79.
• [16] Q. Din, Global behavior of a host-parasitoid model under the constant refuge effect, Appl. Math. Model., 40 (2016), 2815-2826.
• [17] X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32 (2007), 80-94.
• [18] Q. Din, Global stability and Neimark-Sacker bifurcation of a host-parasitoid model, Int. J. Syst. Sci., 48 (2017), 1194-1202.
• [19] Z. He, X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Anal. RWA, 12 (2011), 403-417.
• [20] B. Li, Z. He, Bifurcations and chaos in a two-dimensional discrete Hindmarsh-Rose model, Nonlinear Dyn, 76 (2014), 697-715.
• [21] Z. Jing, J. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos, Solitons & Fractals, 27 (2006), 259-277.
• [22] L. G. Yuan, Q. G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system, Appl. Math. Model., 39 (2015), 2345-2362.
• [23] H. N. Agiza, E. M. ELabbasy, H. El-Metwally, A. A. Elsadany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. RWA, 10 (2009), 116-129.
• [24] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1997.
• [25] A. L. Fradkov, R. J. Evans, Control of chaos: Methods and applications in engineering, Annu. Rev. Control., 29 (2005), 33-56.
• [26] S. Lynch, Dynamical Systems with Applications Using Mathematica, Birkh¨auser, Boston, 2007.
• [27] G. Chen, X. Dong, From Chaos to Order: Perspectives, Methodologies, and Applications, World Scientific, Singapore, 1998.
• [28] Q. Din, O. A. G¨um¨us¸, H. Khalil, Neimark-sacker bifurcation and chaotic behaviour of a modified host–parasitoid model, Zeitschrift f¨ur Naturforschung A, 72(1) (2017), 25-37.
• [29] O. A. G¨um¨us¸, M. Feckan, Stability, Neimark-Sacker bifurcation and chaos control for a prey-predator system with harvesting effect on predator, Miskolc Mathematical Notes, 22(2) (2021), 663-679.
• [30] L. Fei, X. Chen, B. Han, Bifurcation analysis and hybrid control of a discrete-time predator–prey model, Journal of Difference Equations and Applications, 27(1) (2021), 102-117.
• [31] O. A. G¨um¨us¸, A. G. M. Selvam, R. Janagaraj, Neimark-Sacker bifurcation and control of chaotic behavior in a discrete-time plant-herbivore system, Journal of Science and Arts, 22(3) (2022), 549-562.
• [32] A. Q. Khan, T. Khalique, Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka-Volterra model, Mathematical Methods in the Applied Sciences, 43(9) (2020), 5887-5904.
• [33] Q. Zhou, F. Chen, S. Lin, Complex Dynamics Analysis of a Discrete Amensalism System with a Cover for the First Species, Axioms, 11(8)(2022), 365.