Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model
Tóm tắt
A discrete-time Holling-Tanner model with ratio-dependent functional response is examined. We show that the system experiences a flip bifurcation and Neimark-Sacker bifurcation or both together at positive fixed point in the interior of $\mathbb {R}^{2}_{+}$ when one of the model parameter crosses its threshold value. We concentrate our attention to determine the existence conditions and direction of bifurcations via center manifold theory. To validate analytical results, numerical simulations are employed which include bifurcations, phase portraits, stable orbits, invariant closed circle, and attracting chaotic sets. In addition, the existence of chaos in the system is justified numerically by the sign of maximum Lyapunov exponents and fractal dimension. Finally, we control chaotic trajectories exists in the system by feedback control strategy.
Tài liệu tham khảo
Li, Y., Xiao, D.: Bifurcations of a predator-prey system of Holling and Leslie types. Chaos Solit. Fract. 34(2), 606–620 (2007). https://doi.org/10.1016/j.chaos.2006.03.068.
Hsu, S. B., Hwang, T. W.: Global stability for a class of predator–prey systems. SIAM J. Appl. Math. 55, 763–783 (1995).
Gasull, A., Kooij, R. E., Torregrosa, J.: Limit cycles in the Holling-Tanner model. Publ. Mat. 41, 149–167 (1997).
Liang, Z., Pan, H.: Qualitative analysis of a ratio-dependent Holling-Tanner model. J. Math. Anal. Appl. 334, 954–964 (2007).
He, Z. M., Lai, X.: Bifurcation and chaotic behavior of a discrete-time predator-prey system. Nonlinear Anal. Real World Appl. 12, 403–417 (2011).
He, Z. M., Li, B.: Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type. Adv. Differ. Equ. 180 (2014).
Rana, S. M. S.: Bifurcation and complex dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response. Adv. Differ. Equ. 345 (2015). https://doi.org/10.1186/s13662-015-0680-7.
Rana, S. M. S.: Chaotic dynamics and control of discrete ratio-dependent predator-prey system. Discret. Dyn. Nat. Soc., 1–13 (2017). https://doi.org/10.1155/2017/4537450.
Rana, S. M. S., Kulsum, U.: Bifurcation analysis and chaos control in a discrete-time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response. Discret. Dyn. Nat. Soc. (2017). https://doi.org/10.1155/2017/9705985.
Rana, S. M. S.: Bifurcations and chaos control in a discrete-time predator-prey system of Leslie type. J. Appl. Anal. Comput. 9(1), 31–44 (2019). https://doi.org/10.11948/2019.31.
Tan, W., Gao, J., Fan, W.: Bifurcation Analysis and Chaos Control in a Discrete Epidemic System. Discret. Dyn. Nat. Soc. (2015). https://doi.org/10.1155/2015/974868.
Zhao, M., Xuan, Z., Li, C.: Dynamics of a discrete-time predator-prey system. Adv. Differ. Equ. 191 (2016). https://doi.org/10.1186/s13662-016-0903-6.
Zhao, M., Li, C., Wang, J.: Complex dynamic behaviors of a discrete-time predator-prey system. J. Appl. Anal. Comput. 7(2), 478–500 (2017). https://doi.org/10.11948/2017030.
Liu, W., Cai, D.: Bifurcation, chaos analysis and control in a discrete-time predator-prey system. Adv. Differ. Equ. 11 (2019).
Kangalgil, F.: Neimark-Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey. Adv. Differ. Equ. 92 (2019).
Hu, D. P., Cao, H. J.: Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type. Commun. Nonlinear Sci. Numer. Simulat. 22, 702–715 (2015).
Cao, H., Yue, Z., Zhou, Y.: The stability and bifurcation analysis of a discrete Holling-Tanner model. Adv. Differ. Equ. 330 (2013).
Zhao, J., Yan, Y.: Stability and bifurcation analysis of a discrete predator-prey system with modified Holling-Tanner functional response. Adv. Differ. Equ. 402 (2018).
Kuzenetsov, Y. A.: Elements of Applied Bifurcation Theory, 2nd Ed.Springer-Verlag (1998).
Elaydi, S. N.: An Introduction to Difference Equations. Springer-Verlag, New York (2005).
Cartwright, J. H. E.: Nonlinear stiffness Lyapunov exponents and attractor dimension. Phys. Lett. A. 264, 298–304 (1999).
Kaplan, J. L., Yorke, Y. A.: A regime observed in a fluid flow model of Lorenz. Commun. Math. Phys. 67, 93–108 (1979).
Hastings, A., Hom, C. L., Ellner, S., Turchin, P., Godfray, H. C. J.: Chaos in ecology: Is mother nature a strange attractor?. Ann. Rev. Ecol. Evol. Syst. 24, 1–33 (1993).
Nicholson, A. J.: The self-adjustment of populations to change. Cold Spring Harb. Symp. Quant. Biol. 22, 153–173 (1957).