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Chaotic dynamics and control of discrete ratio-dependent predator-prey system. (English) Zbl 1370.92146
Summary: This study examines the complexity of a discrete-time predator-prey system with ratio-dependent functional response. We establish algebraically the conditions for existence of fixed points and their stability. We show that under some parametric conditions the system passes through a bifurcation (flip or Neimark-Sacker). Numerical simulations are presented not only to justify theoretical results but also to exhibit new complex behaviors which include phase portraits, orbits of periods 9, 19, and 26, invariant closed circle, and attracting chaotic sets. Moreover, we measure numerically the Lyapunov exponents and fractal dimension to confirm the chaotic dynamics of the system. Finally, a state feedback control method is applied to control chaos which exists in the system.

MSC:
92D25 Population dynamics (general)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37N25 Dynamical systems in biology
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[1] Lotka, A. J., Elements of Mathematical Biology, (1925), Baltimore, Md, USA: Williams and Wilkins, Baltimore, Md, USA · JFM 51.0416.06
[2] Volterra, V., Variazioni e fluttuazioni del numero diâindividui in specie animali conviventi, Mem. R. Accad. Naz. Dei Lincei, 2, (1926) · JFM 52.0450.06
[3] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, Journal of Theoretical Biology, 139, 3, 311-326, (1989)
[4] Gutierrez, A. P., The physiological basis of ratio-dependent predatorprey theory: a metabolic pool model of Nicholsonâs blowflies as an example, Ecology, 73, 1552-1563, (1992)
[5] Arditi, R.; Ginzburg, L. R.; Akcakaya, H. R., Variation in plankton densities among lakes: a case for ratio-dependent predation models, The American Naturalist, 138, 5, 1287-1296, (1991)
[6] Xiao, D.; Li, W.; Han, M., Dynamics in a ratio-dependent predator-prey model with predator harvesting, Journal of Mathematical Analysis and Applications, 324, 1, 14-29, (2006) · Zbl 1122.34035
[7] Hsu, S.-B.; Hwang, T.-W.; Kuang, Y., A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181, 1, 55-83, (2003) · Zbl 1036.92033
[8] Hsu, S.-B.; Hwang, T.-W.; Kuang, Y., Global analysis of the Michaelis-MENten-type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42, 6, 489-506, (2001) · Zbl 0984.92035
[9] Xiao, D.; Ruan, S., Global dynamics of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 43, 3, 268-290, (2001) · Zbl 1007.34031
[10] He, Z.; Lai, X., Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 12, 1, 403-417, (2011) · Zbl 1202.93038
[11] He, Z.; Li, B. o., Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type, Advances in Difference Equations, 180, (2014) · Zbl 1417.37281
[12] Jing, Z.; Yang, J., Bifurcation and chaos in discrete-time predator-prey system, Chaos, Solitons & Fractals, 27, 1, 259-277, (2006) · Zbl 1085.92045
[13] Liu, X. L.; Xiao, D. M., Complex dynamic behaviors of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32, 1, 80-94, (2007) · Zbl 1130.92056
[14] Rana, S. M. S., Bifurcation and complex dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response, Advances in Difference Equations, 345, (2015) · Zbl 1422.37023
[15] Rana, S. M. S., Bifurcation and complex dynamics of a discrete-time predator-prey system involving group defense, Computational Ecology and Software, 5, 3, 222-238, (2015)
[16] Rana, S. M. S., Bifurcation and complex dynamics of a discrete-time predator-prey system, Computational Ecology and Software, 5, 2, 187-200, (2015)
[17] Tan, W.; Gao, J.; Fan, W., Bifurcation analysis and chaos control in a discrete epidemic system, Discrete Dynamics in Nature and Society, 2015, (2015)
[18] Wang, C.; Li, X., Stability and Neimark-Sacker bifurcation of a semi-discrete population model, The Journal of Applied Analysis and Computation, 4, 419-435, (2014) · Zbl 1323.92193
[19] Zhao, M.; Xuan, Z.; Li, C., Dynamics of a discrete-time predator-prey system, Advances in Difference Equations, 191, (2016)
[20] Zhao, M.; Li, C.; Wang, J., Complex dynamic behaviors of a discrete-time predator-prey system, The Journal of Applied Analysis and Computation, 7, 2, 478-500, (2017)
[21] Chen, G.; Teng, Z.; Hu, Z., Analysis of stability for a discrete ratio-dependent predator-prey system, Indian Journal of Pure and Applied Mathematics, 42, 1, 1-26, (2011) · Zbl 1308.39011
[22] Cheng, L.; Cao, H., Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee effect, Communications in Nonlinear Science and Numerical Simulation, 38, 288-302, (2016)
[23] Xia, Y.; Cao, J.; Lin, M., Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Analysis: Real World Applications, 8, 4, 1079-1095, (2007) · Zbl 1127.39038
[24] Elaydi, S. N., An Introduction to Difference Equations, (1996), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0840.39002
[25] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, (1998), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0914.58025
[26] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, (1983), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0515.34001
[27] Robinson, C., Dynamical Systems: Stability, Symbolic Dynamics and Chaos, (1999), NY, New York, USA: Boca Raton, NY, New York, USA · Zbl 0914.58021
[28] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2, (2003), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 1027.37002
[29] Cartwright, J. H. E., Nonlinear stiffness Lyapunov exponents and attractor dimension, Physics Letters A, 264, 298-304, (1999) · Zbl 0949.37014
[30] Kaplan, J. L.; Yorke, J. A., Preturbulence: a regime observed in a fluid flow model of Lorenz, Communications in Mathematical Physics, 67, 2, 93-108, (1979) · Zbl 0443.76059
[31] Lynch, S., Dynamical Systems with Applications Using Mathematica, (2007), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA
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