| Citation: |
S. M. Sohel Rana. BIFURCATIONS AND CHAOS CONTROL IN A DISCRETE-TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 31-44. doi: 10.11948/2019.31
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BIFURCATIONS AND CHAOS CONTROL IN A DISCRETE-TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE
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Abstract
We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $ \mathbb{R}^{2}_+ $ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.-
Keywords:
- Discrete-time predator-prey system /
- bifurcations /
- chaos /
- Lyapunov exponents /
- feedback control
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References
[1] J. H. E. Cartwright, Stiffness Lyapunov exponents and attractor dimension, Phys. Lett. A., 1999, 264, 298-302. doi: 10.1016/S0375-9601(99)00793-8 [2] S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005. [3] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [4] Z. M. He, Bo. Li, Complex dynamic behavior of a discrete-time predator-prey system of Holling-Ⅲ type, Advances in Difference Equations, 2014, 180. [5] Z. M. He, X. Lai, Bifurcation and chaotic behavior of a discrete-time predatorprey system, Nonlinear Anal. Real World Appl., 2011, 12, 403-417. doi: 10.1016/j.nonrwa.2010.06.026 [6] S. B. Hsu, T.W. Hwang, Global Stability for a Class of Predator-Prey Systems, SIAM J. APPL. MATH., 1995, 55, 763-783. doi: 10.1137/S0036139993253201 [7] D. P. Hu and H. J. Cao, Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type, Commun Nonlinear Sci Numer Simulat, 2015, 22, 702-715. doi: 10.1016/j.cnsns.2014.09.010 [8] J. Huang, S. Ruan, and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, Journal of Differential Equations, 2014, 257, 1721-1752. doi: 10.1016/j.jde.2014.04.024 [9] J. L. Kaplan, Y. A. Yorke, A regime observed in a fluid flow model of Lorenz, Comm. Math. Phys., 1979, 67, 93-108. doi: 10.1007/BF01221359 [10] Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, , 2nd Ed. SpringerVerlag, New York, 1998. [11] S. Lynch, Dynamical Systems with Applications Using Mathematica, Birkhäuser, Boston, 2007. [12] S. M. S. Rana, Bifurcation and complex dynamics of a discrete-time predatorprey system with simplified Monod-Haldane functional response, Advances in Difference Equations, 2015, 345. [13] S. M. S. Rana, Bifurcation and complex dynamics of a discrete-time predatorprey system, Computational Ecology and Software, 2015, 5(2), 187-200. [14] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Ed. Boca Raton, London, New York, 1999. [15] W. Tan, J. Gao, and W. Fan, Bifurcation Analysis and Chaos Control in a Discrete Epidemic System, Discrete Dynamics in Nature and Society, 2015. DOI: 10.1155/2015/974868. [16] C. Wang, X. Li, Stability and Neimark-Sacker bifurcation of a semi-discrete population model, J Applied Analysis and Computation, 2014, 4, 419-435. [17] S. Winggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 2003. [18] M. Zhao, C. Li, and J. Wang, Complex dynamic behaviors of a discrete-time predator-prey system, Journal of Applied Analysis and Computation, 2017, 7, 478-500. DOI:10.11948/2017030. [19] M. Zhao, Z. Xuan, and C. Li, Dynamics of a discrete-time predator-prey system, Advances in Difference Equations, 2016, 191. DOI: 10.1186/s13662-016-0903-6. -
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S. M. Sohel Rana. BIFURCATIONS AND CHAOS CONTROL IN A DISCRETE-TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 31-44. doi: 10.11948/2019.31
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Figure 1. Bifurcation diagrams and maximum Lyapunov exponent of map (1.3) around
$ E_2 $ . (a) Neimark-Sacker bifurcation diagram of map (1.3) in$ (\delta-x) $ plane, (b) NS bifurcation diagram in$ (\delta-y) $ plane, (c) maximum Lyapunov exponents corresponding to (a-b), (d) local amplification corresponding to (a) for$ \delta \in [1.8702, 1.9142] $ (e) Fractal dimension corresponding to (a). The initial value is$ (x_0, y_0) = (0.57, 0.42) $ . -
Figure 2. Phase portraits for various values of
$ \delta $ corresponding to Figure 1(a-b). -
Figure 3. The 3-dimensional bifurcation diagram and maximum Lyapunov exponents of map (1.3) around
$ E_2 $ . (a-b) The 3-dimensional bifurcation diagrams of map (1.3) covering$ \delta \in [1.7, 1.9556], \;\; \beta \in [1.0, 1.3] $ , and$ \alpha = 0.75 $ in$ (\delta-a-x) $ space and$ (\delta-a-y) $ space (c) The 3-dimensional maximum Lyapunov exponents corresponding to (a-b) (d) The 2-dimensional projection onto$ (\delta, a) $ plane. The initial value is$ (x_0, y_0) = (0.57, 0.42) $ . -
Figure 4. Control of chaotic orbits of system (5.1). (a) The bounded region for the stable eigenvalues of the controlled system (5.1) in the
$ (k_1, k_2) $ plane (b-c) The time responses for the states$ x $ and$ y $ of the controlled system (5.1) in the$ (n, x) $ and$ (n, y) $ planes respectively.