The Existence of Li-Yorke Chaos in Certain Predator-Prey System of Difference Equations
DOI:
https://doi.org/10.5644/SJM.18.01.04Keywords:
difference equation, equilibrium, predator-prey system, snap-back repeller, Li-Yorke chaosAbstract
This paper investigates an autonomous predator-prey system of difference equations with three equilibrium points and exhibits chaos in the sense of Li-Yorke in the positive equilibrium point. Numerical simulations are presented to illustrate our results.
Downloads
References
K. T. Alligood, T. Sauer, and J. A. Yorke, CHAOS: An Introduction to Dynamical Systems, Springer, 1996.
X.W. Chen, X.L. Fu, Z.J. Jing, Dynamics in a discrete-time predator-prey system with Allee effect, Acta Math. Appl. Sinica (Engl. Ser.) 29 (2013), 143-164. doi:10.1007/s10255-013-0207-5
Y. Gao, Complex dynamics in a two-dimensional noninvertible map, Chaos, Solitons and Fractals, 39 (2009), 1798-1810.
Y. Gao, W. Feng, and B. Liu, Complex Dynamics in A Financial Model, Pure and Applied Mathematics Quarterly, 8 (2012), no. 3, 589-608.
Y. Gao and B. Liu, Study on the dynamical behaviors of a two-dimensional discrete system, Nonlinear Analysis, 70 (2009), 4209-4216.
Y. Gao, B. Liu, and W. Feng, Bifurcations and Chaos in a Nonlinear Discrete Time Cournot Duopoly Game, Acta Mathematicae Applicatae Sinica, English Series, 30 (2014), no. 4, 951-964. DOI: 10.1007/s10255-014-0435-3
M. Garić-Demirović, S. Moranjkić, M. Nurkanović, and Z. Nurkanović, Stability, Neimark-Sacker Bifurcation, and Approximation of the Invariant Curve of Certain Homogeneous Second-Order Fractional Difference Equation, Discrete Dynamics in Nature and Society, vol. 2020 (2020), Article ID 6254013, 12 pages. https://doi.org/10.1155/2020/6254013
J. Kaplan and J. Yorke, Chaotic behavior of multidimensional difference equations. In Peitgen, H. O.; Walther, H. O. (eds.). Functional Differential Equations and the Approximation of Fixed Points. Lecture Notes in Mathematics. 730. Berlin: Springer. (1979) p. 204–227. ISBN 978-0-387-09518-9.
P. Kloeden and Z. Li, Li-Yorke chaos in higher dimensions: a review, Journal of Difference Equations and Applications, 12 (2006), no. 3-4, 247-269. https://doi.org/10.1080/10236190600574069
M. R. S. Kulenović and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC, Boca Raton, London, 2002.
C.P. Li and G.R. Chen, An Improved Version of the Marotto Theorem, Chaos, Solitons & Fractals, 18 (2003), 69-77. https://doi.org/10.1016/S0960-0779(02)00605-7
T.Y. Li and J. A. Yorke, Period Three Implies Chaos, The American Mathematical Monthly, 82 (1975), 985-992. https://doi.org/10.1080/00029890.1975.11994008
F. R. Marotto, Snap-Back Repellers Imply Chaos in $mathbb{R}^{n}$, Journal of Mathematical Analysis and Applications, 63 (1978), 199-223. https://doi.org/10.1016/0022-247X(78)90115-4
F.R. Marotto, Chaotic Behavior in the Hénon Mapping, Journal of Mathematical Analysis and Applications, 68 (1979), 187-194. https://doi.org/10.1007/BF01418128
F.R. Marotto, On Redefining a Snap-Back Repeller, Chaos, Solitons & Fractals, 25 (2005), 25-28. https://doi.org/10.1016/j.chaos.2004.10.003
Z. Nurkanović, M. Nurkanović, and M. Garić-Demirović, Stability and Neimark--Sacker Bifurcation of Certain Mixed Monotone Rational Second-Order Difference Equation, Qual. Theory Dyn. Syst.} 20 (2021), no. 3, 1-41. https://doi.org/10.1007/s12346-021-00515-4
M. Sandri, Numerical calculation of Lyapunov exponents, The Mathematica Journal, 6 (1996), 78-84.
U. Ufuktepe and S. Kapcak, Applications of Discrete Dynamical Systems with Mathematica, Conference: RIMS, 1909 (2014), 207-216.
M. Zhao, C. Li, and J. Wang, Complex dynamic behaviors of a discrete-time predator-prey system, Journal of Applied Analysis and Computation, 7 (2017), no.2, 478-500
M. Zhao, Z. Xuah and C. Li, Dynamics of a discrete-time predator-prey system, Advances in Difference Equations, 2016, 191 (2016). https://doi.org/10.1186/s13662-016-0903-6.