×

zbMATH — the first resource for mathematics

Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type. (English) Zbl 1331.92125
Summary: A discrete-time predator-prey system of Holling and Leslie type with a constant-yield prey harvesting obtained by the forward Euler scheme is studied in detail. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, phase portraits display new and rich nonlinear dynamical behaviors. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period-\(1, 2, 11, 17, 19, 22\) orbits, attracting invariant cycles, and chaotic attractors of the discrete-time predator-prey system of Holling and Leslie type with a constant-yield prey harvesting. These results demonstrate that the integral step size plays a vital role to the local and global stability of the discrete-time predator-prey system with the Holling and Leslie type after the original continuous-time predator-prey system is discretized.

MSC:
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Huang, J. C.; Gong, Y. J.; Chen, J., Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Int J Bifurcation Chaos, 23, 1350164, (2013) · Zbl 1277.34059
[2] Jing, Z. J.; Jia, Z. Y.; Wang, R. Q., Chaos behavior in the discrete BVP oscillator, Int J Bifurcation Chaos, 12, 619-627, (2002) · Zbl 1069.65141
[3] Jing, Z. J.; Chang, Y.; Guo, B. L., Bifurcation and chaos in discrete Fitzhugh-Nagumo system, Chaos Solitons Fract, 21, 701-720, (2004) · Zbl 1048.37526
[4] Chen, X. W.; Fu, X. L.; Jing, Z. J., Complex dynamics in a discrete-time predator-prey system without allee effect, Acta Math Appl Sin, 29, 355-376, (2013) · Zbl 1275.37038
[5] Sun, H. J.; Cao, H. J., Bifurcation and chaos of a delayed ecological model, Chaos Solitons Fract, 33, 1383-1393, (2007) · Zbl 1138.37342
[6] Cao, H. J.; Wang, C. X.; Sanjuán, M. A.F., Effect of step size on bifurcations and chaos of a map-based BVP oscillator, Int J Bifurcation Chaos, 20, 6, 1789-1795, (2010) · Zbl 1193.37055
[7] Chen, X. W.; Fu, X. L.; Jing, Z. J., Dynamics in a discrete-time predator-prey system with allee effect, Acta Math Appl Sin, 29, 143-164, (2012)
[8] Celik, C.; Duman, O., Allee effect in a discrete-time predator-prey system, Chaos Solitons Fract, 40, 1956-1962, (2009) · Zbl 1198.34084
[9] Hsu, S. B.; Huang, T. W., Global stability for a class of predator-prey system, SIAM J Appl Math, 55, 763-783, (1995) · Zbl 0832.34035
[10] Marotto, F. R., Snap-back repellers imply chaos in \(R^n\), J Math Anal Appl, 63, 1, 199-223, (1978) · Zbl 0381.58004
[11] Marotto, F. R., On redefining a snap-back repeller, Chaos Solitions Fract, 25, 25-28, (2005) · Zbl 1077.37027
[12] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical system, and bifurcation of vector fields, (1983), Springer New York
[13] Kuznetsov, Y. A., Elements of applied bifurcation theory, (1999), Springer New York
[14] Shub, M., Morse-Smale systems, Scholarpedia, 2, 1785, (2007), online
[15] Bielecki, A., Topological conjugacy of discrete time-map and Euler discrete dynamical systems generated by a gradient flow on a two-dimensional compact manifold, Nonlinear Anal, 51, 1293-1317, (2002) · Zbl 1015.37017
[16] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, Texts in applied mathematics, vol. 2, (1990), Springer New York: NY · Zbl 0701.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.