COMPLEX DYNAMIC BEHAVIORS OF A DISCRETE-TIME PREDATOR-PREY SYSTEM

Ming Zhao; Cuiping Li; Jinliang Wang

doi:10.11948/2017030

2017 Volume 7 Issue 2

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Ming Zhao, Cuiping Li, Jinliang Wang. COMPLEX DYNAMIC BEHAVIORS OF A DISCRETE-TIME PREDATOR-PREY SYSTEM[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 478-500. doi: 10.11948/2017030

Citation:

Ming Zhao, Cuiping Li, Jinliang Wang. COMPLEX DYNAMIC BEHAVIORS OF A DISCRETE-TIME PREDATOR-PREY SYSTEM[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 478-500. doi: 10.11948/2017030

COMPLEX DYNAMIC BEHAVIORS OF A DISCRETE-TIME PREDATOR-PREY SYSTEM

LMIB-School of Mathematics and Systems Science, Beihang University, Xueyuanlu 37, Haidian district, Beijing 100191, China

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Abstract

Abstract

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto's chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
- Predator-prey system /
- flip bifurcation /
- Hopf bifurcation /
- Marotto's chaos /
- feedback control
MSC: 39A28;39A33;65P20

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