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Bifurcation and chaotic behavior of a discrete-time predator-prey system. (English) Zbl 1202.93038
Summary: The dynamics of a discrete-time predator-prey system is investigated in the closed first quadrant \(\mathbb R^2_+\). It is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of \(\mathbb R^2_+\) by using a center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits of period 7, 14, 21, 63, 70, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, quasi-periodic orbits and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

MSC:
93B52 Feedback control
34H20 Bifurcation control of ordinary differential equations
49N75 Pursuit and evasion games
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