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Discrete-time dynamics of a system with crowding effect and predator partially dependent on prey. (English) Zbl 1338.92100
Summary: In the present study, the stability and bifurcation analysis of discrete-time predator-prey system with predator partially dependent on prey and crowding effect of predator is examined. Global stability of the system at the fixed points has been discussed. The specific conditions for existence of flip bifurcation and Hopf bifurcation in the interior of $$\mathbb R_+^2$$ have been derived by using a center manifold theorem and bifurcation theory. Numerical simulations have been carried out to show the complex dynamical behavior of the system and to justify our analytic results. In case of flip bifurcation, numerical simulations presented cascade of period-doubling bifurcation in the orbits of period 2, 4, 8, chaotic orbits and stable window of period 9 orbit; whereas in case of Hopf bifurcation, smooth invariant circle bifurcates from the fixed point. The complexity of dynamical behavior is confirmed by computation of Lyapunov exponents.

MSC:
 92D25 Population dynamics (general) 37N25 Dynamical systems in biology
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