Bifurcation and complex dynamics of a discrete-time predator-prey system

In this paper, we investigate the dynamics of a discrete-time predator-prey system of Holling-I type in the closed first quadrant R+2. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of R+2 by using bifurcation theory. It has been found that the dynamical behavior of the model is very sensitive to the parameter values and the initial conditions. Numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamic behaviors, including phase portraits, period-9, 10, 20-orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-20 leading to chaos, quasi-periodic orbits, and sudden disappearance of the chaotic dynamics and attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology