Dynamic complexities in a discrete predator–prey system with lower critical point for the prey

In this paper, a discrete predator–prey system is proposed and analyzed. It is assumed that the prey population has a lower critical point, which is also referred to as extinction threshold. Such behavior has been reported for many flowering plants, many fishes, epidemiology, and so on. The existence and stability of nonnegative fixed points are explored. The conditions for the existence of flip bifurcation and Hopf bifurcation are obtained by using manifold theorem and bifurcation theory. Numerical simulations, including bifurcation diagrams, phase portraits and Maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit other complex dynamics and certain biological phenomena. Complex dynamics include quasi-periodicity, period-doubling bifurcations leading to chaos, chaotic bands with periodic windows, intermittent, supertransient, and so on. Simulations suggest that appropriate growth rate can stabilize the system, but the high growth rate may destabilize the stable system into more complex dynamics. As well, simulations suggest that the system is stable when the lower critical point parameter c is small, but when c increases beyond the critical values, the system changes from quasi-period to collapses. Furthermore, the simulated results are explained according to a biological point of view.