The study of predator–prey system with defensive ability of prey and impulsive perturbations on the predator

Predator–prey system with non-monotonic functional response and impulsive perturbations on the predator is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than the critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. Further, using numerical simulation method the influences of the impulsive perturbations on the inherent oscillation are investigated. With the increasing of the impulsive value, the system displays a series of complex phenomena, which include (1) quasi-periodic oscillating, (2) period-doubling, (3) period-halfing, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis and (6) chaotic bands with periodic windows.