A delayed predator-prey model concerning impulsive spraying pesticides and releasing natural enemies is proposed and investigated, in which both the prey and the predator have a history that takes them through two stages: immature and mature. The global attractiveness of the pest-eradication periodic solution is discussed, and sufficient condition is obtained for the permanence of the system. Further, numerical simulations show that there is a characteristic sequence of bifurcations leading to a chaotic dynamics, which implies that the system with constant periodic impulsive perturbations admits rich and complex dynamics.