Bifurcation and chaos of a pest-control food chain model with impulsive effects

According to biological and chemical control strategy for pest control, we investigate the dynamics of a predator-prey food chain with impulsive effect, periodic releasing natural enemies and spraying pesticide at different fixed times, by using impulsive differential equation. Choose pest birth rate r2 as control parameter, we show that there exists a stable pest-eradication periodic solution when r2 is less than some critical value r2∗ and the system is permanence when r2 is larger than the critical value r2∗. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in prey, middle-predator and top predator. Furthermore, bifurcation diagrams have shown that there exists complexity for the pulsed system including periodic doubling cascade.