A Holling I predator-prey model with mutual interference concerning pest control is proposed and analyzed. The prey and predator are considered to be a pest and a natural enemy, respectively. The model is forced by the addition of periodic impulsive terms representing predator import (biological control) and pesticide application (chemical control) at different fixed moments. By using Floquet theory and small amplitude perturbations, we show the existence and stability of pest-free periodic solutions. Further, we prove that when the stability of pest-free periodic solutions is lost, the system is permanent by using analytic methods of differential equation theory. Numerical solutions are also given, which show that when stability of pest-free periodic solutions is lost, more exotic behavior can occur, such as quasi-periodic oscillation or chaos. We investigate the effect of impulsive perturbations on the unforced continuous system, and find that the forced system has a different dynamical behavior with a different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system. Finally, we compare the validity of the combination of biological control and chemical control with classical methods and conclude that the synthetical strategy is more effective than classical methods if we take effective chemical control.
Bifurcation of nontrivial periodic solutions for a Beddington–DeAngelis interference model with impulsive biological control