Bifurcation and complex dynamics of a two-prey two-predator system concerning periodic biological and chemical control

In this paper, we investigate the dynamic behaviors of a two-prey two-predator system with impulsive effect concerning biological and chemical control strategy–periodic releasing natural enemies and spraying pesticide at different fixed time. By applying the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we prove that there exists a globally asymptotically stable two-prey-eradication periodic solution when the impulsive period is less than some critical value. The conditions for the permanence of the system are given, and meanwhile the conditions for the extinction of one of the two prey species and permanence of the remaining three species are given. Our results suggest a new approach in pest control. The target pest population can be driven to extinction and the non-target pest can be permanent by choosing impulsive period. With the increasing of the predation rate for the super competitor and impulsive period, the system displays complicated behaviors including a sequence of direct and inverse cascades of periodic-doubling, periodic-halfing, chaos, and symmetry breaking bifurcation.