Dynamics of a predator-prey system with pulses

In this paper, we investigate the dynamic behaviors of a Holling II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategy-periodic releasing natural enemies and spraying pesticide at different fixed moment. By using the Floquet theory of linear periodic impulsive equation and small-amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey erad-ication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is larger than some critical value, and meanwhile the conditions for the extinction of one of the two-prey and permanence of the remaining two species are given. Finally, we give numerical simulation, with increasing of 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-halving, chaos and symmetry breaking bifurcation. Our results suggest a new approach in the pest control. It is well known, many insects are beneficial to human, but some insects are harmful to human, only these harmful insects can cause economic damage as their population reaching economic injury level. Controlling harmful insects and other arthropods has become an important issue in recent years, for instance, minimizing losses due to insect pests and insect vectors remains an essential component of the programmes in the Office of Agricultural Entomology in China. Overusing a single control tactic is discouraged to minimize damage to non-target organisms, and to preserve the quality of the environment. Overusing a single control tactic also can lead pest to produce resistance to chemical control (for example, pesticide), it will be more difficult to control pest later. Then biological and chemical control were introduced. Biological control [1–7] is the purposeful introduction one or more natural enemies of an exotic pest, specifically for the purpose of suppressing the abundance of the pest in a new target region to a level at which it no longer causes economic damage. Virtually all insect and mite pests have some natural enemies. Natural enemies are able to play a more active role in suppressing insect pests. Usually, predators feed on not only insect pests but also other insects. There may be more than one pest species – for example, the two species of aphids predominant in small grains: the English grain aphid and the oat-bird cherry aphid. Aphids' high reproductive rate enables their …