Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects

In this paper, we consider the prey-dependent consumption predator–prey (natural enemy–pest) models with age structure for the predator, immature and mature natural enemies are released and pesticide is applied impulsively. We prove that, when the impulsive period is no longer than some threshold, the pest-eradication solution is globally asymptotically stable, or say, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, so we further prove that, when the impulsive period is longer than the threshold, pest population and natural enemy population can coexist, i.e., the system is uniformly permanent. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term into the prey population’s intrinsic growth rate. From two aspects, i.e., when the period of forcing term is same as the impulsive period and when the two periods are different, we illustrate that, the dynamical behaviors of corresponding impulsive system are very complex.