Pest control switching models with instantaneous and non-instantaneous impulsive effects

In this paper, it is considered that the predation rate and transformation rate of natural enemies on pests in different time periods of pesticide action and non-action are different, and after spraying pesticides, the pesticides kill the pests and natural enemies instantaneously, and at the same time bring non-instantaneous residual effects on them. Firstly, an integrated pest control switching model with instantaneous and non-instantaneous impulsive effects at the fixed time is proposed. It assumes that spraying pesticides is more frequently used than releasing natural enemies. The sufficient condition for the global asymptotic stability of the pest-eradication periodic solution is obtained. By numerical simulations, we find that the system we study has many complex dynamics including period-doubling bifurcation, period-halving bifurcation, chaos and multiple attractors. Further, the effects of some key parameters on the threshold for pest eradication are analyzed. The results imply that it is not that the more frequent, the more conductive to pest control, and within each releasing period, applying pesticides too early or too late is not good for pest control. Finally, with the purpose of preventing pests below the economic injury level, a state-dependence pest control switching model with instantaneous and non-instantaneous impulsive effects is investigated numerically. Our results show that the frequency of applying pesticides depends on the key parameters of the established system, such as initial population density, the releasing amount of natural enemies and the period of releasing natural enemies, etc.