Abstract
Considering the effects of the living environment on growth of populations, it is unrealistic to assume that the growth rates of predator and prey are all constants in the models with integrated pest management (IPM) strategies. Therefore, a nonautonomous predator-prey system with impulsive effect is developed and investigated in the present work. In order to determine the optimal application timing of IPM tactics, the threshold value which guarantees the stability of pest-free periodic solution has been obtained firstly. The analytical formula of optimal application timings within a given period for different cases has been obtained such that the threshold value is the smallest, which is the most effective in successful pest control. Moreover, extensively numerical investigations have also been confirmed our main results and the biological implications have been discussed in more detail. The main results can guide the farmer to design the optimal pest control strategies.
Highlights
Many ecologists are becoming increasingly concerned with the questions of pest control and designing the optimal control strategies
It is well known that the growth rate of the species is affected by the living environments, so it is more practical to consider the growth rates of predator and prey as the functions with respect to time t in the models with integrated pest management (IPM) strategies
Nonautonomous predator-prey systems with impulsive effects have been developed and investigated in the literatures [11–13]. Those works mainly focused on the dynamical behavior including the existence and stability of pest-free periodic solutions
Summary
Many ecologists are becoming increasingly concerned with the questions of pest control and designing the optimal control strategies. The mathematical models can help us to design the optimal control tactics and, in particular, help us to predict the density of pest population and to determine optimal application timing of IPM strategies (see [5–8]). Nonautonomous predator-prey systems with impulsive effect have been developed and investigated in [10, 16, 17] Those works mainly focused on the effects of periodic perturbations on the dynamics; the interesting questions concerning the effects of successful pest control, in particular how to apply the IPM tactics in periodic environment and to determine the optimal application timing, have not been addressed in more detail. We assume that only one-impulsive controlling has been applied within a given period, and the optimal application timing has been determined and the analytical formula is provided, under which the threshold value reaches its minimum. We conclude that the main results can help us to design the optimal pest control strategies
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