Abstract
According to resource limitation, a more realistic pest management is that the impulsive control actions should be adjusted according to the densities of both pest and natural enemy in the field, which result in nonlinear impulsive control. Therefore, we have proposed a Beddington–DeAngelis interference predator-prey model concerning integrated pest management with both density-dependent pest and natural enemy population. We find that the pest-eradication periodic solution is globally stable if the impulsive period is less than the critical value by Floquet theorem. The condition of permanent is established, and a stable positive periodic solution appears via a supercritical bifurcation by bifurcation theorem. Finally, in order to investigate the effects of those nonlinear control strategies on the successful pest control, the bifurcation diagrams showed that the model exists with very complex dynamics. Consequently, the resource limitation may result in pest outbreak in complex ways, which means that the pest control strategies should be carefully designed.
Highlights
Since pest outbreak can cause serious economic loss, pest control has been becoming an increasing concern to entomologists and society all over the world
The optimal pest control strategy is that the instantaneous releasing numbers of natural enemies should be adjusted according to the densities of both pest and natural enemy in the field
We have proven that there is a global stability of pest-eradication periodic solution if the impulsive period T < T∗ by using the Floquet theorem and small amplitude perturbation skills, and model (1) is permanent when the period T > T∗
Summary
Since pest outbreak can cause serious economic loss, pest control has been becoming an increasing concern to entomologists and society all over the world. In order to take the resource limitation into the IPM strategy, several predator-prey models with nonlinear impulse have been proposed [29,30,31,32] and mainly focused on establishing the global stability conditions. The nonlinear impulsive function mentioned above is only related to the density of the natural enemy population in the field. Erefore, in order to take the resource limitation into account and to understand how the nonlinear density regulatory factor for the natural enemies affect the dynamics of predator-prey model, we propose the following predatorprey model with Beddington–DeAngelis functional response and nonlinear impulsive control:. 0 ≤ q1, q2 ≤ 1 present survival rate of prey and predator after harvesting or pesticides; q2 ≥ 1 means that the pesticides only affect the pest and an impulsive increase of the predator population density is induced by release of predators.
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