Abstract
<abstract> A type of predator-prey model with refuge and nonlinear pulse feedback control is established. First, we construct the Poincar$ \acute{\rm{e}} $ map of the model and analyse its main properties. Then based on the Poincar$ \acute{\rm{e}} $ map, we explore the existence, uniqueness and global stability of the order-1 periodic solution, and also the existence of order-$ k $($ k \geq $2) periodic solutions of the system. These theoretical analysis gives the relationship between pesticide dosage and spraying cycle and economic threshold, and the relationship between the pesticide dose $ D $ and threshold $ ET $. The results show that choosing the appropriate spraying period and finding the corresponding pesticide dose under the certain economic threshold can control the number of pests and the healthy growth of crops could be controlled. Moreover, we study the influence of refuge on population density. The results show that with the increase of refuge intensity, the population of predators decreases or even becomes extinct. The parameter sensitivity analysis shows that the change of control parameters and pesticide dosage is very sensitive to the critical condition of the stability of the boundary periodic solution. </abstract>
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