Dynamics Analysis Of A Filippov Pest-Natural Enemy System With Time Delay

One of the key factors to improve crop yield is the effective control of pests and pathogens that have great influences on global food security.China is a big agricultural country,and agriculture plays a key role in our economic development.Crop diseases and insect pests can lead to serious crop losses and even production failures,seriously hinder the development of agriculture.Therefore,it is necessary to take measures to reduce the harm of pests to crops.On the other hand,in order to more fully explain how population dynamics depend on the relevant information in the past,it is more reasonable to introduce time delay in predation models considering threshold control strategies.Hence,in this paper,we incorporate time delay into a Filippov pest-natural enemy system with constant releasing rate,where the time delay represents the change in the growth rate of natural enemies before it is released to prey on pests.We investigated the effects of the economic threshold and time delay on system dynamics and pest control,and the results can provide qualitative theoretical reference for pest control.Based on the Integrated Pest Management(IPM)strategies,we propose a delay Filippov pest-natural enemy system.If the pest population does not reach a given economic threshold(ET),the integrated control strategy is not adopted,otherwise,pesticides will be sprayed and natural enemies released to implement IPM strategy measures.Firstly,the existence of positive equilibria between two subsystems is discussed,and study the stability of the equilibria and the existence of local Hopf bifurcations by the distribution of the characteristic roots of the transcendental equation of the Filippov system.Secondly,the existence of the crossing and the sliding segments of the system is analyzed.According to Filippov convex method,the sliding mode dynamics of the system are obtained.Then,the existence and stability of pseudo-equilibrium are studied,and tangent points and boundary equilibria of the system are given.Thirdly,the global dynamics of the model are studied with the economic threshold as the parameter based on three different casesψ<min{x1*,x21*},min{x1*,x21*}<ψ<max{x1*,x21*},and ψ>max{x1*,x21*}.If the economic threshold ψ satisfies ψ>max{x1*,x21*} or the parameters ξ,h and αare chosen such that x21*<x1*,and x21*<ψ<x1*,Filippov control strategy can effectively control the number of pests below the given economic threshold,so as to achieve the purpose of control.While if the parameters ξ,h and a are chosen such that x21*>x1*,and x1*<ψ<x21*,since the final solution depends on the initial value,the solution of different initial values eventually tends to different levels,or below or above the economic threshold,the pest control will be faced with uncertainty.If the selected economic threshold meets ψ<min{x1*,x21*},then the number of pests will eventually exceed the economic threshold,leading to the failure of pest control.Finally,the local and global sliding bifurcations of the system are analyzed by numerical simulations.Our goal in implementing Filippov control strategies is to reduce the number of pests below or equal to the given economic threshold.However,the occurring of buckling and crossing bifurcations switches the number of pests between the economic threshold,which indicates that the introduction of time delay brings great challenges to pest control.This further reveals the importance of time delay,which on the one hand makes the model likely to generate more complex dynamic behavior,on the other hand makes it more difficult to design a successful Filippov control.