Discrete Natural Enemies Of Pests System With Integrated Pest Control Strategy

As we all know, huge losses are caused by pests every year in economy. For this problem, integrated pest management strategy(IPM) is considered to inhibit increase of pests such that economical losses could be reduced. Biological control, as a important component of IPM, is defined to reduce the number of pest population by natural enemies, a familiar implemented process of biological control is raising number of natural enemies by planned released. Another concerned component of IPM is chemical control, periodical spraying insecticide, as one of its universal methods, have remarkable effect on restraint of pest population. In order to describe model of natural enemy-pest with integrated pest management strategy, we should understand the growth rule of pests firstly. Since most kinds of pests' survival generations are not overlap, therefore, we need to use discrete Predator-Pest model to achieve our aim. So in this article, we associate both biological control and chemical control to built new mathematical models, at the same time considering factors of population dispersal, finally discuss effects of biological control, chemical control as well as population dispersal on reduction of pests.Firstly, we consider the discrete natural enemy-pest model, at the same time, we assume that pests and natural enemies could disperse between two patches, pest and natural enemy of the two patches increase in the way of Nicholson-Bailey mode. So we obtain the following model with fi=xni exp(ri=diyni), gi=xni(1-exp(-diyni))+δiyni+τi, parameter ri is the pests' intrinsic growth rate:di is a measure of the natural enemy's predating efficiency:δi denotes the survival of the natural enemy at generation n:τi is the constant-release rate of natural enemy; parameters dij and Dij represent the immigration rate of pests and natural enemy from ith patch to jih patch. For this model, we combine Jury Criterion and numerical simulation to research exis-tence and stability of the model's equilibrium with pest eradication, then discuss how dispersal rates of population affect stability of the model. The results indi-cate:population dispersal has various effects on pest control, for some situations, the population dispersal is beneficial to pest control, but for other situations, it is harmful and may cause outbreak of pests. Further, we extend two-patch-dispersal model to k patches, and obtain threshold conditions for existence and stability of k-patch model by using matrix equation and spectral radius theory.Finally, when considering the affects of periodical spraying pesticide and releas-ing natural enemies on the above model, we obtain the following discrete Nicholson-Bailey model with impulsive effect where the parameters q1,q2 denote the killing rate of pesticide; p1,p2 are natural enemies' releasing proportions; other parameters have the same biological meaning as those before. For the model with impulsive effect, we focus on discussing the existence of the model's boundary steady state by theory, the stability of boundary and inner steady states by numerical simulation, as well as the complexity of system's dynamic behaviors.