Optimal Control Problems Of Predator-Prey System With Allee Effect And Functional Response Function

With the destroy of the global environment increasing,the speed of the species loss is becoming more and more quickly,some researchers of the population biology and the natural biological have shown that if the species of an area is effected by the Allee effect,it will be extincted finally.But according to the observation of actual signs,it has shown that Allee single populations of species is widespread,therefore,from the conservation and development point of view,studying the biological population model with Allee effect is quite meaningful.In ad-dition,since1965,on the basis of experiment,the different types of species,Holling proposed three kinds of different functional response function,respectively is Holling I,Holling II, Holling Ⅲ.they adapt to different types of organisms,the first kind functional reaction func-tion,adapted to the algae,cell type of lower organisms,The second kind functional reaction function of axy/1+bx,adapted to the invertebrates,the third types of functional response function, adapted to the vertebrate.The study of functional response function also is more,the research with Allee effect and functional response function of biological population model has more practical significance.This paper is studied the optimal capture problem with Allee effect and functional response function of predator-prey system,using the theory of ordinary differential.First,we studied the following models with the Allee effect.Through the biological significance concerning the existence of the system equilibrium point,the local asymptotical stability of the positive equilirium in the system is obtained by using Routh-Rurwize criterion.Constructing Lyapunov function to prove the global stability and discussing the existence of the systematic biological economic balance.Using Pontryagin (ponte aurous)maximum principle to study the optimal harvesting problem,thus obtained the optimal harvesting strategy.Second,we studied the predator-prey system with the Allee effect and functional response function: In this paper, we discuss the existence of the positive equilibrium point, the conditions of the local stability,and the existence condition of the economic balance. Finally,using pontrya-gin maximal principle gets the optimal haversting strategy.Third,we considered the economic and price factor of population resources, and tax re-sources available to opertors, discussed the more general proportion with Allee effect and de-pendence on prey on tax model of functional response function:In this chapter, we get the existence and local stability conditions of the positive equilib-rium, using the maximum principle, the optimal tax policy were obtained.when the discount increases indefinitely, eventually the result of the net economic profit is zero.Finally,the paper summarizes the research achievement, and discusses further reasearch problems.