The Optimal Control Problem Of Several Population Models

In the development and using of biological resources, current yields are not only needed, ecosystem stability and balance are but also taken into account to ensure the sustainable using of biological resources, while the economic profits gained by the investment products are also considered. Today, developers usually regard ecological and economic benefits as the principle of development so that biological resources can be developed and sustained by human. In recent decades, development and management of biological resources has always been a very important area of research, many scholars have been doing a number of researches on different issues from different aspects.This paper is divided into three chapters, the specific details are as follows:Chapter I, We introduce the research background of biological mathematics and give some prerequisites.Chapter II, Based on document[1], we study the optimal control problems of predator prey model under the influence of the toxin. We make some improvements in many aspects. We mainly discuss existence of positive equilibrium, local stability conditions and the stability influence of their toxins to the balance. Finally, we analyze the maximum economic benefits and prove it by the Pontryagin's maximum principle. All the improvements make us get the largest economic profit level, we also discuss the effects on the population by the improvements. In the end of the chapter, some numerical simulations are given to illustrate our analytic results.In Chapter III, we discuss local stability in each equilibrium point and the global asymptotic stability of the positive equilibrium point. We also discuss the optimal harvesting problem of fishery resources in the reserve area. A regulatory agency controls exploitation by imposing a tax per unit biomass of the prey species. The objective is to maximize the monetary social benefit as well as conservation of the ecosystem. Finally, some numerical simulations are also given to illustrate our analytic results.In the final, we summarize the innovation of the paper and propose improving direction of the paper.