The Optimal Control Policy For Several Types Of Prey-predator System

With the rapid development of the social economy, the problem of the exploitation andmanagement for biospecies resource has received more and more attention from scholars. Ifwe regarde the ecological benefits and economic benefits as the main principle of exploita-tion, it will make it possible for people to use the biospecies resource durativity. In recentyears, the problem of exploitation and management of biospecies resource is a quite hot issueby many scholars.In this thesis, we analyse the optimal harvesting policy of developed predator-prey sys-tems. The whole thesis is divided into five chapters:In the first chapter, we introduce concisely the historical situation and the present devel-opment of relevant subjects about population dynamics, we also give the main work done inthis paper and some preliminaries.The second chapter deals with the problem of a ratio-dependent prey-predator modelwith stage structure and combined harvesting. The permanent is proved in a certain circum-stances, and we obtain the global asymptotic stability condition of the equilibrium point. Theproblem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.In the third chapter, we propose a stage-structured predator-prey model with time delayand cannibalism. When τ=0, the local asymptotic stability and global asymptotic stabilityconditions of its steady states are analysed. Predator resources are protected from beingover-exploitation by controlling the amount of taxes. We has solved the problem of optimaltaxation policy by using Pontryagin’s maximum principle. When τ=0, we show that bythe corresponding characteristic equations, time delay will cause a stable equilibrium to beunstable, namely, a Hopf bifurcation will take place when time delay passes τ=τ0.In the fourth chapter, we improve the capture function, and establish a Leslie-Gowerpredator-prey model incorporating a prey refuge. The local asymptotic stability and globalasymptotic stability conditions of its steady states are studied. And we then obtain the opti-mal taxation policy. The result will provide some reliable theoretical basis for the develop-ment and utilizetion of biological resources.The fifth chapter deals with the two species nonautonomous predator-prey system withimpulsive effects and feedback control. By applying the method of comparison theorem, weobtain some sufficent conditions to ensure the permanence of the system. These will havevery significant theoretical and practical meaning to keep the population diversity and tosustainable development of ecosystem.