The Problems Of The Stability And The Optimal Harvesting Policy For Three-species Model With Ratio-dependent

Population ecology is an important branch of ecology, and the mathematics method has been will the most widely applied and the most developed mature system by far. In the recent year, with the emerging energy crisis and the human living environment has worsened, more and more scientists have paid attention to the ecological resources. The results of their research for the protection of the resources and the environment in which we live make a great contribution.We know, in the ecological system, any kinds of creature belong to a population (community). In the vast and complex food chain, these species happen a certain relationship each other. The predation phenomenon is most common in our real life. Therefore, the predator-prey system has become the most important and the most research in biological areas of mathematics. Biologists predict trends of population to protect and to develop the ecological resources provide a theoretical basis by analyzing mathematical model of their establishment.The paper studies the optimal harvesting policy of predator-prey system with functional response by using the theory of ordinary differential equations, and gets some main conclusions as follow:Firstly, with Ratio-Dependent functional response is discussed. We find the positive equilibrium conditions of system; and the local asymptotical stability of the positive equilibrium in the system is obtained by using Routh-Hurwitz criterion.Secondly, we capture the prey populations and analyze on a Ratio-Dependent of prey-predator optimal harvesting model, which finds existence conditions and the local asymptotical stability of the positive equilibrium of the system. Finally, using Pontryagin maximal principle gets the optimal harvesting policy.Thirdly, we consider the economic and price factors of the population resources, and tax the resources available to the operators. We study an optimal taxation policy of predator-prey system with more general Beddington-Deangelis function, which gets existence conditions and the local asymptotical stability of the positive equilibrium of the system. Finally, using Pontryagin maximal principle gets the optimal tax policy. We obtain that an infinite discount rate leads to finally result of the net economic revenue tending to zero.Finally, the paper summarizes the research achievement, and discusses further research problems.