The Study Of The Optimal Harvesting Problems For Several Types Of Predator-Prey Systems With Functional Reaction

Currently,the research fields are more widely about development management issues on population resource.In the course of the study,we mainly are based on ecological and economic benefits,in order that resources of biological populations can be effectively developed and uti-lized. We considered the actual problem and created biological mathematical models in order to make qualitative analysis and select the best capture strategy which makes biological resources be under the premise of maintaining the balance of ecosystems and meet the various needs of human beings.Therefore,establishing the optimal harvesting system of biological populations has become a research trend.Establishing the optimal harvest model of biological populations can help the government effectively control the fishermen capture activities,and then maintain ecological balance and stability.This paper is mainly based on the research status of the biological species models at home and abroad in recent years,the research result of development and management of biological populations is regarded as a starting point, which has a strong background and practical applica-tion value,and the following issues are mainly raised through extensive research.The first prob-lem is that we considered several predator systems with the effect of prey refuge and functional response and concluded the existence and stability of the equilibrium point by using qualitative and stability analysis methods of differential equations after the establishment of economic and revenue capture model.The second problem is that we imposed taxes to control development for the unit capture populations (prey and predator) by using Pontryagin maximum principle and obtained the optimal taxation. These results provide a theoretical basis for the effective management of renewable resources for the certain populations. We analyzed the impact of har-vesting effects and refuge effects on asylum local stability and global stability of the system by using eigenvalue method and bounded theory of the system.We obtained a result that a cer-tain percentage of the shelter under certain conditions has stabilization on the system through observation and analysis.It is proved that all positive solutions are uniformly bounded by using the theory of dif-ferential inequalities in this article,the existence of a unique positive equilibrium point of the system is proved,stability of the system is proved by using the Routh-Hurwitz criterion and by using Lyapunov function,and finally the optimal harvest policy is obtained by using Pontryagin maximum principle of the optimal control theory.The following four aspects are discussed and analyzed in the research process in this paper:(1)The optimal harvesting problems of the predator system with the effect of prey refuge and Ivlev functional response. In the second chapter,we studied a class of optimal harvesting problems of predator system with prey refuge and Ivlev functional response.We considered the effect of prey refuge and the simultaneous capture of prey and predator populations.It is proved that all positive solutions of the system is uniformly bound by using the theory of differential inequalities in this chapter,the local asymptotic stability of the equilibrium is proved by using the Routh-Hurwitz criterion,and global stability of the positive equilibrium is proved by constructing Lyapunov function,as well as the optimal capture strategies are obtained.(2)The optimal tax policy of the predator system with the effect of prey refuge and Michaelis-Menten functional response.In the third chapter, we studied the optimal tax policy of a class of predator-prey system with prey refuge and Michaelis-Menten functional response. We considered the effect of prey refuge,the capture of prey populations,and the best tax policy.The local asymptotic stability of the positive equilibrium point is proved by using the Routh-Hurwitz criterion,and global stabil-ity of the positive equilibrium is proved by constructing Lyapunov function and the optimal tax policy is obtained.(3)The optimal tax policy of the predator system with the effect of prey refuge and Holling II functional response.In the fourth chapter, we studied the optimal tax policy of a class of predator system with prey refuge and Holling â…¡ functional response. We considered the effect of prey refuge and the capture of predator populations.The locally asymptotic stability of the positive equilibrium point is proved by using the Routh-Hurwitz criterion,and the optimal tax policy is obtained.(4)The optimal tax policy of the predator system with the effect of prey refuge and Bedding ton-DeAngelis functional response.In the fifth chapter,we studied the optimal tax policy of a class of predator system with prey refuge and Beddington-DeAngelis functional response.We considered the effect of prey refuge and the capture of prey populations.The locally asymptotic stability of the positive equilibrium point is proved by using the Routh-Hurwitz criterion,and the optimal tax policy is obtained.