Leslie-the Gower Predator System With A Prey Refuge Optimal Capture

In the development of human society, how to exploit the renewable resources is alwaysa significant problem and focus.In order to make it possible for people to exploit and userenewable resources for a long time,economic and biological aspects of renewable resourcesmanagement is considered.The biological resources development research is carrying on thequalitative analysis through the establishment mathematical model to forecast the develop-ment of the population and the in?uence of people's development behavior to the population.Besides,establishing corrected capture strategy,which not only can maintain ecosystem bal-ance,but also can meet human needs,so its significance is particularly important.In recentyears,various dynamic models of Leslie or modified Leslie-Gower predator-prey models havebeen proposed and analyzed,but the exploitation of them have been studied a few.So,analysisof a harvested Leslie-Gower predator-prey model incorporating prey refuge through optimalmethods become more and more important and significant.In this paper,according to the study of biological populations model in domestic and for-eign,especially,the development of biological populations with exploitation which is appliedbroadly in reality,we mainly studied the following three bioeconomic models:First,we analyze and formulate a Leslie-Gower predator-prey model incorporating a preyrefuge and a harvesting prey.We study the existence of the non-negative equilibria,obtainthe su?cient conditions of locally asymptotically stable of the equilibria by the Hurwitzcriterion,and we analyze the global stability of the equilibria by constructing appropriateLyapunov functions.Then,we obtain the optimal harvesting effort,the maximum sustainableyield and the corresponding population density.Second,we analyze and formulate a harvested Leslie-Gower predator-prey model incor-porating a prey refuge,both the population are harvested where we use the usual Catch-per-unit-e?ort hypothesis.We study the existence of the non-negative equilibria,obtain the su?-cient conditions of locally asymptotically stable of the equilibria by the Hurwitz criterion,andwe analyze the global stability of the equilibria by constructing appropriate Lyapunov func- tions.Then,the existence of bionomic equilibria of the system is established.Next,the optimalharvesting policy is discussed.The present value of revenues is maximized by using Pontrya-gin's maximal principle.It is established that the zero discounting leads to the maximizationof economic revenue.At last,we deal with the problem of selective harvesting in a Leslie-Gower predator-prey model incorporating a prey refuge.To protect fish population from over exploitation acontrol instrument tax is imposed.The existence of its steady states and their stability arestudied.The problem of optimal taxation policy is then solved by using Pontryagin's maximalprinciple.