Optimal Harvest Problem Of Population System Based On Fertility Control And Dependent On Size-Structure In Periodic Environment

In order to protect the ecosystem and develop the sustainable economy,it is necessary to establish the population system model and study the related control problems.There are many differences in species,such as size,age,sex,and so on.Due to these differences,individual behavioral abilities also vary greatly.At this time,it is necessary for us to build different models for these structural differences and analyze their optimal control problem,which has great practical significance.Up to now,many scholars have studied the control of population system dependent on age structure,and achieved remarkable results.Compared with age structure,scale structure can be used more widely.Based on this,this paper studies the optimal birth rate control problem for two kinds of multiple-population systems dependent on scale structure.It is found that the influence of “internal environment”on individual life parameters cannot be ignored,so in chapter 4,the optimal input rate control problem of a class of competitive population dependent on age hierarchy structure is analyzed.This paper is divided into the following four chapters:The first chapter is the introduction.Through reading relevant literature,the research background and practical significance of this paper are summarized.The first part analyzes the research status of this issue at home and abroad,and understands the theoretical achievements that scholars have made,as well as the fields that have not been involved at present,so as to grasp the key points for research.The last part summarizes the definitions,theorems and lemmas that will be used in the analysis model.The content of this chapter lays a foundation for the research of this paper.In the second chapter,it is assumed that the mortality of individual population is not only due to the natural death of individual,but also depends on the total amount of individual and the effect of sterility agent.The existence and uniqueness of the solution of the system are proved by using the method of characteristic lines and Banach fixed point theorem.The continuous dependence of the system solution on the control variable is proved by comparison principle.The necessity of the existence of optimal strategy is analyzed by using the definition of normal cone.The third chapter in the case of control the birth rate,the establishment of a class depends on the scale structure of species population model,first of all,by using the fixed point theorem is proved that the system is the only solution,then push through rigorous theoretical analysis to the continuous dependence of control variables,the last conjugate system is derived and starting from the characteristics of the structure of normal cone export control is the necessary condition of optimal harvest.Chapter 4 discusses a competitive population model in which there is a hierarchical system based on age within the population.Firstly,the existence of the optimal solution is proved by using the relative compactness and the extreme sequence method,and the necessity of the existence of the optimal control is proved by constructing the conjugate system and using the definition of the normal cone.