Optimal Harvesting Control For Several Classes Of Multi-Population Systems With Size-Structure

Human beings and nature are interdependent,and the healthy and balanced ecosystem is closely related to each species.In order to protect the sustainable development of the ecosystem and economy,it is necessary to establish and analyze population system models,and to study related control problems.In the study of population system control problem,optimal harvesting control has very important ecological and economic significance.At present,many scholars have done a lot of research on the control of age-dependent population systems,and have achieved fruitful results.Compared with age-dependent,the application value of size-structure is more extensive.Therefore,this paper studies optimal harvesting control for several classes of multi-population systems with size-structure.The paper is mainly divided into the following four chapters:The first chapter is the introduction.The research background and significance of the paper is briefly introduced.The current research status on the optimal control of population systems with age-dependent and size-structure is summarized.The preliminary knowledge related to the paper is also introduced.In the second chapter,the optimal harvest rate control problem for a predator-prey system with size-structure is studied.The existence and uniqueness of the solution is proved by using the fixed point theorem.Then,the adjoint system is derived and the necessary conditions for optimal harvesting control are obtained by means of tangent and normal cones.In the third chapter,the optimal control of a prey-predator population system with size-structure in periodic environments is studied.The size growth rate considered in this chapter is a function of size and time.Firstly,the formal solution of the system is showed by using the characteristic line and the constant change method.Then,the existence and uniqueness of the system solution is proved by applying the knowledge of volterra integral equation and operator spectral radius.The existence of the optimal solution of optimal control is proved by applying Mazur theorem.Then the adjoint system is derived and the necessary conditions for optimal control are obtained by means of tangent and normal cones.In the fourth chapter,the optimal harvesting control for a class of size-structured n-dimensional food chain model is studied.Firstly,the existence,uniqueness and continuous dependence of the system solutions are proved by using comparison principle and fixed point theorem.Then the existence of the optimal solution of optimal control is proved by applying Mazur theorem.Finally,the adjoint system is derived and the necessary conditions for optimal harvesting control are obtained by means of tangent and normal cones.Under this condition,the total economic benefits obtained by individuals in the harvesting population can be maximized.