Model Analysis And Control Problems Of A Hierarchical Age-structured Competitive System

The difference of dominance rank of individuals within a biological population is widely observed,which has an important impact on individual life parameters and population evolution behavior.We propose a class of new hierarchical model for the evolution of two interacting age-structured populations,which is a system of integro-partial differential equations with global feedback boundary conditions and may describe the competition relation.Based upon a group of natural conditions,by means of differential equations,integral equations,functional analysis and MATLAB,we investigate dynamical properties and control problems.All the theoretical and experimental results provide the necessary conditions for the practical application of the model.The research body consists of chapters 2,3 and 4.The second chapter focuses on the dynamic behavior of the hierarchical age-structured competition system.In section 1,we propose a class of new hierarchical model.In section 2,we introduce the normalized model and basic assumptions.Section 3 analyzes the well-posedness of the model,the existence and uniqueness of solutions are proved by means of fixed-point theorem.Moreover,we show that the solution is non-negative and the continuous dependence of the solution on the initial age distribution is established.In section 4,we prove the existence of positive equilibrium state.In section 5,we discuss the stability of zero equilibrium state.In section 6,we propose a numerical method for solving the model and show its convergence.The numerical simulation of population evolution is carried out with MATLAB to verify the effectiveness of the numerical method.The third chapter is devoted to studies of the optimal control problem of the hierarchical age-structured competition system.Section 1 presents the basic model and assumptions.In section 2,considering the harvesting factor of human to population system,we propose the maximum yield problem,and prove the existence of optimal solution.In section 3,the minimum deviation problem,the minimum principle and the existence and uniqueness of the optimal strategy are discussed.In section 4,based on the model in section 3,the maximum profit problem is proposed and the maximum principle is derived.In addition,the influence of some parameters on these control problems is studied by numerical method.In section 5,the initial distribution control problem of the system is treated.The optimality conditions are derived and the existence of unique optimal solution proved.The fourth chapter is concerned with the approximate controllability of the system.In section 1,the research status of controllability is described.In section 2,the basic model and basic hypothesis of hierarchical competitive system are introduced.In section3,basing on a controllability result of linear system,we prove that the nonlinear system is approximately controllable by means of a fixed point theorem for multi-valued mappings.