Analysis And Control Of A Population Model Integrating Spatial Diffusion And Age Hierarchy

Because individuals of animal populations often have hierarchical status differences,face different competitive environments within a population,and they are often in motion,a population model covering hierarchical structure and spatial distribution is proposed in this dissertation,which is an infinite dimensional system composed of a second-order nonlinear partial differential equation with integral boundary conditions.Using functional analysis,variational principles,differential and integral inequalities and MATLAB tools,we study the existence and uniqueness of non-negative solutions to the model,the principle of comparison,the continuity of solutions on initial distributions and distributed controls,two types of optimal control problems(optimal harvesting and optimal initial distribution).Both theoretical analysis and numerical method are given,and some numerical experiments are carried out by MATLAB.This dissertation is divided into three chapters.The first chapter describes the modeling background and the research overview of related fields;the second and third chapters are the core of this dissertation.In chapter 2,the dynamics of the diffusive hierarchical population model is investigated.The section 1 describes the population model and assumptions on the model parameters.In section 2,the existence and uniqueness of nonnegative solutions of the model are proved by means of fixed point method,and the comparison principle and separable solutions are derived.In section 3,an algorithm is given to approximate the solutions to the model,the convergence of the algorithm is established,and numerical simulations of the population model are carried out by using MATLAB.In section 4,the asymptotic behaviors of the population in some simple cases are discussed.In chapter 3,two optimal control problems of population system are researched.Section 1 is devoted to introductive remarks.In section 2,the maximum benefits problem is considered.The existence of optimal strategies is shown by using Aubin’s lemma and extremum sequences.The characteristic description of optimal controls is derived by means of a normal cone and an adjoint system.Numerical experiments are carried out on the equilibrium system.In section 3,optimal control of initial distributions is studied;i.e.,choose controls maximizing the difference between the total economic values at the terminal time and the control costs.The maximum principle is derived,the existence and unique optimal policy is proved via Fatou and Riese lemmas and Ekeland variational principle.Finally,the effects of model parameters on the optimal strategy are presented with MATLAB.