Analysis And Control Of Hierarchical Age-Structured Population Model With Periodic Rates

The structures,functions and dynamic changes of ecological system have di?erent manifestations in di?erent levels in nature,which is an ordered whole with multi-level hierarchical structures.On the other hand,there are hierarchy di?erences among individuals within the population,and the hierarchy of age is the most intuitive.Due to seasonal changes and other cyclical factors,the individual life parameters of the population often experience periodic oscillations.In this dissertation,we will propose a kind of age hierarchical population model with periodic parameters.The mathematical theory is used for system analysis and control.The former is the basis,and the latter uses the conclusion to implement reasonable transformation of the population.The results of this dissertation also provide scientific theoretical support for practical application except academic significance.The research content of this dissertation consists of chapters 2,3 and 4.The second chapter mainly analyzes the well posedness of the hierarchical agestructured system in periodic environment.In section 1,we propose an age hierarchical population model in which individual life parameters oscillate periodically due to environmental factors.Furthermore,we give the basic assumptions according to actual background and theoretical analysis needs.In section 2,the existence,uniqueness,nonnegativity and boundedness of periodic solutions to the model are established by means of fixed point approach.In section 3,an upwind di?erence scheme is proposed to approximate the solutions for the model,and the convergence of the algorithm is proved.Finally,we present some test examples,and numerical simulation is carried out by using MATLAB.The third chapter mainly discusses the optimal control problem of population system in the regulation period.In Section 1,our discussion builds on the system described in chapter 2,and the basic assumptions for relevant parameters are established.In Section 2,we consider the harvest control of the population,and propose an optimal harvest problem.The well posedness conclusion is given,and the existence of optimal control of the system is proved by using Mazur’s.In addition,we derive the maximum principle,then proceed numerical simulations.In Section 3,we discusse the initial distributed control problem of the system.The optimal strategy is carefully depicted via the characteristics of the tangent cone and normal cone,and an adjoint system.Moreover,Ekeland’s variational principle is used to show the existence of unique optimal strategy.Finally,the numerical simulation results are presented.The fourth chapter mainly studies approximate controllability and a class of optimal control problems based on boundary control.In section 1,a kind of nonlinear model with control variables on the boundary is proposed.Then we give the basic assumptions.In section 2,the continuity of the population densities with respect to the control is established by characteristics method and Gronwall’s inequality.In section 3,we prove that the population system is approximately controllable by means of a controllability result for linear systems and Ky Fan-Glicksberg fixed-point theorem for set-valued mappings.In section 4,for a specific minimum cost model,Ekeland’s variational principle is used to show the existence of unique optimal strategy,and then the optimal strategy is carefully depicted via an adjoint system and a normal vector.Finally,some numerical experiments display the feasibility of the boundary control policy.