Stability And Optimal Harvesting Strategy For Hierarchical Population Models With Age-structure

In the field of ecology,the dynamics of biological populations is a basic research topic.The evolution of populations and development trends can be revealed to a large extent by studying population dynamics.Since the twentieth century,due to the increase of human activities,ecological balance,Biodiversity has been severely damaged.For ecological stability and sustainable development,more and more scholars have begun to study population control issues(such as system controllability,optimal control issues,etc.).From the last century to the present,based on The research on age-structured population models has achieved quite a lot of results.However,ecological studies have shown that for many populations(such as forests,large marine fish,etc.),individuals of all ages within the population have different life parameters for individuals of a certain age.The impact is different.Based on such considerations,a hierarchical age structure model is established.Based on the hierarchical age structure model,this paper studies its stability and optimal harvesting strategy.The main work of this article is as follows;In Chapter 2,a hierarchical age-structured population model was established and analyzed,and the situation in which new individuals migrated from the external environment was also considered.First,a nonlinear model was established with partial differential equations and differentialintegral equations,and the Related assumptions.Then the formal solution of the model is obtained by the characteristic line method,and the related properties of the formal solution are discussed using Banach fixed point theorem and Gronwall inequality.Chapter 3 discusses the equilibrium state of the hierarchical age structure population model.First,the existence and uniqueness of the solution of the state system is proved by the feature line method and the comparison principle;then through linearization,the existence of positive equilibrium is given;The stable and unstable conditions of the positive equilibrium state are out.Chapter 4 studies the optimal harvesting problem of population models with hierarchical age structure.First,the basic model and the relevant assumptions of the model parameters are given;then the Banach fixed point theorem and Gronwall inequality are used to prove the well-posedness and boundness of the state system solution And the existence and uniqueness of the solution of the conjugate system;then,with the help of Mazur's theorem,the existence of the optimal control is proved;finally,the tangent cone method is applied to give the optimality condition,and the uniqueness of the optimal control is proved.