Dynamical Analysis And Optimal Control Of A Hierarchical Size-structured Population Model With Delay

For a long time,the research on biological populations has attracted much attention,not only because of its important economic and environmental values,but also as an important object of ecology.Since the rank difference of individuals in populations is widely observed,the establishment of appropriate models can help to understand and predict the evolution of populations.There are mainly three kinds of structured population models: age-structured models,size-structured models and hierarchical models.The first two kinds of models are well studied,while the last is still the research hotspot of scholars.Among the study results of hierarchical models,the work on related control problems is rare.In this dissertation,a continuous hierarchical size-structured model is proposed and studied.The main results are divided into two parts: Chapters 2 and 3,which are roughly stated as follows:In Chapter 2,the dynamic behaviors of the population are analyzed.In Section 1,the population model is proposed,which is a nonlinear integro-partial functional differential equation with a boundary condition including individual incubation time delay.In Section 2,the well-posedness of the model is analyzed,i.e.,the existence,uniqueness and boundedness of the non-negative solutions are proved by using the freezing coefficients method,the fixed point principle and the piecewise time domain analysis.In sections 3and 4,the steady states of the model are concerned,and the existence of the nontrivial equilibria is proved.In the fifth section,a numerical method is presented and two examples are given,and the MATLAB is used to simulate the evolution of the population to verifys the effectiveness of the algorithm.In Chapter 3,three kinds of optimal control problems-optimal harvesting problem,optimal input control problem and optimal initial distribution problem are discussed,respectively in sections 1 to 3.For each kind of problem,the existence of the optimal solution is proved and the optimality conditions are derived by a normal cone and an adjoint system.