Behavior Analysis And Optimal Control Of Biological Population Dynamical Systems With Size-structure

Age-structured population models are a traditional tool for mathematical modeling and control in biology and demography. The study of population systems are of great significance for species conservation, management of renewable resources, containment of pests and parasites, and epidemic intervention. Recently, size-structured models have attracted much attention and a number of research results appeared. By size we mean a set of continuous indexes related to target individuals, such as mass, length, diameter, volume, maturity, or any other quantity displaying their physiologic or statistical characteristics. Consequently, age structure is only a special kind of size structure. Generally speaking, size is more intuitive and easier to measure than age, and size-structured models better describe the dynamics of populations than age-structured ones, especially for marine invertebrates (e.g., Barnacles and Corals) and many poikilothermal animal species (e.g., fishes and snakes). On the other hand, size-structured models are expected to play more important role in the exploitation of renewable resources.This dissertation is concerned with several classes of size-structured population systems, for which we investigate dynamical properties (e.g., existence, uniqueness, non-negativeness, boundedness, stability of stationary solutions, continuous dependence on model parameters, etc.) and control problems such as optimal harvesting and optimal birth control. By means of (linear and nonlinear) functional analytic approach (e.g., semigroups theory, Mazur's theorem, Ekeland variational principle, etc.), differential equations, integral equations, and modern control theory, we obtain some theoretical results, which provide a solid ground for the practical applications of the involved models.The principal works of this dissertation are as follows:It is undoubted that the survival of individuals of population is dependent on resources (e.g., foods). Therefore, we formulate and analyze a size-structured population model with resources-dependence and inflow in chapter 2. In section 2.1, we propose the basic model, which is a nonlinear hybrid system of ordinary and partial differential and integral equations, and present a method to solve the positive stationary of the basic system. Then in section 2.2, we linearize the nonlinear system and derive some regularity properties for the linearized system by means of the semigroups theory, following that we deduce the characteristic equation and establish some conditions for stability and instability of the stationary solution in section 2.3. Section 2.4 consists of two examples and their computer simulations, which are used to show the effectiveness of the theoretical results. Finally, we extend this model to the multi-resources-dependent situations. Chapter 3 is devoted to the study on the optimal control problems of size-structured population models. Section 3.1 focuses on an optimal harvesting problem. We firstly state the control problem and assumptions. Then by making use of the Banach fixed point theorem and Bellman's inequality, we prove well-posedness and boundedness of solutions to the basic system, and derive existence of a unique solution to the adjoint system. Next we show the existence of optimal controls by means of Mazur's theorem, and establish optimality conditions of first order in the form of maximum principle via normal cone technique. Finally we demonstrate the uniqueness of optimal control. Section 3.2 handles a control problem with birth rate as control variable. Firstly, the basic model and assumptions are proposed. Then we show existence and uniqueness of solutions both to the state system and the adjoint system, and make several estimates, which will be useful in the derivation of the uniqueness of optimal control. Furthermore, we establish first order necessary conditions of optimality via tangent-normal cone techniques, and prove the existence of a unique optimal control by means of the Ekeland variational principle. Finally, we establish an example and its numerical result. Our conclusions in this chapter cover the corresponding results of age-structured population models. Although our index and the underlying model are simple, we still get an insight that the price factor of individuals plays a key role in the structure of the optimal controller.