Analysis And Control Of A Nonlinear Size-Structured Population Model With Elastic Growth

Biological population dynamics is one of the fundamental research topics in ecology, which displays the evolutional law of a population to a certain extent. Population ecol-ogy is a very important branch of ecology, whose basic object is population composed of individuals. Vital parameters of individuals directly affect the individuals growth and evolutional trend of the population., and the evolution of a population is actually a process of the interaction between the population and the environment. Besides, with the develop-ment of human concern with ecological balance and sustainable development, more and more scholars devote to study of population-related control problems, such as stability of equilibria and optimal harvesting. Many research results have been established for age-structured population models since the last century. Studies in ecology show that the body size is more significant than age of an individual for many species(e.g. forests and fishes). Therefore, more and more results about size-structured population models appear.The aim of this dissertation is the analysis of a class of size-dependent population model with elastic growth, and the main concern includes:dynamical behaviors (e.g. ex-istence and uniqueness of solutions to the model, boundedness and non-negativeness, and stability of equilibria), and optimal harvesting in steady states. Furthermore, we present a numerical scheme to approximate the population density and prove its convergence. The main tools used in the dissertation are differential and integral equations, functional anal-ysis, linear algebra and control theory. The results obtained can be used as scientific basis to the applications of the model.The principal works of this dissertation are as follows:The chapter 2 proposes the model, and shows the well-posedness by a coupled upper and lower solutions, provides a criterion for the stability of equilibria by means of char-acteristic equation. The chapter 3 sees a upwind difference scheme for an approximate solution of the size distribution, and the convergence is demonstrated. Finally, we treat an optimal harvesting problem in which the effort is required to keep the balance of the pop-ulation. Existence of unique optimal policies has been verified and optimality conditions derived out.