Stability And Optimal Harvesting For A Size-structed Population Model With Weighted Sizes

The dynamics of biological populations based on individual structural diferences isone of the basic research topics. It plays a very important role in the evolution of pop-ulations. Population ecology is an significant part of ecology and the individual vitalparameters are the basis of population ecology research. Because the whole population’sgrowth or reduction, persistence or extinction, development or degradation are afected bythe individual’s birth, growth, reproduction and death processes. And ecological environ-ment afects the evolution of the population by changing the individual life process. Onthe other hand, for the concern with ecological balance and resource development, schol-ars pay more and more attention to the study of control problems of population dynamicalsystems. As far as the age distribution of population is concerned, many research resultshave been obtained since1960’s. However, ecological studies show that, in the process ofevolution, the body size of the individuals are more important than age for many species,such as trees, fish, etc. Therefore, population models with size structure have attractedwide attention in recent years.The purpose of this dissertation is to study two kinds of population model with sizestructure and weighted population size, and investigate their dynamics, such as the ex-istence, uniqueness, non-negativeness, boundedness and the continuous dependence ofsolutions on the control variables, the stability of the equilibrium, and the optimal harvest-ing control problem. We use some tools in functional analysis (e.g. Mazur’s theoremand Ekeland’s variational principle), diferential equations, integral equations and moderncontrol theory to obtain some theoretical results, which provide theoretical basis for thepractical application of the models.The principal works of this dissertation are as follows:The second chapter discusses the well-posedness of a class of nonlinear size-structuredpopulation model, including the existence, uniqueness of the solution, continuous depen-dence of solutions on control variables.The third chapter is devoted to the study of existence and stability of steady statesof a nonlinear population model. We incorporate newborn individual migration into themodel and get some conditions for stability of equilibrium. The fourth chapter analyzes an optimal harvesting problem of nonlinear populationmodels, the structure of the optimal strategy is carefully described.