Quantitative Analysis And Control Problems Of Several Discrete Size-Structured Population Models

As its name implies,biomathematics is a joint discipline between biology and mathematics,which uses mathematical methods to study and solve biological problems.Mathematical ecology is an important branch of ecology.Population is a basic unit of species in nature,and a basic subject of ecological research.It is composed of all individuals in same species,inhabits in a common space area,and has the potential of reproduction,independent features,structures and functions.In order to protect the ecological environment and biological diversity,to maintain the ecological balance and to utilize the renewable resources reasonably,we need to analyze the evolutional laws of populations.For this purpose,the scholars all over the world have established a large number of mathematical models,including continuous and discrete types.The discrete population models have their advantages,which are suitable for describing the evolution of the populations in small size and non-overlapping generations.Since the development of discrete population model proposed by Leslie in 1945,the researchers have paid more and more attention at discrete type models.Studies on discrete population models would be helpful for prediction of the long-term evolution of the population,and,on the other hand,for the development of scientific management strategy of resources exploitation,such as some optimal harvesting problems based on the population persistence.In this dissertation,we mainly discuss some discrete population models with size-structure,which is divided into two kinds: linear and nonlinear.For the linear model,the optimal harvesting policy of the population in steady states,and the controllability and feedback control of the system are our main concerns;while for the nonlinear models,the existence and stability of the model equilibrium,and the optimal harvesting problem are mainly studied.The main tools used are matrix theory,discrete system control theory and numerical analysis,etc..In the second chapter,a class of linear size-structured model is established,which is based on consideration of the phenomenon of the delayed growth.The optimal harvesting problem of a class of equilibrium conditions is studied,and the optimal solution is obtained by using Lagrange multiplier method.Then we analyze the controllability and stabilization of the population system,and present the feedback control method.In the third chapter,a nonlinear model is discussed,which is mainly based on the observation of group-crossing growth.We show that the model solutions are non-negative and bounded bymeans of the point dissipation theorem and primitive matrix.Then,the existence and stability of equilibriums are discussed.The nonlinear model in the fourth chapter is based on the linear one in the second chapter.Because of the sincere difficulty,we just deal with the situations from system with two size groups.The optimal harvesting strategy is proposed,and the effects of vital parameters on the optimal harvesting efforts are analyzed.Finally,we use MATLAB to carry out some numerical simulations,which verify the theoretical results.