Several Kinds Of Optimal Harvesting Control Of Population Systems With Spatial Diffusion And Age-structure

Nowadays, it is self-evident to the importance of ecosystem. As a main subsystem of ecosystem, the study of population system referring to dynamical behavior and opti-mal control becomes necessary. It is not only because mathematic model can be used to describe and study the process of internal mechanism in biological system. More impor-tantly, it has great ecological value to the study of population system such as maintaining the ecological balance and protecting biological diversity, controlling population intrusion,optimizing the ecological environment, preventing and controlling epidemic, finally using the renewable resources scientifically and so on.The paper studied the optimal control issues of multi-population diffusion system considering the toxin, age, density and more. The author analyzed the issues referring to dynamical and optimal control. The dynamical behavior assumes the existence and uniqueness of solution, non-negative boundedness, stability and continuous dependence of the solution of control variables. Controlling problems have the minimum cost and maximum profit. The main mathematical tools used in the analysis include the real function and functional analysis, differential equations and optimal control theory and so on. This paper aims to offer some practical theoretical basis for the practical application of the model.The main contents of the paper are as following:The chapter two builds up and analyzes the optimal harvesting control issues of a predator-prey population system with functional response function under the influence of toxin. The first section sets the model and gives some basic assumptions. The second proves the existence and uniqueness of solution, non-negative boundedness through the operator semigroup theory. The first order necessary optimality condition is derived by means the technique of tangent-normal cones. The forth uses the Lagrange function to get the second order optimal condition of the optimal harvesting policy by judging the negative derivation of Hessian matrix.The chapter three discussed the optimal harvesting control issues of the three popula-tions with Beddington-DeAngelis functional response in which the two preys are compete.The first section sets the model and gives some basic assumptions. The second proves the existence, uniqueness, non-negative boundedness and continuous dependence of the solu-tion of control variables. The third section gets the optimal conditions of controlling issues by using technique of tangent-normal cones. The forth section proves the uniqueness of the optimal control pair by using Ekeland variational principal.In the fourth chapter, we consider the optimal harvesting control problem for a class of nonlinear diffusion systems with age-structure. The rate of birth and death are nonlinearly dependent on the total size of the population. It reflects that the living environment and congestion degree have practical influence on the dynamic process. The first section built up model and gave some preliminary results. The second proved the existence of the optimal harvesting by using the Mazur theory. The third derived the necessary conditions of optimal harvesting control by using the technique of tangent-normal cones. The forth got the criteria for the stability of equilibrium through obtaining characteristic equation of equilibrium in the process of handling linearization system.