Optimal Control Problems For Several Population Models With Size-Structure

It is well known that there are many structural differences among population individuals,such as age,body size,gender and genes,which in turn affect the dynamic behavior of the population.As a result,it is necessary to establish and analyze population models with structures.Long-term ecological researches show that for many populations(such as forests,fish,etc.),size of an individual has a strong influence upon dynamical processes like its fertility,mortality,predation ability and competitive ability,which in turn affect the dynamics of the population as a whole.Here by size we mean some indices displaying the physiological or statistical characteristics of population individuals.Sizes can be mass,length,diameter,volume and maturity.Note that age is only a special kind of size and size is one of the most important variables to describe population dynamics.As a result,modeling population dynamics with size structure has been an active theme in mathematical biology.The dissertation is considered with several population models with size-structure.By means of the functional analysis,differential-integral equations,we investigate the dynamic behavior of the models(including the existence of a unique non-negative bounded solution and the continuous dependence of solutions on parameters,etc.),and using modern control theory,we investigate the optimal control problems(including the optimal harvest control,optimal birth control and optimal contraception control).Some theoretical results obtained in this paper provide scientific theoretical basis for the practical application of the models(optimal exploitation of fish resources and optimal control of vermin population).The second and third Chapters mainly discuss the optimal harvesting problems for size-structured models of fish.The fourth,fifth and sixth Chapters mainly investigate the optimal control problems for size-structured population models that describe the optimal management of the vermin.For the fish,only a fraction of the eggs produced by mature fish can convert to fry and a large amount of fry is needed in the process of artificial breeding.Motivated by these considerations,in Chapters 2 and 3,we investigate the optimal exploitation problems of fish with size-structure.In Chapter 2,we discuss the optimal harvesting problem for a size-structured model of the fish resources.In the modeling,it is assumed that the amount of fry put by the fisherman is a known function.The control variable is the harvest effort,which appears in the main equation.First,we show the existence of a unique non-negative solution of the system,and give a comparison principle.Next,we prove the existence of optimal policies by using Mazur theorem and comparison principle.Then,we obtain the optimality conditions by using normal cones.Finally,numerical analysis is carried out.In Chapter 3,we investigate the optimal harvesting problem for a nonlinear sizestructured model of the fish resources.In the modeling,it is assumed that the amount of artificial stocking fry depends on the size of the fish population.The objective functional not only includes the benefits from harvesting the fish and the cost of controls,but also includes the cost of restocking fish fry and the cost of fish feed.First,we prove the existence and uniqueness of the nonnegative bounded solution of the system by fixed-point theorem.Then,necessary conditions for optimality are established via the normal cone technique.Moreover,the existence of a unique optimal policy is proved via fixed-point reasoning and Ekeland variational principle.Finally,numerical analysis is carried out.As for the vermin,decreasing its reproductive rate instead of using the chemical drugs has been suggested as a promising means for managing the impact of overabundant species.Chapters 4,5 and 6 establish and analyze the optimal control problems for the vermin population models with size structure.In Chapter 4,we investigate the optimal birth control problem for a nonlinear vermin population model with size structure.The control variable is the fertility rate,which appears in the boundary condition.First,we obtain the existence of a unique non-negative solution by considering the separable solution of the model.We also illustrate the continuous dependence of the population density on initial value and control parameter by the inequality theories.For the least cost-size problem,a feedback control strategy is presented by using normal cone and the existence of a unique optimal policy is proved via fixed-point reasoning and Ekeland variational principle.Chapters 5 and 6 establish and analyze the optimal contraception control problems for the vermin population models with size-structure.The basic principle is that the female sterilant is used to reduce the reproductive ability of the vermin population in order to control the number of the vermin.The control variable measures the average amount of female sterilant eaten by a single individual,which appears in the principle equation and the boundary condition as well.In Chapter 5,we investigate the optimal contraception control problem for a sizestructured vermin population model.In the modeling,we assume that the female sterilant can lead to extra death of the vermin.The existence of a unique non-negative solution is established by using the Banach fixed-point theorem.The existence of optimal strategy isderived by Mazur theorem and the optimality conditions are obtained by means of normal cone techniques.Finally,numerical analysis is carried out.In chapter 6,we investigate the optimal contraception control for a nonlinear sizestructured vermin population model.In the modeling,we assume that the mortality of the vermin individual depends not only on the intrinsic but also the effects of population size and the female sterilant.First,we obtain the existence of a unique non-negative solution by considering the separable solution of the mode.Then,the existence of an optimal control strategy is proved via compactness and extremal sequence.Next,optimality conditions are established by the use of tangent-normal cone technique and adjoint system.Moreover,numerical analysis is carried out.