Analysis And Exact Null Control Of A Class Of Structured Population Dynamics

European grape moth (EGVM) has been the most serious wine pest in Europe, North Africa, and even some Asian countries. It has a significant impact on the vineyards. A physiologically structured multistage population model is then pre-sented to study the dynamics of one of the most important grapevine insect pests. In this thesis, we mainly make a qualitative analysis and exact null control for a class of structured population dynamical systems.Firstly, we consider a multistage, physiologically age structured dynamics sys-tem, with adult moths diffusing around the vineyard. Growth function of the pop-ulation at each stage is modeled considering the climatic variations and the grape variety, which depends on the physiological age, and allows us to model great vari-ability of growth within a cohort. Based on the contraction fixed point principle, the existence and uniqueness of solutions for the model can be obtained. Then we also prove the existence of a global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, the theory of compact operators and the Krasnoselskii’s fixed point theorem are used to prove the existence of steady states.Next, it is meaningful to think about the control problem of this Lobesia bo-trana model. First of all, we investigate the exact null controllability of an age-dependent life cycle dynamics with nonlocal transition processes arising as boundary conditions. Assume that the four stages of this system:egg, larva, female moth and male moth are all in static station. We obtain the null controllability for the pest by acting on eggs in a small age interval. The main method is based on the derivation of estimations for the adjoint variables related to an optimal control problem. Then we apply a fixed point theorem to draw the conclusion that the population of egg except the small enough age groups to zero at a certain moment in the future, using an age-and time-dependent control of eggs.Inspired by the above result, it is necessary to consider the control problem for the Lobesia botrana population dynamics system, while the adult moths can be diffusive. Therefore, we describe a control by a removal of egg and larva population, and also on female moths in a small region. Specifically, we transform the nonlocal term into a local one, create one optimal control problem, and get the backward adjoint system related to the original system. Then combining some estimations and the Carleman inequality for the local backward system related to an optimal control problem, and choosing a control corresponding to a fixed point of a multi-valued function, the null controllability for female moths in a nonempty open sub-domain at a given time except a small enough age interval can be proved.