The Study Of Some Predator-pery Models With Impulsive Control And Periodic Coefficient

Biomathematics is a new borderline subject between biology and mathematics. It has four subfields which are mathematical ecology, biostatistics, quantitative ge-netics and mathematical medicine. Mathematical ecology, a fast developing subfield, quantitatively studies the change process of ecological systems with mathematical methods and ecological models. It mainly studies population ecology which explores the relationship between populations and the environments, the interaction among populations, centering on the spatial distribution of populations and the change law of quantity. People have established various models to describe the competitive, predator-prey, and cooperative relationship among the populations as well as the environment. Studying these models can help us to predict the developing process of the populations, to regulate and to control population system, to serve the protec-tion of natural ecological environment, and to promote the coordinated development of human society economy and nature environment.In the early research of population dynamics, the development of population system was generally modeled by an ordinary differential equation system, and the state of the model was determined by the parameters of the system and the time. However, there is no independent population in nature, and population may be af-fected by various outside factors resulting in a sudden change of the system variables or the system law. In these cases, an ordinary differential equation is not capable of describing this phenomenon, while an impulsive differential equation works. The theory of impulsive ordinary differential equation emerged in1960’s, and has been greatly developed since then. It has been used to describe population dynamical systems, to optimize biological resources, and to control pest ecosystems etc.In this paper, we use the theory of impulsive differential equations and popula-tion dynamical system to study several predator-prey models with impulsive control, and mainly study the globally asymptotically stability of prey-free periodic solution, the existence of the periodic solution and globally asymptotically stability, and the permanence of the system. Furthermore, we investigate the complicated dynamics behaviors of the system by numerical analysis method. The results not only enrich the theory of impulsive differential equations but also have a wide application and solve some practical problems, and have important practical significance.The following problems are discussed and results are obtained in this thesis:(1) We studied a class of general periodic Lotka-Volterra impulsive system. Using the comparison principle of impulsive differential system, Floquet theory and some special analysis techniques, we established the sufficient conditions that ensure existence of the periodic solution, and analyzed globally asymptotic stability of the system. Furthermore, we apply the general results obtained to a predator-prey system, obtaining the existence and uniqueness of positive periodic solution to the system. Some relevant nature of extinction of population species is further discussed.(2) We studied a class of predator-prey model with impulse control. First, using Floquet theory, we obtained that the system has a unique pest-free periodic solution which is globally asymptotically stable under certain condition. Then, using the fixed point theorem and Lyapunov function, we obtained the sufficient conditions which ensure existence of the periodic solution and globally asymptotic stability of the system. Finally, we verified the theoretical results using numerical simulation.(3) We studied the persistence of a class of predator-prey model with two kinds of impulse in a period. In order to control the increase of pests, it is necessary to spray pesticides and to put on the natural enemy at different time. First, us-ing Floquet theory and the comparison theorem of impulsive differential equations, we obtained the condition under which the pest-free periodic solution is globally asymptotically stable. Then, using the comparison theorem of impulsive differen-tial equations and some special analysis technique, we proved that the system is permanent when the conditions of pest-free periodic solution do not hold.Since many changes in nature are periodic, this paper mainly studies the pop-ulation models with periodic coefficient and impulsive control. It is more significant to study the models with periodic coefficient than constant coefficients, the study is of course more difficult and more complicated, and the results can be better applied in practice. The results obtained about the existence of periodic solution for the general Lotka-Volterra system and the globally asymptotical property have theoret-ical significance to impulsive differential system study. The results of the existence and stability of solutions, of the existence of positive periodic solutions and of the permanence of the system have some practical significance.