Asymptotic Properties Of Some Impulsive Competition Systems

Many processes in nature and in science and technology have the following characters:following smooth change in a relative long time, due to some natural or mankind perturbation, a process may suddenly change at some moments. Compar-ing to the whole process, the duration of the perturbation and the sudden change is negligible. It is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. Thus impulsive differential equations, that is, differential equations involving impulse effects, can well describe these processes. In early stud-ies of population dynamics, researchers often used classical differential equations to describe the growth of population. However, ordinary differential equations cannot accurately describe the processes with impulsive effects. Therefore, it is necessary using impulsive differential equation to study population dynamics.Since the1960’s, the study of the theory of impulsive ordinary differential equation has widely attracted the attention from researchers in ecology and mathe-matics. This greatly promoted the theoretical and application research of impulsive differential equations. In recent years, impulsive differential equation is introduced to study the population dynamics. Biomathematics is a borderline subject of math-ematics and biology, agriculture as well as medicine. It uses mathematical methods and techniques to solve practical problems in above subjects. Using impulsive differ-ential equation, we can transform a complex biological problem into a mathematical problem. Studying the property of an impulsive differential equation can help us understand the developing law of a biological system quantitatively or qualitatively.In this thesis, we use the theory about the impulsive differential equation and time delay differential equation to study the dynamical behavior of some periodic population competitive systems with impulse, focusing on the existence and unique-ness of the periodic solutions of impulsive differential equations, the persistence of the systems, and the global attraction. Our results can enrich the study of impulsive differential equation, can help make decision on practical biological problems, and thus have some practical significance.The main results of this thesis are as follows:(1) We studied a class of two-dimensional periodic Lotka-Volterra competition system with impulsive effects. The impulse condition in the model describes the proportional harvesting or stocking to the population at fixed time. We mainly studied the global asymptotical property. First, using the comparison principle of impulsive differential equation, we studied the property of corresponding one-dimensional impulsive differential equation, obtained boundedness of the solution. Based on this, we further studied the asymptotical property of the solution using some analytical techniques. Then we discussed the existence and uniqueness of the positive periodic solution. Finally, using Floquet theory, we studied the relevant properties of extinction and semi-trivial periodic solutions.(2) We studied the existence and uniqueness of positive periodic solutions to a class of two-dimensional delay competition system with impulsive effects in the periodic environment. Firstly, by using the comparison principle and some analysis techniques, we studied the properties of corresponding one-dimensional differential equation, obtained the boundedness and asymptotic properties of the system. Then, by constructing a continuous operator and using Brouwer fixed point principle, we obtained the existence and uniqueness of the positive periodic solutions, and con-sidered asymptotic properties of the system.(3) A model of periodic Gilpin-Ayala competitive system with impulse and time delay is studied. This model can depict the impulsive stocking or harvesting effects as well as the developing process affected by time delay factor. Firstly, corresponding one-dimensional system without delay are studied, and the properties of impulsive systems with delay are obtained by using the periodic property. The global asymp-totic properties of the system are studied by using Lyapunov-Razumikhin method. Then under some conditions the existence and uniqueness is obtained.(4) We studied a model of two population competition system with stage struc-ture and impulse in polluted environment. Using the related theory of impulsive differential equations and delay differential equation, we studied the positivity and boundedness of the solutions. On this basis, we studied the extinction condition-s of one population of two populations, and obtain the sufficient conditions that guarantee the persistence of the system.