The Existence Of The Positive Periodic Solution To Some Impulsive Competition Systems

Impulsive differential equations have recently been applied to population ecosys-tems more and more extensively. Comparing with differential equations, impulsive differential equations can model some ecological phenomena more exactly. In popu-lation dynamics, one may easily visualize situations in nature where abrupt changes such as harvesting, disasters and instantaneous stocking may occur. If the effects of instantaneous changes of the population densities are taken into consideration, the mathematical models are described naturally by impulsive differential equations.The theory of impulsive ordinary differential equation started in the sixties of the twentieth century, and it was recently an object of intensive investigations. It is an important subject in recent years applying the impulsive differential equation to describe and study the developing process of the ecological system. Biomathematics is a borderline science of biology and mathematics aiming at researching and solving biological problems with mathematical method. Using the impulsive differential sys-tem models to describe and study the ecosystem is not only the theoretical basis to qualitative analysis and quantitative research to the ecological system, but also the technology to solve the practical problem of ecosystem, optimize the management of ecology systems and improve the ecological and economic benefits. Thus, the more and more attentions are given to the study of impulsive differential ecosystems.The following problems are discussed and results are obtained in this thesis:(1) The existence of the positive periodic solution is studied for a class of Lotka-Volterra competition model with impulsive harvesting. The harvesting function is supposed to be linear, that includes both cases of the proportional and constant har-vesting. The corresponding impulsive logistic equation is discussed firstly, and based on the results obtained for logistic equation, a monotone-iterative scheme is estab-lished to obtain the existence of the positive periodic solutions for the competition system. The competition model established is new and the method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic solution.(2) The two classes of impulsive competition models with delay are studied, the existence of the positive periodic solutions and the persistence are discussed. The corresponding impulsive logistic equation is discussed firstly, and based on the results obtained for logistic equation, a monotone-iterative scheme is established to obtain the existence of the positive periodic solutions for the competition system. Moreover, we also show that the system is permanent under the certain conditions. Our results improve and extend some known results.(3) An optimal harvesting control problem of a biological species is considered. The species is harvested at fixed moments for economic profit. The purpose is to control the harvesting effort so as to maximize the profit which is the difference between economic revenue and cost, assuming that the cost in the payoff functional depends quadratically on the control variable. We show the existence of an optimal impulsive harvest control, and under certain conditions, we characterize the optimal control in terms of an impulsive optimality system. For the optimality system, using the method of upper and lower solutions and its associated iterative scheme, monotone sequences closing in to all appropriate solutions of optimality system can be constructed.All the problems involved in this thesis are foundational in impulsive differential equations theory and its applications. The results not only enrich and improve impulsive differential equations in theory, but also are expected to be applied to real ecosystems.