Asymptotic Analysis On Solution Of Several Classes Of Biomathematics Models

Singular perturbation theory and method is a very active and expanding discipline. Its development has gone through more than a century. It contains rich content. In the deepening of the study, the increasing continuously of results, the singular perturbation has become an important branch of applied mathematics. It plays a very important role in the exploration of nonlinear phenomena in various fields. Now singular perturbation obtained the very good development, and there are a lot of singular perturbation methods to obtain the development, including matching method, synthetic expansion method, the method of multiple scales and so on.This paper studies singular perturbation method and perturbation theory used in Mathematical Biology. Then, the asymptotic properties of solutions of biological models is discussed. The main contents are as follows:1. The brief introduction of study on the method of the main research object, mathematical biology, establishing a population model, the singular perturbation theory development history, and the application in Biomathematics;2. Elaborate the necessary a priori knowledge;3. A class of single population Logistic models with slow variation and initial value problems are studied. In real world situation, the parameters in the Logistic models may vary with time. By using synthetic method, the formal asymptotic solution is constructed. Under the conditions, it is proved that the uniform validity of the asymptotic solution. Then the error between the approximate solution and the exact solution via the way of upper and lower solution is estimated;4. A class of Lotka-Volterra equations with slow variation and initial values problems are studied. We apply matching method to obtain the asymptotic solution. Under appropriate conditions, the uniform validity of the asymptotic solution is proved. Then, via the way of upper and lower solutions, the error between the asymptotic approximating solution and the exact solution is estimated;5. A class of epidemic models are studied. Considering the infection rate 0 ??1, the perturbation solution is constructed. Then, it is proved that the uniformly valid asymptotic solution, the error between the asymptotic approximating solution and the exact solution is estimated;...