The Partial Stability For Ordinary Differential Equations And The Stability For The Ecological Systems Of Reaction-diffusion Type

This thesis for master degree studies the partial stability for ordinary differential equations and the stability and the existence of periodic solutions for ecological systems of reaction-diffusion type. It consists of four parts. 1. We present a new approach to dealing with the stability, asymptotic stability and exponential stability with respect to partial variables for time-varying linear systems. We give constructive algebraic criterion without introducing the difficulty of constructing Lyapunov functions.2. Using techniques of Lyapunov functions and inner products, we extend the approach and the results in Part I and give criterion on the stability, asymptotic stability and exponential stability with respect to partial variables for time-varying nonlinear systems.3. We construct Lyapunov functions with separable variables and prove results on global stability with respect to partial variables for nonlinear and non-autonomous systems with separable variables. The results extend those of autonomous systems to the case for non-autonomous systems.4. The stability and the existence of periodic solutions for ecological systems of reaction-diffusion type are discussed by using of Smoller invariant domains principle,Lyapunov functional method,comparison principle and fixed point theorem.