Lengyel-epstein Equation With Reaction - Diffusion - Migration Equation

Lengyel-Epstein equation is Proposed from the Modeling experiments on the CIMA reaction, while the Reaction-Diffusion-Advection model is derived from bio-mathematics to the Population competitive model. The main work of this paper is studies the stability of two types of equations with different values of the parameters. Specifically, for Lengyel-Epstein equation, we get a rage of a, when a satisfy the conditions, the solutions of the Lengyel-Epstein equation are all global asymptotic stability; For the Reaction-Diffusion-Advection model, through the analysis of com-petition model, we get the Stability of the model when the parameters are different.This paper is organized as follows:In the first chapter, we make a simple introduction to "Lengyel-Epstein equation" and "reaction-diffusion-advection model". Some notations and preliminaries relate to the paper are described in the second chapter. we consider the in-fluence of the feed concentration of activator in the Lengyel-Epstein reaction-diffusion system. By constructing a proper Lyapunov function, we show that when the feed concentration is small enough, the constant equilibrium solution of the Lengyel-Epstein reaction-diffusion system is globally asymptotically stable. We also show that all solutions converge uniformly to the con-stant equilibrium solution. Finally, in the fourth chapter, we study a competing species model for tow organisms with different feed inhabiting a spatially heterogeneous environment. by using the perturbation analysis, we get some results.