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Lengyel-Epstein方程与反应—扩散—迁徙方程
【作者】 段祖宁；
【导师】 徐本龙；
【作者基本信息】 上海师范大学 ， 基础数学， 2010， 硕士
【摘要】 Lengyel-Epstein方程的提出来自于对CIMA反应实验的建模,而反应-扩散-迁徙模型则来自于生物数学中对种群竞争的建模。本文主要研究了这两类方程中不同的参数取值对方程解的稳定性的影响。具体的说,对于Lengyel-Epstein方程,本文将给出一个关于a的取值范围,使得方程在满足这个条件时,方程的解具有全局渐近稳定性；对于反应-扩散-迁徙模型,本文主要通过对竞争模型的分析,得到了方程中含有的参数在取值不同时,对模型稳定性的影响。本论文的内容安排如下：第一章是对Lengyel-Epstein方程和反应-扩散-迁徙模型做一个简单的叙述；第二章则介绍了与本论文相关的一些预备知识；第三章讨论Lengyel-Epstein方程中注入催化剂的浓度对常数解稳定性的影响,通过构造适当的Lyapunov函数,证明了当注入催化剂的浓度不是很大时,Lengyel-Epstein方程的常数解是全局渐进稳定的,对任意的初值,方程解最终一致收敛到这个常数解。第四章则主要研究了在空间资源分布不均匀的环境下。含有两个竞争种群的反应-扩散-迁徙模型,竞争结果对模型中各个参数的依赖性.通过对各个参数的扰动分析,得到了一些结果.
【Abstract】 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.