对高阶自催化反应扩散模型和Langford推广扩散模型的时空斑图研究进行计算机辅助分析。利用图灵分支理论、Hopf分支理论、Hassard中心流形理论和规范型计算方法，建立单重特征处的图灵分支和Hopf分支的存在性及稳定性；综合运用隐函数定理、Lyapunov-Schmidt约化方法和奇异性理论，建立二重特征值处的平衡态分支、Turing-Hopf分支、奇异性静态和动态分支，给出分支结构和斑图稳定性判定条件；基于Maple和Matlab平台，建立辅助分析两类反应扩散模型平衡点稳定性、Hopf分支和Turing分支产生的条件推导及分支处振幅方程建立的机械化算法，在空间一维和二维情况下研发两类反应扩散模型分支图、斑图、相图绘制的GUI软件包，分析斑图结构及演化过程。期望所得研究成果为其他反应扩散型的深入研究提供有效依据并起到重要指导意义，进一步丰富计算机符号计算的研究内容。

This project is mainly concerned with the computer-aided analysis on the spatio-temporal patterns of an autocatalysis reaction-diffusion model with high order and the generlized Langford diffusion model. By Turing bifurcation theory, Hopf bifurcation theory, Hassard center manifold theory and normal form method, the existence and stability of Turing bifurcation and Hopf bifurcation at the simple eigenvalue are obtained; By implicit function theorem, Lyapunov-Schmidt technique and singularity theory, the steady-state bifurcation at the double eigenvalue, Turing-Hopf bifurcation and singularity bifurcation are established. Based on Maple and Matlab, the mechanization algorithms are designed for the stability of the equilibrium, the conditions of Hopf bifurcation and Turing bifurcation, and the corresponding amplitude equation. GUI software package is finally made for depicting the bifurcation graph, spatio-temporal pattern and phase diagram. Pattern structure and evolution process are investigated and revealed. This project can provide effective tools for further researches on other reaction-diffusion models and play an important guiding role. Meanwhile, it will further enrich the research content of computer symbolic computation.