Development of reaction-diffusion systems - Studies of singular limit methods -

反应扩散系统在自然科学的许多领域都有涉及。在这个项目中，我们发展了奇异极限程序和一般数值方法，以研究这类系统中的模式，形式，界面和奇异性的动力学。下面，我们展示了我们在这个项目中所获得的结果：（1）利用奇异极限方法导出竞争扩散系统的自由边界问题（2）利用奇异极限方法定性研究生物系统的尖峰解。(3)斜梯度反应扩散系统的稳定性分析。(4)（5）反应扩散方程组的行波解的奇异摄动方法（6）高维空间中平均曲率的运动分析。(7)扩散-趋化系统中聚集模式的分析（8）反应-扩散系统中复杂时空模式的数值模拟（9）反应-扩散系统自适应有限元方法的发展

Reaction-diffusion systems have been encountered in many fields of natural sciences. In this project, we develop singular limit procedures and comtempolarily numerical methods in order to investigate dynamics of patterns, forms, interfaces and singularities in such systems. In the below, we show our results obtained in this project:(1) Derivation of a free boundary problem from competition-diffusion systems by using singular limit procedures(2) Qualitative study of spiky solutions in biological systems by singular limit procedures.(3) Stability analysis of skew-gradient reaction-diffusion systems.(4) Interaction of traveling pulses and spots by using dynamical system theory(5) Traveling wave solutions of reaction diffusion systems by using singular perturbation methods(6) analysis of motion of mean curvatures in higher dimensional spaces.(7) Analysis of aggregating patterns in diffusion-chemotaxis systems(8) Numerical simulations of complex spatio-temporal patterns in reaction-diffusion systems(9) Development of adoptive FEM methods to reaction-diffusion systems