Mathematical study for pattern formation on localized moving area in reaction-diffusion systems

本文研究了反应扩散型系统在受限运动区域内的模式形成问题。列色岗型降水系统就是这种模式形成的典型代表。众所周知，由于凝胶浓度的差异或其性质的不同，列色岗型沉淀的最终模式可能会有所不同。此外，期望对系统的数学理解可以导致对自组织机制的控制。在研究中，我们用所提出的模式对烈色岗型降水进行了多次模拟，并观察了最终的模式与外界噪声的关系。虽然，对于以数学分析为目标的收缩，遗憾的是，我们并没有得到令人满意的结果，但是，根据我们从模拟中得到的建议，我们想得出结论，未来的发展是有可能的。

In this study, we dealt a pattern formation problem which is developed in restricted moving area of reaction-diffusion type system. As an example of such pattern formation, the Liesegang type precipitation system is famous. It is well known that the final patterns of the Liesegang type precipitation may differ due to the gel concentration difference or its nature. Furthermore, it is expected that the mathematical understanding of the system can be lead to control the self-organization mechanism. In the study, we did several simulations with our proposed model for the Liesegang type precipitation, and we witnessed the relation between the final pattern and the external noise. Although, when it came to the contraction which goal is mathematical analysis, unfortunately, we could not obtain satisfactory result, however, we would like to conclude that there is a potential of future development based on the suggestion we got from the simulation.