Pattern formation in reaction diffusion systems - theory and applications

非线性偏微分方程的一个共同特征是有序结构的自发形成。 我研究一类特殊的偏微分方程，反应扩散方程。 在这些系统中，扩散与非线性反应项的结合常常导致高度局域化的空间结构的形成。 这样的系统来自发育生物学、生态学、化学和光学等科学领域的模型。 目前，我主要在四个不同的领域工作。 最成熟的研究是细胞内信号传导的数学模型，其中促进信号分子产生的酶被固定在细胞内的特定位置。 另一个项目考虑在蛋白质合成的数学模型中添加延迟，以模拟蛋白质合成中某些步骤所需的长时间。 第三个方案考虑平面上不同格网布置结构的相对稳定性。 最后一个主题考虑了几个生态模型产生的偏微分方程。 这一领域的数学研究有着悠久的历史，可以追溯到阿兰·图灵的论文《形态发生的化学基础》（1951年）。 它现在是一个充满活力的研究领域，世界各地的团体都在研究这些系统。 最近的实验证据已经验证了许多数学结果，以及提供新的问题进行建模和数学分析。 这项研究的应用来自许多不同的科学领域，包括发育生物学，生态学，微波焊接和激光。 在理解图案的自发形成方面的任何进展都将影响广泛的领域。

A common feature of nonlinear partial differential equations is the spontaneous formation of ordered structures. I study a particular class of such partial differential equations, reaction diffusion equations. In these systems the combination of diffusion with nonlinear reaction terms often results in the formation of highly localized spatial structures. Such systems come from models in such areas of science as developmental biology, ecology, chemistry and optics. In particular I am currently working in four distinct areas. The most mature investigation is a mathematical model of intracellular signaling in which the enzymes promoting the production of signal molecules are fixed to specific locations within the cell. Another project considers adding delay to the mathematical models of protein synthesis to mimic the long times required for some of the steps in protein synthesis. The third project considers the relative stability of structures arranged in different lattices on the plane. The final topic considers several partial differential equations resulting from ecological models. This area of mathematical investigation has a long history dating back to Alan Turing's paper "A Chemical Basis for Morphogenesis" (1951). It is now a vibrant area of research with groups all over the world studying these systems. Recently experimental evidence has validated many of the mathematical results as well as providing new problems to be modeled and analyzed mathematically. Applications for this research come from many different areas of science including developmental biology, ecology, microwave welding and lasers. Any advances in understanding the spontaneous formations of patterns would impact a wide range of fields.