Pattern formation in reaction diffusion systems - theory and applications

反应扩散系统中的图案形成 - 理论与应用

• 批准号: 312604-2011
• 负责人: Iron, David
• 金额: $ 0.8万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571782
• 关键词: Pattern; formation; reaction; diffusion; systems

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.

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，即反应扩散方程。 在这些系统中，扩散与非线性反应项的组合通常会导致高度局部化的空间结构的形成。 这些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。 具体来说，我目前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。 第三个项目考虑了平面上不同格子排列的结构的相对稳定性。 最后一个主题考虑由生态模型得出的几个偏微分方程。 这一数学研究领域有着悠久的历史，可以追溯到艾伦·图灵的论文“形态发生的化学基础”（1951）。 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。 最近的实验证据验证了许多数学结果，并提供了需要数学建模和分析的新问题。 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。 在理解模式的自发形成方面的任何进展都会影响广泛的领域。