Concentration Phenomena in Pattern Formation

研究摘要倪伟明本奖项旨在研究各种扩散相关机制的数学分析。P.I.和他的合作者研究了Gierer和Meinhardt基于图灵在1952年提出的著名思想——扩散驱动的不稳定性——在形态发生中提出的活化剂-抑制剂系统的尖刺层稳态。他们的分析利用了两种化学物质的扩散速率之间的差距，并获得了各种情况下的稳定性结果。多维子集上的集中现象研究取得了进展。此外，最近还描述了相应动力学系统的完整动力学。然而，原始扩散系统的动力学仍远未完全了解。对于空间非均匀的情况，这些扩散系统的稳定性可能与它们的自治系统非常不同，这将被研究。在一个稍微不同的方向上，已经注意到仅扩散速率之间的差距不足以产生模式。因此，“交叉扩散”的概念，引入了理论生物学家在1979年建模隔离现象在种群动力学，将系统地研究。近年来，交叉扩散系统也被用于模拟奇异现象，包括细菌菌落的树突生长。它们在最高阶项上都是非线性和强耦合的，因此在数学上是非常具有挑战性的。这些扩散和交叉扩散系统以及它们的影子系统将在自主和非自主情况下进行研究。他们的定性行为和稳定性也将被研究。这个奖项是为了继续一个研究项目，试图理解，在数学上严格的方式，各种扩散相关机制的现象和影响。这些结果将改善应用科学中更复杂和/或现实现象的建模，以及创造新的和重要的数学。本研究将研究在自主和非自主情况下交叉扩散系统及其影子系统中发生的各种“浓度现象”所产生的模式形成，以及它们的稳定性。特别的例子包括图灵模式，如Gierer-Meinhardt的激活剂-抑制剂系统，水螅在形态发生中的再生现象模型，模拟细菌菌落树突状生长的非线性扩散系统，以及考虑到种间种群压力的Lotka-Volterra竞争系统。

Concentration Phenomena in Pattern Formation Abstract of Proposed ResearchWei-Ming NiThis award is to investigate the mathematical analysis of various diffusion-related mechanisms. The P.I. and his collaborators have studied the spike-layer steady states for an activator-inhibitor system in morphogenesis proposed by Gierer and Meinhardt based on the celebrated idea - diffusion-driven instability - of Turing in 1952. Their analysis exploits the gap between the diffusion rates for the two chemical substances, and stability results in various cases have also been obtained. Progress has been made for concentration phenomena on multi-dimensional subsets. Moreover, the complete dynamics of the corresponding kinetic systems has recently been described. However, the dynamics of the original diffusion system is still far from being fully understood. For the spatially inhomogeneous case, stability properties of these diffusion systems can be very different from their autonomous counterparts and this will be investigated. In a slightly different direction it has been noted that the gap between the diffusion rates alone is insufficient to create patterns. Thus, the notion of "cross-diffusion," introduced by theoretical biologists in 1979 in modeling segregation phenomena in population dynamics, will be systematically studied. Cross-diffusion systems have also been used in recent years to model singular phenomena including dendritic growth of bacteria colonies. They are both nonlinear and strongly-coupled in highest order terms, so they are very challenging mathematically. These diffusion and cross-diffusion systems as well as their shadow systems will be studied in both autonomous and non-autonomous cases. Their qualitative behavior as well as their stability properties will also be investigated..This award is to continue a research project that tries to understand, in a mathematically rigorous manner, the phenomena and effects of various diffusion-related mechanisms. The results should improve the modeling of more complicated and/or realistic phenomena in applied sciences, as well as in creating new and significant mathematics. This investigation will study pattern formation resulting from various "concentration phenomena" occurring in cross-diffusion systems, and their shadow systems, in both autonomous and non-autonomous cases, as well as their stability properties. Particular examples include Turing patterns, as in Gierer-Meinhardt's activator-inhibitor systems, models of the regeneration phenomena of hydra in morphogenesis, a nonlinear diffusion system modeling dendritic growth of bacteria colonies, and Lotka-Volterra competition systems with inter-specific population pressures taken into consideration.