Concentration Phenomena in Pattern Formation

图案形成中的集中现象提出研究摘要倪卫明该奖项是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1952年图灵的著名思想-扩散驱动的不稳定性-提出的形态发生中的激活剂-抑制系统的尖峰层稳态。他们的分析利用了这两种化学物质的扩散速率之间的差距，并在各种情况下获得了稳定性结果。多维子集上的集中现象已经取得了进展。此外，最近还描述了相应动力学系统的完整动力学。然而，人们对原始扩散系统的动态仍远未完全了解。对于空间不均匀的情况，这些扩散系统的稳定性性质可能与它们的自治系统非常不同，这一点将被研究。在略有不同的方向上，已经注意到仅扩散速率之间的差距不足以产生图案。因此，理论生物学家在1979年提出的“交叉扩散”概念将被系统地研究，该概念被用来模拟种群动力学中的隔离现象。近年来，交叉扩散系统也被用来模拟奇异现象，包括细菌菌落的树枝状生长。它们都是非线性的，并且在最高阶项上是强耦合的，所以它们在数学上非常具有挑战性。这些扩散和交叉扩散系统以及它们的影子系统将在自治和非自治情况下进行研究。他们的定性行为以及他们的稳定性属性也将被调查。这个奖项是为了继续一个研究项目，试图以数学严格的方式理解各种与扩散相关的机制的现象和影响。这些结果应该会改进应用科学中更复杂和/或更现实的现象的建模，以及在创造新的和重要的数学方面。这项研究将研究在自治和非自治情况下，交叉扩散系统及其影子系统中出现的各种“集中现象”所导致的图案形成及其稳定性。具体的例子包括图灵模式，如Gierer-Meinhardt的激活剂-抑制剂系统中的图灵模式，形态发生过程中水合物再生现象的模型，模拟细菌菌落树枝生长的非线性扩散系统，以及考虑了物种间种群压力的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.