Concentration Phenomena in Pattern Formation

图案形成中的集中现象拟研究摘要倪伟明该奖项旨在研究各种扩散相关机制的数学分析。私家侦探1952年，Gierer和Meinhardt基于Turing的扩散驱动不稳定性思想，提出了形态发生中激活剂-抑制剂系统的尖峰层定态。他们的分析利用了两种化学物质的扩散速率之间的差距，并且还获得了各种情况下的稳定性结果。对多维子集上的集中现象的研究已经取得了进展。此外，最近已经描述了相应的动力学系统的完整动力学。然而，原始扩散系统的动力学仍然远未被完全理解。对于空间非齐次的情况下，这些扩散系统的稳定性可以是非常不同的自治同行，这将被调查。在稍微不同的方向上，已经注意到，仅扩散速率之间的差距不足以产生图案。因此，“交叉扩散”的概念，介绍了理论生物学家在1979年在模拟隔离现象的种群动态，将系统地研究。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，并且在最高阶项中是强耦合的，因此它们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阴影系统将在自治和非自治的情况下进行研究。它们的定性行为以及它们的稳定性也将被研究。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。这项调查将研究图案的形成所造成的各种“浓度现象”发生在交叉扩散系统，其阴影系统，在自治和非自治的情况下，以及它们的稳定性。具体的例子包括图灵模式，如在Gierer-Meinhardt的激活剂-抑制剂系统中，水螅在形态发生中再生现象的模型，细菌菌落树突状生长的非线性扩散系统模型，以及考虑种间种群压力的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.