Diffusion and Cross-Diffusion in Pattern Formation

图案形成中的扩散和交叉扩散

• 批准号: 9988635
• 负责人: Wei-Ming Ni
• 金额: $ 25.2万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988635&HistoricalAwards=false
• 关键词: Diffusion; Cross; Pattern; Formation

AbstractNiProfessor Ni plans to continue his research in understandingmathematically the effects of various diffusion-related mechanisms. Using calculus of variations and perturbation techniques, Professor Ni andhis collaborators have established the spike-layer solutions for anactivator-inhibitor system in morphogenesis by exploiting the gap betweenthe diffusion rates of the activator and inhibitor. Stability of suchsolutions with single peak in one dimension is also proved. However, toundrestand the complete dynamics, multipeak solutions in multi- dimensionsand their stability properties are very important and not yet fullyunderstood. They are currently under investigation. It should be notedthat the gap between the diffusion rates alone is not sufficient increating patterns, as is shown by the Lotka-Volterra competition-diffusionsystem in the weak competition case. Thus, from the point of view ofpattern formation, the notion of "cross-diffusion" was introduced in 1979by theoretical biologists. Cross-diffusion systems are both nonlinear andstrongly-coupled in the highest order terms, and are thereforemathematically challenging. Professor Ni and his collaborators proposesto study the effect of cross-diffusion: First, to obtain necessary andsufficient conditions for cross-diffusion rates to create patterns, andthen to investigate their qualitative behavior as well as their stabilityproperties. Professor Ni plans to continue his research in understanding, in amathematically rigorous manner, the phenomena and effects of variousdiffusion-related mechanisms and hopefully thereby to have some impact in bothimproving our ability in modeling more complicated or realistic situation inapplied sciences, as well as creating new and significant mathematics. Inthis project, from the point of view of pattern formation, Professor Niintends to investigate the various "concentration phenomena" in diffusion orcross-diffusion systems. These, in particular, include Turing patterns inchemical reactions (e.g. the CIMA reaction), Gierer-Meinhardt's activator-inhibitor systems (in modeling the regeneration phenomena of hydra) inmorphogenesis, and the Lotka-Volterra competitions with cross-diffusion (inmodeling segregation phenomena) in population dynamics.

倪教授计划继续他的研究，在数学上理解各种扩散相关机制的影响。 利用变分法和微扰技术，倪教授及其合作者利用激活剂和抑制剂扩散速率之间的差距，建立了形态发生中激活剂-抑制剂系统的钉层解。 证明了一维单峰解的稳定性。 然而，为了建立完整的动力学，多维多峰解及其稳定性是非常重要的，但尚未完全理解。他们目前正在接受调查。 在弱竞争情形下，Lotka-Volterra竞争-扩散系统表明，仅扩散速率之间的差距并不是充分的增长模式。 因此，从模式形成的角度来看，理论生物学家在1979年引入了“交叉扩散”的概念。 交叉扩散系统在最高阶项上是非线性的、强耦合的，因此在数学上具有挑战性。 倪教授和他的合作者提出研究交叉扩散效应：首先，得到交叉扩散率产生模式的充要条件，然后研究它们的定性行为以及它们的稳定性。 倪教授计划继续他的研究，在数学上严谨的方式，理解各种扩散相关机制的现象和影响，并希望由此产生一些影响，提高我们的能力，在应用科学中建模更复杂或现实的情况，以及创造新的和有意义的数学。 在这个项目中，倪教授打算从图案形成的角度来研究扩散或交叉扩散系统中的各种“集中现象”。 这些，特别是，包括图灵模式在化学反应（如CIMA反应），Gierer-Meinhardt的激活剂-抑制剂系统（在模拟水螅再生现象）在形态发生，和Lotka-Volterra竞争与交叉扩散（在模拟隔离现象）在人口动态。