Diffusion and Cross-Diffusion in Pattern Formation

倪教授计划继续他的研究，从数学上理解各种扩散相关机制的影响。利用变分和微扰技术，倪教授和他的合作者利用激活剂和抑制剂的扩散速率之间的差距，建立了形态发生中钝化剂-抑制剂系统的尖峰层解。证明了这类一维单峰解的稳定性。然而，要理解完整的动力学，多维的多峰解及其稳定性性质是非常重要的，目前还没有完全了解。他们目前正在接受调查。值得注意的是，在弱竞争情形下，Lotka-Volterra竞争扩散系统表明，仅有扩散速率之间的差距并不是充分的增量模式。于是，从图案形成的角度出发，理论生物学家于1979年提出了“交叉扩散”的概念。交叉扩散系统既是非线性的，又是最高阶强耦合的，因此在数学上具有挑战性。倪教授和他的合作者建议研究交叉扩散的影响：首先，获得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，以惊人的严谨方式理解各种扩散相关机制的现象和影响，从而有望在提高我们对应用科学中更复杂或更现实情况的建模能力以及创造新的和有意义的数学方面产生一些影响。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”。其中尤其包括化学反应中的图灵模式(例如CIMA反应)，形态发生中的Gierer-Meinhardt的激活剂-抑制系统(在模拟水蛇的再生现象中)，以及种群动力学中具有交叉扩散的Lotka-Volterra竞争(在模拟分离现象中)。

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.