Diffusion and Cross-Diffusion in Pattern Formation

摘要Ni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。利用变分法和摄动技术，Ni教授和他的合作者通过利用激活剂和抑制剂的扩散速率之间的差距，建立了激活剂-抑制剂系统形态发生的尖峰层解决方案。并证明了这种单峰解在一维上的稳定性。然而，要了解完整的动力学过程，多维的多峰解及其稳定性是非常重要的，但尚未完全了解。他们目前正在接受调查。应该注意的是，扩散率之间的差距本身并不是充分的增长模式，正如弱竞争情况下Lotka-Volterra竞争-扩散系统所示。因此，从模式形成的角度来看，1979年理论生物学家提出了“交叉扩散”的概念。交叉扩散系统在最高阶项上是非线性和强耦合的，因此在数学上具有挑战性。倪教授和他的合作者提议研究交叉扩散的影响：首先，获得交叉扩散速率形成图案的必要和充分条件，然后研究它们的定性行为和稳定性。倪教授计划继续他的研究，以严谨的数学方式理解各种扩散相关机制的现象和影响，希望借此对提高我们在应用科学中模拟更复杂或现实情况的能力，以及创造新的和重要的数学产生一些影响。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，特别是包括化学反应中的图灵模式（例如CIMA反应），形态发生中的Gierer-Meinhardt的活化剂-抑制剂系统（用于模拟水螅再生现象），以及种群动力学中的Lotka-Volterra竞争与交叉扩散（用于模拟隔离现象）。

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.