(I) Technical Description:
Professor Ni plans to continue his research in understanding mathematically the effects of various diffusion-related mechanisms. Using variational approach and techniques in classical analysis, Professor Ni and his collaborators have established 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. The method exploits the gap between the diffusion rates for the two chemical substances, and stability results in one space dimension have also been obtained. Very recently, progress has been made for concentration phenomena on multi-dimensional subsets. However, the complete dynamics is still far from being fully understood. 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, deserves systematic studies. Cross-diffusion systems, which have also been used in recent years to model singular phenomena including dendritic growth of bacteria colonies, are both nonlinear and strongly-coupled in highest order terms, thus are mathematically very challenging. Professor Ni and his collaborators propose to study the effects of cross-diffusion by first obtain necessary and sufficient conditions for cross-diffusion to help create patterns, and then to investigate their qualitative behavior as well as their stability properties.
(II) Non-technical Descriptions:
Professor Ni plans to continue his research in understanding, in a mathematically rigorous manner, the phenomena and effects of various diffusion-related mechanisms and hopefully thereby to have some impact in both improving our ability in modeling more complicated and/or realistic phenomena in applied sciences, as well as in creating new and significant mathematics. In this proposal, from the viewpoint of pattern formation, Professor Ni intends to investigate the various "concentration phenomena" in diffusion and/or cross-diffusion systems. These, in particular, include Turing patterns in chemical reactions (e.g. the CIMA reaction), Gierer-Meinhardt's activator-inhibitor systems in modeling the regeneration phenomena of hydra in morphogenesis, a nonlinear diffusion system modeling dendritic growth of bacteria colonies, and the Lotka-Volterra competition systems with inter-specific population pressures taken into considerations."
