Diffusion and Cross-Diffusion in Pattern Formation

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Abstract Ni Professor Ni plans to continue his research in understanding mathematically the effects of various diffusion-related mechanisms. Using calculus of variations and perturbation techniques, Professor Ni and his collaborators have established the spike-layer solutions for an activator-inhibitor system in morphogenesis by exploiting the gap between the diffusion rates of the activator and inhibitor. Stability of such solutions with single peak in one dimension is also proved. However, to undrestand the complete dynamics, multipeak solutions in multi- dimensions and their stability properties are very important and not yet fully understood. They are currently under investigation. It should be noted that the gap between the diffusion rates alone is not sufficient in creating patterns, as is shown by the Lotka-Volterra competition-diffusion system in the weak competition case. Thus, from the point of view of pattern formation, the notion of "cross-diffusion" was introduced in 1979 by theoretical biologists. Cross-diffusion systems are both nonlinear and strongly-coupled in the highest order terms, and are therefore mathematically challenging. Professor Ni and his collaborators proposes to study the effect of cross-diffusion: First, to obtain necessary and sufficient conditions for cross-diffusion rates to create patterns, and then to investigate their qualitative behavior as well as their stability properties. 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 or realistic situation in applied sciences, as well as creating new and significant mathematics. In this project, from the point of view of pattern formation, Professor Ni intends to investigate the various "concentration phenomena" in diffusion 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, and the Lotka-Volterra competitions with cross-diffusion (in modeling segregation phenomena) in population dynamics.