Concentration Phenomena in Pattern Formation

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Concentration Phenomena in Pattern Formation Abstract of Proposed Research Wei-Ming Ni This award is to investigate the mathematical analysis of various diffusion-related mechanisms. The P.I. and his collaborators have studied 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. Their analysis exploits the gap between the diffusion rates for the two chemical substances, and stability results in various cases have also been obtained. Progress has been made for concentration phenomena on multi-dimensional subsets. Moreover, the complete dynamics of the corresponding kinetic systems has recently been described. However, the dynamics of the original diffusion system is still far from being fully understood. For the spatially inhomogeneous case, stability properties of these diffusion systems can be very different from their autonomous counterparts and this will be investigated. In a slightly different direction 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, will be systematically studied. Cross-diffusion systems have also been used in recent years to model singular phenomena including dendritic growth of bacteria colonies. They are both nonlinear and strongly-coupled in highest order terms, so they are very challenging mathematically. These diffusion and cross-diffusion systems as well as their shadow systems will be studied in both autonomous and non-autonomous cases. Their qualitative behavior as well as their stability properties will also be investigated.. This award is to continue a research project that tries to understand, in a mathematically rigorous manner, the phenomena and effects of various diffusion-related mechanisms. The results should improve the modeling of more complicated and/or realistic phenomena in applied sciences, as well as in creating new and significant mathematics. This investigation will study pattern formation resulting from various "concentration phenomena" occurring in cross-diffusion systems, and their shadow systems, in both autonomous and non-autonomous cases, as well as their stability properties. Particular examples include Turing patterns, as in Gierer-Meinhardt's activator-inhibitor systems, models of the regeneration phenomena of hydra in morphogenesis, a nonlinear diffusion system modeling dendritic growth of bacteria colonies, and Lotka-Volterra competition systems with inter-specific population pressures taken into consideration.