Pattern formation in reaction diffusion systems - theory and applications

A common feature of nonlinear partial differential equations is the spontaneous formation of ordered structures. I study a particular class of such partial differential equations, reaction diffusion equations. In these systems the combination of diffusion with nonlinear reaction terms often results in the formation of highly localized spatial structures. Such systems come from models in such areas of science as developmental biology, ecology, chemistry and optics.In particular I am currently working in four distinct areas. The most mature investigation is a mathematical model of intracellular signaling in which the enzymes promoting the production of signal molecules are fixed to specific locations within the cell. Another project considers adding delay to the mathematical models of protein synthesis to mimic the long times required for some of the steps in protein synthesis. The third project considers the relative stability of structures arranged in different lattices on the plane. The final topic considers several partial differential equations resulting from ecological models.This area of mathematical investigation has a long history dating back to Alan Turing's paper "A Chemical Basis for Morphogenesis" (1951). It is now a vibrant area of research with groups all over the world studying these systems. Recently experimental evidence has validated many of the mathematical results as well as providing new problems to be modeled and analyzed mathematically. Applications for this research come from many different areas of science including developmental biology, ecology, microwave welding and lasers. Any advances in understanding the spontaneous formations of patterns would impact a wide range of fields.