Notes on the Turing Instability and Chemical Instabilities

In these notes, I discuss in more detail the nonlinear evolution equations and their linear stability analysis for chemicals that react and diffuse in solutions. Historically, such a linear stability analysis of a uniform state was first carried out in 1952 by Alan Turing [9]. He suggested the radical and highly stimulating idea that reaction and diffusion of chemicals in an initially uniform state could explain morphogenesis, how biological patterns arise during growth. Although reaction-diffusion systems are perhaps the easiest to study mathematically of the many experimental systems considered in this book, they have the drawback that quantitative comparisons with experiment remain difficult. The reason is that many chemical reactions involve short-lived intermediates in small concentrations that go undetected, so that the corresponding evolution equations are incomplete. Still, reaction diffusion systems are such a broad and important class of nonequilibrium systems—prevalent in biology, chemistry, ecology, and engineering—that a detailed discussion is worthwhile.