In Section 2.2 of the previous chapter we used a simplified mathematical model Eq. (2.3) to identify the assumptions, mathematical issues, and insights associated with the linear stability analysis of a stationary uniform nonequilibrium state. In this chapter, we would like to discuss a realistic set of evolution equations and the linear stability analysis of their uniform states, using the example of chemicals that react and diffuse in solutions (see Fig. 1.18). Historically, such a linear stability analysis of a uniform state was first carried out in 1952 by Alan Turing [106]. 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.

The chapter is divided into two halves. In the first part, we introduce the simple model put forward by Turing, and give a careful analysis of the instability of the uniform states. In the second part, we apply these ideas to realistic models of experimental systems.