This chapter begins the discussion of another important application of the diffusion equation, which is the modeling of pattern formation in biology. In this regard, you should know that the development of complicated patterns and structures in biological systems is currently under intense study. In particular, it is clear now that developmental patterns arise because

• Cells in an embryo react differently to different concentrations of certain molecules.

• Moreover, these concentrations are definitely not constant over the embryo.

Current investigations seek to determine answers to the following questions: First, which molecules are relevant to the formation of a particular structure? Next, how do the cells react to these molecules? Finally, how do the concentrations of the relevant molecules come to vary across the embryo?

The last question suggests a model of pattern formation that uses a diffusion equation. However, even if diffusion is the appropriate model, the story as uncovered so far seems anything but simple. (For example, often there are a multitude of molecules involved that affect each other's concentration through complicated feedback loops. When fully sorted out, the picture will be dazzling.)

In any event, here is a pertinent question: What is the simplest diffusion model that exhibits complicated spatial patterns? In particular, can a complex pattern arise from the varying concentration of a single chemical species? For example, can the varying concentration of a single chemical give a tiger stripes?