Section 1

Table of Contents

Section 3


2. Reaction-diffusion models in 2D

Historically, the first model of morphogenesis was proposed by Alan Turing [Tur1952], and is known as the reaction-diffusion model. The model operates in a plane, as shown in Plate 1.

Plate 1 : Principle of the reaction-diffusion model

Each point of this plane is characterized by two numbers, representing concentrations of substances (morphogens) a and b. A system of coupled partial differential equations describes changes of these concentrations over time. The substances diffuse and react with each other.

In the equations, the reaction components are captured by functions f and g, and the diffusion components are represented by the remaining terms. The original intent of the reaction-diffusion model was to explain the ``breakdown of symmetry and homogeneity'' or the emergence of a pattern in an originally homogenous medium.

An example of this process, using a variant of the reaction-diffusion model proposed by Young [You1984], is shown in Plate 2.

  Plate 2: Pattern emergence

In a sequence of steps, the areas of high concentration (yellow) become clustered, producing a pattern of light blotches in a dark background.

Plate 3 illustrates the effect of varying model parameters on the final pattern. In nature, the pattern on the right-hand side can be found, for example, in feathers of some birds, and the middle pattern can be found in the rabbit fish (as noticed and modeled by Camazine [Cam1993]).

  Plate 3: Activator-inhibitor patterns

The idea of using reaction-diffusion models for image synthesis purposes was introduced at Siggraph 1991 by Witkin and Kass [Wit1991], and Turk [Tur1991], who applied it to generate synthetic animal coat patterns.