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 (see caption). 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 (see caption). 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 (see caption) 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]). 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.
Section 1
Table of Contents
Section 3