Reaction-diffusion models are mathematical models used to describe the behavior and interplay of reactions and diffusion in various systems, such as in chemical and biological processes. They combine the effects of chemical reactions and diffusion to see how patterns change over time and space. Because these models can create complex patterns that are beyond imagination from simple rules, it is often used in the context of computer graphics and generative art as well.

反応拡散(Reaction-diffusion)モデルは、化学や生物学的なプロセスなど様々なシステムの振る舞いや、反応と拡散の相互作用を記述するために使われる数学モデルです。 化学反応と拡散の効果を組み合わせて、パターンが時間と空間にわたってどのように変化するかを調べます。簡単なルールから思ってもみないような複雑なパターンを形成することができるので、コンピュータグラフィックスやジェネラティブアートの文脈でもよく用いられます。

Gray-Scott model


The Gray-Scott model is the most famous reaction-diffusion model. It was proposed by James P. Gray and Scott K. Scott in 1983. Since it is widely used, you may have encountered it before, to the point of becoming a cliché (so much so that I hesitated to add yet another article).


Before discussing the model in detail, let's see what it looks like. Run the demo below. Did you see the black square spread, forming a clover-like shape, and eventually transforming into something that looks like a flower or worms? The Gray-Scott model is known for producing a variety of complex patterns depending on its parameters.


Break down of the model


The Gray-Scott model is a model that describes the interaction and diffusion of two imaginary "substances". It is like dropping two different chemicals into water and observing their reaction. Let’s call these substances as V and U. Several things happen simultaneously.


  1. Diffusion: Just as ink dropped into a glass of water will spread out, substances U and V will also spread out over time.
  2. Reaction: U and V react with each other. When they meet, U turns into V. The rate of the reaction depends on the concentrations of both substances.
  3. Feed: We keep feeding U into the system to compensate for the amount lost during the reaction. The rate becomes faster when there is less U and slower when there is more.