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# Turing patterns

Rajouté le Thursday 10 February 2022

Alan Turing was the first to propose a model to account for the very large diversity of patterns in nature, such as animal coats. This model is based on a "reaction-difusion equation" of the form(*)

$\begin{cases} \frac{\partial u}{\partial t}=f(u,v)+A \nabla^2 u\\\frac{\partial v}{\partial t}=g(u,v)+B \nabla^2 v\end{cases}$

where $u(x,y,t)$ is the concentration at point $(x,y)$ and at time $t$ of the activator (which color the skin), and $v(x,y,t)$ is that of the inhibitor (which prevents the activator from being expressed). The positive coefficients $A,B$ are the diffusion coefficients.

In the digital experiment below, we have a portion of the plan $(x,y)$ and we have taken $f(u,v)=u(v-1)-12$, $g(u,v)=16-uv$. What is represented is the minimum of $u(x,y,t)$, in black, and its maximum, in red.

(*) $\nabla^2$ is the Laplacian operator: $\nabla^2 u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}$.

To integrate this simulation into your own web pages:

<iframe style="overflow: hidden;" src="http://experiences.math.cnrs.fr/simulations/sysdyn-Morphogenese-en/index.html" height="600" width="800" frameborder="0" scrolling="no"></iframe>

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