What next?

Turing’s patterns

It is obvious that many natural phenomena are not captured by differential equations because there are changes both in time and space. Mathematically, this means that we have to consider partial differential equations. In the so-called reaction-diffusion equations, there are local reactions in which the substances are transformed into each other and can be degraded, and diffusion which causes the substances to spread out in space. Such equations display a wide range of behaviors, such as traveling waves, front propagation (biological invasion), self-organized patterns (spots and stripes in animal skin, vegetal pattern in arid regions, etc), and so forth and so on.

In 1952, Alan Turing was the first to propose a simple reaction-diffusion model to account for the development of spots like the ones appearing on the pelage of certain animals. The general model he proposed describes the interaction between two chemicals he called `morphogens’ which diffuse at different rates. One serves as activator to express a unique characteristic, like a leopard’s spot, and the other acts as an inhibitor, which can suppress the activator’s expression. The model is 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 $u=u(x,y,t)$ is the concentration of the activator, $v=v(x,y,t)$ that of the inhibitor. The functions $f,g$ describe the local reactions. It is not important to know their explicit expressions here. The terms $\nabla^2 u, \nabla^2 u$ describe the diffusion of the morphogens. The parameter equation_4.pdf is the diffusion coefficient of the activator, $B$ that of the inhibitor.

In the digital experiment, you can produce a variety of natural-looking textures by tuning the parameters $A$ and $B$, thereby making for instance the activator diffusing more rapidly than the inhibitor. What you visualize is the concentration of the activator.