The English mathematician, logician and computer scientist, Alan Turing, devised an insightful paper relating to aspects of mathematical biology. The paper ultimately implied that reaction diffusion equations may be utilised by specific organisms to generate a pattern during the early stages of their life. In doing so, he formulated a number of differential equations, which were formed on the basis of chemicals known as morphogens. Morphogens are the chemical interactions responsible for generating patterns; one hinders the production of itself and the other chemical (called Inhibitor) and one increases production of both chemicals (Activator).
Nevertheless, I shall not delve too much into the biological aspect of Turing's findings, for this is, of course, a website dedicated to the study of mathematics. I have provided several links below, for all those biology enthusiasts.
https://www.livescience.com/950-leopard-spots.html
https://www.lakeforest.edu/live/news/7986-the-turing-mechanism-how-did-the-leopard-get-his
As I was saying, the concentrations of the activator and inhibitor are given by A(x,t) and I(x,t) individually. Over a period of time, these concentrations gradually change, due to reactions and diffusion.
Therefore, the change of concentration over time for activators would be given by $\frac{\delta A}{\delta t} = f(A,I) + \frac{\delta ^2 A}{\delta x^2}.$
The change of concentration with respect to time for an inhibitor, on the other hand, would be represented in an equation as: $ \frac{\delta I}{\delta t} = g(A,I) + d\frac{\delta ^2 I}{\delta x^2}.$
The first term on the very right refers to the quantity of activator being produced, and f(A,I) is a second derivative describing how much the activator is changing. As you may be conscious of the equation for the concentration of an inhibitor contains a d, whereas for an activator, it does not. The d is essentially the diffusion coefficient, implying that the inhibitors diffuse at a faster rate than the activators.
However, I severely lack the intellectual capacity to comprehend these partial differential equations, and would quite appreciate it if someone could provide me an insight.