Turing’s “The Chemical Basis of Morphogenesis”

Reaction-Diffusion Model: Two morphogens – one activator and one inhibitor which has larger diffusion coefficients.

$\displaystyle \frac{d X_{r}}{d t}=f\left(X_{r}, Y_{r}\right)+\mu\left(X_{r+1}-2 X_{r}+X_{r-1}\right)$

$\displaystyle \frac{d Y_{r}}{d t}=g\left(X_{r}, Y_{r}\right)+v\left(Y_{r+1}-2 Y_{r}+Y_{r-1}\right)$

Use Fourier transform on the space variables to separate $X_r$ and then we a linear ODE in two variables.