To illustrate the mechanism of pattern formation, consider the hypothetical activator-inhibitor reaction sequence:

(R1)

(R2)

(R3)

(R4)

(R5)

and .

From the expression of the reaction rate of , , the species inhibits the activator production, hence its name: the larger its concentration, the lower the production rate of .

Introducing the variables and , the governing equations of the reaction-diffusion system in 2D can be written in the following nondimensional form:

,

,

with Set . For the formation of spatial patterns, the diffusion rates of activator and inhibitor should be very different: set the diffusion coefficients to be and .

For the numerical solution of the system of ODEs, assume (1) periodic boundary conditions as well as (2) the initial conditions:

,

,

where and are numbers taken randomly in the interval .

The Chebyshev orthogonal collocation method applied with nodes in both spatial directions transforms the system of two coupled nonlinear PDEs into a system of 392 nonlinear coupled first-order ordinary differential equations. This system of ODEs is solved using the built-in Mathematica command NDSolve. The snapshots show the results for the 2D inhibitor concentration distribution at times and at . It is interesting to note, in particular, the emergence of Turing patterns similar to leopard spots in the concentration distribution at .