Gierer–Meinhardt pattern formation

Another Turing system is the Gierer–Meinhardt model, given by

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u+a+\frac{u^2}{v}-bu,\\ \pd{v}{t}&=D\nabla^2v+ u^2-cv,\end{aligned}\]

where we take $a,b,c>0$ and $D>1$.

Note that the colour scale here is fixed, but that changing parameters will lead to solutions with different maxima and minima, so using the auto-snap feature under ColourAuto snap may be advised.

Stripes stability

We can observe the instability of stripe patterns in this model by choosing initial conditions which become stripes along one direction. We set

\[u(0,x,y) = 1+\cos\left(\frac{n\pi x}{L}\right), \quad v(0,x,y) = 1,\]

with $n$ an integer.

Saturation leads to stripes

A common way to obtain stripe-like patterns in this model is to consider saturation of the self-activation term ($u^2/v$ in the equation above). In this case we have the model

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u+a+\frac{u^2}{v(1+Ku^2)}-bu,\\ \pd{v}{t}&=D\nabla^2v+ u^2-cv,\end{aligned}\]

where $K>0$ is a saturation constant.

For very large values of $K$, the system will not admit Turing patterns, and for very small values it will behave as in the above spot-forming model. However, for intermediate values of $K$ one can get labyrinthine patterns.