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 $D>1$.

  • Load the interactive simulation

  • Changing any of the parameters can lead to different solutions, though this system generically favours spot-like patterns.

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.

  • Load the simulation

  • Change the value of $n$ and restart the simulation by pressing . In each case a different number of initial stripes will evolve into some number of stripes, but they should persist indefinitely.

  • Now click on or near a stripe to destabilise it into spots. Note that the brush value, found in Brush is set to $1.01u$, so that it is a relatively small perturbation of the current solution.

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.

  • Observe the patterns in this stripey simulation
  • Try increasing or decreasing the size of $K$ to observe how this influences spot/stripe selection in the system. Note that the colour scale is changing to match solution maxima and minima.