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 $D>1$.
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Load the interactive simulation
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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 → Colour → Auto 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.
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Load the simulation
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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.
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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.