Schnakenberg pattern formation

Next we’ll consider a classical reaction–diffusion system which forms Turing patterns,

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

where we need $D>1$ to form patterns, and typically take $a,b>0$.

• Load the Schnakenberg simulation

• Click within the box to visualise a pulse of a population, which will then spread out as a planar wave leaving patterns behind it.

• You can change the diffusion coefficients to effectively change the size of the domain (the diffusion coefficients will scale like $1/L^2$ where $L$ is the domain size, so decreasing both diffusion coefficients by $100$ will effectively simulate a domain $10$ times larger). As the patterns have approximately fixed wavelengths, this should lead to a different number of pattern elements.

• With $D=100$, the system forms spot-like patterns. If you reduce to $D=30$, instead stripe-like patterns will be formed.

• The homogeneous equilibrium can undergo Hopf bifurcations for small values of $1 > b > a \geq 0$. In this regime, one can find Turing patterns, homogeneous oscillations, and complex spatiotemporal interactions of Turing and Hopf instabilities. One example is this interactive simulation, but you can find others by tweaking the parameters.

• Check out this growing simulation for a crude implementation of the Schnakenberg system on a growing domain with Dirichlet boundary conditions.

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