Mathematical biology
Reaction–crossdiffusion systems
We now consider an example of a crossdiffusion system based on the following reaction kinetics:
\[\begin{aligned}\pd{u}{t}&=\vnabla\cdot(D_{uu}\vnabla u+D_{uv}\vnabla v)+au+u^2v,\\ \pd{v}{t}&=\vnabla\cdot(D_{vu}\vnabla u+D_{vv}\vnabla v)+bu^2v,\end{aligned}\]which is a crossdiffusion version of the Schnakenberg model.

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The default parameters create localised inverted spots (sometimes called ‘dark solitons’) wherever the perturbation is, but these do not seem to propagate patterns in any direction.

Setting the value of $b=1$ gives pattern formation closer to the Schnakenberg system observed before, though note that the selfdiffusion terms are equal.

Finally taking $b=0.1$ allows for spatiotemporal behaviors, as the homogeneous equilibrium is then well into a Hopf regime.