We now consider an example of a cross-diffusion system based on the following reaction kinetics:

\[\begin{aligned}\pd{u}{t}&=\vnabla\cdot(D_{uu}\vnabla u+D_{uv}\vnabla v)+a-u+u^2v,\\ \pd{v}{t}&=\vnabla\cdot(D_{vu}\vnabla u+D_{vv}\vnabla v)+b-u^2v,\end{aligned}\]

which is a cross-diffusion version of the Schnakenberg model.

  • Load the interactive simulation

  • 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 self-diffusion terms are equal.

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