Reaction–cross-diffusion systems

Biology | Parabolic | Patterns

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

$$\begin{array}{rl}\frac{\partial u}{\partial t}& =\mathbf{\nabla }\cdot (D_{uu}\mathbf{\nabla }u+D_{uv}\mathbf{\nabla }v)+a-u+u^{2}v,\\ \frac{\partial v}{\partial t}& =\mathbf{\nabla }\cdot (D_{vu}\mathbf{\nabla }u+D_{vv}\mathbf{\nabla }v)+b-u^{2}v,\end{array}$$

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 behaviours, as the homogeneous equilibrium is then well into a Hopf regime.

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