Brusselator pattern formation

Biology | Parabolic | Patterns

Another Turing system is the Brusselator, given by

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u+a-(b+1)u+u^2v,\\ \pd{v}{t}&=D\nabla^2v+ bu-u^2v,\end{aligned}\]

where we take $a,b>0$.

Hyperbolic Brusselator & Turing–Wave instabilities

One can show that two–species reaction–diffusion systems can only ever have Turing–like instabilities with real growth rates. In contrast, hyperbolic reaction–diffusion systems (or systems with more than two species) allow for Turing–Wave (or sometimes Wave or Turing–Hopf) instabilities. Such instabilities lead to spatial eigenfunctions that grow and oscillate, typically giving rise to spatiotemporal dynamics. Here we consider a hyperbolic version of the Brusselator given by

\[\begin{aligned}\tau\pdd{u}{t}+\pd{u}{t}&=D\nabla^2 u+a-(b+1)u+u^2v,\\ \tau\pdd{v}{t}+\pd{v}{t}&=\nabla^2v+ bu-u^2v,\end{aligned}\]

where there are two new terms proportional to $\tau$ and we have instead put the diffusion ratio $D$ on the $u$ equation. The normal Turing instabilities will occur for $D<1$, but new Turing–Wave instabilities may occur for $D>1$, so we set $D=2$.

For more details on such systems and their generalisations, take a look at this paper.