Pattern Formation in the Gray-Scott Reaction-Diffusion System

Description of the System

The presented system describes the dynamics of two interacting variables U(x,t) and V(x,t) defining a density distribution of two chemical substances in two dimensions. The modeled chemical system can be scetched in the following way:

The evolution of both variables over time are defined through two first-order differential equations:

Here, the -terms correspond to the reaction , to the reaction and and to the inflow and outflow respectively. The dynamic of the inert substance P does not influence the dynamics of U and V and therefore does not have to be considered. The terms and describe the diffusion of the substances in the volume (or, to be precise, in the area, since the system is modeled in two dimensions in this case).

The Applet



To understand the dynamics of the system, one can first look at the dynamics without the diffusive terms:

The fixed points of the system can be found to be

The Jacobian of has eigenvalues -F and -K and is therefore always stable.

The two other two fixed points only exist, if the term in the square root is positive:

Looking at , one finds, that given the existence of this fixed point, it is always a saddle:

on the other hand, is never a saddle, and by looking at the trace of , one finds that this fixed points changes from stable to unstable if .

Furthermore, one can also observe a transition between a stable node and a rotating focus, but this transition is of no importance for the occurence of patterns.

One can map the parameter values at which the types of patterns listed in the applet occur onto the phase-space together with the separatrices just described:

It turns out that patterns form mostly in the regime of only the trivial (1,0)-fixpoint but close to the separatrix defining a saddle-node bifurcation. The chaotic behaviour can be found in the regime enclosed by both lines.

Turing instability is often considered to be responsible for pattern formation, but cannot be applied to this case, since one cannot find a Diffusive term which would make the Jacobian

of the trivial fixed point unstable if it where added.