Gray–Scott model

Chaos | Parabolic | Patterns | Waves

A reaction–diffusion system heavily studied for its complex dynamics is the Gray–Scott system, given by

$$\begin{array}{rl}\frac{\partial u}{\partial t}& ={\mathrm{\nabla }}^{2}u+u^{2}v-(a+b)u,\\ \frac{\partial v}{\partial t}& =D{\mathrm{\nabla }}^{2}v-u^{2}v+a(1-v),\end{array}$$

where we take $D=2$ and only vary $a,b>0$. This model has a wide range of behaviours, shown in another WebGL simulator that partially inspired VisualPDE.

• Load the interactive simulation to explore the system.

A famous 1993 paper on this model explored a range of the parameters $a$ and $b$ to classify different behaviours, and many people have hence made these parameters depend linearly on $x$ and $t$ to see all of this behaviour at once.

• This classification simulation explores this approach, with $a$ depending on $y$ in the range $a\in [0,0.07]$ and $b$ depending on $x$ in the range $b\in [0.02,0.068]$.

Building from the previous simulation, we can rescale the heterogeneity to still be monotonic, but to use up more of the domain to see different dynamical regimes.

• Explore this in this rescaled simulation, where we also plot the variable $v$ instead by default (you can click on to change this to plot $u$ instead). Here is another version of this rescaled model which also includes a porous-medium diffusion term (that is, ${\mathrm{\nabla }}^2{u}^m$), which radically alters the parameter space as $m$ is increased from 1.

Interestingly, the value of $D=2$ used gives a very rich parameter space, whereas making $D$ smaller reduces the regions of patterned behaviour, and taking $D$ larger increases it at the cost of making things more stationary and more spot-like for most of the parameter domain.

Furthermore, when $D=1$ the system no longer supports stationary patterns, but does exhibit waves similar to the spiral waves in the equal-diffusion case of the cyclic competition models.

Below we’ve listed some parameter combinations that give rise to different and interesting behaviours.

One of our favourites is the moving spots simulation, which exhibits spots bobbing around.

• Initiate this motion and then increase $b$ slowly to about $b=0.56$. The spots become sparse and start exhibiting strange diversions in their motions.

$a$	$b$	Description
0.037	0.06	Labyrinthine
0.03	0.062	Spots
0.025	0.06	Pulsating spots
0.078	0.061	Worms
0.039	0.058	Holes
0.026	0.051	Spatiotemporal chaos
0.034	0.056	Intermittent chaos/holes
0.014	0.054	Moving spots (glider-like)
0.018	0.051	Small waves
0.014	0.045	Big waves
0.062	0.061	U-skate world

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