Stochastic pattern formation

We consider a stochastic version of the Gray–Scott model given by

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

where $W_t$ is an approximation of a Brownian sheet, representing noise in both space and time, so its derivative is understood in the sense of stochastic calculus.

Load the interactive stochastic reaction–diffusion model
The model is initially deterministic ($\sigma=0$), so the initial condition simply spreads out to make a homogeneous state across the domain. Clicking indicates that this homogeneous equilibrium is stable, even to large perturbations.
Increasing the value of $\sigma$ in the range $(0.1,0.2)$, we can observe that noise destabilises this equilibrium, forming stripe and spot-like patterns. This is an example of stochastic resonance in reaction-diffusion systems. Smaller values of $\sigma$ will select for stripes, and larger for spots.
Further increasing $\sigma$ to the value of $0.3$ or so will destroy these patterns, as the noise eventually destabilises everything. This is sometimes known as stochastic extinction or random attractor collapse.

Wave propagation through a random medium

Next we consider a version of the inhomogeneous wave equation where the diffusion coefficient is a random function of space,

\[\pdd{u}{t} = \vnabla \cdot \left[\eta(\v{x},\sigma) \vnabla u \right],\]

where $\eta(\v{x},\sigma)$ is a smoothed spatial random variable approximated as a Gaussian with mean $1$ and variance which scales with $\sigma$.

Load the interactive random wave equation
Click to initiate a wave at some point in the domain.
The default noise level is $\sigma=0.5$. If you change this value and restart the simulation by pressing , you can observe deterministic wave propagation for small values of $\sigma$, and noisier wave behaviour for larger values.

Stochastic vegetation waves

Consider a stochastic version of the Klausmeier model in 1D of the form:

\[\begin{aligned} \pd{n}{t} &= \pdd{n}{x} + w {n}^{2}-m n+\sigma \diff{W_t}{t} \left[\frac{n}{1+n}\right],\\ \pd{w}{t} &= \pdd{w}{x} + a-w-w {n}^{2}+V \pd{w}{x}. \end{aligned}\]

Load the interactive stochastic Klausmeier model
The parameter $\sigma$ represents the strength of the noise, so increasing or decreasing it will lead to more stochastic or more deterministic behaviour, respectively.
The noise function should scale correctly independently of the geometry and dimension. You can see how it behaves in 2D by pressing → Domain and change Dimension from 1 to 2. Playing with $\sigma$ as well as other parameters, and clicking to perturb the waves, shows that irregularity of the waves can occur in different ways with stochastic forcing.

Numerical health warning

Randomness and stochastic forcing can lead to less regularity and stability of numerical schemes.

Warning: the method only works for the forward Euler timestepping scheme! All other timestepping schemes will not scale properly with the timestep.

In implementing our random noise terms, we have taken

\[\diff{W_t}{t} \propto \frac{1}{\sqrt{\dt \, \dx^N}}\xi(t,\v{x}),\]

with $\dt$ and $\dx$ the space and time steps respectively, and $N$ is the number of dimensions. The variable $\xi$ represents (for each spatial point and every time step) an independently distributed normal random variable with mean $0$ and variance $1$. The scaling in time is to preserve the Brownian increment scaling of $W_t \propto \sqrt{\dt}$ after the multiplication by $\dt$ from the forward Euler stepping. Similarly, we take $\eta$ to scale the same way in space, but it will not scale with time. We also mollify $\eta$ by running a diffusion smoothing on it for a short time.