Consequences of imperfect mixing the Gray-Scott model

We study an autocatalytic reaction-diffusion scheme, the Gray-Scott model, when the mixing processes do not homogenize the reactants. Starting from the master equation, we derive the resulting coupled, nonlinear, stochastic partial differential equations that rigorously include the spatiotemporal fluctuations resulting from the interplay between the reaction and mixing processes. The fields are complex and depend on correlated complex noise terms. We implement a method to solve for these complex fields numerically and extract accurate information about the system evolution and stationary states under different mixing regimes. Through this example, we show how the reaction-induced fluctuations interact with the temporal nonlinearities, leading to results that differ significantly from the mean-field (perfectly mixed) approach. This procedure can be applied to an arbitrary nonlinear reaction diffusion scheme.

Figure 1

(Color online) Each row of squares shows, from left to right, the time evolution of $\mathrm{Re}u(\widehat{x},\widehat{y},\tau )$ for a different simulation. The initial condition, at $\tau =0$, is shown on the leftmost image, see text for details. (a) and (b) have the same initial condition and reaction parameters. (a) If $ϵ\to 0$, which corresponds to the perfect mixing regime, then the system evolves to the uniform absorbing state $R$ (here shown at $\tau =50$). (b) As $ϵ$ increases, in the imperfect mixing regime, the system evolves to a new active state with globular replicating structures $(\tau =50)$. Equivalently, for a new initial condition and parameter set, (c) evolves in the perfect mixing regime to the uniform active state $B_{-}$ $(\tau =300)$, whereas (d), in the imperfect mixing regime, evolves to a new active state $(\tau =300)$.

Figure 2

(a) After a transient time, in the perfect mixing limit $ϵ=0.5$, the average of the real part of the field tends to the mean-field value. However, in the imperfect mixing regime, $ϵ=2$, these averages deviate significantly from the mean-field solution and show oscillatory behavior. (b) The averaged imaginary parts of the fields are always negligible since the averages correspond to physical, real, densities.