Figure 15.

Snapshots at $t=0,t=1,t=5$ and $t=20$ of the spatial-extended system (74). Spatially periodic boundary conditions are used here. The parameters are given in (75). In all the three simulations in (a–c), the initial values for both u and v are the same. (a) model simulation of (74) with diffusion terms ${\nabla }^{2}u$ and ${\nabla }^{2}v$ but the noise coefficient $f\left(u\right)=0$; (b) model simulation of (74) without diffusion terms ${\nabla }^{2}u={\nabla }^{2}v=0$. Therefore, the system is spatially decoupled. However, a stochastic noise $f\left(u\right)=0.05u/\sqrt{1+u^2}$ is added to the system; (c) simulations with both the diffusion terms ${\nabla }^{2}u$ and ${\nabla }^{2}v$ and the stochastic noise $f\left(u\right)=0.05u/\sqrt{1+u^2}$.