Imagine I have a system describing two species concentrations $x$ and $y$ defined on a ring of $N$ cells, given by \begin{align} \dot{x}_j&=F_x(x_j,\{x_{r}\},y_j,\{y_{r}\})\\ \dot{y}_j&=F_y(x_j,\{x_{r}\},y_j,\{y_{r}\}) \end{align} for each cell $1\leq j\leq N$, where $\{x_{r}\}$ and $\{y_{r}\}$ are the sets of all the variables of cells that are not cell $j$, and $F_x,F_y$ are general non-linear functions.
For example, consider the system \begin{align} \dot{x}_j&=F_x(x_j,x_{j-1},x_{j+1},y_j,y_{j-1},y_{j+1})\\ \dot{y}_j&=F_y(x_j,x_{j-1},x_{j+1},y_j,y_{j-1},y_{j+1}) \end{align} where we assume dependence on the species concentrations in the immediate neighbours of cell $j$.
Assume we perturb this system around the homogeneous steady state and assume we converge to a patterned solution (I'm skipping many steps here, but essentially we may look at the fastest growing modes, upon a Fourier transform, in order to predict the final pattern, please check this for a brief introduction on linear stability analysis).
Is it possible, via linear stability analysis, to predict the speed or time taken to develop a pattern? I have been trying to look for references and methods in the literature, but with no success so far. Naturally this depends a lot on the shapes of $F_x$ and $F_y$, but I'm wondering what can, in general, be said about the patterning speed (or to reach a certain threshold value).