I'd just like to know if the following model has received any attention:

A state at discrete time $t$ consists of a function $S_t:{\Bbb Z}^2\rightarrow S^1$.

So view each cell $c$ (element of ${\Bbb Z}^2$) as having eight neighbors in the usual way.

The update rule: set $S_{t+1}(c)$ equal that value $S_{t}(c')$, for $c'$ a neighbor of $c$, occurring nearest to $S_t(c)$ when moving clockwise around $S^1$.

Computer simulation suggests that an random initial state quickly evolves to a pattern reminiscent of lichens provided one represents locations on $S^1$ with shades of gray (despite the unfortunate spurious discontinuity). In time some of the "lichens" (distinguished visually one from another by their prevailing shade) grow while others eventually get absorbed.