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.

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 (but not coincident with) $S_t(c)$ 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.