In a continuous cellular automaton, each cell has a gray level between [0,1] as its state.
In the cellular automaton shown, the transition function computes the state of the cell s, at evolution step t+1, as the fractional part of the mean of its neighborhood's gray levels at evolution step t plus a constant k.
This rule is described on page 159 of Wolfram's NKS
See Mathematica notebook for more details and demo.
k = 0.009, t = 500, initial condition = "one cell"
k = 0.009, t = 500, initial condition = "random"