``Eater'' rules
There is a close affinity between Zhabotinsky type rules and the class of ``eaters,'' defined as those for which the central cell takes on the identity of one of its neighbors. The usual arrangement is to rank the states in some order, then close the list to form a cycle. In a traditional form, ``scissors cut paper, paper covers stone, stone breaks scissors.'' An automaton results when each cell suffers the ravages of any neighbor in a position to affect it. A stone cell becomes paper if there is a paper cell roundabout; meanwhile the paper cell suffers a fate dependent upon its own neighbors.
In the CAM/PC , the greatest variety comes from the von Neumann neighborhood, whose cells can have as many as four states; however this limits the number of neighbors to four. A much greater number of states produces a more interesting automaton, but four is still sufficient for many interesting studies.
CAMEX still does not have a submenu entitled ``eaters,'' but the rule can be incorporated into just about any type of automaton which has a sufficient number of states; generally, the more the better. Consequently several of the submenus, for which this automaton makes sense, include options to generate the eater rule without having to edit the rule table.
The evolution of an eater rule progresses through stages.
Given a large number of states, there is a small probability that a cell will be affected by one of its neighbors. Initially the evolution consists of the formation of small droplets; whenever a cell actually changes its state, there is a chance that it can feed on neighbors which were previously unattractive. Similarly, the growth of any region consisting of a single state increases the number of additional cells which it might devour.
Once the droplets have begun to coalesce, two new things begin to happen; uniform regions start to collide, chains of cells close into cycles. The chain may even have been present initially; if the number of states is small, there may be nuclei where the states are clumped together.
Whenever chains become established, they generate spirals which sweep out larger and larger areas, because cells in the spiral having the appropriate phase will eventually pass by any cell sitting on the edge of the spiral and capture it.
In the end, regions which do not interact further will establish themselves, unless there are spirals present. A single spiral will go on spinning; with two or more, they can steal cells from each other, provoking a duel which will only end with the eventual dominance of one or the other.
On the other hand, they may just continue to coexist, circling around each other forever.
Introducing eater rules into cellular automata is more successful than trying to do such things as discretize Laplace's equation. In the latter case, the value (that is, state) of a cell should be the average of its neighbors.
Programming such a rule leads to an evolution which quickly settles down to an average value; if the number of states is small, it is impossible to get interesting evolutions in which boundary conditions are maintained. The reason is simply that few states cannot maintain a gradual variation between appreciably differing values on the boundary.
Applications to nonlinear differential equations, such as those that describe Zhabotinsky reactions, can be more successful whenever the solutions cluster around well defined values with definite criteria for the solution to jump between them.
Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx