Eventually number-conserving cellular automata

Although it is undecidable whether a one-dimentional cellular automaton obeys a given conservation law over its limit set, it is however possible to obtain sufficient conditions to be satisfied by a one-dimentional cellular automaton to be eventually number-conserving. We present a preliminary study of two-input one-dimensional cellular automation rules called eventually number-conserving cellular automation rules whose limit sets, reached after a number of time steps of the order of the cellular automation size, consist of states having a constant number of active sites. In particular, we show how to find rules having given limit sets satisfying a conservation rule. Viewed as models of systems of interacting particles, these rules obey a kind of Darwinian principle by iether annihilating unnecessary particles or creating necessary ones.