Eventually Number-Conserving Cellular Automata

We present a preliminary study of a new class of two-input cellular automata called eventually number-conserving cellular automata characterized by the property of evolving after a finite number of time steps to states whose number of active sites remains constant. Eventually number-conserving cellular automata are models of open systems of interacting particles, that is, system of particles interacting with the external world, The particle aspect of eventually number-conserving cellular automata can be emphasized by the motion representation of the cellular automaton evolution rule. This new class of cellular automata contains, as strict subclasses, number-conserving cellular automata, monotone cellular automata, and cellular automata emulating number-conserving ones. Our main objective is to show that they are not what one might naively think they are.