Leah Edelstein-Keshet in her 1993 article Cellular automata approaches to biological modelling writes:
We do not believe that CA should be viewed as a replacement for rigorous mathematical models. Once it has been established that a CA implementation of one's hypothesis produces the desired results, then one must proceed towards deriving a traditional mathematical model. For then and only then is it possible to bring to bear tools from analysis such as stability theory, bifurcation theory and analysis theory. Only in very rare cases has it been possible to prove the existence of some particular behaviour for a CA.
How has our repertoire of tools designed to prove things about a discrete set of rules, such as cellular automata, changed since then? Does simulation remain our only approach?