On the derivation of approximations to cellular automata models and the assumption of independence

•We explore a cellular automata with motility and proliferation events.•We attempt to derive equations for the mean behaviour of the system.•Different limiting regimes can have significant effects if not chosen carefully.•Continuous time allows for easier derivations with less assumptions.•The independence assumption is key to accurately modelling the systems behaviour.

Cellular automata are discrete agent-based models, generally used in cell-based applications. There is much interest in obtaining continuum models that describe the mean behaviour of the agents in these models. Previously, continuum models have been derived for agents undergoing motility and proliferation processes, however, these models only hold under restricted conditions. In order to narrow down the reason for these restrictions, we explore three possible sources of error in deriving the model. These sources are the choice of limiting arguments, the use of a discrete-time model as opposed to a continuous-time model and the assumption of independence between the state of sites. We present a rigorous analysis in order to gain a greater understanding of the significance of these three issues. By finding a limiting regime that accurately approximates the conservation equation for the cellular automata, we are able to conclude that the inaccuracy between our approximation and the cellular automata is completely based on the assumption of independence.