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By discretising space into compartments and letting system dynamics begoverned by the reaction-diffusion master equation, it is possible to deriveand simulate a stochastic model of reaction and diffusion on an arbitrarydomain. However, there are many implementation choices involved in thisprocess, such as the choice of discretisation and method of derivation of thediffusive jump rates, and it is not clear a priori how these affect modelpredictions. To shed light on this issue, in this work we explore how a varietyof discretisations and method for derivation of the diffusive jump rates affectthe outputs of stochastic simulations of reaction-diffusion models, inparticular using Turing's model of pattern formation as a key example. Weconsider both static and uniformly growing domains and demonstrate that, whileonly minor differences are observed for simple reaction-diffusion systems,there can be vast differences in model predictions for systems that includecomplicated reaction kinetics, such as Turing's model of pattern formation. Ourwork highlights that care must be taken in using the reaction-diffusion masterequation to make predictions as to the dynamics of stochasticreaction-diffusion systems.

Effects of different discretisations of the Laplacian upon stochastic simulations of reaction-diffusion systems on both static and growing domains