  Reaction-diffusion PDEs and particle-based stochastic reaction-diffusion
(PBSRD) models are commonly-used approaches for modeling the spatial dynamics
of chemical and biological systems. Standard reaction-diffusion PDE models
ignore the underlying stochasticity of spatial transport and reactions, and are
often described as appropriate in regimes where there are large numbers of
particles in a system. Recent studies have proven the rigorous large-population
limit of PBSRD models, showing the resulting mean-field models (MFM) correspond
to non-local systems of partial-integro differential equations. In this work we
explore the rigorous relationship between standard reaction-diffusion PDE
models and the derived MFM. We prove that the former can be interpreted as an
asymptotic approximation to the later in the limit that bimolecular reaction
kernels are short-range and averaging. As the reactive interaction length scale
approaches zero, we prove the MFMs converge at second order to standard
reaction-diffusion PDE models. In proving this result we also establish local
well-posedness of the MFM model in time for general systems, and global
well-posedness for specific reaction systems and kernels. Finally, we
illustrate the agreement and disagreement between the MFM, SM and the
underlying particle model for several numerical examples.
