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Reaction-diffusion PDEs and particle-based stochastic reaction-diffusion(PBSRD) models are commonly-used approaches for modeling the spatial dynamicsof chemical and biological systems. Standard reaction-diffusion PDE modelsignore the underlying stochasticity of spatial transport and reactions, and areoften described as appropriate in regimes where there are large numbers ofparticles in a system. Recent studies have proven the rigorous large-populationlimit of PBSRD models, showing the resulting mean-field models (MFM) correspondto non-local systems of partial-integro differential equations. In this work weexplore the rigorous relationship between standard reaction-diffusion PDEmodels and the derived MFM. We prove that the former can be interpreted as anasymptotic approximation to the later in the limit that bimolecular reactionkernels are short-range and averaging. As the reactive interaction length scaleapproaches zero, we prove the MFMs converge at second order to standardreaction-diffusion PDE models. In proving this result we also establish localwell-posedness of the MFM model in time for general systems, and globalwell-posedness for specific reaction systems and kernels. Finally, weillustrate the agreement and disagreement between the MFM, SM and theunderlying particle model for several numerical examples.

How reaction-diffusion PDEs approximate the large-population limit of stochastic particle models