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

TL;DR

This paper establishes a rigorous mathematical connection between stochastic particle models, mean-field models, and classical reaction-diffusion PDEs, showing how PDEs approximate stochastic systems in certain limits.

Contribution

It proves that reaction-diffusion PDEs are asymptotic approximations to mean-field models derived from stochastic particle systems, with second-order convergence as reaction kernels become short-range.

Findings

Reaction-diffusion PDEs approximate mean-field models in the short-range reaction limit.

Second-order convergence of MFMs to PDEs as interaction length scales go to zero.

Global well-posedness established for specific reaction systems.

Abstract

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.