Hybrid approaches for multiple-species stochastic reaction-diffusion models

TL;DR

This paper introduces a hybrid modeling scheme that couples stochastic reaction-diffusion models with PDE-based mean field models across different regions, enabling efficient and accurate simulations of systems with varying local entity counts.

Contribution

The authors develop a novel coupling scheme for stochastic and PDE models that handles multiple interfaces and conserves particles, improving simulation speed and accuracy.

Findings

The hybrid scheme accurately captures stochastic effects like extinction.

It significantly reduces computational time compared to pure stochastic models.

The method effectively manages multiple dynamic interfaces.

Abstract

Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. This way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.