The Spatial Regime Conversion Method

TL;DR

The paper introduces the spatial regime conversion method (SRCM), a hybrid modeling framework that adaptively combines stochastic and deterministic approaches for efficient reaction-diffusion system simulations, capturing stochastic effects with improved efficiency.

Contribution

The SRCM extends the regime conversion method to spatial systems, enabling dynamic switching between stochastic and deterministic models based on local concentrations.

Findings

Accurately captures stochastic features in reaction-diffusion systems.

Offers significant computational efficiency improvements over fully stochastic models.

Validated on multiple one-dimensional test systems.

Abstract

We present the spatial regime conversion method (SRCM), a novel hybrid modelling framework for simulating reaction-diffusion systems that adaptively combines stochastic discrete and deterministic continuum representations. Extending the regime conversion method (RCM) to spatial settings, the SRCM employs a discrete reaction-diffusion master equation (RDME) representation in regions of low concentration and continuum partial differential equations (PDEs) where concentrations are high, dynamically switching based on local thresholds. This enables efficient and accurate simulation of systems in which stochasticity plays a key role but is not required uniformly across the domain. We specify the full mathematical formulation of the SRCM, including conversion reactions, hybrid kinetic rules, and consistent numerical updates. The method is validated across several one-dimensional test systems, including simple diffusion from a region of high concentration, the formation of a morphogen gradient, and the propagation of FKPP travelling waves. Results show that the SRCM captures key stochastic features while offering substantial gains in computational efficiency over fully stochastic models.