Hybrid methods in reaction-diffusion equations

TL;DR

This paper reviews hybrid modeling approaches for reaction-diffusion systems, combining stochastic and PDE methods, and introduces recent advancements to address limitations, especially in multiscale cell population models.

Contribution

It presents new extensions of hybrid schemes to multiscale cell population models, improving their accuracy and applicability.

Findings

Hybrid schemes effectively couple stochastic and PDE models.

Recent methods improve accuracy in heterogeneous systems.

Extensions enable multiscale modeling of cell populations.

Abstract

Simulation of stochastic spatially-extended systems is a challenging problem. The fundamental quantities in these models are individual entities such as molecules, cells, or animals, which move and react in a random manner. In big systems, accounting for each individual is inefficient. If the number of entities is large enough, random effects are negligible, and often partial differential equations (PDEs) are used in which the fluctuations are neglected. When the system is heterogeneous, so that the number of individuals is large in certain regions and small in others, the PDE description becomes inaccurate in certain regions. To overcome this problem, the so-called hybrid schemes have been proposed that couple a stochastic description in parts of the domain with its mean field limit in the others. In this chapter, we review the different formulations of this approach and our recent contributions to overcome several of the limitations of previous schemes, including the extension of the concept to multiscale models of cell populations.