# Hybrid approaches for multiple-species stochastic reaction-diffusion models

Research output: Contribution to journal › Article › peer-review

## Standard

**Hybrid approaches for multiple-species stochastic reaction-diffusion models.** / Spill, Fabian; Spill, Fabian; Guerrero, Pilar; Alarcon, Tomas; Maini, Philip K.; Byrne, Helen.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Journal of Computational Physics*, vol. 299, pp. 429-445. https://doi.org/10.1016/j.jcp.2015.07.002

## APA

*Journal of Computational Physics*,

*299*, 429-445. https://doi.org/10.1016/j.jcp.2015.07.002

## Vancouver

## Author

## Bibtex

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## RIS

TY - JOUR

T1 - Hybrid approaches for multiple-species stochastic reaction-diffusion models

AU - Spill, Fabian

AU - Spill, Fabian

AU - Guerrero, Pilar

AU - Alarcon, Tomas

AU - Maini, Philip K.

AU - Byrne, Helen

PY - 2015/10/5

Y1 - 2015/10/5

N2 - 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. In 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.

AB - 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. In 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.

KW - Fisher-Kolmogorov equation

KW - Hybrid model

KW - Lotka-Volterra equation

KW - Reaction-diffusion system

KW - Stochastic model

KW - Monte Carlo (MC) simulation

KW - Optimization

UR - http://www.scopus.com/inward/record.url?scp=84937952936&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2015.07.002

DO - 10.1016/j.jcp.2015.07.002

M3 - Article

AN - SCOPUS:84937952936

VL - 299

SP - 429

EP - 445

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -