title: Hybrid approaches for multiple-species stochastic reaction-diffusion models.
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. 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.
authors.affiliation: Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA ; Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.
authors.forename: Fabian
authors.initials: F
authors.lastname: Spill
authors.affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
authors.forename: Pilar
authors.initials: P
authors.lastname: Guerrero
authors.affiliation: Centre de Recerca Matematica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain ; Departament de Matemàtiques, Universitat Atonòma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
authors.forename: Tomas
authors.initials: T
authors.lastname: Alarcon
authors.affiliation: Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
authors.forename: Philip K
authors.initials: PK
authors.lastname: Maini
authors.affiliation: Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK ; Computational Biology Group, Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK.
authors.forename: Helen
authors.initials: H
authors.lastname: Byrne
keywords: Fisher–Kolmogorov equation; Hybrid model; Lotka–Volterra equation; Reaction–diffusion system; Stochastic model