Limits...
Hybrid approaches for multiple-species stochastic reaction-diffusion models.

Spill F, Guerrero P, Alarcon T, Maini PK, Byrne H - J Comput Phys (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

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.

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.

No MeSH data available.


Series of curves showing how the wave speed of the stochastic model, the mean field model (i.e. the finite difference discretisation of the PDE) and hybrid models for thresholds of Θ = 10 and 16⁎h changes as the lattice spacing varies. We note that the wave speed of the hybrid model, with a fixed threshold, converges to that of the stochastic model, whereas the hybrid model where the threshold is adjusted with the lattice spacing does not. The other parameters are Ω = 80⁎h, D = λ = 1, and all stochastic and hybrid results are obtained from averaging 1024 different simulations.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4554296&req=5

fg0030: Series of curves showing how the wave speed of the stochastic model, the mean field model (i.e. the finite difference discretisation of the PDE) and hybrid models for thresholds of Θ = 10 and 16⁎h changes as the lattice spacing varies. We note that the wave speed of the hybrid model, with a fixed threshold, converges to that of the stochastic model, whereas the hybrid model where the threshold is adjusted with the lattice spacing does not. The other parameters are Ω = 80⁎h, D = λ = 1, and all stochastic and hybrid results are obtained from averaging 1024 different simulations.

Mentions: Fig. 3 shows the dependence of the wave speed on the lattice constant. We compare the stochastic model against the mean field model, i.e. the finite difference discretisation of the PDE with the same lattice constant, and the hybrid model. Densities are fixed by adjusting . Likewise, , but we also compare to the hybrid model with a fixed threshold . The wave speed c is calculated by observing that the change of the total number of particles in time, averaged over all simulations, , should be proportional to c. We approximate the wave speed by comparing after fixed time intervals , obtaining . The finite difference model converges, as expected, to as , even though every finite lattice spacing will still result in some visible dispersion after a long time. The stochastic model has a significantly lower wave speed and the wave speed of the hybrid model, with , converges to that for the stochastic model. However, if , then the hybrid model does not appear to converge to the stochastic model.


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

Spill F, Guerrero P, Alarcon T, Maini PK, Byrne H - J Comput Phys (2015)

Series of curves showing how the wave speed of the stochastic model, the mean field model (i.e. the finite difference discretisation of the PDE) and hybrid models for thresholds of Θ = 10 and 16⁎h changes as the lattice spacing varies. We note that the wave speed of the hybrid model, with a fixed threshold, converges to that of the stochastic model, whereas the hybrid model where the threshold is adjusted with the lattice spacing does not. The other parameters are Ω = 80⁎h, D = λ = 1, and all stochastic and hybrid results are obtained from averaging 1024 different simulations.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4554296&req=5

fg0030: Series of curves showing how the wave speed of the stochastic model, the mean field model (i.e. the finite difference discretisation of the PDE) and hybrid models for thresholds of Θ = 10 and 16⁎h changes as the lattice spacing varies. We note that the wave speed of the hybrid model, with a fixed threshold, converges to that of the stochastic model, whereas the hybrid model where the threshold is adjusted with the lattice spacing does not. The other parameters are Ω = 80⁎h, D = λ = 1, and all stochastic and hybrid results are obtained from averaging 1024 different simulations.
Mentions: Fig. 3 shows the dependence of the wave speed on the lattice constant. We compare the stochastic model against the mean field model, i.e. the finite difference discretisation of the PDE with the same lattice constant, and the hybrid model. Densities are fixed by adjusting . Likewise, , but we also compare to the hybrid model with a fixed threshold . The wave speed c is calculated by observing that the change of the total number of particles in time, averaged over all simulations, , should be proportional to c. We approximate the wave speed by comparing after fixed time intervals , obtaining . The finite difference model converges, as expected, to as , even though every finite lattice spacing will still result in some visible dispersion after a long time. The stochastic model has a significantly lower wave speed and the wave speed of the hybrid model, with , converges to that for the stochastic model. However, if , then the hybrid model does not appear to converge to the stochastic model.

Bottom Line: 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.

View Article: PubMed Central - PubMed

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.

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.

No MeSH data available.