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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 plots comparing the travelling waves profile generated by the stochastic model in the column to the left and the hybrid model with a threshold of Θ = 25 on the right (simulation time t = 60). As the carrying capacity Ω increases, Ω = 10,25,100. Each plot shows a single realisation as well as the mean of 256 realisations of the stochastic or hybrid model, respectively. Results for the corresponding PDE (20) are also shown. The other parameter values are D = λ = 1, , h = 1.
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fg0020: Series of plots comparing the travelling waves profile generated by the stochastic model in the column to the left and the hybrid model with a threshold of Θ = 25 on the right (simulation time t = 60). As the carrying capacity Ω increases, Ω = 10,25,100. Each plot shows a single realisation as well as the mean of 256 realisations of the stochastic or hybrid model, respectively. Results for the corresponding PDE (20) are also shown. The other parameter values are D = λ = 1, , h = 1.

Mentions: Fig. 2 compares travelling wave solutions generated from the stochastic model, the hybrid model (with ) and the PDE. In each plot we present a single realisation and the mean of 256 realisations of the stochastic (Fig. 2(a), (c), (e)) and hybrid models (Fig. 2(b), (d), (f)) together with the numerical solution of the corresponding PDE. We fix , , , and allow to vary Ω. We note that the travelling wave speeds for the stochastic and hybrid models are slower than those of the PDE, and the speed increases with Ω. Furthermore, the relative noise, i.e. the fluctuation of a single stochastic or hybrid realisation about the mean, decreases as Ω increases. Finally, the wave front of the PDE appears to be steeper compared to the wave front of the mean of 256 realisations both in the stochastic and hybrid models compared to the PDE, and the steepness increases with Ω. This is explained as the different realisations of the stochastic model can have different speeds, hence the average broadens the wave front.


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 plots comparing the travelling waves profile generated by the stochastic model in the column to the left and the hybrid model with a threshold of Θ = 25 on the right (simulation time t = 60). As the carrying capacity Ω increases, Ω = 10,25,100. Each plot shows a single realisation as well as the mean of 256 realisations of the stochastic or hybrid model, respectively. Results for the corresponding PDE (20) are also shown. The other parameter values are D = λ = 1, , h = 1.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

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

fg0020: Series of plots comparing the travelling waves profile generated by the stochastic model in the column to the left and the hybrid model with a threshold of Θ = 25 on the right (simulation time t = 60). As the carrying capacity Ω increases, Ω = 10,25,100. Each plot shows a single realisation as well as the mean of 256 realisations of the stochastic or hybrid model, respectively. Results for the corresponding PDE (20) are also shown. The other parameter values are D = λ = 1, , h = 1.
Mentions: Fig. 2 compares travelling wave solutions generated from the stochastic model, the hybrid model (with ) and the PDE. In each plot we present a single realisation and the mean of 256 realisations of the stochastic (Fig. 2(a), (c), (e)) and hybrid models (Fig. 2(b), (d), (f)) together with the numerical solution of the corresponding PDE. We fix , , , and allow to vary Ω. We note that the travelling wave speeds for the stochastic and hybrid models are slower than those of the PDE, and the speed increases with Ω. Furthermore, the relative noise, i.e. the fluctuation of a single stochastic or hybrid realisation about the mean, decreases as Ω increases. Finally, the wave front of the PDE appears to be steeper compared to the wave front of the mean of 256 realisations both in the stochastic and hybrid models compared to the PDE, and the steepness increases with Ω. This is explained as the different realisations of the stochastic model can have different speeds, hence the average broadens the wave front.

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.