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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.


Simulations of the spatial Lotka–Volterra model with parameters , a = 1, b = 0.1, c = 1, L = 20, . We plot the spatial average  of the number of predators, and likewise prey, in a box over time for three realisations of the stochastic model, (a), (c), (e), and three realisations of the hybrid model, (b), (d), (f), with a threshold of Θ = 10. These three realisations of each model show the three qualitatively different outcomes, namely, oscillatory solutions, extinction of both species or extinction of predators and subsequent blow-up of prey.
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fg0080: Simulations of the spatial Lotka–Volterra model with parameters , a = 1, b = 0.1, c = 1, L = 20, . We plot the spatial average of the number of predators, and likewise prey, in a box over time for three realisations of the stochastic model, (a), (c), (e), and three realisations of the hybrid model, (b), (d), (f), with a threshold of Θ = 10. These three realisations of each model show the three qualitatively different outcomes, namely, oscillatory solutions, extinction of both species or extinction of predators and subsequent blow-up of prey.

Mentions: Fig. 7 shows how the number of predators and prey at the two boundaries (, ) vary in time for the mean field model. We note that during the first few oscillations both predator and prey populations are close to zero. In the corresponding stochastic model, the population can only be integer-valued, so a value below 1 in the deterministic model indicates that extinction of the population is likely. At later times, we observe regular oscillations with minima significantly above 0, and conclude that if extinction were to occur in the stochastic model, it would most likely happen at early times. Fig. 8 confirms these expectations. Fig. 8(a) shows that, for the stochastic model, the spatial average of the number of predators (or prey) may exhibit oscillations similar to those of the deterministic model. However, extinction of either population can also occur. Fig. 8(c) shows that if the prey die out first, then, necessarily, the predators will also die out. On the other hand, if the predators die out first, then prey numbers will blow up (see Fig. 8(e)). Figs. 8(b), (d), (f) confirm that the hybrid model can reproduce each of these three qualitatively different scenarios. Since statistics for the mean and standard deviation are not meaningful if the prey population blows up, we compare the frequency of extinction events in the stochastic and hybrid models.


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

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

Simulations of the spatial Lotka–Volterra model with parameters , a = 1, b = 0.1, c = 1, L = 20, . We plot the spatial average  of the number of predators, and likewise prey, in a box over time for three realisations of the stochastic model, (a), (c), (e), and three realisations of the hybrid model, (b), (d), (f), with a threshold of Θ = 10. These three realisations of each model show the three qualitatively different outcomes, namely, oscillatory solutions, extinction of both species or extinction of predators and subsequent blow-up of prey.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

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

fg0080: Simulations of the spatial Lotka–Volterra model with parameters , a = 1, b = 0.1, c = 1, L = 20, . We plot the spatial average of the number of predators, and likewise prey, in a box over time for three realisations of the stochastic model, (a), (c), (e), and three realisations of the hybrid model, (b), (d), (f), with a threshold of Θ = 10. These three realisations of each model show the three qualitatively different outcomes, namely, oscillatory solutions, extinction of both species or extinction of predators and subsequent blow-up of prey.
Mentions: Fig. 7 shows how the number of predators and prey at the two boundaries (, ) vary in time for the mean field model. We note that during the first few oscillations both predator and prey populations are close to zero. In the corresponding stochastic model, the population can only be integer-valued, so a value below 1 in the deterministic model indicates that extinction of the population is likely. At later times, we observe regular oscillations with minima significantly above 0, and conclude that if extinction were to occur in the stochastic model, it would most likely happen at early times. Fig. 8 confirms these expectations. Fig. 8(a) shows that, for the stochastic model, the spatial average of the number of predators (or prey) may exhibit oscillations similar to those of the deterministic model. However, extinction of either population can also occur. Fig. 8(c) shows that if the prey die out first, then, necessarily, the predators will also die out. On the other hand, if the predators die out first, then prey numbers will blow up (see Fig. 8(e)). Figs. 8(b), (d), (f) confirm that the hybrid model can reproduce each of these three qualitatively different scenarios. Since statistics for the mean and standard deviation are not meaningful if the prey population blows up, we compare the frequency of extinction events in the stochastic and hybrid models.

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.