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.


Simulations of the spatial Lotka–Volterra Model with parameters , a = 2, b = 0.1, c = 3, , h = 0.2 and initial values N(k,t = 0)=50, M(k,t = 0)=5 for . We compare (a)–(b), the stochastic model, to (c)–(d), the hybrid model with thresholds Θ = 10, and (e)–(f) with Θ = 25. The figures on the left show the spatial profile at time t = 4.1 of the number of predators and prey of a single realisation as well as the mean of 256 different realisations, whereas the figures on the right show the time evolution of the spatial average of numbers of prey in a single realisation and the mean of realisations. The corresponding PDE solution is shown in Fig. 5.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

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

fg0060: Simulations of the spatial Lotka–Volterra Model with parameters , a = 2, b = 0.1, c = 3, , h = 0.2 and initial values N(k,t = 0)=50, M(k,t = 0)=5 for . We compare (a)–(b), the stochastic model, to (c)–(d), the hybrid model with thresholds Θ = 10, and (e)–(f) with Θ = 25. The figures on the left show the spatial profile at time t = 4.1 of the number of predators and prey of a single realisation as well as the mean of 256 different realisations, whereas the figures on the right show the time evolution of the spatial average of numbers of prey in a single realisation and the mean of realisations. The corresponding PDE solution is shown in Fig. 5.

Mentions: The domain of length is divided into boxes, so . Initially it contains a spatially homogeneous distribution of prey and predators so that , . The model parameters are fixed so that , , , . With Neumann boundary conditions, equations (24) remain spatially homogeneous at all times, and both populations oscillate in time. Typical results are presented in Fig. 5, and show that the peak in prey numbers is followed by a peak in the number of predators. For this choice of parameter values the minimum number of individuals of either species is always sufficiently large that extinction in the stochastic, spatial model is almost impossible. Corresponding results for the stochastic and hybrid models for two choices of the threshold values ( and ) are shown in Fig. 6. The column on the left shows the spatial profile of the number of predators and prey in a given box at time , both for a single realisation and the mean of 256 different realisations. Comparing either predator and prey numbers, we note that the means for the stochastic and hybrid models appear similar for both values of Θ (see Fig. 6(a), (c), (e)), and are fairly homogeneous, whereas single realisations of either model differ markedly. As expected, the profile of predator and prey numbers in the stochastic model is noisy throughout the domain, whereas noise is suppressed in the hybrid model when the population numbers exceed the threshold. We observe the predator numbers in Fig. 6(c) are above the threshold and hence smoothly distributed in space, but they are not homogeneous, in contrast to the profile of the mean.


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 = 2, b = 0.1, c = 3, , h = 0.2 and initial values N(k,t = 0)=50, M(k,t = 0)=5 for . We compare (a)–(b), the stochastic model, to (c)–(d), the hybrid model with thresholds Θ = 10, and (e)–(f) with Θ = 25. The figures on the left show the spatial profile at time t = 4.1 of the number of predators and prey of a single realisation as well as the mean of 256 different realisations, whereas the figures on the right show the time evolution of the spatial average of numbers of prey in a single realisation and the mean of realisations. The corresponding PDE solution is shown in Fig. 5.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

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

fg0060: Simulations of the spatial Lotka–Volterra Model with parameters , a = 2, b = 0.1, c = 3, , h = 0.2 and initial values N(k,t = 0)=50, M(k,t = 0)=5 for . We compare (a)–(b), the stochastic model, to (c)–(d), the hybrid model with thresholds Θ = 10, and (e)–(f) with Θ = 25. The figures on the left show the spatial profile at time t = 4.1 of the number of predators and prey of a single realisation as well as the mean of 256 different realisations, whereas the figures on the right show the time evolution of the spatial average of numbers of prey in a single realisation and the mean of realisations. The corresponding PDE solution is shown in Fig. 5.
Mentions: The domain of length is divided into boxes, so . Initially it contains a spatially homogeneous distribution of prey and predators so that , . The model parameters are fixed so that , , , . With Neumann boundary conditions, equations (24) remain spatially homogeneous at all times, and both populations oscillate in time. Typical results are presented in Fig. 5, and show that the peak in prey numbers is followed by a peak in the number of predators. For this choice of parameter values the minimum number of individuals of either species is always sufficiently large that extinction in the stochastic, spatial model is almost impossible. Corresponding results for the stochastic and hybrid models for two choices of the threshold values ( and ) are shown in Fig. 6. The column on the left shows the spatial profile of the number of predators and prey in a given box at time , both for a single realisation and the mean of 256 different realisations. Comparing either predator and prey numbers, we note that the means for the stochastic and hybrid models appear similar for both values of Θ (see Fig. 6(a), (c), (e)), and are fairly homogeneous, whereas single realisations of either model differ markedly. As expected, the profile of predator and prey numbers in the stochastic model is noisy throughout the domain, whereas noise is suppressed in the hybrid model when the population numbers exceed the threshold. We observe the predator numbers in Fig. 6(c) are above the threshold and hence smoothly distributed in space, but they are not homogeneous, in contrast to the profile of the mean.

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.