Particle distributions at for pure diffusion of a 1000 particle spike obtained using the SPLA and Eq. (15). The initial delta function is shown as a dashed line. The cross-sectional area is set such that the total concentration over the domain is . The SPLA simulation was performed using the -selection procedure of Table II (SPLA-SB).
(a) Average numbers of simulation steps and (b) average CPU times vs total particle number for pure diffusion of a function until using the SPLA-RB, SPLA-SB, and NSM. In each case, the particle number was changed by varying the cross-sectional area , while maintaining a constant concentration of over the domain. All results are averaged over 500 simulation runs performed on an Intel Core 2 Duo, 2.13 GHz machine with 2 Gbytes of RAM.
Means and standard deviations of the particle number over the entire domain at for pure diffusion of a particle function using the SPLA-SB and the NSM. In both cases, results are from 500 simulation runs. The dotted lines constitute an envelope of twice the standard deviation about the SPLA-SB mean.
Solution of the one-dimensional Fisher’s Eq. (16). The initial condition is shown by the dashed lines. The deterministic trajectory (blue) is shown at , the time at which the solution reaches half its saturation value at . A stochastic trajectory (red) is shown at
(a) Average numbers of simulation steps and (b) average CPU times vs for simulations until of the reaction-diffusion system (17) using various methods. In each case, the particle number is changed by varying the cross-sectional area while maintaining a constant concentration of within each . All results are averaged over 500 simulation runs performed on an Intel Core 2 Duo, 2.13 GHz machine with 2 Gbytes of RAM.
Time steps calculated during individual simulation runs of the reaction-diffusion system (17) with using various methods.
Percent deviations between mean wave velocities obtained from various leaping methods and the NSM for varying system sizes . (Inset) Convergence of to the analytical solution (Ref. 63) with increasing system size. All results are based on 500 leaping and NSM simulation runs. Note that the apparent discrepancies between the NSM and the SPLA and Marquez-Lago methods at small particle numbers are simply due to random sampling error [also see Fig. 8(c)].
Kolmogorov distances at obtained using (a) Marquez-Lago, (b) Rossinelli, and (c) SPLA. The -axis corresponds to position within the domain and the -axis to system size (i.e., ). All results are based on 500 leaping and NSM simulation runs. The NSM self-distance is subtracted from the Kolmogorov distances and negative values are clipped to zero. The black line is the mean position of the wavefront for different system sizes. We can infer from this that the simulation error arises mainly at the propagating wavefront.
Snapshots of the Gray–Scott reaction-diffusion system (19) at obtained using (a) Marquez-Lago, (b) Rossinelli, (c) the full SPLA-SB, and (d) Eq. (18). The concentration of (plotted above) ranges from (blue) to (red). The features present in regions I–III are compared for different simulation methods. All simulations are performed with the parameters , , , and .
Spatial versions of the -selection formulas of Cao et al. (Ref. 19), as modified in Ref. 31. One calculation is required for each reaction and outgoing diffusion event in . Note that in Eq. (23), is the smallest possible nonzero value of ( for elementary reactions).
Spatial versions of the -selection formulas of Cao et al. (Ref. 19). One calculation is required for each species in . Note that in Eq. (29), the parameter depends on the types of events species participates in. See Ref. 19 for formulas applicable to elementary event types, Ref. 31 for simplified versions of these, and Ref. 56 for extensions to selected nonelementary events.
Article metrics loading...
Full text loading...