Figure 4.—

Long-term independence of the coalescent process on the sampling scheme. This skyline plot illustrates the convergence of the continuous-habitat coalescent to a process that is equivalent to the standard neutral coalescent with a constant effective population size. For each of the three curves, 100,000 replicates of the coalescent process were simulated for a sample of 25 sequences. The parameters used for the simulations were N = 10,000, ρ = 100, σ = 0.3, in a square habitat measuring 10 × 10. The mean waiting time for each coalescent event was converted into a value corresponding to the population size that would give that expected time in a panmictic population. This inverse rate is plotted on the y-axis against the average time in the past at which this coalescent event occurred. In the sampling scheme labeled “1 Location,” the 25 samples are drawn from a 5 × 5 grid of equally spaced points at the center of the habitat [location (5, 5)]. Grid spacing is 0.1, corresponding to the minimum distance separating two individuals in a two-dimensional lattice with ρ = 100. In the “5 Locations” scheme, 5 samples are drawn from a vertical cross with spacing 0.1 between samples. There are five such crosses centered on (2, 2), (2, 8), (5, 5), (8, 2), and (8, 8). In the “20 Locations” scheme, samples are drawn from a centered 5 × 5 grid with spacing of 2 [locations (1, 1), (1, 3), etc.]. These three different sampling schemes all converge on the same process, and coalescent events > ∼600 generations in the past are independent of the original sampling locations. The fact that all three converge to a horizontal line indicates that this long-term process can be approximated by Kingman's coalescent with time rescaled by the effective population size. Variances have been omitted from this and other figures for clarity of presentation, but the variance of the waiting time for each coalescent event is close to the square of the mean, consistent with the exponential distribution of waiting times expected under the SNCM.