Table 3 Comparison of the effects of different scenarios of demographic change on the probability of ultimate fixation when the initial frequency is y = 0.1
DescriptionProbability of Ultimate FixationError of Numerical Result (%)
ScenariosNumerical ResultAveraged Kimura ResultSimulation ResultRelative to the Averaged Kimura ResultRelative to the Simulation Result
Reference case: Constant population size, N = 100−0.0010.08300.08300.08530.02.7
0.0000.10000.10000.10440.04.2
0.0100.33580.33580.33650.00.2
Reference case: Constant population size, N = 200−0.0010.06760.06800.06890.61.9
0.0000.09950.10000.09870.50.8
0.0100.55080.55090.55130.00.1
Figure 4 (i): Population size discontinuously increases from N = 100 to N = 200 at t = 100−0.0010.07470.07500.07130.44.8
0.0000.09960.10000.10000.40.4
0.0100.38410.38410.38660.00.6
Figure 4 (ii): Population size continuously increases−0.0010.07310.07340.07110.42.8
0.0000.09960.10000.09980.40.2
0.0100.40290.40300.40100.00.5
Figure 4 (iii): Population size discontinuously increases from N = 100 to N = 200 at t = 200−0.0010.07630.07710.07681.00.7
0.0000.09970.10000.09520.34.7
0.0100.36380.36380.37070.01.9
  • This table compares the effects of different scenarios of demographic change on the probability of ultimate fixation when the initial frequency is y = 0.1. It includes two reference cases (populations of constant size) and three cases where the population size changes over time, which are illustrated in Figure 4. For the long-time numerical calculations (column “Numerical result” in the table), we fixed the ratio α, Equation 5, at the value α = 1000 and determined the probability associated with bin K at a sequence of progressively smaller values of the spacing of discrete frequencies, ε. Extrapolating a straight line through the data yielded the values given in the table (cf. Figure 3). The values in the column “Averaged Kimura result,” were obtained using the approach in Appendix C; these values used the entire distribution of the frequency and provide some evidence of its numerical accuracy. The simulations for this table were made within the framework of a Wright–Fisher model (Fisher 1930; Wright 1931). In such a framework, selection is treated as a deterministic process, and only the process of population thinning in the life cycle, corresponding to the random sampling of individuals without regard to type (i.e., random genetic drift) is treated stochastically. The simulation results were obtained from 104 replicate populations and simulations were continued until all populations either fixed or lost the A allele.