Table 2 Results for the probability of ultimate fixation at a locus subject to genic selection when the effective population size is Ne = 100
4NesInitial frequency, yProbability of ultimate fixationRelative error (%)
Numerical result, c(∞)Kimura’s result
−10.050.02970.02980.3
0.100.06080.06120.7
00.050.04960.05000.8
0.100.09930.10000.7
100.050.39340.3935< 0.1
0.100.63210.6321< 0.1
  • This table gives results for the probability of ultimate fixation at a locus subject to genic selection when the effective population size is Ne = 100. It covers different values of the initial frequency, y, and different values of the strength of selection, s. The results were obtained from (i) the numerical scheme of this work, in the limit of long times, and (ii) Kimura’s expression for the probability of fixation, Equation 9. The final column of the table contains the magnitude of the error of the numerical approach, relative to Kimura’s result. For the column containing the numerical results, we fixed the ratio α, Equation 5, at the value α = 500 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). This procedure leads to a value of c(t), which is the probability of fixation by time t. The value adopted for t was such that the sum c(t)+ (probability of loss by time t) was greater than 0.999. Theoretically, and in accordance with our numerically findings, this sum increases monotonically with t, hence the probability of ultimate fixation, c(∞), should differ from c(t) by <10−3.