Table 1  Illustrative numerical results for the probability of fixation when the composite quantity S(t) = 4Ne(t)s(t) changes with time
Data setMethodTN0Ns0sS0SPfixCost/rep
1Direct simulation2050500.0050.0501100.068712
2Finite T simulation2050500.0050.0501100.06894
3Direct simulation20505000.0050.0051100.0817115
4Finite T simulation20505000.0050.0051100.08189
5Direct simulation20050500.0050.0501100.032011
6Finite T simulation20050500.0050.0501100.032110
7Direct simulation200505000.0050.0051100.045151
8Finite T simulation200505000.0050.0051100.044718
9Direct simulation205005000.0050.050101000.067218
10Finite T simulation205005000.0050.050101000.06714
11Direct simulation2050050000.0050.005101000.0814192
12Finite T simulation2050050000.0050.005101000.08128
13Direct simulation2005005000.0050.050101000.029814
14Finite T simulation2005005000.0050.050101000.030211
15Direct simulation20050050000.0050.005101000.042890
16Finite T simulation20050050000.0050.005101000.042517
  • We take S(t) to start at the positive value S0 at time t = 0 and then to linearly increase to the value S = 10S0 by time T and then remain constant at the value S for all times larger than T. Two different methods are used to estimate the probability of fixation when initially there is only a single copy of an A allele: (i) direct simulation, where we “follow” each replicate population until either fixation or loss occurs and (ii) simulations based on Equation 4, where we follow each replicate population for, at most, T generations. The column labeled “Cost/rep” gives the mean number of generations a replicate population was followed in a simulation. In the simulations, 5 × 105 replicate populations were used, and we adopted a Wright–Fisher model where the only place in the life cycle where randomness occurs is in the thinning of the number of individuals to Ne = N adults. The initial values of Ne and s are N0 and s0, while the final values are N and s; data sets 1, 2, 5, 6, 9, 10, 13, and 14 correspond to Ne fixed and s changing with time. It is evident from the table that there are differences in the fixation probability, depending on whether Ne changed with time, at fixed s, or s changed with time at fixed Ne.