size $N=10,000$ for a subset of these parameters, in order to make sure that our results were robust against a change in population size. Because of the sizable amount of CPU time needed to carry out 100,000 replicates for $N=10,000$, we could however not study this case exhaustively.

![\[figure:fig1\] Probability of fixation as a function of time interval $\Delta t$ ($N=1000$, $s_{1}=0.1$, $s_{2}=0.5$, $u=0.0$). Solid lines represent the prediction according to the logistic growth model Eq. (\[tanh-ansatz\]). a: Probability of fixation of genotype $1$ (less beneficial mutation), $\pi_{1}$. b: Probability of fixation of genotype $2$ (more beneficial mutation), $\pi_{2}$. c: Overall probability of fixation, $\pi$.](Figure1-new.eps){width="8cm"}

In order to keep track of the evolutionary history of each mutant, we marked the initial mutants 1 and 2 with two distinct inheritable neutral markers. In that way, we could distinguish wild type sequences that were descendants from mutants 1 or 2 from the wild type sequences that were originally present. We continued all simulations until all individuals in the population were either unmarked, marked as descendants of mutant 1, or marked as descendants of mutant 2.

Figure \[figure:fig1\] shows the probability of ultimate fixation of mutants 1 and 2, $\pi_1$ and $\pi_2$, and 