$\pi_{1}$ and $\pi_{2}$ have smaller values than in the absence of mutations. The phenomenological description still works well, and appears to correctly take into account the effects of mutation. The overall fixation probability $\pi$ is again independent of $\Delta t$.

![\[figure:figtime\] Time to fixation as a function of the time interval between beneficial mutants $\Delta t$ ($N=1000$, $s_{1}=0.1$, $s_{2}=0.5$, $u=0.0$). a: Time to fixation of genotype 1, $T_{1}$. b: Time to fixation of genotype 2, $T_{2}$.](Figure3.eps){width="9cm"}

In Figure \[figure:figtime\] we show the time to fixation of mutant $1$, $T_{1}$, and the time to fixation of mutant $2$, $T_{2}$, as functions of $\Delta t$. The parameter values are the same as those of Figure \[figure:fig1\]. In Figure \[figure:figtime\]a, we see that $T_{1}$ is approximately constant for all values of $\Delta t$ except for the range $0 \leq \Delta t \leq 200$. For negative $\Delta t$ the fixation of mutant 1 occurs only when mutant 2 has been eliminated. Therefore, $T_{1}$ corresponds to the result for the fixation time of a mutant with selective advantage $s_{1}$ in a homogeneous population with wild-type individuals only. The same is true for large positive $\Delta t$ because there mutant 1 has enough time to reach 