1 has only a realistic chance of proliferating and going to fixation if mutant 2 is not present in the population. As long as $\Delta
t\lesssim0$, mutant 2 will either go to fixation relatively unscathed from the later appearance of mutant 1, or it will be lost to drift, in which case mutant 1 will have its turn. Likewise, for $\Delta t$ larger than the time to fixation of mutant 1, $T_1$, we expect that $\pi_i(\Delta t)=\pi_i(\infty)$, because $T_1$ generations after the introduction of mutant 1, mutant 2 will either find a population in which mutant 1 has already gone to fixation, or one in which it has been lost to drift. For $0\lesssim\Delta t\lesssim T_1$, we expect that $\pi_1(\Delta t)$ smoothly increases from $\pi_1(-\infty)$ to $\pi_1(\infty)$, while $\pi_2(\Delta t)$ smoothly decreases from $\pi_2(-\infty)$ to $\pi_2(\infty)$. These considerations suggest a sigmoidal form for $\pi_i(\Delta
t)$. We use a logistic growth model to describe the smooth transition from $\pi_i(-\infty)$ to $\pi_i(\infty)$ within the range $0\lesssim\Delta t\lesssim
T_1$: $$\label{tanh-ansatz}
 \pi_i(\Delta t)=\pi_i(-\infty) + \frac{\pi_i(\infty)-\pi_i(-\infty)}
    {1+e^{-\gamma_1\left(\Delta
t-T_1/2\right)}}\;,$$ where $\gamma_1$ is the fitness advantage of mutant 1 at finite mutation rate (see the definition following Eq. (\[pieq\]) below). This expression appears to be an acceptable 