heterogeneous genetic background [@barton95; @johnson2002], and also to investigate the process of fixation in a viral quasispecies [@wilke2003].

Theoretical analysis
====================

Probability of ultimate fixation
--------------------------------

In the Appendix, we derive equations for the probability of fixation $P(s,u)$ and time to fixation $T(s,u)$ of an individual mutant with selective advantage $s$ and mutation rate $u$. For two mutants, we are mostly interested in the probability of ultimate fixation, that is, the probability that a mutant reaches fixation and is not subsequently replaced by the other mutant. In the following, we will denote the probability that mutant $i$ reaches ultimate fixation under the condition that the two mutants are introduced $\Delta t$ generations apart as $\pi_i(\Delta t)$. We can calculate $\pi_i(\Delta t)$ for the two limiting cases $\Delta t\rightarrow -\infty$ and $\Delta t\rightarrow \infty$, and present a phenomenological description for intermediate $\Delta t$.

For $\Delta t\rightarrow -\infty$, that is, when mutant 2 arises much earlier than mutant 1, we have $$\begin{aligned}
\label{p1minusinf}
\pi_1(-\infty) &= [1-P(s_2,u)]P(s_1,u)\,,\\\label{p2minusinf}
\pi_2(-\infty) &= P(s_2,u)\,.\end{aligned}$$ Since $s_2>s_1$, mutant 1 can reach fixation only when mutant 2 has not reached fixation. For $\Delta t\rightarrow \infty$, we have to consider the possibility that mutant 1 reaches fixation first, but is 