later replaced by mutant 2. Therefore, we find $$\begin{aligned}
\label{p1inf}
\pi_1(\infty) &= P(s_1,u)\Big[1-P\Big(\frac{1+s_2}{(1+s_1)(1-u)}-1,u\Big)\Big]\,,\\\label{p2inf}
\pi_2(\infty) &= P(s_1,u)P\Big(\frac{1+s_2}{(1+s_1)(1-u)}-1,u\Big)+[1-P(s_1,u)]P(s_2,u)\,,\end{aligned}$$ where $P(\frac{1+s_2}{(1+s_1)(1-u)}-1,u)$ is the probability that mutant 2 goes to fixation after mutant 1 has already reached fixation. Since $P(\frac{1+s_2}{(1+s_1)(1-u)}-1,u)$ is always smaller than or equal to $P(s_2,u)$, we have $\pi_1(-\infty)\leq \pi_1(\infty)$ and $\pi_2(-\infty)\geq
\pi_2(\infty)$. In other words, both mutants have a higher probability of fixation when they are introduced first than when they are introduced second.

For intermediate values of $\Delta t$, there is no theory that enables us to derive a simple expression for $\pi_i(\Delta t)$ (but see @barton95). We cannot use branching process theory, because it assumes that the presence of the invading mutants does not influence the mean fitness (which they do for intermediate $\Delta
t$). Also, diffusion theory becomes unwieldy when there are more than 2 different sequence types. Nevertheless, we have found that we can develop a phenomenological description of the competition of two mutants as follows. We know that $\pi_i(t)$ must reach the two limiting values $\pi_i(\infty)$ and $\pi_i(-\infty)$ for sufficiently large positive or negative $\Delta t$. Moreover, as long as $\Delta
t<0$, we do not expect $\pi_i(\Delta t)$ to be very different from $\pi_i(-\infty)$, because mutant 