e of origin.

In evolutionary dynamics, the probability that a mutation occurring at one of a population’s individuals eventually spreads through the entire population is known as the mutation’s fixation probability, $\rho$. In an otherwise homogeneous population, the value of $\rho$ depends on the ratio $r$ of the mutant’s fitness to that of the other individuals, and it is the interplay between $\rho$ and $r$ that determines the effectiveness of natural selection on the evolution of the population, given its size. In essence, highly correlated $\rho$ and $r$ lead to a prominent role of natural selection in driving evolution; random drift takes primacy, otherwise [@n06].

Let $P$ be a population of $n$ individuals and, for individual $i$, let $P_i$ be any nonempty subset of $P$ that excludes $i$. We consider the evolution of $P$ according to a sequence of steps, each of which first selects $i\in P$ randomly in proportion to $i$’s fitness, then selects $j\in P_i$ randomly in proportion to some weighting function on $P_i$, and finally replaces $j$ by an offspring of $i$ having the same fitness as $i$.

When $P$ is a homogeneous population of fitness $1$ (except for a randomly chosen mutant, whose fitness is initially set to $r\neq 1$), $P_i=P\setminus\{i\}$ [^1], and moreover the weighting function on every $P_i$ is a constant (thus choosing $j\in P_i$ occurs uniformly at random), this sequence of steps is known as the Moran process [@m58]. In this setting, evolution can be modeled by a simple discrete-time Markov chain, of states $0,1,\ldots,n$, in which state $s$ indicates the existence of $s$ individuals of fitness $r$, the others $n-s$ having fitness $1$.

In this chain, states $0$ and $n$ are absorbing and all others are transient. If $s$ is a transient state, then it is possible either to move from $s$ to $s+1$ or $s-1$, with probabilities $p$ and $q$, respectively, such that $p/q=r$, or to remain at state $s$ with probability $1-p-q$. When $r>1$ (an advantageous mutation), the evolution of the syste