Figure 1.

(A) Illustration of how our model resamples allele frequencies from generation to generation. From a given frequency $X\left(t\right)$ of mutants in generation t, the frequency $x\left(t+1\right)$ of the next generation is produced in two steps. First, each individual gets an offspring number drawn from a given offspring distribution. This step generates random numbers $U_i^{\left(m\right)}$ for the mutant family sizes, and $U_i^{\left(w\right)}$ for the wild-type family sizes (a couple of large family sizes are indicated in the illustration). After this first step, the total population size will strongly deviate from the starting size N (middle column is much higher than the left column). Thus, we binomially sample with replacement from this intermediate population exactly N individuals (right column has the same size as left column). The resulting fraction of mutants is the mutant frequency $X\left(t+1\right)$ in generation $t+1.$ (B) A total of 10 sample paths are shown for a population size $N={10}^9$ and an offspring distribution with density $p\left(u\right)\sim u^{-2}.$ All sample paths start with the same initial frequency 0.5. The time axis is measured in units of $\text{ln}\left(N\right),$ which turns out to be the coalescence timescale (for large N).