In 1951 William Feller published the solution to the diffusion equation

u_t(t, x) = ½{xu(t, x)}_xx - ?{xu(t, x)}_x

for an initial condition u(0, x) = δ(x − x0). The equation models the growth of a

population of independently reproducing individuals, and manifests as the infinite

population limit of a continuous-time birth-death process in which the birth and

death rates both become infinite while their difference remains equal to α.

The Feller diffusion is a useful population genetics model for explaining the distribution

of genetic differences across a population in terms of genetic mutations

that have happened in ancestral lineages. I will give a pedagogical description of

how the ancestral tree of a random sample of individuals (the so-called “coalescent

tree”) can be interpreted as a stochastic process running backwards in time from

the present. Once this process is determined, the distribution of allele frequencies

resulting from neutral mutations can be calculated by summing over all possible coalescent

trees and the locations within those trees at which a mutation has occurred.