  We consider two finite population Markov chain models, the two-island
Wright-Fisher model with mutation, and the seed-bank model with mutation.
Despite the relatively simple descriptions of the two processes, the the exact
form of their stationary distributions is in general intractable. For each of
the two models we provide two approximation theorems with explicit upper bounds
on the distance between the stationary distributions of the finite population
Markov chains, and either the stationary distribution of a two-island diffusion
model, or the beta distribution. We show that the order of the bounds, and
correspondingly the appropriate choice of approximation, depends upon the
relative sizes of mutation and migration. In the case where migration and
mutation are of the same order, the suitable approximation is the two-island
diffusion model, and if migration dominates mutation, then the weighted average
of both islands is well approximated by a beta random variable. Our results are
derived from a new development of Stein's method for the stationary
distribution of the two-island diffusion model for the weak migration results,
and utilising the existing framework for Stein's method for the Dirichlet
distribution.
