## When Can One Detect Overdominant Selection
in the Infinite Alleles Model?

*Paul Joyce, Stephen M. Krone, and Thomas G. Kurtz*
**Abstract**

One of the goals of this paper is to show that the
infinite alleles model with overdominant
selection ``looks like'' the neutral infinite
alleles model when the selection intensity and
mutation rate get large together. This rather
surprising behavior was noticed by Gillespie
(1999) in simulations. To make rigorous and
refine Gillespie's observations, we analyze the
limiting behavior of the likelihood ratio of the
stationary distributions for the model under
selection and neutrality, as the mutation rate
and selection intensity go to infinity together
in a specified manner. In particular, we show
that the likelihood ratio tends to one as the
mutation rate goes to infinity, provided the
selection intensity is a multiple of the mutation
rate raised to a power less than 3/2. (Gillespie's simulations
correspond to the power 1.) This
implies that we cannot distinguish between the
two models in this setting. Conversely, if the
selection intensity grows like a multiple of the
mutation rate raised to a power greater than 3/2,
selection can be detected; i.e., the likelihood
ratio tends to 0 under neutrality and infinity under
selection. We also determine the nontrivial limit distributions
in the case of the critical exponent 3/2. We further analyze the limiting
behavior when the exponent is less than 3/2 by determining the rate
at which the likelihood ratio converges to one and by developing results
for the distributions of finite samples.