The Deterministic Limit of the Moran Model: a Uniform Central Limit Theorem

#### F. Cordero

2017, v.23, №2, 313-324

ABSTRACT

We consider a Moran model with two allelic types, mutation and selection. In this work, we study the behaviour of the proportion of fit individuals when the size of the population tends to infinity, without any rescaling of parameters or time. We first prove that the latter converges, uniformly in compacts in probability, to the solution of an ordinary differential equation, which is explicitly solved. Next, we study the stability properties of its equilibrium points. Moreover, we show that the fluctuations of the proportion of fit individuals, after a proper normalisation, satisfy a uniform central limit theorem in $[0,\infty)$. As a consequence, we deduce the convergence of the corresponding stationary distributions.

Keywords: Moran model; mutation-selection models; limit of large populations; central limit theorem; density dependent populations