  We study a generalized discrete-time multi-type Wright-Fisher population
process. The mean-field dynamics of the stochastic process is induced by a
general replicator difference equation. We prove several results regarding the
asymptotic behavior of the model, focusing on the impact of the mean-field
dynamics on it. One of the results is a limit theorem that describes sufficient
conditions for an almost certain path to extinction, first eliminating the type
which is the least fit at the mean-field equilibrium. The effect is explained
by the metastability of the stochastic system, which under the conditions of
the theorem spends almost all time before the extinction event in a
neighborhood of the equilibrium. In addition, to limit theorems, we propose a
variation of Fisher's maximization principle, fundamental theorem of natural
selection, for a completely general deterministic replicator dynamics and study
implications of the deterministic maximization principle for the stochastic
model.
