  Epochal dynamics, in which long periods of stasis in an evolving population
are punctuated by a sudden burst of change, is a common behavior in both
natural and artificial evolutionary processes. We analyze the population
dynamics for a class of fitness functions that exhibit epochal behavior using a
mathematical framework developed recently. In the latter the approximations
employed led to a population-size independent theory that allowed us to
determine optimal mutation rates. Here we extend this approach to include the
destabilization of epochs due to finite-population fluctuations and show that
this dynamical behavior often occurs around the optimal parameter settings for
efficient search. The resulting, more accurate theory predicts the total number
of fitness function evaluations to reach the global optimum as a function of
mutation rate, population size, and the parameters specifying the fitness
function. We further identify a generalized error threshold, smoothly bounding
the two-dimensional regime of mutation rates and population sizes for which
evolutionary search operates efficiently.
