  The dynamics of adaptation is difficult to predict because it is highly
stochastic even in large populations. The uncertainty emerges from number
fluctuations, called genetic drift, arising in the small number of particularly
fit individuals of the population. Random genetic drift in this evolutionary
vanguard also limits the speed of adaptation, which diverges in deterministic
models that ignore these chance effects. Several approaches have been developed
to analyze the crucial role of noise on the expected dynamics of adaptation,
including the mean fitness of the entire population, or the fate of newly
arising beneficial deleterious mutations. However, very little is known about
how genetic drift causes fluctuations to emerge on the population level,
including fitness distribution variations and speed variations. Yet, these
phenomena control the replicability of experimental evolution experiments and
are key to a truly predictive understanding of evolutionary processes. Here, we
develop an exact approach to these emergent fluctuations by a combination of
computational and analytical methods. We show, analytically, that the infinite
hierarchy of moment equations can be closed at any arbitrary order by a
suitable choice of a dynamical constraint. This constraint regulates (rather
than fixes) the population size, accounting for resource limitations. The
resulting linear equations, which can be accurately solved numerically, exhibit
fluctuation-induced terms that amplify short-distance correlations and suppress
long-distance ones. Importantly, by accounting for the dynamics of
sub-populations, we provide a systematic route to key population genetic
quantities, such as fixation probabilities and decay rates of the genetic
diversity.
