I consider an asexual population undergoing mutation, selection and drift.
Theories of adaptation of these systems often rely on stationarity assumptions:
stationary variance of a trait distribution in quantitative genetics,
stationary regime of xation in stochastic theories of tness dynamics. Yet,
these stationary regimes ignore the impact of initial standing variance in
tness, and may not be reached in the early phase of adaptive response to
an environmental challenge. They may even not exist in the presence of
context-dependent mutation and abrupt changes. I will present an alternative
approach to model these dynamics as a stochastic PDE followed by the
generating function of the tness distribution. Analytical solutions can be
found when drift is neglected (I will discuss when this is reasonable), providing
predictions for mutation-selection balance as well as adaptation. I
will apply these results to the problem of evolutionary rescue, focusing on
the contribution of de novo mutation vs standing variance from a population
previously at equilibrium.