  We investigate the evolutionary dynamics of a population structured in
phenotype, subjected to trait dependent selection with a linearly moving
optimum and an asexual mode of reproduction. Our model consists of a non-local
and non-linear parabolic PDE. Our main goal is to measure the history of traits
when the population stays around an equilibrium. We define an ancestral process
based on the idea of neutral fractions. It allows us to derive quantitative
information upon the evolution of diversity in the population along time.
First, we study the long-time asymptotics of the ancestral process. We show
that the very few fittest individuals drive adaptation. We then tackle the
adaptive dynamics regime, where the effect of mutations is asymptotically
small. In this limit, we provide an interpretation for the minimizer of some
related optimization problem, an Hamilton Jacobi equation, as the typical
ancestral lineage. We check the theoretical results against individual based
simulations.
