  We investigate a simple quantitative genetics model subjet to a gradual
environmental change from the viewpoint of the phylogenies of the living
individuals. We aim to understand better how the past traits of their ancestors
are shaped by the adaptation to the varying environment. The individuals are
characterized by a one-dimensional trait. The dynamics -- births and deaths --
depend on a time-changing mortality rate that shifts the optimal trait to the
right at constant speed. The population size is regulated by a nonlinear
non-local logistic competition term. The macroscopic behaviour can be described
by a PDE that admits a unique positive stationary solution. In the stationary
regime, the population can persist, but with a lag in the trait distribution
due to the environmental change. For the microscopic (individual-based)
stochastic process, the evolution of the lineages can be traced back using the
historical process, that is, a measure-valued process on the set of continuous
real functions of time. Assuming stationarity of the trait distribution, we
describe the limiting distribution, in large populations, of the path of an
individual drawn at random at a given time $T$. Freezing the non-linearity due
to competition allows the use of a many-to-one identity together with
Feynman-Kac's formula. This path, in reversed time, remains close to a simple
Ornstein-Uhlenbeck process. It shows how the lagged bulk of the present
population stems from ancestors once optimal in trait but still in the tail of
the trait distribution in which they lived.
