  We consider a stochastic individual-based population model with competition,
trait-structure affecting reproduction and survival, and changing environment.
The changes of traits are described by jump processes, and the dynamics can be
approximated in large population by a non-linear PDE with a non-local mutation
operator. Using the fact that this PDE admits a non-trivial stationary
solution, we can approximate the non-linear stochastic population process by a
linear birth-death process where the interactions are frozen, as long as the
population remains close to this equilibrium. This allows us to derive, when
the population is large, the equation satisfied by the ancestral lineage of an
individual uniformly sampled at a fixed time $T$, which is the path constituted
of the traits of the ancestors of this individual in past times $t\leq T$. This
process is a time inhomogeneous Markov process, but we show that the time
reversal of this process possesses a very simple structure (e.g.
time-homogeneous and independent of $T$). This extends recent results where the
authors studied a similar model with a Laplacian operator but where the methods
essentially relied on the Gaussian nature of the mutations.
