  We are interested in the dynamic of a structured branching population where
the trait of each individual moves according to a Markov process. The rate of
division of each individual is a function of its trait and when a branching
event occurs, the trait of a descendant at birth depends on the trait of the
mother. We prove a law of large numbers for the empirical distribution of
ancestral trajectories. It ensures that the empirical measure converges to the
mean value of the spine which is a time-inhomogeneous Markov process describing
the trait of a typical individual along its ancestral lineage. Our approach
relies on ergodicity arguments for this time-inhomogeneous Markov process. We
apply this technique on the example of a size-structured population with
exponential growth in varying environment.
