  We construct a pathwise formulation for a multi-type age-structured
population dynamics, which involves an age-dependent cell replication and
transition of gene- or phenotypes. By employing the formulation, we derive a
variational representation of the stationary population growth rate; the
representation comprises a trade-off relation between growth effects and a
single-cell intrinsic dynamics described by a semi-Markov process. This
variational representation leads to a response relation of the stationary
population growth rate, in which statistics on a retrospective history work as
the response coefficients. These results contribute to predicting and
controlling growing populations based on experimentally observed cell-lineage
information.
