  During their lifetimes, individuals in populations pass through different
states, and the notion of an occupancy time describes the amount of time an
individual spends in a given set of states. Questions related to this idea were
studied in a recent paper by Roth and Caswell for cases where the environmental
conditions are constant. However, it is truly important to consider the case
where environments are changing randomly or in directional way through time, so
the transition probabilities between different states change over time,
motivating the use of time-dependent stage-structured models.
  Using absorbing inhomogenous Markov chains and the discrete-time
McKendrick--von F{\"o}rster equation, we derive explicit formulas for the
occupancy time, its expectation, and its higher-order moments for
stage-structured models with time-dependent transition rates. We apply our
approach to study a time-dependent model of the Southern Fulmar, and obtain
insights into how the number of breeding attempts depends on external
conditions that vary through time.
