Occupancy times for time-dependent stage-structured models

TL;DR

This paper develops explicit formulas for occupancy times in stage-structured models with time-dependent transition rates, extending previous work to environments that change over time, and applies it to a seabird population.

Contribution

It introduces a novel analytical framework for occupancy times in time-dependent models using Markov chains and the McKendrick--von F{"o}rster equation, addressing environmental variability.

Findings

Derived formulas for occupancy times and moments in time-dependent models

Applied model to Southern Fulmar, revealing effects of environmental changes on breeding

Provided insights into how external conditions influence population dynamics

Abstract

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.