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 Foerster 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. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. We apply our approach to study a specific 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.
Highlights
The study of structured populations in ecology often is focused on organisms with a life history with distinct yearly events
A seminal contribution to this work was that of Leslie in the 1940’s (Leslie 1945) who represented the dynamics of age-structured populations using a discrete-time model formulated as a vector with entries representing the number of individuals at different, spaced, ages, and a matrix representing yearly transitions with the entries in the top row representing births and the sub-diagonal elements representing the year to year survival probabilities
We have presented extensions to the analysis of occupancy times for states in a compartmental model such as a stage-structured demographic model from the case of constant transition rates through time previously analyzed by Roth and Caswell Roth and Caswell (2018) to the important case of time-varying transition rates
Summary
The study of structured populations in ecology often is focused on organisms with a life history with distinct yearly events. Ecologists typically think of stage-structured modeling approaches as focusing on the numbers of individuals in different states, a different interpretation is possible if we break up the model into two pieces: one representing reproduction, and the other representing transitions among states not involving reproduction. This separation of the description of reproduction from the description of transitions between states by organisms An important feature to include in demographic models for ecological populations is possible changes in the environment through time, representing either short or long term environmental fluctuations In this case, the resulting demographic models lead to inhomogeneous Markov chains. We will demonstrate how to calculate and interpret these quantities with a specific application, given by breeder states of the Southern Fulmar bird population
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