Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1319–1347

# How old is this bird? The age distribution under some phase sampling schemes

Article

## Abstract

In this paper, we use a finite-state continuous-time Markov chain with one absorbing state to model an individual’s lifetime. Under this model, the time of death follows a phase-type distribution, and the transient states of the Markov chain are known as phases. We then attempt to provide an answer to the simple question “What is the conditional age distribution of the individual, given its current phase”? We show that the answer depends on how we interpret the question, and in particular, on the phase observation scheme under consideration. We then apply our results to the computation of the age pyramid for the endangered Chatham Island black robin Petroica traversi during the monitoring period 2007–2014.

### Keywords

Phase-type distribution Transient Markov chain Age distribution Petroica traversi

### Mathematics Subject Classification

60J27 92B05 92D25

## Notes

### Acknowledgements

The authors are supported by the Australian Research Council Laureate Fellowship FL130100039. The first author has also conducted part of the work under the Discovery Early Career Researcher Award DE150101044. Field-based research on black robins from 2007–2014 was funded by a New Zealand Foundation for Research, Science and Technology fellowship to MM (UOCX0601), and the University of Canterbury, the Brian Mason Scientific and Technical Trust and the Mohamed bin Zayed Species Conservation Fund.

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## Authors and Affiliations

• Sophie Hautphenne
• 1
• 2
• Melanie Massaro
• 3
• Peter Taylor
• 1
1. 1.School of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
2. 2.Institute of MathematicsEcole polytechnique fédérale de LausanneLausanneSwitzerland
3. 3.Institute of Land, Water and Society, School of Environmental SciencesCharles Sturt UniversityAlburyAustralia