Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1319–1347 | Cite as

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

  • Sophie HautphenneEmail author
  • Melanie Massaro
  • Peter Taylor


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.


Phase-type distribution Transient Markov chain Age distribution Petroica traversi 

Mathematics Subject Classification

60J27 92B05 92D25 



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.


  1. Aalen OO (1995) Phase-type distributions in survival analysis. Scand J Stat 22(4):447–463Google Scholar
  2. Butler D, Merton D (1992) The Black Robin: saving the world’s most endangered bird. Oxford University Press, AucklandGoogle Scholar
  3. Carbonell F, Jimenez JC, Pedroso LM (2008) Computing multiple integrals involving matrix exponentials. J Comput Appl Math 213(1):300–305MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cubrinovska I, Massaro M, Hale ML (2016) Assessment of hybridisation between the endangered Chatham Island black robin (Petroica traversi) and the Chatham Island tomtit (Petroica macrocephala chathamensis). Conserv Genet 17:259–265CrossRefGoogle Scholar
  5. Gavrilov LA, Gavrilova NS (1991) The biology of life span: a quantitative approachGoogle Scholar
  6. Hautphenne S, Latouche G (2012) The Markovian binary tree applied to demography. J Math Biol 64(7):1109–1135MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hautphenne S, Massaro M, Turner K (2017) Fitting Markovian binary trees using global and individual demographic data. Submitted for publication. Preprint available on
  8. Kennedy ES, Grueber CE, Duncan RP, Jamieson IG (2014) Severe inbreeding depression and no evidence of purging in an extremely inbred wild species–the Chatham Island black robin. Evolution 68(4):987–995CrossRefGoogle Scholar
  9. Lin XS, Liu X (2007) Markov aging process and phase-type law of mortality. N Am Actuar J 11(4):92–109MathSciNetCrossRefGoogle Scholar
  10. Massaro M, Sainudiin R, Merton D, Briskie JV, Poole AM, Hale ML (2013) Human-assisted spread of a maladaptive behavior in a critically endangered bird. PloS ONE 8(12):e79066CrossRefGoogle Scholar
  11. Massaro M, Stanbury M, Briskie JV (2013) Nest site selection by the endangered black robin increases vulnerability to predation by an invasive bird. Anim Conserv 16(4):404–411CrossRefGoogle Scholar
  12. Neuts MF (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Dover Publications, MineolazbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  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

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