The reconstructed tree in the lineage-based model of protracted speciation

Journal of Mathematical Biology volume 70, pages367–397(2015)Cite this article

Abstract

A popular line of research in evolutionary biology is the use of time-calibrated phylogenies for the inference of diversification processes. This requires computing the likelihood of a given ultrametric tree as the reconstructed tree produced by a given model of diversification. Etienne and Rosindell in Syst Biol 61(2):204–213, (2012) proposed a lineage-based model of diversification, called protracted speciation, where species remain incipient during a random duration before turning good species, and showed that this can explain the slowdown in lineage accumulation observed in real phylogenies. However, they were unable to provide a general likelihood formula. Here, we present a likelihood formula for protracted speciation models, where rates at which species turn good or become extinct can depend both on their age and on time. Our only restrictive assumption is that speciation rate does not depend on species status. Our likelihood formula utilizes a new technique, based on the contour of the phylogenetic tree and first developed by Lambert in Ann Probab 38(1):348–395, (2010). We consider the reconstructed trees spanned by all extant species, by all good extant species, or by all representative species, which are either good extant species or incipient species representative of some good extinct species. Specifically, we prove that each of these trees is a coalescent point process, that is, a planar, ultrametric tree where the coalescence times between two consecutive tips are independent, identically distributed random variables. We characterize the common distribution of these coalescence times in some, biologically meaningful, special cases for which the likelihood reduces to an elegant analytical formula or becomes numerically tractable.

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Acknowledgments

AL was financially supported by grant MANEGE ‘Modèles Aléatoires en Écologie, Génétique et Évolution’ 09-BLAN-0215 of ANR (French national research agency). AL also thanks the Center for Interdisciplinary Research in Biology (Collège de France) for funding. HM was funded by the CNRS and ANR grant ECOEVOBIO-CHEX2011. RSE was financially supported by the Netherlands Organisation for Scientific Research (NWO) through a VIDI grant.

Author information

Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires CNRS UMR 7599, UPMC Univ Paris 06, Paris, France

    Amaury Lambert

  2. Center for Interdisciplinary Research in Biology CNRS UMR 7241, Collège de France, Paris, France

    Amaury Lambert

  3. Center for Applied Mathematics CNRS UMR 7641, École Polytechnique, Paris, France

    Hélène Morlon

  4. Institut de Biologie CNRS UMR 8197, École Normale Supérieure, Paris, France

    Hélène Morlon

  5. Community and Conservation Ecology, Centre for Ecological and Evolutionary Studies, University of Groningen, Box 11103, 9700 CC, Groningen, The Netherlands

    Rampal S. Etienne

Authors
  1. Amaury Lambert
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  2. Hélène Morlon
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  3. Rampal S. Etienne
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Corresponding author

Correspondence to Amaury Lambert.

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Lambert, A., Morlon, H. & Etienne, R.S. The reconstructed tree in the lineage-based model of protracted speciation. J. Math. Biol. 70, 367–397 (2015). https://doi.org/10.1007/s00285-014-0767-x

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Keywords

  • Phylogeny
  • Reconstructed tree
  • Protracted speciation
  • Multitype branching process
  • Coalescent point process
  • Splitting tree
  • Birth–death process
  • Lévy process
  • Scale function

Mathematics Subject Classification (2000)

  • Primary 60J80
  • Secondary 92D15
  • 60J85
  • 92D25
  • 92D40
  • 60G51
  • 60G55