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A critical branching process model for biodiversity

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

David Aldous  and
Lea Popovic
David Aldous*
Affiliation:
University of California, Berkeley
Lea Popovic*
Affiliation:
University of Minnesota
*
∗ Postal address: Department of Statistics, University of California, 367 Evans Hall #3860, Berkeley, CA 94720-3860, USA. Email address: aldous@stat.berkeley.edu
∗∗ Current address: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA.
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Abstract

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We study the following model for a phylogenetic tree on n extant species: the origin of the clade is a random time in the past whose (improper) distribution is uniform on (0,∞); thereafter, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on there being the prescribed number n of species at the present time. We study various mathematical properties of this model as n→∞: namely the time of origin and of the most recent common ancestor, the pattern of divergence times within lineage trees, the time series of the number of species, the total number of extinct species, the total number of species ancestral to the extant ones, and the ‘local’ structure of the tree itself. We emphasize several mathematical techniques: the association of walks with trees; a point process representation of lineage trees; and Brownian limits.

Keywords

BiodiversityBrownian excursioncontour processcritical branching processgenealogylocal weak convergencephylogenetic treepoint process

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 60J65: Brownian motion 92D15: Problems related to evolution
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 1094 - 1115
Copyright
Copyright © Applied Probability Trust 2005 

References

Aldous, D. J. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Prob. 1, 228–266.CrossRefGoogle Scholar
Aldous, D. J. (1993). The continuum random tree. III. Ann. Prob. 21, 248–289.CrossRefGoogle Scholar
Aldous, D. J. and Steele, J. M. (2004). The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (Encyclopaedia Math. Sci. 110), ed. Kesten, H., Springer, Berlin, pp. 1–72.Google Scholar
Dembo, A. and Zeitouni, O. (1992). Large Deviations and Applications, 2nd edn. Jones and Bartlett, Boston, MA.Google Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi+147.Google Scholar
Ewens, W. J. (1979). Mathematical Population Genetics. Springer, Berlin.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
Geiger, J. (2000). Poisson point processes in size-biased Galton–Watson trees. Electron. J. Prob. 5, 12pp.CrossRefGoogle Scholar
Hey, J. (1992). Using phylogenetic trees to study speciation and extinction. Evolution 46, 627–640.CrossRefGoogle ScholarPubMed
Krikun, M., Popovic, L. and Aldous, D. J. (2005). Stochastic models for phylogenetic trees on higher order taxa. In preparation.Google Scholar
Le Gall, J.-F. (1989). Marches aléatoires, mouvement brownien et processus de branchement. In Séminaire de Probabilités XXIII (Lecture Notes Math. 1372), Springer, Berlin, pp. 258–274.CrossRefGoogle Scholar
Möhle, M. (2000). Ancestral processes in population genetics. J. Theoret. Biol. 204, 629–638.CrossRefGoogle ScholarPubMed
Nee, S., May, R. M. and Harvey, P. H. (1994). The reconstructed evolutionary process. Philos. Trans. R. Soc. London B 344, 305–311.Google ScholarPubMed
Neveu, J. and Pitman, J. (1989). Renewal property of the extrema and tree property of a one-dimensional Brownian motion. In Séminaire de Probabilités XXIII (Lecture Notes Math. 1372), Springer, Berlin, pp. 239–247.CrossRefGoogle Scholar
Page, R. D. M. and Holmes, E. C. (1998). Molecular Evolution: A Phylogenetic Approach. Blackwell Science, Oxford.Google Scholar
Pitman, J. (2002). Combinatorial stochastic processes. Tech. Rep. 621, Department of Statistics, University of California, Berkeley.Google Scholar
Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Prob. 14, 2120–2148.CrossRefGoogle Scholar
Wollenberg, K., Arnold, J. and Avise, J. C. (1996). Recognizing the forest for the trees: testing temporal patterns of cladogenesis using a null model of stochastic diversification. Molec. Biol. Evol. 13, 833–849.CrossRefGoogle Scholar
Yule, G. U. (1924). A mathematical theory of evolution, based on the conclusions of Dr J. C. Willis. Philos. Trans. R. Soc. London B 213, 21–87.Google Scholar