Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-11T10:43:41.751Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence to the coalescent in populations of substantially varying size

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Peter Jagers  and
Serik Sagitov
Peter Jagers*
Affiliation:
Chalmers University of Technology and Göteborg University
Serik Sagitov*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden.
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.

Keywords

Coalescentexchangeabilitypopulation genetics

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 92D15: Problems related to evolution 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 368 - 378
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Adv. Appl. Prob. 6, 260–290.Google Scholar
Cannings, C. (1975). The latent roots of certain Markov chains arising in genetics: a new approach. II. Further haploid models. Adv. Appl. Prob. 7, 264–282.CrossRefGoogle Scholar
Donnelly, P., and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Prob. 27, 166–205.CrossRefGoogle Scholar
Donnelly, P. and Tavaré, S. (1995). Coalescents and genealogical structure under neutrality. Annu. Rev. Genet. 29, 401–421.Google Scholar
Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury, Belmont, CA.Google Scholar
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterisation and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Griffiths, R. C. and Tavaré, S. (1994). Sampling theory for neutral alleles in a varying environment. Phil. Trans. R. Soc. London B 344, 403–410.Google Scholar
Iizuka, M. (2001). The effective size of fluctuating populations. Theoret. Pop. Biol. 59, 281–286.Google Scholar
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1997). Discrete Multivariate Distributions. John Wiley, New York.Google Scholar
Kaj, I., and Krone, S. M. (2003). The coalescent process in a population with stochastically varying size. J. Appl. Prob. 40, 33–48.Google Scholar
Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds Koch, G. and Spizzichino, F., North Holland, Amsterdam, pp. 97–112.Google Scholar
Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Stastical Science (J. Appl. Prob. Spec. Vol 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Möhle, M. (1998). Robustness results for the coalescent. J. Appl. Prob. 35, 438–447.CrossRefGoogle Scholar
Möhle, M. (2002). The coalescent in population models with time-inhomogeneous environment. Stoch. Process. Appl. 97, 199–227.Google Scholar
Möhle, M., and Sagitov, S. (2003). Coalescent patterns in exchangeable diploid population models. J. Math. Biol. 47, 337–352.Google Scholar
Nordborg, M., and Krone, S. (2002). Separation of time scales and convergence to the coalescent in structured populations. In Modern Developments in Theoretical Population Genetics, eds Slatkin, M. and Veuille, M., Oxford University Press, pp. 194–232.CrossRefGoogle Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar
Sagitov, S. (2003). Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Prob. 40, 839–854.CrossRefGoogle Scholar
Swoffold, D. L., Olsen, G. J., Waddell, P. J., and Hillis, D. M. (1996). Phylogeny reconstruction. In Molecular Systematics, 2nd edn, eds Hillis, D. M., Morritz, C. and Mabel, B. K., Sinauer, Sunderland, MA, pp. 407–514.Google Scholar