Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T12:01:33.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

M. Möhle
M. Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

Keywords

Cannings modeldualityexchangeabilitygeneralized Stirling numberinversion formulamulti-allele modelneutralityprinciple of inclusion and exclusionSilvester's sieve formula

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 92D10: Genetics 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 713 - 731
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] 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
[2] 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
[3] Felsenstein, J. (1971). The rate of loss of multiple alleles in finite haploid populations. Theoret. Pop. Biol. 2, 391–403.CrossRefGoogle ScholarPubMed
[4] Gladstien, K. (1976). Loss of alleles in a haploid population with varying environment. Theoret. Pop. Biol. 10, 383–394.Google Scholar
[5] Gladstien, K. (1977). Haploid populations subject to varying environment: the characteristic values and the rate of loss of alleles. SIAM J. Appl. Math. 32, 778–783.Google Scholar
[6] Gladstien, K. (1977). Subdivided populations: the characteristic values and rate of loss of alleles. J. Appl. Prob. 14, 241–248.Google Scholar
[7] Gladstien, K. (1978). The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM J. Appl. Math. 34, 630–642.Google Scholar
[8] Huillet, T. and Möhle, M. (2009). Duality and asymptotics for a class of nonneutral discrete Moran models. J. Appl. Prob. 46, 866–893.Google Scholar
[9] Karlin, S. and McGregor, J. (1965). Direct product branching processes and related induced Markoff chains. I. Calculations of rates of approach to homozygosity. In Proc. Internat. Res. Sem., Springer, New York, pp. 111–145.Google Scholar
[10] Kimura, M. (1957). Some problems of stochastic processes in genetics. Ann. Math. Statist. 28, 882–901.Google Scholar
[11] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
[12] Möhle, M. (1999). The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5, 761–777.Google Scholar
[13] Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.CrossRefGoogle Scholar
[14] Wright, S. (1931). Evolution in Mendelian populations. Genetics 16, 97–159.Google Scholar