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Coalescence theory for a general class of structured populations with fast migration

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

O. Hössjer
O. Hössjer*
Affiliation:
Stockholm University
*
∗ Postal address: Department of Mathematics, Stockholm University, SE 106 91, Stockholm, Sweden. Email address: [email protected]
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Abstract

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In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population size N tends to ∞, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scale c. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions for the coalescent effective population size N/c are obtained.

Keywords

Age-structured populationcoalescence theoryeffective population sizegeographical substructureweak convergence

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 1027 - 1047
Copyright
Copyright © Applied Probability Trust 2011 

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
Caswell, H. (2001). Matrix Population Models, 2nd edn. Sinauer, Sunderland, MA.Google Scholar
Cenik, C. and Wakeley, J. (2010). Pacific salmon and the coalescent effective population size. PLoS One, 5, e13019.Google Scholar
Engen, S., Lande, R. and Saether, B.-E. (2005a). Effective size of a fluctuating age-structured population. Genetics 170, 941–954.Google Scholar
Engen, S., Lande, R., Saether, B.-E. and Weimerskirch, H. (2005b). Extinction in relation to demographic and environmental stochasticity in age-structured models. Math. Biosci. 195, 210–227.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Ewens, W. J. (1982). On the concept of effective population size. Theoret. Pop. Biol. 21, 373–378.Google Scholar
Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction, 2nd edn. Springer, New York.Google Scholar
Felsenstein, J. (1971). Inbreeding and variance effective numbers in populations with overlapping generations. Genetics 68, 581–597.CrossRefGoogle ScholarPubMed
Fu, Y. X. (1997). Coalescent theory for a partially selfing population. Genetics 146, 1489–1499.CrossRefGoogle ScholarPubMed
Herbots, H. M. (1997). The structured coalescent. In Progress in Population Genetics and Human Evolution (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 87), eds. Donnelly, P. and Tavaré, S.. Springer, New York, pp. 231–255.Google Scholar
Jagers, P. and Sagitov, S. (2004). Convergence to the coalescent in populations of substantially varying size. J. Appl. Prob. 41, 368–378.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.CrossRefGoogle Scholar
Kaj, I., Krone, S. M. and Lascoux, M. (2001). Coalescent theory for seed bank models. J. Appl. Prob. 38, 285–300.Google Scholar
Kingman, J. F. C. (1982a). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
Kingman, J. F. C. (1982b). On the genealogy of large populations. In Essays in Statistical 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. (1998a). A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing. Adv. Appl. Prob. 30, 493–512.Google Scholar
Möhle, M. (1998b). Coalescent results for two-sex population models. Adv. Appl. Prob. 30, 513–520.Google Scholar
Möhle, M. (1998c). Robustness results for the coalescent. J. Appl. Prob. 35, 438–447.Google Scholar
Möhle, M. (2000). Ancestral processes in population genetics - the coalescent. J. Theoret. Biol. 204, 629–638.CrossRefGoogle ScholarPubMed
Möhle, M. and Sagitov, S. (2003). Coalescent patterns in diploid exchangeable population models. J. Math. Biol. 47, 337–352.Google Scholar
Naglyaki, T. (1980). The strong-migration limit in geographically structured populations. J. Math. Biol. 9, 101–114.Google Scholar
Nordborg, M. and Donnelly, P. (1997). The coalescent process with selfing. Genetics 146, 1185–1195.Google Scholar
Nordborg, M. and Krone, S. M. (2002). Separation of time scales and convergence to the coalescent in structured populations. In Modern Development in Theoretical Population Genetics, eds Slatkin, M. and Veuille, M., Oxford University Press, pp. 194–232.CrossRefGoogle Scholar
Notohara, M. (1990). The coalescent and the genealogical process in geographically structured populations. J. Math. Biol. 29, 59–75.Google Scholar
Orrive, M. E. (1993). Effective population size in organisms with complex life-histories. Theoret. Popul. Biol. 44, 316–340.CrossRefGoogle Scholar
Pollack, E. (2010). Coalescent theory for a monoecious random mating population with a varying size. J. Appl. Prob. 47, 41–57.CrossRefGoogle Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar
Sagitov, S. and Jagers, P. (2005). The coalescent effective size of age-structured populations. Ann. Appl. Prob. 15, 1778–1797.Google Scholar
Sampson, K. Y. (2006). Structured coalescent with nonconservative migration. J. Appl. Prob. 43, 351–362.Google Scholar
Sano, A., Shimizu, A. and Iizuka, M. (2004). Coalescent process with fluctuating population size and its effective size. Theoret. Pop. Biol. 65, 39–48.Google Scholar
Sjödin, P. et al. (2005). On the meaning and existence of an effective population size. Genetics 169, 1061–1070.Google Scholar
Waples, R. S. (2002). Definition and estimation of effective population size in the conservation of endangered species. In Population Viability Analysis, eds Beissinger, S. R. and McCullogh, D. R., University of Chicago Press, pp. 147–168.Google Scholar
Wright, S. (1943). Isolation by distance. Genetics 28, 114–138.CrossRefGoogle ScholarPubMed