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Population genetics models with skewed fertilities: a forward and backward analysis. (English) Zbl 1367.92074
Stoch. Models 27, No. 3, 521-554 (2011); correction ibid. 28, No. 3, 527-532 (2012).
Summary: Discrete population genetics models with unequal (skewed) fertilities are considered, with an emphasis on skewed versions of Cannings models, conditional branching process models in the spirit of Karlin and McGregor, and compound Poisson models. Three particular classes of models with skewed fertilities are investigated, the Wright-Fisher model, the Dirichlet model, and the Kimura model. For each class the asymptotic behavior as the total population size \(N\) tends to infinity is investigated for power law fertilities and for geometric fertilities. This class of models can exhibit a rich variety of sub-linear or even constant effective population sizes. Therefore, the models are not necessarily in the domain of attraction of the Kingman coalescent. For a substantial range of the parameters, discrete-time coalescent processes with simultaneous multiple collisions arise in the limit.

MSC:
92D10 Genetics and epigenetics
92D25 Population dynamics (general)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:
[1] Berestycki N., Journal of Statistical Physics 127 pp 381– (2007) · Zbl 1126.82013 · doi:10.1007/s10955-006-9261-1
[2] Caballero A., Heredity 73 pp 657– (1994) · doi:10.1038/hdy.1994.174
[3] Cannings C., I. Haploid models. Advances in Applied Probability 6 pp 260– (1974) · Zbl 0284.60064 · doi:10.2307/1426293
[4] Cannings C., II. Further haploid models. Advances in Applied Probability 7 pp 264– (1975) · Zbl 0299.92008
[5] Comtet L., Analyse Combinatoire 1 (1970) · Zbl 0221.05001
[6] Eldon B., Genetics 172 pp 2621– (2006) · doi:10.1534/genetics.105.052175
[7] Ethier S.N., Markov Processes (1986) · doi:10.1002/9780470316658
[8] Ewens W.J., Theoretical Population Biology 3 pp 87– (1972) · Zbl 0245.92009 · doi:10.1016/0040-5809(72)90035-4
[9] Gladstien K., SIAM Journal of Applied Mathematics 34 pp 630– (1978) · Zbl 0386.92004 · doi:10.1137/0134050
[10] Handa K., Bernoulli 15 pp 1082– (2010) · Zbl 1255.60020 · doi:10.3150/08-BEJ180
[11] Huillet T., Far East Journal of Theoretical Statistics 24 pp 1– (2008)
[12] Karlin S., Proceeding of the National Academy & Sciences USA 51 pp 598– (1964) · Zbl 0129.30504 · doi:10.1073/pnas.51.4.598
[13] Karlin , S. ; McGregor , J. Direct product branching processes and related Markov chains. I. Calculations of rates of approach to homozygosity . Proceeding of the International Research Seminar . Springer : Berlin , pp. 111 – 145 , 1965 .
[14] Kingman J.F.C., Stochastic Process and their Applications 13 pp 235– (1982) · Zbl 0491.60076 · doi:10.1016/0304-4149(82)90011-4
[15] Kolchin V.F., Random Mappings (1986)
[16] Liggett T.M., Interacting Particle Systems (1985)
[17] Maruyama T., Stochastic Problems in Population Genetics (1977) · Zbl 0377.92007
[18] McSweeney J.K., Advances in Applied Probability 40 pp 1002– (2008) · Zbl 1165.60006 · doi:10.1239/aap/1231340162
[19] Möhle M., Journal of Applied Probability 35 pp 438– (1998) · Zbl 0913.60022 · doi:10.1239/jap/1032192859
[20] Möhle M., Bernoulli 5 pp 761– (1999) · Zbl 0942.92020 · doi:10.2307/3318443
[21] Möhle M., Advances in Applied Probability 32 pp 983– (2000) · Zbl 1002.92015 · doi:10.1239/aap/1013540343
[22] Möhle M., Stochastic Processes and their Applications 120 pp 2159– (2010) · Zbl 1214.60037 · doi:10.1016/j.spa.2010.07.004
[23] Möhle M., Journal of Applied Probability 47 pp 713– (2010) · Zbl 1210.60078 · doi:10.1239/jap/1285335405
[24] Möhle M., Annals of Probability 29 pp 1547– (2001) · Zbl 1013.92029 · doi:10.1214/aop/1015345761
[25] Möhle M., Journal of Mathematical Biology. 47 pp 337– (2003) · Zbl 1054.92039 · doi:10.1007/s00285-003-0218-6
[26] Moran P.A.P., Australian Journal of Biological Sciences 12 pp 1– (1959)
[27] Pitman E.J.G., Journal of the Australian Mathematical Society 8 pp 422– (1968)
[28] Pitman , J.Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875, 13–35, 2006.
[29] Sagitov S., Journal of Applied Probability 40 pp 839– (2003) · Zbl 1052.92044 · doi:10.1239/jap/1067436085
[30] Sagitov S., Annals of Applied Probability 15 pp 1778– (2005) · Zbl 1073.92034 · doi:10.1214/105051605000000223
[31] Schweinsberg J., Electronic Journal of Probability 5 (2000) · Zbl 0959.60065 · doi:10.1214/EJP.v5-68
[32] Schweinsberg J., Stochastic Processes and their Applications 106 pp 107– (2003) · Zbl 1075.60571 · doi:10.1016/S0304-4149(03)00028-0
[33] Steutel F.W., Annals of Probability 7 pp 893– (1979) · Zbl 0418.60020 · doi:10.1214/aop/1176994950
[34] Wakeley J., Genetics 181 pp 341– (2009) · doi:10.1534/genetics.108.092460
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