## On the stationary distribution of the block counting process for population models with mutation and selection.(English)Zbl 1415.92145

Summary: We consider two population models subject to the evolutionary forces of selection and mutation, the Moran model and the $${\Lambda}$$-Wright-Fisher model. In such models the block counting process traces back the number of potential ancestors of a sample of the population at present. Under some conditions the block counting process is positive recurrent and its stationary distribution is described via a linear system of equations. In this work, we first characterise the measures $${\Lambda}$$ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. Next, we solve the linear system of equations corresponding to the Moran model. For the $${\Lambda}$$-Wright-Fisher model we show that the probability generating function associated to the stationary distribution of the block counting process satisfies an integro differential equation. We solve the latter for the Kingman model and the star-shaped model.

### MSC:

 92D25 Population dynamics (general) 92D15 Problems related to evolution
Full Text:

### References:

 [1] Appell, P., Sur les fonctions hypergéométriques de deux variables, J. Math. Pures Appl., 8, 173-216 (1882) · JFM 14.0375.01 [2] Baake, E.; Cordero, F.; Hummel, S., A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent, J. Math. Biol., 77, 3, 795-820 (2018) · Zbl 1408.92015 [3] Baake, E.; Lenz, U.; Wakolbinger, A., The common ancestor type distribution of a Λ-Wright-Fisher process with selection and mutation, Electron. Commun. Probab., 21, 59, 1-16 (2016) · Zbl 1346.60131 [4] Baake, E.; Wakolbinger, A., Lines of descent under selection, J. Stat. Phys., 172, 1, 156-174 (2018) · Zbl 1396.92055 [5] Berestycki, N., Recent Progress in Coalescent Theory, Ensaios Matemáticos, vol. 16 (2009), Sociedade Brasileira de Matemática: Sociedade Brasileira de Matemática Rio de Janeiro · Zbl 1204.60002 [6] Bertoin, J.; Le Gall, J.-F., Stochastic flows associated to coalescent processes, Probab. Theory Related Fields, 126, 2, 261-288 (2003) · Zbl 1023.92018 [7] Birkner, M.; Blath, J., Measure-valued diffusions, coalescents and genetic inference, (Blath, J.; Mörters, P.; Scheutzow, M., Trends in Stochastic Analysis (2009), Cambridge University Press: Cambridge University Press Cambridge), 329-364 · Zbl 1170.92021 [8] J. Blath, E. Buzzoni, A. González Casanova, M. Wilke-Berenguer, Structural properties of the seed bank and the two-island diffusion, preprint, 2018.; J. Blath, E. Buzzoni, A. González Casanova, M. Wilke-Berenguer, Structural properties of the seed bank and the two-island diffusion, preprint, 2018. [9] Bolthausen, E.; Sznitman, A.-S., On Ruelle’s probability cascades and an abstract cavity method, Comm. Math. Phys., 197, 2, 247-276 (1998) · Zbl 0927.60071 [10] Cordero, F., Common ancestor type distribution: a Moran model and its deterministic limit, Stochastic Process. Appl., 127, 2, 590-621 (2017) · Zbl 1353.92066 [11] Cordero, F., The deterministic limit of the Moran model: a uniform central limit theorem, Markov Process. Related Fields, 23, 2, 313-324 (2017) · Zbl 1379.92035 [12] Corless, R. M.; Gonnet, G. H.; Hare, D. E.G.; Jeffrey, D. J.; Knuth, D. E., On the Lambert W function, Adv. Comput. Math., 5, 1, 329-359 (1996) · Zbl 0863.65008 [13] Crow, J. F.; Kimura, M., Some genetic problems in natural populations, Proc. Third Berkeley Symp. Math. Statist. Prob., 4, 1-22 (1956) · Zbl 0071.35603 [14] Der, R.; Epstein, C.; Plotkin, J. B., Generalized population models and the nature of genetic drift, Theor. Popul. Biol., 80, 2, 80-99 (2011) · Zbl 1297.92051 [15] Der, R.; Epstein, C.; Plotkin, J. B., Dynamics of neutral and selected alleles when the offspring distribution is skewed, Genetics, 191, 4, 1331-1344 (2012) [16] Donnelly, P.; Kurtz, T. G., Particle representations for measure-valued population models, Ann. Probab., 27, 1, 166-205 (1999) · Zbl 0956.60081 [17] Drmota, M.; Iksanov, A.; Moehle, M.; Roesler, U., Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent, Stochastic Process. Appl., 117, 1404-1421 (2007) · Zbl 1129.60069 [18] Durrett, R., Probability Models for DNA Sequence Evolution (2008), Springer: Springer New York · Zbl 1311.92007 [19] Estrada, R.; Kanwal, R. P., The Carleman type singular integral equations, SIAM Rev., 29, 2, 263-290 (1987) · Zbl 0624.45002 [20] Etheridge, A., Some Mathematical Models from Population Genetics (2011), Springer: Springer Heidelberg · Zbl 1320.92003 [21] Etheridge, A. M.; Griffiths, R. C., A coalescent dual process in a Moran model with genic selection, Theor. Popul. Biol., 75, 4, 320-330 (2009) · Zbl 1213.92038 [22] Etheridge, A. M.; Griffiths, R. C.; Taylor, J. E., A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit, Theor. Popul. Biol., 78, 2, 77-92 (2010) · Zbl 1403.92173 [23] Ethier, S. N.; Kurtz, T. G., Markov Processes: Characterization and Convergence (1986), Wiley: Wiley New York · Zbl 0592.60049 [24] Fearnhead, P., The common ancestor at a nonneutral locus, J. Appl. Probab., 39, 1, 38-54 (2002) · Zbl 1001.92037 [25] Foucart, C., The impact of selection in the Λ-Wright-Fisher model, Electron. Commun. Probab., 18, 72, 1-10 (2013) · Zbl 1337.60179 [26] Gaiser, F.; Möhle, M., On the block counting process and the fixation line of exchangeable coalescents, ALEA Lat. Am. J. Probab. Math. Stat., 13, 2, 809-833 (2016) · Zbl 1346.60124 [27] González Casanova, A.; Spanò, D., Duality and fixation in Ξ-Wright-Fisher processes with frequency-dependent selection, Ann. Appl. Probab., 28, 1, 250-284 (2018) · Zbl 1391.92037 [28] Griffiths, R. C., The Λ-Fleming-Viot process and a connection with Wright-Fisher diffusion, Adv. Appl. Probab., 46, 4, 1009-1035 (2014) · Zbl 1305.60038 [29] Hausdorff, F., Summationsmethoden und momentfolgen. I, Math. Z., 9, 1-2, 74-109 (1921) · JFM 48.2005.01 [30] Hénard, O., The fixation line in the Λ-coalescent, Ann. Appl. Probab., 25, 5, 3007-3032 (2015) · Zbl 1325.60124 [31] Herriger, P.; Möhle, M., Conditions for exchangeable coalescents to come down from infinity, ALEA Lat. Am. J. Probab. Math. Stat., 9, 2, 637-665 (2012) · Zbl 1277.60122 [32] Jansen, S.; Kurt, N., On the notion(s) of duality for Markov processes, Probab. Surv., 11, 59-120 (2014) · Zbl 1292.60077 [33] Kaj, I.; Krone, S. M.; Lascoux, M., Coalescent theory for seed bank models, J. Appl. Probab., 38, 2, 285-300 (2001) · Zbl 0989.92017 [34] Kimura, M., On the probability of fixation of mutant genes in a population, Genetics, 47, 713-719 (1962) [35] Kingman, J. F.C., The coalescent, Stochastic Process. Appl., 13, 3, 235-248 (1982) · Zbl 0491.60076 [36] Kluth, S.; Hustedt, T.; Baake, E., The common ancestor process revisited, Bull. Math. Biol., 75, 11, 2003-2027 (2013) · Zbl 1310.92039 [37] Koopmann, B.; Müller, J.; Tellier, A.; Živković, D., Fisher-Wright model with deterministic seed bank and selection, Theor. Popul. Biol., 114, 29-39 (2017) · Zbl 1369.92067 [38] Krone, S. M.; Neuhauser, C., Ancestral processes with selection, Theor. Popul. Biol., 51, 3, 210-237 (1997) · Zbl 0910.92024 [39] Kurtz, T. G., Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probab., 7, 49-58 (1970) · Zbl 0191.47301 [40] Lebedev, N. N., Special Functions and Their Applications (1972), Dover Publications: Dover Publications New York, revised edition, translated from the Russian and edited by Richard A. Silverman · Zbl 0271.33001 [41] Lenz, U.; Kluth, S.; Baake, E.; Wakolbinger, A., Looking down in the ancestral selection graph: a probabilistic approach to the common ancestor type distribution, Theor. Popul. Biol., 103, 27-37 (2015) · Zbl 1342.92141 [42] Liggett, T. M., Continuous Time Markov Processes. An Introduction (2010), American Mathematical Society: American Mathematical Society Providence · Zbl 1205.60002 [43] Möhle, M., Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models, Adv. Appl. Probab., 32, 4, 983-993 (2000) · Zbl 1002.92015 [44] Möhle, M., Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent, J. Appl. Probab., 51A, 87-97 (2014) · Zbl 1328.60173 [45] Möhle, M., On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity, ALEA Lat. Am. J. Probab. Math. Stat., 11, 141-159 (2014) · Zbl 1341.60088 [46] Neuhauser, C.; Krone, S. M., The genealogy of samples in models with selection, Genetics, 145, 2, 519-534 (1997) [47] Pitman, J., Coalescents with multiple collisions, Ann. Probab., 27, 4, 1870-1902 (1999) · Zbl 0963.60079 [48] Pitman, J., Combinatorial Stochastic Processes, Lecture Notes in Mathematics, vol. 1875 (2006), Springer: Springer Berlin · Zbl 1103.60004 [49] Pokalyuk, C.; Pfaffelhuber, P., The ancestral selection graph under strong directional selection, Theor. Popul. Biol., 87, 25-33 (2013) · Zbl 1296.92217 [50] Sagitov, S., The general coalescent with asynchronous mergers of ancestral lines, J. Appl. Probab., 36, 4, 1116-1125 (1999) · Zbl 0962.92026 [51] Taylor, J. E., The common ancestor process for a Wright-Fisher diffusion, Electron. J. Probab., 12, 28, 808-847 (2007) · Zbl 1127.60079 [52] Tricomi, F. G., Integral Equations (1957), Intercience Publishers: Intercience Publishers London · Zbl 0078.09404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.