Alberti, F.; Baake, E.; Letter, I.; Martínez, S. Solving the migration-recombination equation from a genealogical point of view. (English) Zbl 1465.92072 J. Math. Biol. 82, No. 5, Paper No. 41, 27 p. (2021). Summary: We consider the discrete-time migration-recombination equation, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration-recombination equation. As a consequence, one obtains an explicit solution of the nonlinear dynamics, simply in terms of powers of the transition matrix of the Markov chain. The limiting and quasi-limiting behaviour of the Markov chain are investigated, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time. MSC: 92D15 Problems related to evolution 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) Keywords:migration-recombination equation; ancestral recombination graph; duality; labelled partitioning process; quasi-stationarity; Haldane linearisation PDFBibTeX XMLCite \textit{F. Alberti} et al., J. Math. Biol. 82, No. 5, Paper No. 41, 27 p. (2021; Zbl 1465.92072) Full Text: DOI arXiv References: [1] Baake E, Baake M (2003) An exactly solved model for mutation, recombination and selection. Can J Math 55: 3-41 and Erratum 60: 264-265 (2008) · Zbl 1231.92052 [2] Baake, E.; Baake, M., Haldane linearisation done right: solving the nonlinear recombination equation the easy way, Discrete Contin Dyn Syst A, 36, 6645-6656 (2016) · Zbl 1353.92064 [3] Baake, E.; Esser, M.; Probst, S., Partitioning, duality, and linkage disequilibria in the Moran model with recombination, J Math Biol, 73, 161-197 (2016) · Zbl 1359.92080 [4] Baake E, Baake M (2020) Ancestral lines under recombination. In: Baake E, Wakolbinger A (eds) Probabilistic structures in evolution. EMS Press, Berlin (in press). arXiv:2002.08658 · Zbl 1056.92040 [5] Baake E, Baake M, Salamat M (2016) The general recombination equation in continuous time and its solution. Discrete Contin Dyn Syst A 36: 63-95 and Erratum and addendum 36:2365-2366 (2016) · Zbl 1326.34080 [6] Baake, E.; von Wangenheim, U., Single-crossover recombination and ancestral recombination trees, J Math Biol, 68, 1371-1402 (2014) · Zbl 1284.92063 [7] Bhaskar, A.; Song, YS, Closed-form asymptotic sampling distributions under the coalescent with recombination for an arbitrary number of loci, Adv Appl Probab, 44, 391-407 (2012) · Zbl 1241.92054 [8] Bürger, R., Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration, J Math Biol, 58, 939-978 (2009) · Zbl 1204.92050 [9] Christiansen, FB, Population genetics of multiple loci (1999), Chichester: Wiley, Chichester · Zbl 0941.92019 [10] Collet, P.; Martínez, S.; San Martín, J., Quasi-stationary distributions. Markov chains, diffusions and dynamical systems (2013), Berlin: Springer, Berlin · Zbl 1261.60002 [11] Darroch, JN; Seneta, E., On quasi-stationary distribution in absorbing discrete-time finite Markov chains, J Appl Probab, 2, 88-100 (1965) · Zbl 0134.34704 [12] Durrett, R., Probability models for DNA sequence evolution (2008), New York: Springer, New York · Zbl 1311.92007 [13] Griffiths, RC; Marjoram, P., Ancestral inference from samples of DNA sequences with recombination, J Comput Biol, 3, 479-502 (1996) [14] Griffiths, RC; Marjoram, P.; Donnelly, P.; Tavaré, S., An ancestral recombination graph, Progress in population genetics and human evolution, 257-270 (1997), New York: Springer, New York · Zbl 0893.92020 [15] McHale, D.; Ringwood, GA, Haldane linearisation of baric algebras, J Lond Math Soc, 28, 17-26 (1983) · Zbl 0515.17010 [16] Hudson, RR, Properties of a neutral allele model with intragenic recombination, Theor Popul Biol, 23, 183-201 (1983) · Zbl 0505.62090 [17] Jansen, S.; Kurt, N., On the notion(s) of duality for Markov processes, Probab Surv, 11, 59-120 (2014) · Zbl 1292.60077 [18] Karlin, S.; Taylor, HM, A first course in stochastic processes (1975), San Diego: Academic Press, San Diego · Zbl 0315.60016 [19] Lambert A, Miró Pina V, Schertzer E (2020) Chromosome painting: how recombination mixes ancestral colors. Ann Appl Probab (online first) [20] Liggett, TM, Continuous time Markov processes: an introduction (2010), Providence: American Mathematical Society, Providence [21] Lyubich, YI, Mathematical structures in population genetics (1992), Berlin: Springer, Berlin [22] Martínez S (2017) A probabilistic analysis of a discrete-time evolution in recombination. Adv Appl Math 91:115-136; and Corrigendum 110:403-411 (2019) · Zbl 1469.92078 [23] Matsen, FA; Wakeley, J., Convergence to the island-model coalescent process in populations with restricted migration, Genetics, 172, 701-708 (2006) [24] Nagylaki, T., Introduction to theoretical population genetics (1992), Berlin: Springer, Berlin · Zbl 0839.92011 [25] Nagylaki, T.; Hofbauer, J.; Brunovský, P., Convergence of multilocus systems under weak epistasis or weak selection, J Math Biol, 38, 103-133 (1999) · Zbl 0981.92019 [26] Notohara, M., The coalescent and the genealogical process in geographically structured populations, J Math Biol, 29, 59-75 (1990) · Zbl 0726.92014 [27] Slade, PF; Wakeley, J., The structured ancestral selection graph and the many-demes limit, Genetics, 169, 1117-1131 (2005) [28] von Wangenheim, U.; Baake, E.; Baake, M., Single-crossover recombination in discrete time, J Math Biol, 60, 727-760 (2010) · Zbl 1208.92050 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.