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Stationary frequencies and mixing times for neutral drift processes with spatial structure. (English) Zbl 1407.92095
Summary: We study a general setting of neutral evolution in which the population is of finite, constant size and can have spatial structure. Mutation leads to different genetic types (traits), which can be discrete or continuous. Under minimal assumptions, we show that the marginal trait distributions of the evolutionary process, which specify the probability that any given individual has a certain trait, all converge to the stationary distribution of the mutation process. In particular, the stationary frequencies of traits in the population are independent of its size, spatial structure and evolutionary update rule, and these frequencies can be calculated by evaluating a simple stochastic process describing a population of size one (i.e. the mutation process itself). We conclude by analysing mixing times, which characterize rates of convergence of the mutation process along the lineages, in terms of demographic variables of the evolutionary process.

92D15 Problems related to evolution
92D25 Population dynamics (general)
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[1] Wright S. (1931) Evolution in Mendelian populations. Genetics 16, 97-159.
[2] Maruyama T. (1970) Effective number of alleles in a subdivided population. Theor. Popul. Biol. 1, 273-306. (doi:10.1016/0040-5809(70)90047-x) · Zbl 0255.92006
[3] Maruyama T. (1970) On the fixation probability of mutant genes in a subdivided population. Genet. Res. 15, 221-225. (doi:10.1017/s0016672300001543)
[4] Maruyama T. (1974) A simple proof that certain quantities are independent of the geographical structure of population. Theor. Popul. Biol. 5, 148-154. (doi:10.1016/0040-5809(74)90037-9)
[5] Nagylaki T. (1980) The strong-migration limit in geographically structured populations. J. Math. Biol. 9, 101-114. (doi:10.1007/bf00275916) · Zbl 0427.92012
[6] Nagylaki T. (1992) Introduction to theoretical population genetics. Berlin, Germany: Springer. · Zbl 0839.92011
[7] Lieberman E, Hauert C, Nowak MA. (2005) Evolutionary dynamics on graphs. Nature 433, 312-316. (doi:10.1038/nature03204)
[8] Ohtsuki H, Hauert C, Lieberman E, Nowak MA. (2006) A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502-505. (doi:10.1038/nature04605)
[9] Tarnita CE, Antal T, Ohtsuki H, Nowak MA. (2009) Evolutionary dynamics in set structured populations. Proc. Natl Acad. Sci. USA 106, 8601-8604. (doi:10.1073/pnas.0903019106)
[10] Traulsen A, Hauert C, De Silva H, Nowak MA, Sigmund K. (2009) Exploration dynamics in evolutionary games. Proc. Natl Acad. Sci. USA 106, 709-712. (doi:10.1073/pnas.0808450106) · Zbl 1202.91029
[11] Loewe L, Hill WG. (2010) The population genetics of mutations: good, bad and indifferent. Phil. Trans. R. Soc. B 365, 1153-1167. (doi:10.1098/rstb.2009.0317)
[12] Hauert C, Imhof L. (2012) Evolutionary games in deme structured, finite populations. J. Theor. Biol. 299, 106-112. (doi:10.1016/j.jtbi.2011.06.010) · Zbl 1337.91021
[13] Kimura M. (1983) The neutral theory of molecular evolution. Cambridge, UK: Cambridge University Press.
[14] Lynch M, Hill WG. (1986) Phenotypic evolution by neutral mutation. Evolution 40, 915-935. (doi:10.2307/2408753)
[15] Ochman H. (2003) Neutral mutations and neutral substitutions in bacterial genomes. Mol. Biol. Evol. 20, 2091-2096. (doi:10.1093/molbev/msg229)
[16] Nei M. (2005) Selectionism and neutralism in molecular evolution. Mol. Biol. Evol. 22, 2318-2342. (doi:10.1093/molbev/msi242)
[17] Bloom JD, Romero PA, Lu Z, Arnold FH. (2007) Neutral genetic drift can alter promiscuous protein functions, potentially aiding functional evolution. Biol. Direct 2, 17. (doi:10.1186/1745-6150-2-17)
[18] Antal T, Ohtsuki H, Wakeley J, Taylor PD, Nowak MA. (2009) Evolution of cooperation by phenotypic similarity. Proc. Natl Acad. Sci. USA 106, 8597-8600. (doi:10.1073/pnas.0902528106) · Zbl 1355.92145
[19] Antal T, Traulsen A, Ohtsuki H, Tarnita CE, Nowak MA. (2009) Mutation-selection equilibrium in games with multiple strategies. J. Theor. Biol. 258, 614-622. (doi:10.1016/j.jtbi.2009.02.010) · Zbl 1402.91039
[20] Tarnita CE, Ohtsuki H, Antal T, Fu F, Nowak MA. (2009) Strategy selection in structured populations. J. Theor. Biol. 259, 570-581. (doi:10.1016/j.jtbi.2009.03.035) · Zbl 1402.91064
[21] Nowak MA, Tarnita CE, Antal T. (2009) Evolutionary dynamics in structured populations. Phil. Trans. R. Soc. B 365, 19-30. (doi:10.1098/rstb.2009.0215)
[22] Tarnita CE, Wage N, Nowak MA. (2011) Multiple strategies in structured populations. Proc. Natl Acad. Sci. USA 108, 2334-2337. (doi:10.1073/pnas.1016008108)
[23] Gokhale CS, Traulsen A. (2011) Strategy abundance in evolutionary many-player games with multiple strategies. J. Theor. Biol. 283, 180-191. (doi:10.1016/j.jtbi.2011.05.031) · Zbl 1397.91064
[24] Wu B, García J, Hauert C, Traulsen A. (2013) Extrapolating weak selection in evolutionary games. PLoS Comput. Biol. 9, e1003381. (doi:10.1371/journal.pcbi.1003381)
[25] Wu B, Traulsen A, Gokhale CS. (2013) Dynamic properties of evolutionary multi-player games in finite populations. Games 4, 182-199. (doi:10.3390/g4020182) · Zbl 1314.91039
[26] Allen B, Tarnita CE. (2014) Measures of success in a class of evolutionary models with fixed population size and structure. J. Math. Biol. 68, 109-143. (doi:10.1007/s00285-012-0622-x) · Zbl 1280.92045
[27] Catalán P, Seoane JM, Sanjuán MAF. (2015) Mutation-selection equilibrium in finite populations playing a Hawk-Dove game. Commun. Nonlinear Sci. Numer. Simul. 25, 66-73. (doi:10.1016/j.cnsns.2015.01.012)
[28] Zhang Y, Liu A, Sun C. (2016) Impact of migration on the multi-strategy selection in finite group-structured populations. Sci. Rep. 6, 35114. (doi:10.1038/srep35114)
[29] Moran PAP. (1958) Random processes in genetics. Math. Proc. Cambridge Philos. Soc. 54, 60-71. (doi:10.1017/s0305004100033193) · Zbl 0091.15701
[30] Taylor J. (2007) The common ancestor process for a Wright-Fisher diffusion. Elect. J. Probab. 12, 808-847. (doi:10.1214/ejp.v12-418) · Zbl 1127.60079
[31] Birky CW, Walsh JB. (1988) Effects of linkage on rates of molecular evolution. Proc. Natl Acad. Sci. USA 85, 6414-6418. (doi:10.1073/pnas.85.17.6414)
[32] Sigmund K. (2010) The calculus of selfishness. Princeton, NJ: Princeton University Press. · Zbl 1189.91010
[33] Nowak M, Sigmund K. (1990) The evolution of stochastic strategies in the Prisoner’s dilemma. Acta Appl. Math. 20, 247-265. (doi:10.1007/bf00049570) · Zbl 0722.90092
[34] Geritz SAH, Kisdi É, Meszéna G, Metz JAJ. (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12, 35-57. (doi:10.1023/a:1006554906681)
[35] Dieckmann U, Doebeli M. (1999) On the origin of species by sympatric speciation. Nature 400, 354-357. (doi:10.1038/22521)
[36] Killingback T, Doebeli M, Knowlton N. (1999) Variable investment, the Continuous Prisoner’s Dilemma, and the origin of cooperation. Proc. R. Soc. Lond. B 266, 1723-1728. (doi:10.1098/rspb.1999.0838)
[37] Killingback T, Doebeli M. (2002) The Continuous Prisoner’s Dilemma and the evolution of cooperation through reciprocal altruism with variable investment. Am. Nat. 160, 421-438. (doi:10.1086/342070)
[38] Dieckmann U, Metz JAJ. (2006) Surprising evolutionary predictions from enhanced ecological realism. Theor. Popul. Biol. 69, 263-281. (doi:10.1016/j.tpb.2005.12.001) · Zbl 1117.92044
[39] Imhof LA, Nowak MA. (2009) Stochastic evolutionary dynamics of direct reciprocity. Proc. R. Soc. B 277, 463-468. (doi:10.1098/rspb.2009.1171)
[40] Doebeli M. (2011) Adaptive diversification (MPB-48). Princeton, NJ: Princeton University Press.
[41] Nowak M. (1990) Stochastic strategies in the Prisoner’s Dilemma. Theor. Popul. Biol. 38, 93-112. (doi:10.1016/0040-5809(90)90005-G) · Zbl 0726.92015
[42] Nowak M, Sigmund K. (1993) A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364, 56-58. (doi:10.1038/364056a0)
[43] Roberts G, Sherratt TN. (1998) Development of cooperative relationships through increasing investment. Nature 394, 175-179. (doi:10.1038/28160)
[44] Wahl LM, Nowak MA. (1999) The continuous Prisoner’s dilemma. I. Linear reactive strategies. J. Theor. Biol. 200, 307-321. (doi:10.1006/jtbi.1999.0996)
[45] Wahl LM, Nowak MA. (1999) The continuous Prisoner’s dilemma. II. Linear reactive strategies with noise. J. Theor. Biol. 200, 323-338. (doi:10.1006/jtbi.1999.0997)
[46] Oechssler J, Riedel F. (2001) Evolutionary dynamics on infinite strategy spaces. Econ. Theory 17, 141-162. (doi:10.1007/pl00004092) · Zbl 0982.91002
[47] van Veelen M, Spreij P. (2008) Evolution in games with a continuous action space. Econ. Theory 39, 355-376. (doi:10.1007/s00199-008-0338-8) · Zbl 1166.91008
[48] Cleveland J, Ackleh AS. (2013) Evolutionary game theory on measure spaces: well-posedness. Nonlinear Anal. Real World Appl. 14, 785-797. (doi:10.1016/j.nonrwa.2012.08.002) · Zbl 1254.91048
[49] Cheung M-W. (2014) Pairwise comparison dynamics for games with continuous strategy space. J. Econ. Theory 153, 344-375. (doi:10.1016/j.jet.2014.07.001) · Zbl 1309.91021
[50] Cheung M-W. (2016) Imitative dynamics for games with continuous strategy space. Games Econ. Behav. 99, 206-223. (doi:10.1016/j.geb.2016.08.003) · Zbl 1394.91039
[51] Allen B, Sample C, Dementieva Y, Medeiros RC, Paoletti C, Nowak MA. (2015) The molecular clock of neutral evolution can be accelerated or slowed by asymmetric spatial structure. PLoS Comput. Biol. 11, e1004108. (doi:10.1371/journal.pcbi.1004108)
[52] Durrett R. (2009) Probability: theory and examples. Princeton, NJ: Princeton University Press.
[53] Huber ML. (2016) Perfect simulation. London, UK: Chapman and Hall.
[54] Allen B, McAvoy A. (2018)A mathematical formalism for natural selection with arbitrary spatial and genetic structure. (http://arxiv.org/abs/1806.04717)
[55] Knoll AH, Nowak MA. (2017) The timetable of evolution. Sci. Adv. 3, e1603076. (doi:10.1126/sciadv.1603076)
[56] Ewens WJ. (2004) Mathematical population genetics. New York, NY: Springer.
[57] Imhof LA, Nowak MA. (2006) Evolutionary game dynamics in a Wright-Fisher process. J. Math. Biol. 52, 667-681. (doi:10.1007/s00285-005-0369-8) · Zbl 1110.92028
[58] Der R, Epstein CL, Plotkin JB. (2011) Generalized population models and the nature of genetic drift. Theor. Popul. Biol. 80, 80-99. (doi:10.1016/j.tpb.2011.06.004) · Zbl 1297.92051
[59] Levin D, Peres Y, Wilmer E. (2008) Markov chains and mixing times. Providence, RI: American Mathematical Society.
[60] Montenegro R, Tetali P. (2005) Mathematical aspects of mixing times in Markov chains. Found. Trends Theoret. Comput. Sci. 1, 237-354. (doi:10.1561/0400000003) · Zbl 1193.68138
[61] Knapp AW. (2006) Basic algebra. Boston, MA: Birkhäuser.
[62] Taylor PD, Day T, Wild G. (2007) From inclusive fitness to fixation probability in homogeneous structured populations. J. Theor. Biol. 249, 101-110. (doi:10.1016/j.jtbi.2007.07.006)
[63] Débarre F. (2017) Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations. J. Theor. Biol. 420, 26-35. (doi:10.1016/j.jtbi.2017.02.027) · Zbl 1370.92103
[64] Fearnhead P. (2002) The common ancestor at a nonneutral locus. J. Appl. Prob. 39, 38-54. (doi:10.1017/s0021900200021495) · Zbl 1001.92037
[65] van Nimwegen E, Crutchfield JP, Huynen M. (1999) Neutral evolution of mutational robustness. Proc. Natl Acad. Sci. USA 96, 9716-9720. (doi:10.1073/pnas.96.17.9716)
[66] Doebeli M, Hauert C, Killingback T. (2004) The evolutionary origin of cooperators and defectors. Science 306, 859-862. (doi:10.1126/science.1101456)
[67] Wakano JY, Iwasa Y. (2012) Evolutionary branching in a finite population: deterministic branching versus stochastic branching. Genetics 193, 229-241. (doi:10.1534/genetics.112.144980)
[68] Haccou P, Jagers P, Vatutin VA. (2005) Branching processes: variation growth and extinction of populations. Cambridge, UK: Cambridge University Press. · Zbl 1118.92001
[69] Lambert A. (2006) Probability of fixation under weak selection: a branching process unifying approach. Theor. Popul. Biol. 69, 419-441. (doi:10.1016/j.tpb.2006.01.002) · Zbl 1121.92051
[70] Faure M, Schreiber SJ. (2014) Quasi-stationary distributions for randomly perturbed dynamical systems. Ann. Appl. Probab. 24, 553-598. (doi:10.1214/13-aap923) · Zbl 1334.60137
[71] Hamza K, Jagers P, Klebaner FC. (2015) On the establishment, persistence, and inevitable extinction of populations. J. Math. Biol. 72, 797-820. (doi:10.1007/s00285-015-0903-2) · Zbl 1334.60179
[72] Huang W, Hauert C, Traulsen A. (2015) Stochastic game dynamics under demographic fluctuations. Proc. Natl Acad. Sci. USA 112, 9064-9069. (doi:10.1073/pnas.1418745112)
[73] McAvoy A, Fraiman N, Hauert C, Wakeley J, Nowak MA. (2018) Public goods games in populations with fluctuating size. Theor. Popul. Biol. 121, 72-84. (doi:10.1016/j.tpb.2018.01.004) · Zbl 1397.92588
[74] Krone SM, Neuhauser C. (1997) Ancestral processes with selection. Theor. Popul. Biol. 51, 210-237. (doi:10.1006/tpbi.1997.1299) · Zbl 0910.92024
[75] Neuhauser C. (1999) The ancestral graph and gene genealogy under frequency-dependent selection. Theor. Popul. Biol. 56, 203-214. (doi:10.1006/tpbi.1999.1412) · Zbl 0956.92026
[76] Slade PF, Wakeley J. (2005) The structured ancestral selection graph and the many-demes limit. Genetics 169, 1117-1131. (doi:10.1534/genetics.104.032276)
[77] Kingman JFC. (1982) On the genealogy of large populations. J. Appl. Probab. 19, 27-43. (doi:10.2307/3213548)
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