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Measures of success in a class of evolutionary models with fixed population size and structure. (English) Zbl 1280.92045
Summary: We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the G.R. Price equation [see Ann. Hum. Genet. 35, 485–490 (1972; Zbl 0231.92006)] and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth-death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

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.)
37N25 Dynamical systems in biology
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[1] Allen B, Traulsen A, Tarnita CE, Nowak MA (2012) How mutation affects evolutionary games on graphs. J Theor Biol 299:97-105. doi:10.1016/j.jtbi.2011.03.034 · Zbl 1337.91016
[2] Antal, T; Ohtsuki, H; Wakeley, J; Taylor, PD; Nowak, MA, Evolution of cooperation by phenotypic similarity, Proc Natl Acad Sci, 106, 8597-8600, (2009) · Zbl 1355.92145
[3] 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
[4] Barbour AD (1976) Quasi-stationary distributions in markov population processes. Adv Appl Prob, pp 296-314 · Zbl 0337.60069
[5] Broom, M; Hadjichrysanthou, C; Rychtář, J, Evolutionary games on graphs and the speed of the evolutionary process, Proc R Soc A Math Phys Eng Sci, 466, 1327-1346, (2010) · Zbl 1202.91027
[6] Broom M, Rychtár J (2008) An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc R Soc A Math Phys Eng Sci 464:2609-2627. doi:10.1098/rspa.2008.0058 · Zbl 1152.92341
[7] Cannings, C, The latent roots of certain Markov chains arising in genetics: a new approach, i. haploid models, Adv Appl Prob, 6, 260-290, (1974) · Zbl 0284.60064
[8] Cattiaux, P; Collet, P; Lambert, A; Martinez, S; Méléard, S, Quasi-stationary distributions and diffusion models in population dynamics, Ann Prob, 37, 1926-1969, (2009) · Zbl 1176.92041
[9] Champagnat, N; Ferrière, R; Méléard, S, Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models, Theor Popul Biol, 69, 297-321, (2006) · Zbl 1118.92039
[10] Collet, P; Martínez, S; Méléard, S, Quasi-stationary distributions for structured birth and death processes with mutations, Prob Theory Relat Fields, 151, 191-231, (2011) · Zbl 1245.92062
[11] Cox, J, Coalescing random walks and voter model consensus times on the torus in \(\mathbb{Z}^d\), Ann Prob, 17, 1333-1366, (1989) · Zbl 0685.60100
[12] Cox, JT; Durrett, R; Perkins, EA, Rescaled voter models converge to super-Brownian motion, Ann Prob, 28, 185-234, (2000) · Zbl 1044.60092
[13] Cressman R (1992) The stability concept of evolutionary game theory: a dynamic approach. Springer, Berlin · Zbl 0763.92006
[14] Darroch, JN; Seneta, E, On quasi-stationary distributions in absorbing discrete-time finite Markov chains, J Appl Prob, 2, 88-100, (1965) · Zbl 0134.34704
[15] Dieckmann, U; Law, R, The dynamical theory of coevolution: a derivation from stochastic ecological processes, J Math Biol, 34, 579-612, (1996) · Zbl 0845.92013
[16] Diekmann, O; Gyllenberg, M; Huang, H; Kirkilionis, M; Metz, JAJ; Thieme, HR, On the formulation and analysis of general deterministic structured population models ii. nonlinear theory, J Math Biol, 43, 157-189, (2001) · Zbl 1028.92019
[17] Diekmann O, Gyllenberg M, Metz J (2007) Physiologically structured population models: towards a general mathematical theory. In: Takeuchi Y, Iwasa Y, Sato K (eds) Mathematics for ecology and environmental sciences. Springer, Berlin, Heidelberg Biological and Medical Physics, Biomedical Engineering, pp 5-20
[18] Diekmann, O; Gyllenberg, M; Metz, JAJ; Thieme, HR, On the formulation and analysis of general deterministic structured population models i. linear theory, J Math Biol, 36, 349-388, (1998) · Zbl 0909.92023
[19] Durinx, M; Metz, JAJ; Meszéna, G, Adaptive dynamics for physiologically structured population models, J Math Biol, 56, 673-742, (2008) · Zbl 1146.92027
[20] Ewens WJ (1979) Mathematical population genetics. Springer, New York · Zbl 0422.92011
[21] Falconer DS (1981) Introduction to quantitative genetics. Longman, London
[22] Fehl K, van der Post DJ, Semmann D (2011) Co-evolution of behaviour and social network structure promotes human cooperation. Ecol Lett doi:10.1111/j.1461-0248.2011.01615.x · Zbl 1106.92056
[23] Fisher RA (1930) The genetical theory of natural selection. Clarendon Press, Oxford · JFM 56.1106.13
[24] Fu, F; Hauert, C; Nowak, MA; Wang, L, Reputation-based partner choice promotes cooperation in social networks, Phys Rev E, 78, 026117, (2008)
[25] Gardner, A; West, SA; Wild, G, The genetical theory of kin selection, J Evol Biol, 24, 1020-1043, (2011)
[26] Geritz, SAH; Kisdi, E; Meszéna, G; Metz, JAJ, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evol Ecol, 12, 35-57, (1997)
[27] Grafen A (2000) Developments of the Price equation and natural selection under uncertainty. Proc R Soc London Ser B Biol Sci 267:1223
[28] Gross T, Blasius B (2008) Adaptive coevolutionary networks: a review. J R Soc Interface 5:259-271. doi:10.1098/rsif.2007.1229
[29] Gyllenberg M, Silvestrov D (2008) Quasi-stationary phenomena in nonlinearly perturbed stochastic systems. Walter de Gruyter, Berlin · Zbl 1175.60002
[30] Hauert, C; Doebeli, M, Spatial structure often inhibits the evolution of cooperation in the snowdrift game, Nature, 428, 643-646, (2004)
[31] Hofbauer, J; Sigmund, K, Adaptive dynamics and evolutionary stability, Appl Math Lett, 3, 75-79, (1990) · Zbl 0709.92015
[32] Hofbauer J, Sigmund K (1998) Evolutionary games& replicator dynamics. Cambridge University Press, Cambridge · Zbl 0914.90287
[33] Hofbauer, J; Sigmund, K, Evolutionary game dynamics, Bull Am Math Soc, 40, 479-520, (2003) · Zbl 1049.91025
[34] Holley, R; Liggett, T, Ergodic theorems for weakly interacting infinite systems and the voter model, Annals Probability, 3, 643-663, (1975) · Zbl 0367.60115
[35] Imhof, LA; Nowak, MA, Evolutionary game dynamics in a wright-Fisher process, J Math Biol, 52, 667-681, (2006) · Zbl 1110.92028
[36] Iosifescu M (1980) Finite Markov processes and their applications. Wiley, New York · Zbl 0436.60001
[37] Kimura, M, Diffusion models in population genetics, J Appl Prob, 1, 177-232, (1964) · Zbl 0134.38103
[38] Kingman JFC (1982) The coalescent. Stochastic processes and their applications 13:235-248. doi:10.1016/0304-4149(82)90011-4
[39] Koralov L, Sinai Y (2007) Theory of probability and random processes. Springer, Berlin · Zbl 1181.60004
[40] Lessard, S; Ladret, V, The probability of fixation of a single mutant in an exchangeable selection model, J Math Biol, 54, 721-744, (2007) · Zbl 1115.92046
[41] Lieberman, E; Hauert, C; Nowak, MA, Evolutionary dynamics on graphs, Nature, 433, 312-316, (2005)
[42] Lynch M, Walsh B (1998) Genetics and analysis of quantitative traits. Sinauer, Sunderland
[43] Marshall, JA, Group selection and kin selection: formally equivalent approaches, Trends Ecol Evol, 26, 325-332, (2011)
[44] Maynard Smith, J; Price, GR, The logic of animal conflict, Nature, 246, 15-18, (1973) · Zbl 1369.92134
[45] Metz JAJ, de Roos AM (1992) The role of physiologically structured population models within a general individual-based modelling perspective. In: L DD, A GL, G HT (eds) Individual-based models and approaches in ecology: populations, communities, and ecosystems. Chapman& Hall, London, pp 88-111
[46] Metz JAJ, Geritz SAH, Meszéna G, Jacobs FA, van Heerwaarden JS (1996) Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: van Strien SJ, Lunel SMV (eds) Stochastic and spatial structures of dynamical systems. KNAW Verhandelingen. Afd., Amsterdam, pp 183-231 · Zbl 0134.38103
[47] Mihoc, G, On general properties of dependent statistical variables, Bull Math Soc Roumaine Sci, 37, 37-82, (1935) · JFM 61.1295.03
[48] Moran PAP (1958) Random processes in genetics. In: Proceedings of the Cambridge Philosophical Society, vol 54, p 60 · Zbl 0091.15701
[49] Nathanson, C; Tarnita, C; Nowak, M, Calculating evolutionary dynamics in structured populations, PLoS Comp Biol, 5, e1000615, (2009)
[50] Nowak MA (2006a) Evolutionary dynamics. Harvard University Press, Cambridge · Zbl 1115.92047
[51] Nowak, MA, Five rules for the evolution of cooperation, Science, 314, 1560-1563, (2006)
[52] Nowak, MA; May, RM, Evolutionary games and spatial chaos, Nature, 359, 826-829, (1992)
[53] Nowak, MA; Sasaki, A; Taylor, C; Fudenberg, D, Emergence of cooperation and evolutionary stability in finite populations, Nature, 428, 646-650, (2004)
[54] Nowak, MA; Tarnita, CE; Antal, T, Evolutionary dynamics in structured populations, Philos Trans R Soc B Biol Sci, 365, 19, (2010)
[55] Nowak, MA; Tarnita, CE; Wilson, EO, The evolution of eusociality, Nature, 466, 1057-1062, (2010) · Zbl 1206.60079
[56] Ohtsuki, H; Hauert, C; Lieberman, E; Nowak, MA, A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441, 502-505, (2006)
[57] Pacheco JM, Traulsen A, Nowak MA (2006) Active linking in evolutionary games. J Theor Biol 243: 437-443. doi:10.1016/j.jtbi.2006.06.027 · Zbl 1189.91034
[58] Pacheco, JM; Traulsen, A; Nowak, MA, Coevolution of strategy and structure in complex networks with dynamical linking, Phys Rev Lett, 97, 258103, (2006)
[59] Perc, M; Szolnoki, A, Coevolutionary games-a mini review, BioSystems, 99, 109-125, (2010)
[60] Price, GR, Selection and covariance, Nature, 227, 520-521, (1970)
[61] Price, GR, Extension of covariance selection mathematics, Ann Human Genet, 35, 485-490, (1972)
[62] Queller D (1992) A general model for kin selection. Evolution 376-380
[63] Queller, DC, Expanded social fitness and hamilton’s rule for kin, kith, and kind, Proc Natl Acad Sci, 108, 10792-10799, (2011)
[64] Rand, DG; Arbesman, S; Christakis, NA, Dynamic social networks promote cooperation in experiments with humans, Proc Natl Acad Sci, 108, 19193-19198, (2011)
[65] Rice, S, A stochastic version of the price equation reveals the interplay of deterministic and stochastic processes in evolution, BMC Evol Biol, 8, 262, (2008)
[66] Rice, SH; Papadopoulos, A, Evolution with stochastic fitness and stochastic migration, PLoS ONE, 4, e7130, (2009)
[67] Roca, CP; Cuesta, JA; Sánchez, A, Effect of spatial structure on the evolution of cooperation, Phys Rev E, 80, 046106, (2009)
[68] Rousset, F; Ronce, O, Inclusive fitness for traits affecting metapopulation demography, Theor Popul Biol, 65, 127-141, (2004) · Zbl 1106.92056
[69] Santos, FC; Pacheco, JM, Scale-free networks provide a unifying framework for the emergence of cooperation, Phys Rev Lett, 95, 98104, (2005)
[70] Santos, FC; Santos, MD; Pacheco, JM, Social diversity promotes the emergence of cooperation in public goods games, Nature, 454, 213-216, (2008)
[71] Shakarian, P; Roos, P; Johnson, A, A review of evolutionary graph theory with applications to game theory, Biosystems, 107, 66-80, (2012)
[72] Simon B (2008) A stochastic model of evolutionary dynamics with deterministic large-population asymptotics. J Theor Biol 254:719-730 · Zbl 1400.92381
[73] Sonin I (1999) The state reduction and related algorithms and their applications to the study of markov chains, graph theory, and the optimal stopping problem. Adv Math 145:159-188. doi:10.1006/aima.1998.1813 · Zbl 0953.60064
[74] Sood, V; Antal, T; Redner, S, Voter models on heterogeneous networks, Phys Rev E, 77, 41121, (2008)
[75] Sood, V; Redner, S, Voter model on heterogeneous graphs, Phys Rev Lett, 94, 178701, (2005)
[76] Szabó, G; Fáth, G, Evolutionary games on graphs, Phys Rep, 446, 97-216, (2007)
[77] Szolnoki, A; Perc, M; Szabó, G, Diversity of reproduction rate supports cooperation in the prisoner’s dilemma game on complex networks, Eur Phys J B Condens Matter Complex Syst, 61, 505-509, (2008) · Zbl 1189.91034
[78] Tarnita, CE; Antal, T; Ohtsuki, H; Nowak, MA, Evolutionary dynamics in set structured populations, Proc Natl Acad Sci, 106, 8601, (2009)
[79] 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
[80] Tarnita, CE; Wage, N; Nowak, MA, Multiple strategies in structured populations, Proc Natl Acad Sci, 108, 2334-2337, (2011)
[81] Taylor, C; Fudenberg, D; Sasaki, A; Nowak, MA, Evolutionary game dynamics in finite populations, Bull Math Biol, 66, 1621-1644, (2004) · Zbl 1334.92372
[82] Taylor, P; Lillicrap, T; Cownden, D, Inclusive fitness analysis on mathematical groups, Evolution, 65, 849-859, (2011)
[83] Taylor PD, Day T, Wild G (2007a) Evolution of cooperation in a finite homogeneous graph. Nature 447: 469-472
[84] Taylor, PD; Day, T; Wild, G, From inclusive fitness to fixation probability in homogeneous structured populations, J Theor Biol, 249, 101-110, (2007)
[85] Taylor, PD; Jonker, LB, Evolutionary stable strategies and game dynamics, Math Biosci, 40, 145-156, (1978) · Zbl 0395.90118
[86] Traulsen, A; Pacheco, JM; Nowak, MA, Pairwise comparison and selection temperature in evolutionary game dynamics, J Theor Biol, 246, 522-529, (2007)
[87] Baalen, M; Rand, DA, The unit of selection in viscous populations and the evolution of altruism, J Theor Biol, 193, 631-648, (1998)
[88] Veelen, M, On the use of the price equation, J Theor Biol, 237, 412-426, (2005)
[89] Veelen, M; García, J; Sabelis, MW; Egas, M, Group selection and inclusive fitness are not equivalent; the price equation vs. models and statistics, J Theor Biol, 299, 64-80, (2012) · Zbl 1337.92156
[90] Wakeley J (2009) Coalescent Theory: an introduction. Roberts& Co, Greenwood Village · Zbl 1366.92001
[91] Weibull JW (1997) Evolutionary game theory. MIT Press, Cambridge
[92] Woess W (2009) Denumerable Markov chains: generating functions, boundary theory, random walks on trees. European Mathematical Society, Zürich · Zbl 1219.60001
[93] Wu, B; Zhou, D; Fu, F; Luo, Q; Wang, L; Traulsen, A; Sporns, O, Evolution of cooperation on stochastic dynamical networks, PLoS One, 5, 1560-1563, (2010)
[94] Zhou, D; Qian, H, Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics, Phys Rev E, 84, 031907, (2011)
[95] Zhou, D; Wu, B; Ge, H, Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics, J Theor Biol, 264, 874-881, (2010) · Zbl 1406.91061
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