# zbMATH — the first resource for mathematics

A mathematical formalism for natural selection with arbitrary spatial and genetic structure. (English) Zbl 1409.92162
Summary: We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

##### MSC:
 92D10 Genetics and epigenetics 92D15 Problems related to evolution 91A22 Evolutionary games 91A43 Games involving graphs 60J85 Applications of branching processes
Full Text:
##### References:
 [1] Adlam, B.; Chatterjee, K.; Nowak, M., Amplifiers of selection, Proc R Soc A Math Phys Eng Sci, 471, 20150,114, (2015) · Zbl 1371.92093 [2] Akçay, E.; Cleve, J., There is no fitness but fitness, and the lineage is its bearer, Philos Trans R Soc B Biol Sci, 371, 20150,085, (2016) [3] Allen, B.; Nowak, MA, Games on graphs, EMS Surv Math Sci, 1, 113-151, (2014) · Zbl 1303.91040 [4] Allen, B.; Tarnita, CE, Measures of success in a class of evolutionary models with fixed population size and structure, J Math Biol, 68, 109-143, (2014) · Zbl 1280.92045 [5] Allen, B.; Traulsen, A.; Tarnita, CE; Nowak, MA, How mutation affects evolutionary games on graphs, J Theor Biol, 299, 97-105, (2012) · Zbl 1337.91016 [6] Allen, B.; Nowak, MA; Dieckmann, U., Adaptive dynamics with interaction structure, Am Nat, 181, e139-e163, (2013) [7] Allen, B.; Sample, C.; Dementieva, Y.; Medeiros, RC; Paoletti, C.; Nowak, MA, The molecular clock of neutral evolution can be accelerated or slowed by asymmetric spatial structure, PLoS Comput Biol, 11, e1004,108, (2015) [8] Allen, B.; Lippner, G.; Chen, YT; Fotouhi, B.; Momeni, N.; Yau, ST; Nowak, MA, Evolutionary dynamics on any population structure, Nature, 544, 227-230, (2017) [9] 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 [10] Antal, T.; Traulsen, A.; Ohtsuki, H.; Tarnita, CE; Nowak, MA, Mutation-selection equilibrium in games with multiple strategies, J Theor Biol, 258, 614-622, (2009) · Zbl 1402.91039 [11] Benaïm M, Schreiber SJ (2018) Persistence and extinction for stochastic ecological difference equations with feedbacks. arXiv preprint arXiv:1808.07888 [12] Blume, LE, The statistical mechanics of strategic interaction, Games Econ Behav, 5, 387-424, (1993) · Zbl 0797.90123 [13] 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 [14] Bürger R (2000) The mathematical theory of selection, recombination, and mutation. Wiley, London · Zbl 0959.92018 [15] Bürger, R., A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait, J Math Biol, 50, 355-396, (2005) · Zbl 1062.92047 [16] Cattiaux, P.; Collet, P.; Lambert, A.; Martinez, S.; Méléard, S.; San Martín, J., Quasi-stationary distributions and diffusion models in population dynamics, Ann Probab, 37, 1926-1969, (2009) · Zbl 1176.92041 [17] Cavaliere, M.; Sedwards, S.; Tarnita, CE; Nowak, MA; Csikász-Nagy, A., Prosperity is associated with instability in dynamical networks, J Theor Biol, 299, 126-138, (2012) · Zbl 1337.91018 [18] 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 [19] Chen, YT, Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs, Ann Appl Probab, 23, 637-664, (2013) · Zbl 1267.91019 [20] Chen, YT, Wright-Fisher diffusions in stochastic spatial evolutionary games with death-birth updating, Ann Appl Probab, 28, 3418-3490, (2018) · Zbl 1404.60148 [21] Chen, YT; McAvoy, A.; Nowak, MA, Fixation probabilities for any configuration of two strategies on regular graphs, Sci Rep, 6, 181, (2016) [22] Chotibut, T.; Nelson, DR, Population genetics with fluctuating population sizes, J Stat Phys, 167, 777-791, (2017) · Zbl 1370.92090 [23] Cohen, D., Optimizing reproduction in a randomly varying environment, J Theor Biol, 12, 119-129, (1966) [24] Constable, GW; Rogers, T.; McKane, AJ; Tarnita, CE, Demographic noise can reverse the direction of deterministic selection, Proc Natl Acad Sci, 113, e4745-e4754, (2016) [25] Cox, JT, Coalescing random walks and voter model consensus times on the torus in $${\mathbb{Z}}^d$$, Ann Probab, 17, 1333-1366, (1989) · Zbl 0685.60100 [26] Cox JT, Durrett R, Perkins EA (2013) Voter model perturbations and reaction diffusion equations. Asterisque 349 · Zbl 1277.60004 [27] Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper and Row, New York · Zbl 0246.92003 [28] Cvijović, I.; Good, BH; Jerison, ER; Desai, MM, Fate of a mutation in a fluctuating environment, Proc Natl Acad Sci, 112, e5021-e5028, (2015) [29] Débarre, F., Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations, J Theor Biol, 420, 26-35, (2017) · Zbl 1370.92103 [30] Débarre, F.; Hauert, C.; Doebeli, M., Social evolution in structured populations, Nat Commun, 5, 4409, (2014) [31] Dercole F, Rinaldi S (2008) Analysis of evolutionary processes: the adaptive dynamics approach and its applications. Princeton University Press, Princeton · Zbl 1305.92001 [32] Dieckmann, U.; Doebeli, M., On the origin of species by sympatric speciation, Nature, 400, 354-357, (1999) [33] 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 [34] 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 [35] 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 [36] Diekmann, O.; Gyllenberg, M.; Metz, J.; Takeuchi, Y. (ed.); Iwasa, Y. (ed.); Sato, K. (ed.), Physiologically structured population models: towards a general mathematical theory, 5-20, (2007), Berlin [37] Doebeli, M.; Ispolatov, Y.; Simon, B., Towards a mechanistic foundation of evolutionary theory, eLife, 6, e23,804, (2017) [38] Durinx, M.; Metz, JAJ; Meszéna, G., Adaptive dynamics for physiologically structured population models, J Math Biol, 56, 673-742, (2008) · Zbl 1146.92027 [39] Durrett, R., Spatial evolutionary games with small selection coefficients, Electron J Probab, 19, 1-64, (2014) · Zbl 1414.91051 [40] Eshel, I.; Feldman, MW; Bergman, A., Long-term evolution, short-term evolution, and population genetic theory, J Theor Biol, 191, 391-396, (1998) [41] Ewens WJ (2004) Mathematical population genetics 1: theoretical introduction, vol 27, 2nd edn. Springer, New York · Zbl 1060.92046 [42] Faure, M.; Schreiber, SJ, Quasi-stationary distributions for randomly perturbed dynamical systems, Ann Appl Probab, 24, 553-598, (2014) · Zbl 1334.60137 [43] Fisher R (1930) The genetical theory of natural selection. Clarendon Press, Oxford · JFM 56.1106.13 [44] Fotouhi, B.; Momeni, N.; Allen, B.; Nowak, MA, Conjoining uncooperative societies facilitates evolution of cooperation, Nat Hum Behav, 2, 492-499, (2018) [45] Fudenberg, D.; Imhof, LA, Imitation processes with small mutations, J Econ Theory, 131, 251-262, (2006) · Zbl 1142.91342 [46] 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) [47] Gould, SJ; Lloyd, EA, Individuality and adaptation across levels of selection: How shall we name and generalize the unit of Darwinism?, Proc Natl Acad Sci, 96, 11,904-11,909, (1999) [48] Gyllenberg, M.; Parvinen, K., Necessary and sufficient conditions for evolutionary suicide, Bull Math Biol, 63, 981-993, (2001) · Zbl 1323.92141 [49] Gyllenberg M, Silvestrov D (2008) Quasi-stationary phenomena in nonlinearly perturbed stochastic systems. Walter de Gruyter, Berlin · Zbl 1175.60002 [50] Haccou, P.; Iwasa, Y., Establishment probability in fluctuating environments: a branching process model, Theor Popul Biol, 50, 254-280, (1996) · Zbl 0867.92013 [51] Haccou P, Jagers P, Vatutin VA (2005) Branching processes: variation, growth, and extinction of populations. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511629136 · Zbl 1118.92001 [52] Haldane, J., A mathematical theory of natural and artificial selection. Part I, Trans Camb Philos Soc, 23, 19-41, (1924) [53] Hammerstein, P., Darwinian adaptation, population genetics and the streetcar theory of evolution, J Math Biol, 34, 511-532, (1996) · Zbl 0845.92014 [54] Holley, RA; Liggett, TM, Ergodic theorems for weakly interacting infinite systems and the voter model, Ann Probab, 3, 643-663, (1975) · Zbl 0367.60115 [55] Hull, DL, Individuality and selection, Annu Rev Ecol Syst, 11, 311-332, (1980) [56] Ibsen-Jensen, R.; Chatterjee, K.; Nowak, MA, Computational complexity of ecological and evolutionary spatial dynamics, Proc Natl Acad Sci, 112, 15,636-15,641, (2015) [57] Jeong, HC; Oh, SY; Allen, B.; Nowak, MA, Optional games on cycles and complete graphs, J Theor Biol, 356, 98-112, (2014) [58] Kemeny JG, Snell JL (1960) Finite Markov chains, vol 356. van Nostrand, Princeton · Zbl 0089.13704 [59] Kimura, M., Diffusion models in population genetics, J Appl Probab, 1, 177-232, (1964) · Zbl 0134.38103 [60] Kimura, M.; etal., Evolutionary rate at the molecular level, Nature, 217, 624-626, (1968) [61] Kingman, JFC, The coalescent, Stoch Processes Appl, 13, 235-248, (1982) · Zbl 0491.60076 [62] Korolev, KS, The fate of cooperation during range expansions, PLoS Comput Biol, 9, e1002,994, (2013) [63] Kussell, E.; Leibler, S., Phenotypic diversity, population growth, and information in fluctuating environments, Science, 309, 2075-2078, (2005) [64] Lambert, A., Probability of fixation under weak selection: a branching process unifying approach, Theor Popul Biol, 69, 419-441, (2006) · Zbl 1121.92051 [65] Lehmann, L.; Rousset, F., Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment, Theor Popul Biol, 76, 35-51, (2009) · Zbl 1213.92041 [66] Lehmann, L.; Mullon, C.; Akcay, E.; Cleve, J., Invasion fitness, inclusive fitness, and reproductive numbers in heterogeneous populations, Evolution, 70, 1689-1702, (2016) [67] 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 [68] Lessard, S.; Soares, CD, Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection, J Math Biol, 77, 229-259, (2018) · Zbl 1392.92078 [69] Leturque, H.; Rousset, F., Dispersal, kin competition, and the ideal free distribution in a spatially heterogeneous population, Theor Popul Biol, 62, 169-180, (2002) · Zbl 1101.92307 [70] Lewontin, RC, The units of selection, Annu Rev Ecol Syst, 1, 1-18, (1970) [71] Lieberman, E.; Hauert, C.; Nowak, M., Evolutionary dynamics on graphs, Nature, 433, 312-316, (2005) [72] Lindholm, AK; Dyer, KA; Firman, RC; Fishman, L.; Forstmeier, W.; Holman, L.; Johannesson, H.; Knief, U.; Kokko, H.; Larracuente, AM; etal., The ecology and evolutionary dynamics of meiotic drive, Trends Ecol Evol, 31, 315-326, (2016) [73] Maciejewski, W., Reproductive value in graph-structured populations, J Theor Biol, 340, 285-293, (2014) [74] Malécot G (1948) Les Mathématiques de l’Hérédité. Masson et Cie, Paris [75] McAvoy, A.; Hauert, C., Structure coefficients and strategy selection in multiplayer games, J Math Biol, 72, 203-238, (2016) · Zbl 1409.91038 [76] McAvoy A, Adlam B, Allen B, Nowak MA (2018a) Stationary frequencies and mixing times for neutral drift processes with spatial structure. Proc Ro Soc A Math Phys Eng Sci. https://doi.org/10.1098/rspa.2018.0238 · Zbl 1407.92095 [77] McAvoy, A.; Fraiman, N.; Hauert, C.; Wakeley, J.; Nowak, MA, Public goods games in populations with fluctuating size, Theor Popul Biol, 121, 72-84, (2018) · Zbl 1397.92588 [78] Metz, JAJ; Roos, AM; DeAngelis, DL (ed.); Gross, LA (ed.); Hallam, TG (ed.), The role of physiologically structured population models within a general individual-based modelling perspective, 88-111, (1992), London [79] Metz, JA; Geritz, SA, Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective, J Math Biol, 72, 1011-1037, (2016) · Zbl 1337.92151 [80] Metz, J.; Nisbet, R.; Geritz, S., How should we define ‘fitness’ for general ecological scenarios?, Trends Ecol Evol, 7, 198-202, (1992) [81] Metz, JAJ; Geritz, SAH; Meszéna, G.; Jacobs, FA; Heerwaarden, JS; Strien, SJ (ed.); Lunel, SMV (ed.), Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction, 183-231, (1996), Amsterdam [82] Nowak, MA; May, RM, Evolutionary games and spatial chaos, Nature, 359, 826-829, (1992) [83] Nowak, MA; Sasaki, A.; Taylor, C.; Fudenberg, D., Emergence of cooperation and evolutionary stability in finite populations, Nature, 428, 646-650, (2004) [84] Nowak, MA; Tarnita, CE; Antal, T., Evolutionary dynamics in structured populations, Philos Trans R Soc B Biol Sci, 365, 19, (2010) [85] Nowak, MA; Tarnita, CE; Wilson, EO, The evolution of eusociality, Nature, 466, 1057-1062, (2010) [86] 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) [87] Okasha S (2006) Evolution and the levels of selection. Oxford University Press, Oxford [88] Pacheco, JM; Traulsen, A.; Nowak, MA, Active linking in evolutionary games, J Theor Biol, 243, 437-443, (2006) [89] Pacheco, JM; Traulsen, A.; Nowak, MA, Coevolution of strategy and structure in complex networks with dynamical linking, Phys Rev Lett, 97, 258,103, (2006) [90] Parsons, TL; Quince, C., Fixation in haploid populations exhibiting density dependence I: the non-neutral case, Theor Popul Biol, 72, 121-135, (2007) · Zbl 1123.92020 [91] Parsons, TL; Quince, C., Fixation in haploid populations exhibiting density dependence II: the quasi-neutral case, Theor Popul Biol, 72, 468-479, (2007) · Zbl 1147.92027 [92] Parsons, TL; Quince, C.; Plotkin, JB, Some consequences of demographic stochasticity in population genetics, Genetics, 185, 1345-1354, (2010) [93] Parvinen, K.; Seppänen, A., On fitness in metapopulations that are both size-and stage-structured, J Math Biol, 73, 903-917, (2016) · Zbl 1360.92078 [94] Pavlogiannis, A.; Tkadlec, J.; Chatterjee, K.; Nowak, MA, Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory, Commun Biol, 1, 71, (2018) [95] Pelletier, F.; Clutton-Brock, T.; Pemberton, J.; Tuljapurkar, S.; Coulson, T., The evolutionary demography of ecological change: linking trait variation and population growth, Science, 315, 1571-1574, (2007) [96] Peña, J.; Wu, B.; Arranz, J.; Traulsen, A., Evolutionary games of multiplayer cooperation on graphs, PLoS Comput Biol, 12, e1005,059, (2016) [97] Perc, M.; Szolnoki, A., Coevolutionary games—a mini review, BioSystems, 99, 109-125, (2010) [98] Philippi, T.; Seger, J., Hedging one’s evolutionary bets, revisited, Trends Ecol Evol, 4, 41-44, (1989) [99] Price, GR, Selection and covariance, Nature, 227, 520-521, (1970) [100] Rand, DA; Wilson, HB; McGlade, JM, Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics, Philos Trans R Soc B Biol Sci, 343, 261-283, (1994) [101] Roth, G.; Schreiber, SJ, Persistence in fluctuating environments for interacting structured populations, J Math Biol, 69, 1267-1317, (2013) · Zbl 1351.37216 [102] Roth, G.; Schreiber, SJ, Pushed beyond the brink: Allee effects, environmental stochasticity, and extinction, J Biol Dyn, 8, 187-205, (2014) [103] Rousset, F.; Billiard, S., A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal, J Evol Biol, 13, 814-825, (2000) [104] Sample, C.; Allen, B., The limits of weak selection and large population size in evolutionary game theory, J Math Biol, 75, 1285-1317, (2017) · Zbl 1378.91029 [105] Sandler, L.; Novitski, E., Meiotic drive as an evolutionary force, Am Nat, 91, 105-110, (1957) [106] Santos, FC; Pacheco, JM, Scale-free networks provide a unifying framework for the emergence of cooperation, Phys Rev Lett, 95, 98,104, (2005) [107] Schoener, TW, The newest synthesis: understanding the interplay of evolutionary and ecological dynamics, Science, 331, 426-429, (2011) [108] Schreiber, SJ; Benaïm, M.; Atchadé, KAS, Persistence in fluctuating environments, J Math Biol, 62, 655-683, (2010) · Zbl 1232.92075 [109] Silvestrov D, Silvestrov S (2017) Nonlinearly perturbed semi-Markov processes. Springer, Cham · Zbl 1404.60003 [110] Simon, B.; Fletcher, JA; Doebeli, M., Towards a general theory of group selection, Evolution, 67, 1561-1572, (2013) [111] Starrfelt, J.; Kokko, H., Bet-hedging—a triple trade-off between means, variances and correlations, Biol Rev, 87, 742-755, (2012) [112] Szabó, G.; Fáth, G., Evolutionary games on graphs, Phys Rep, 446, 97-216, (2007) [113] Tarnita, CE; Taylor, PD, Measures of relative fitness of social behaviors in finite structured population models, Am Nat, 184, 477-488, (2014) [114] Tarnita, CE; Antal, T.; Ohtsuki, H.; Nowak, MA, Evolutionary dynamics in set structured populations, Proc Natl Acad Sci, 106, 8601-8604, (2009) [115] Tarnita, CE; Ohtsuki, H.; Antal, T.; Fu, F.; Nowak, MA, Strategy selection in structured populations, J Theor Biol, 259, 570-581, (2009) · Zbl 1402.91064 [116] Tarnita, CE; Wage, N.; Nowak, MA, Multiple strategies in structured populations, Proc Natl Acad Sci, 108, 2334-2337, (2011) [117] Tavaré, S., Line-of-descent and genealogical processes, and their applications in population genetics models, Theor Popul Biol, 26, 119-164, (1984) · Zbl 0555.92011 [118] Taylor, PD, Allele-frequency change in a class-structured population, Am Nat, 135, 95-106, (1990) [119] Taylor, PD; Frank, SA, How to make a kin selection model, J Theor Biol, 180, 27-37, (1996) [120] Taylor, P.; Day, T.; Wild, G., From inclusive fitness to fixation probability in homogeneous structured populations, J Theor Biol, 249, 101-110, (2007) [121] Taylor, PD; Day, T.; Wild, G., Evolution of cooperation in a finite homogeneous graph, Nature, 447, 469-472, (2007) [122] Traulsen, A.; Hauert, C.; Silva, H.; Nowak, MA; Sigmund, K., Exploration dynamics in evolutionary games, Proc Natl Acad Sci, 106, 709-712, (2009) · Zbl 1202.91029 [123] Uecker, H.; Hermisson, J., On the fixation process of a beneficial mutation in a variable environment, Genetics, 188, 915-930, (2011) [124] Cleve, J., Social evolution and genetic interactions in the short and long term, Theor Popul Biol, 103, 2-26, (2015) · Zbl 1342.92147 [125] Veelen, M., On the use of the price equation, J Theor Biol, 237, 412-426, (2005) [126] Wakano, JY; Nowak, MA; Hauert, C., Spatial dynamics of ecological public goods, Proc Natl Acad Sci, 106, 7910-7914, (2009) [127] Wakano, JY; Ohtsuki, H.; Kobayashi, Y., A mathematical description of the inclusive fitness theory, Theor Popul Biol, 84, 46-55, (2013) · Zbl 1275.92082 [128] Wakeley J (2009) Coalescent theory: an introduction. Roberts & Company Publishers, Greenwood Village · Zbl 1366.92001 [129] Wardil, L.; Hauert, C., Origin and structure of dynamic cooperative networks, Sci Rep, 4, 5725, (2014) [130] Waxman, D., A unified treatment of the probability of fixation when population size and the strength of selection change over time, Genetics, 188, 907-913, (2011) [131] Williams GC (1966) Adaptation and natural selection: a critique of some current evolutionary thought. Princeton University Press, Princeton [132] Wu, B.; Zhou, D.; Fu, F.; Luo, Q.; Wang, L.; Traulsen, A., Evolution of cooperation on stochastic dynamical networks, PLoS ONE, 5, e11,187, (2010) [133] Wu, B.; Gokhale, CS; Wang, L.; Traulsen, A., How small are small mutation rates?, J Math Biol, 64, 803-827, (2012) · Zbl 1260.91025 [134] Wu, B.; Traulsen, A.; Gokhale, CS, Dynamic properties of evolutionary multi-player games in finite populations, Games, 4, 182-199, (2013) · Zbl 1314.91039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.