×

Multilocus selection in subdivided populations. I: Convergence properties for weak or strong migration. (English) Zbl 1204.92050

Summary: The dynamics and equilibrium structure of a deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is irreducible and aperiodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Various limiting cases are derived in which one or more of the three evolutionary forces, selection, migration, and recombination, are weak relative to the others.
Two are particularly interesting. If migration and recombination are strong relative to selection, the dynamics can be conceived as a perturbation of the so-called weak-selection limit, a simple dynamical system for suitably averaged allele frequencies. Under nondegeneracy assumptions on this weak-selection limit which are generic, every equilibrium of the full dynamics is a perturbation of an equilibrium of the weak-selection limit and has the same stability properties. The number of equilibria is the same in both systems, equilibria in the full (perturbed) system are in quasi-linkage equilibrium, and differences among allele frequencies across demes are small. If migration is weak relative to recombination and epistasis is also weak, then every equilibrium is a perturbation of an equilibrium of the corresponding system without migration, has the same stability properties, and is in quasi-linkage equilibrium. In both cases, every trajectory converges to an equilibrium, thus no cycling or complicated dynamics can occur.

MSC:

92D15 Problems related to evolution
37N25 Dynamical systems in biology
92D10 Genetics and epigenetics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akin E (1979) The geometry of population genetics. Lecture Notes in Biomath, vol 31. Springer, Berlin · Zbl 0437.92016
[2] Akin E (1982) Cycling in simple genetic systems. J Math Biol 13: 305–24 · Zbl 0484.92009
[3] Akin E (1993) The general topology of dynamical systems. American Mathematical Society, Providence · Zbl 0781.54025
[4] Barton NH (1999) Clines in polygenic traits. Genet Res 74: 223–36
[5] Bennett JH (1954) On the theory of random mating. Ann Eugen 18: 311–17
[6] Bulmer MG (1972) Multiple niche polymorphism. Am Nat 106: 254–57
[7] Bürger R (2000) The mathematical theory of selection, recombination, and mutation. Wiley, Chichester · Zbl 0959.92018
[8] Cannings C (1971) Natural selection at a multiallelic autosomal locus with multiple niches. J Genet 60: 255–59
[9] Christiansen FB (1975) Hard and soft selection in a subdivided population. Am Nat 109: 11–6
[10] Christiansen FB (1999) Population genetics of multiple loci. Wiley, Chichester · Zbl 0941.92019
[11] Christiansen FB, Feldman M (1975) Subdivided populations: a review of the one- and two-locus deterministic theory. Theor Popul Biol 7: 13–8 · Zbl 0293.92018
[12] Conley C (1978) Isolated invariant sets and the Morse index. NSF CBMS Lecture Notes, vol 38. American Mathematical Society, Providence · Zbl 0397.34056
[13] Deakin MAB (1966) Sufficient conditions for genetic polymorphism. Am Nat 100: 690–92
[14] Dempster ER (1955) Maintenance of genetic heterogeneity. Cold Spring Harbor Symp Quant Biol 20: 25–2
[15] Deuflhard P, Bornemann F (1995) Numerical mathematics II. Integration of ordinary differential equations. de Gruyter, Berlin · Zbl 0856.65080
[16] Ewens WJ (1969) Mean fitness increases when fitnesses are additive. Nature 221: 1076
[17] Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York · Zbl 0155.23101
[18] Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Ind Univ Math J 21: 193–26 · Zbl 0246.58015
[19] Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7: 355–69 · JFM 63.1111.04
[20] Gantmacher FR (1959) The theory of matrices, vol II. Chelsea, New York · Zbl 0085.01001
[21] Garay BM (1993) Discretization and some qualitative properties of ordinary differential equations about equilibria. Acta Math Univ Comenianae 62: 249–75 · Zbl 0822.34042
[22] Garay BM, Hofbauer J (1997) Chain recurrence and discretization. Bull Austral Math Soc 55: 63–1 · Zbl 0883.34015
[23] Geiringer H (1944) On the probability theory of linkage in Mendelian heredity. Ann Math Stat 15: 25–7 · Zbl 0063.01560
[24] Haldane JBS (1930) A mathematical theory of natural and artificial selection Part VI. Isolation. Proc Camb Phil Soc 28: 224–48
[25] Haldane JBS (1948) The theory of a cline. J Genet 48: 277–84
[26] Hastings A (1981) Stable cycling in discrete-time genetic models. Proc Natl Acad Sci USA 78: 7224–225 · Zbl 0466.92010
[27] Hirsch MW, Pugh C, Shub M (1977) Invariant Manifolds, Lecture Notes in Mathematics, vol 583. Springer, Berlin · Zbl 0355.58009
[28] Hofbauer J, Iooss G (1984) A Hopf bifurcation theorem of difference equations approximating a differential equation. Monatsh Math 98: 99–13 · Zbl 0546.58037
[29] Karlin S (1977) Gene frequency patterns in the Levene subdivided population model. Theor Popul Biol 11: 356–85 · Zbl 0357.92022
[30] Karlin S (1982) Classification of selection-migration structures and conditions for a protected polymorphism. Evol Biol 14: 61–04
[31] Karlin S, Campbell RB (1980) Selection-migration regimes characterized by a globally stable equilibrium. Genetics 94: 1065–084
[32] Karlin S, McGregor J (1972a) Application of method of small parameters to multi-niche population genetics models. Theor Popul Biol 3: 186–08 · Zbl 0262.92006
[33] Karlin S, McGregor J (1972b) Polymorphism for genetic and ecological systems with weak coupling. Theor Popul Biol 3: 210–38 · Zbl 0262.92007
[34] Kolmogoroff A, Pretrovsky I, Piscounoff N (1937) Étude de l’équation de la diffusion avec croissance de la quantite de matiére et son application à un problème biologique. (French) Bull. Univ. Etat Moscou, Ser. Int., Sect. A, Math. et Mecan. 1, Fasc. 6:1–5 · Zbl 0018.32106
[35] Kun LA, Lyubich YuI (1980) Convergence to equilibrium in a polylocus polyallele population with additive selection. Probl Inform Transmiss 16: 152–61 · Zbl 0495.92008
[36] Kruuk LEB, Baird SJE, Gale KS, Barton NH (1999) A comparison of multilocus clines maintained by environmental selection or by selection against hybrids. Genetics 153: 1959–971
[37] Levene H (1953) Genetic equilibrium when more than one ecological niche is available. Am Nat 87: 331–33
[38] Li CC (1955) The stability of an equilibrium and the average fitness of a population. Am Nat 89: 281–95
[39] Li W-H, Nei M (1974) Stable linkage disequilibrium without epistasis in subdivided populations. Theor Popul Biol 6: 173–83 · Zbl 0292.92006
[40] Lyubich YuI (1971) Basic concepts and theorems of evolutionary genetics of free populations. Russ Math Surv 26: 51–23
[41] Lyubich YuI (1992) Mathematical structures in population genetics. Springer, Berlin
[42] Nagylaki T (1992) Introduction to theoretical population genetics. Springer, Berlin · Zbl 0839.92011
[43] Nagylaki T (1993) The evolution of multilocus systems under weak selection. Genetics 134: 627–47
[44] Nagylaki T (2008) Polymorphism in multiallelic migration-selection models with dominance (submitted) · Zbl 1213.92046
[45] Nagylaki T, Hofbauer J, Brunovský P (1999) Convergence of multilocus systems under weak epistasis or weak selection. J Math Biol 38: 103–33 · Zbl 0981.92019
[46] Nagylaki T, Lou Y (2001) Patterns of multiallelic poylmorphism maintained by migration and selection. Theor Popul Biol 59: 297–13 · Zbl 1043.92025
[47] Nagylaki T, Lou Y (2006) Evolution under the multiallelic Levene model. Theor Popul Biol 70: 401–11 · Zbl 1112.92047
[48] Nagylaki T, Lou Y (2007) Evolution under multiallelic migration-selection models. Theor Popul Biol 72: 21–0 · Zbl 1125.92045
[49] Nagylaki T, Lou Y (2008) The dynamics of migration-selection models. In: Friedman A (ed) Tutorials in Mathematical Biosciences IV. Lecture Notes in Mathematics, vol 1922. Springer, Berlin, pp 119–72 · Zbl 1300.92059
[50] Prout T (1968) Sufficient conditions for multiple niche polymorphism. Am Nat 102: 493–96
[51] Reiersøl O (1962) Genetic algebras studied recursively and by means of differential operators. Math Scand 10: 25–4 · Zbl 0286.17006
[52] Seneta E (1981) Non-negative matrices and Markov Chains. Springer, New York · Zbl 0471.60001
[53] Slatkin M (1975) Gene flow and selection in two-locus systems. Genetics 81: 787–02
[54] Wright S (1931) Evolution in Mendelian populations. Genetics 16: 97–59
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.