2014, 19(4): 883-959. doi: 10.3934/dcdsb.2014.19.883

A survey of migration-selection models in population genetics

1. 

Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Received  September 2013 Revised  January 2014 Published  April 2014

This survey focuses on the most important aspects of the mathematical theory of population genetic models of selection and migration between discrete niches. Such models are most appropriate if the dispersal distance is short compared to the scale at which the environment changes, or if the habitat is fragmented. The general goal of such models is to study the influence of population subdivision and gene flow among subpopulations on the amount and pattern of genetic variation maintained. Only deterministic models are treated. Because space is discrete, they are formulated in terms of systems of nonlinear difference or differential equations. A central topic is the exploration of the equilibrium and stability structure under various assumptions on the patterns of selection and migration. Another important, closely related topic concerns conditions (necessary or sufficient) for fully polymorphic (internal) equilibria. First, the theory of one-locus models with two or multiple alleles is laid out. Then, mostly very recent, developments about multilocus models are presented. Finally, as an application, analysis and results of an explicit two-locus model emerging from speciation theory are highlighted.
Citation: Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883
References:
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show all references

References:
[1]

A. Akerman and R. Bürger, The consequences of gene flow for local adaptation and differentiation: a two-locus two-deme model,, \emph{J. Math. Biol.}, 68 (2014), 1135. doi: 10.1007/s00285-013-0660-z.

[2]

E. Akin, The Geometry of Population Genetics,, Lect. Notes Biomath. 31, (1979).

[3]

E. Akin, Cycling in simple genetic systems,, J. Math. Biol., 13 (1982), 305. doi: 10.1007/BF00276066.

[4]

E. Akin, The General Topology of Dynamical Systems,, Amer. Math. Soc., (1993).

[5]

L. Altenberg, Resolvent positive linear operators exhibit the reduction phenomenon,, Proc. Natl. Acad. Sci., 109 (2012), 3705. doi: 10.1073/pnas.1113833109.

[6]

C. Bank, R. Bürger, and J. Hermisson, The limits to parapatric speciation: Dobzhansky-Muller incompatibilities in a continent-island model,, Genetics, 191 (2012), 845. doi: 10.1534/genetics.111.137513.

[7]

N. H. Barton, Clines in polygenic traits,, Genetical Research, 74 (1999), 223. doi: 10.1017/S001667239900422X.

[8]

N. H. Barton, What role does natural selection play in speciation?, Phil. Trans. R. Soc. B, 365 (2010), 1825. doi: 10.1098/rstb.2010.0001.

[9]

N. H. Barton and M. Turelli, Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects,, Amer. Natur., 178 (2011). doi: 10.1086/661246.

[10]

L. E. Baum and J. A. Eagon, An inequality with applications to statistical estimation for probability functions of Markov processes and to a model for ecology,, Bull. Amer. Math. Soc., 73 (1967), 360. doi: 10.1090/S0002-9904-1967-11751-8.

[11]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (1994). doi: 10.1137/1.9781611971262.

[12]

M. G. Bulmer, Multiple niche polymorphism,, Amer. Natur., 106 (1972), 254. doi: 10.1086/282765.

[13]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation,, Wiley, (2000).

[14]

R. Bürger, Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration,, J. Math. Biol., 58 (2009), 939. doi: 10.1007/s00285-008-0236-5.

[15]

R. Bürger, Multilocus selection in subdivided populations II. Maintenance of polymorphism and weak or strong migration,, J. Math. Biol., 58 (2009), 979. doi: 10.1007/s00285-008-0237-4.

[16]

R. Bürger, Polymorphism in the two-locus Levene model with nonepistatic directional selection,, Theor. Popul. Biol., 76 (2009), 214.

[17]

R. Bürger, Evolution and polymorphism in the multilocus Levene model with no or weak epistasis,, Theor. Popul. Biol., 78 (2010), 123.

[18]

R. Bürger, Some mathematical models in evolutionary genetics,, in The Mathematics of Darwin's Legacy (eds. F. A. C. C. Chalub and J. F. Rodrigues), (2011), 67. doi: 10.1007/978-3-0348-0122-5_4.

[19]

R. Bürger and A. Akerman, The effects of linkage and gene flow on local adaptation: A two-locus continent-island model,, Theor. Popul. Biol., 80 (2011), 272.

[20]

C. Cannings, Natural selection at a multiallelic autosomal locus with multiple niches,, J. Genetics, 60 (1971), 255. doi: 10.1007/BF02984168.

[21]

B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics,, Roberts & Co, (2010).

[22]

F.B. Christiansen, Sufficient conditions for protected polymorphism in a subdivided population,, Amer. Natur., 108 (1974), 157. doi: 10.1086/282896.

[23]

F. B. Christiansen, Hard and soft selection in a subdivided population,, Amer. Natur., 109 (1975), 11. doi: 10.1086/282970.

[24]

F. B. Christiansen, Population Genetics of Multiple Loci,, Wiley, (1999).

[25]

C. Conley, Isolated invariant sets and the Morse index,, NSF CBMS Lecture Notes 38, (1978).

[26]

M. A. B. Deakin, Sufficient conditions for genetic polymorphism,, Amer. Natur., 100 (1966), 690. doi: 10.1086/282462.

[27]

M. A. B. Deakin, Corrigendum to genetic polymorphism in a subdivided population,, Australian J. Biol. Sci., 25 (1972), 213.

[28]

E. R. Dempster, Maintenance of genetic heterogeneity,, Cold Spring Harbor Symp. Quant. Biol., 20 (1955), 25. doi: 10.1101/SQB.1955.020.01.005.

[29]

W. J. Ewens, Mean fitness increases when fitnesses are additive,, Nature, 221 (1969). doi: 10.1038/2211076a0.

[30]

W. J. Ewens, Mathematical Population Genetics,, 2nd edition, (2004).

[31]

W. J. Ewens, What changes has mathematics made to the Darwinian theory?, in The Mathematics of Darwin's Legacy (eds. F. A. C. C. Chalub & J. F. Rodrigues), (2011), 7. doi: 10.1007/978-3-0348-0122-5_2.

[32]

E. A. Eyland, Moran's island model,, Genetics, 69 (1971), 399.

[33]

M. W. Feldman, Equilibrium studies of two locus haploid populations with recombination,, Theor. Popul. Biol., 2 (1971), 299. doi: 10.1016/0040-5809(71)90022-0.

[34]

W. Feller, An Introduction to Probability Theory and Its Applications,, vol. I, (1968).

[35]

R. A. Fisher, The correlation between relatives on the supposition of Mendelian inheritance,, Trans. Roy. Soc. Edinburgh, 52 (1918), 399. doi: 10.1017/S0080456800012163.

[36]

R. A. Fisher, The Genetical Theory of Natural Selection,, Clarendon Press, (1930).

[37]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[38]

S. Friedland and S. Karlin, Some inequalities for the spectral radius of nonnegative matrices and applications,, Duke Math. J., 42 (1975), 459.

[39]

H. Geiringer, On the probability theory of linkage in Mendelian heredity,, Ann. Math. Stat., 15 (1944), 25. doi: 10.1214/aoms/1177731313.

[40]

K. p. Hadeler and D. Glas, Quasimonotone systems and convergence to equilibrium in a population genetic model,, J. Math. Anal. Appl., 95 (1983), 297. doi: 10.1016/0022-247X(83)90108-7.

[41]

J. B. S. Haldane, A mathematical theory of natural and artificial selection. Part VI. Isolation,, Proc. Camb. Phil. Soc., 28 (1930), 224. doi: 10.1017/S0305004100015450.

[42]

J. B. S. Haldane, The Causes of Evolution,, Longmans, (1992).

[43]

J. B. S. Haldane, The theory of a cline,, J. Genetics, 48 (1948), 277. doi: 10.1007/BF02986626.

[44]

G. H. Hardy, Mendelian proportions in a mixed population,, Science, 28 (1908), 49. doi: 10.1007/BF01990610.

[45]

A. Hastings, Simultaneous stability of $D=0$ and $D\ne0$ for multiplicative viabilities at two loci: An analytical study,, J. Theor. Biol., 89 (1981), 69. doi: 10.1016/0022-5193(81)90180-6.

[46]

A. Hastings, Stable cycling in discrete-time genetic models,, Proc. Natl. Acad. Sci. USA, 78 (1981), 7224. doi: 10.1073/pnas.78.11.7224.

[47]

M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I: Limit sets,, SIAM J. Math. Anal., 13 (1982), 167. doi: 10.1137/0513013.

[48]

J. Hofbauer, An index theorem for dissipative semiflows,, Rocky Mountain J. Math., 20 (1990), 1017. doi: 10.1216/rmjm/1181073059.

[49]

J. Hofbauer and G. Iooss, A Hopf bifurcation theorem of difference equations approximating a differential equation,, Monatsh. Math., 98 (1984), 99. doi: 10.1007/BF01637279.

[50]

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