# American Institute of Mathematical Sciences

June  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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##### 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. Google Scholar [2] E. Akin, The Geometry of Population Genetics,, Lect. Notes Biomath. 31, (1979). Google Scholar [3] E. Akin, Cycling in simple genetic systems,, J. Math. Biol., 13 (1982), 305. doi: 10.1007/BF00276066. Google Scholar [4] E. Akin, The General Topology of Dynamical Systems,, Amer. Math. Soc., (1993). Google Scholar [5] L. Altenberg, Resolvent positive linear operators exhibit the reduction phenomenon,, Proc. Natl. Acad. Sci., 109 (2012), 3705. doi: 10.1073/pnas.1113833109. Google Scholar [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. Google Scholar [7] N. H. 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Google Scholar [13] R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation,, Wiley, (2000). Google Scholar [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. Google Scholar [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. Google Scholar [16] R. Bürger, Polymorphism in the two-locus Levene model with nonepistatic directional selection,, Theor. Popul. Biol., 76 (2009), 214. Google Scholar [17] R. Bürger, Evolution and polymorphism in the multilocus Levene model with no or weak epistasis,, Theor. Popul. Biol., 78 (2010), 123. Google Scholar [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. Google Scholar [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. Google Scholar [20] C. Cannings, Natural selection at a multiallelic autosomal locus with multiple niches,, J. Genetics, 60 (1971), 255. doi: 10.1007/BF02984168. Google Scholar [21] B. Charlesworth and D. Charlesworth, Elements of Evolutionary Genetics,, Roberts & Co, (2010). Google Scholar [22] F.B. Christiansen, Sufficient conditions for protected polymorphism in a subdivided population,, Amer. Natur., 108 (1974), 157. doi: 10.1086/282896. Google Scholar [23] F. B. Christiansen, Hard and soft selection in a subdivided population,, Amer. Natur., 109 (1975), 11. doi: 10.1086/282970. Google Scholar [24] F. B. Christiansen, Population Genetics of Multiple Loci,, Wiley, (1999). Google Scholar [25] C. 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Eyland, Moran's island model,, Genetics, 69 (1971), 399. Google Scholar [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. Google Scholar [34] W. Feller, An Introduction to Probability Theory and Its Applications,, vol. I, (1968). Google Scholar [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. Google Scholar [36] R. A. Fisher, The Genetical Theory of Natural Selection,, Clarendon Press, (1930). Google Scholar [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. Google Scholar [38] S. Friedland and S. Karlin, Some inequalities for the spectral radius of nonnegative matrices and applications,, Duke Math. J., 42 (1975), 459. Google Scholar [39] H. 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