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:
[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. Barton, Clines in polygenic traits,, Genetical Research, 74 (1999), 223. doi: 10.1017/S001667239900422X. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

M. G. Bulmer, Multiple niche polymorphism,, Amer. Natur., 106 (1972), 254. doi: 10.1086/282765. 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. Conley, Isolated invariant sets and the Morse index,, NSF CBMS Lecture Notes 38, (1978). Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

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

[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. Google Scholar

[50]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems,, University Press, (1988). Google Scholar

[51]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, University Press, (1998). Google Scholar

[52]

S. Karlin, Gene frequency patterns in the Levene subdivided population model,, Theor. Popul. Biol., 11 (1977), 356. doi: 10.1016/0040-5809(77)90018-1. Google Scholar

[53]

S. Karlin, Classification of selection-migration structures and conditions for a protected polymorphism,, Evol. Biol., 14 (1982), 61. Google Scholar

[54]

S. Karlin and R. B. Campbell, Selection-migration regimes characterized by a globally stable equilibrium,, Genetics, 94 (1980), 1065. Google Scholar

[55]

S. Karlin and M. W. Feldman, Simultaneous stability of $D=0$ and $D\ne0$ for multiplicative viabilities at two loci,, Genetics, 90 (1978), 813. Google Scholar

[56]

S. Karlin and J. McGregor, Application of method of small parameters to multi-niche population genetics models,, Theor. Popul. Biol., 3 (1972), 186. doi: 10.1016/0040-5809(72)90026-3. Google Scholar

[57]

S. Karlin and J. McGregor, Polymorphism for genetic and ecological systems with weak coupling,, Theor. Popul. Biol., 3 (1972), 210. doi: 10.1016/0040-5809(72)90027-5. Google Scholar

[58]

J. F. C. Kingman, An inequality in partial averages,, Quart. J. Math., 12 (1961), 78. doi: 10.1093/qmath/12.1.78. Google Scholar

[59]

A. Kolmogoroff, I. Pretrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantite de matiére et son application à un problème biologique,, Bull. Univ. Etat Moscou, 1 (1937), 1. Google Scholar

[60]

J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conf. Ser. Appl. Math. 25, (1976). Google Scholar

[61]

H. Levene, Genetic equilibrium when more than one ecological niche is available,, Amer. Natur., 87 (1953), 331. doi: 10.1086/281792. Google Scholar

[62]

S. Lessard, Fisher's fundamental theorem of natural selection revisited,, Theor. Pop. Biol., 52 (1997), 119. doi: 10.1006/tpbi.1997.1324. Google Scholar

[63]

R. C. Lewontin and K.-I. Kojima, The evolutionary dynamics of complex polymorphisms,, Evolution, 14 (1969), 458. doi: 10.2307/2405995. Google Scholar

[64]

C. C. Li, The stability of an equilibrium and the average fitness of a population,, Amer. Natur., 89 (1955), 281. doi: 10.1086/281893. Google Scholar

[65]

C. C. Li, Fundamental theorem of natural selection,, Nature, 214 (1967), 505. doi: 10.1038/214505a0. Google Scholar

[66]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics,, J. Diff. Eqs., 181 (2002), 388. doi: 10.1006/jdeq.2001.4086. Google Scholar

[67]

Y. Lou, T. Nagylaki and W.-M. Ni, An introduction to migration-selection PDE models,, Disc. Cont. Dyn. Syst. A, 33 (2013), 4349. doi: 10.3934/dcds.2013.33.4349. Google Scholar

[68]

Yu. I. Lyubich, Basic concepts and theorems of evolutionary genetics of free populations,, Russ. Math. Surv., 26 (1971), 51. doi: 10.1070/RM1971v026n05ABEH003829. Google Scholar

[69]

Yu. I. Lyubich, Mathematical Structures in Population Genetics,, Springer, (1992). doi: 10.1007/978-3-642-76211-6. Google Scholar

[70]

J. Maynard Smith, Genetic polymorphism in a varied environment,, Amer. Natur., 104 (1970), 487. doi: 10.1086/282683. Google Scholar

[71]

T. Nagylaki, Selection in One- and Two-Locus Systems,, Lect. Notes Biomath. 15, (1977). Google Scholar

[72]

T. Nagylaki, The diffusion model for migration and selection,, in Some Mathematical Questions in Biology (ed. A. Hastings), (1989), 55. Google Scholar

[73]

T. Nagylaki, Introduction to Theoretical Population Genetics,, Berlin, (1992). doi: 10.1007/978-3-642-76214-7. Google Scholar

[74]

T. Nagylaki, The evolution of multilocus systems under weak selection,, Genetics, 134 (1993), 627. Google Scholar

[75]

T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance,, Theor. Popul. Biol., 75 (2009), 239. doi: 10.1016/j.tpb.2009.01.004. Google Scholar

[76]

T. Nagylaki, Evolution under the multilocus Levene model without epistasis,, Theor. Popul. Biol., 76 (2009), 197. doi: 10.1016/j.tpb.2009.07.003. Google Scholar

[77]

T. Nagylaki, J. Hofbauer and P. Brunovský, Convergence of multilocus systems under weak epistasis or weak selection,, J. Math. Biol., 38 (1999), 103. doi: 10.1007/s002850050143. Google Scholar

[78]

T. Nagylaki and Y. Lou, Patterns of multiallelic poylmorphism maintained by migration and selection,, Theor. Popul. Biol., 59 (2001), 297. Google Scholar

[79]

T. Nagylaki and Y. Lou, Multiallelic selection polymorphism,, Theor. Popul. Biol., 69 (2006), 217. doi: 10.1016/j.tpb.2005.09.003. Google Scholar

[80]

T. Nagylaki and Y. Lou, Evolution under the multiallelic Levene model,, Theor. Popul. Biol., 70 (2006), 401. doi: 10.1016/j.tpb.2006.03.002. Google Scholar

[81]

T. Nagylaki and Y. Lou, Evolution under multiallelic migration-selection models,, Theor. Popul. Biol., 72 (2007), 21. doi: 10.1016/j.tpb.2007.02.005. Google Scholar

[82]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models,, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), (2008), 119. doi: 10.1007/978-3-540-74331-6_4. Google Scholar

[83]

A. Novak, The number of equilibria in the diallelic Levene model with multiple demes,, Theor. Popul. Biol., 79 (2011), 97. doi: 10.1016/j.tpb.2010.12.002. Google Scholar

[84]

S. Peischl, Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes,, Theor. Popul. Biol., 78 (2010), 12. doi: 10.1016/j.tpb.2010.03.006. Google Scholar

[85]

G.R. Price, Selection and covariance,, Nature, 227 (1970), 520. doi: 10.1038/227520a0. Google Scholar

[86]

T. Prout, Sufficient conditions for multiple niche polymorphism,, Amer. Natur., 102 (1968), 493. doi: 10.1086/282562. Google Scholar

[87]

W.B. Provine, The Origins of Theoretical Population Genetics,, Chicago Univ. Press, (1971). Google Scholar

[88]

D. Roze and F. Rousset, Multilocus models in the infinite island model of population structure,, Theor. Popul. Biol., 73 (): 529. doi: 10.1016/j.tpb.2008.03.002. Google Scholar

[89]

D. Rutschman, Dynamics of the two-locus haploid model,, Theor. Popul. Biol., 45 (1994), 167. doi: 10.1006/tpbi.1994.1009. Google Scholar

[90]

E. Seneta, Non-negative Matrices,, 2nd ed., (1981). doi: 10.1007/0-387-32792-4_6. Google Scholar

[91]

S. Shahshahani, A new mathematica framework for the study of linkage and selection,, Memoirs Amer. Math. Soc., 211 (1979). doi: 10.1090/memo/0211. Google Scholar

[92]

M. Spichtig and T. J. Kawecki, The maintenance (or not) of polygenic variation by soft selection in a heterogeneous environment,, Amer. Natur., 164 (2004), 70. doi: 10.1086/421335. Google Scholar

[93]

B. Star, R. J. Stoffels and H. G. Spencer, Single-locus polymorphism in a heterogeneous two-deme model,, Genetics, 176 (2007), 1625. doi: 10.1534/genetics.107.071639. Google Scholar

[94]

B. Star, R.J. Stoffels, and H.G. Spencer, Evolution of fitnesses and allele frequencies in a population with spatially heterogeneous selection pressures,, Genetics, 177 (2007), 1743. doi: 10.1534/genetics.107.079558. Google Scholar

[95]

C. Strobeck, Haploid selection with $n$ alleles in $m$ niches., Amer. Natur., 113 (1979), 439. doi: 10.1086/283401. Google Scholar

[96]

Yu. M. Svirezhev, Optimality principles in population genetics,, in Studies in Theoretical Genetics (in Russian), (1972), 86. Google Scholar

[97]

G. S. van Doorn and U. Dieckmann, The long-term evolution of multilocus traits under frequency-dependent disruptive selection,, Evolution, 60 (2006), 2226. doi: 10.1111/j.0014-3820.2006.tb01860.x. Google Scholar

[98]

J. Wakeley, Coalescent Theory: An Introduction,, Roberts & Company Publishers, (2008). Google Scholar

[99]

W. Weinberg, Über den Nachweis der Vererbung beim Menschen,, Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg, 64 (1908), 368. Google Scholar

[100]

W. Weinberg, Über Vererbungsgesetze beim Menschen,, Zeitschrift für induktive Abstammungs- und Vererbungslehre, 1 (1909), 377. doi: 10.1007/BF01975801. Google Scholar

[101]

T. Wiehe and M. Slatkin, Epistatic selection in a multi-locus Levene model and implications for linkage disequilibrium,, Theor. Popul. Biol., 53 (1998), 75. doi: 10.1006/tpbi.1997.1342. Google Scholar

[102]

S. Wright, Evolution in Mendelian populations,, Genetics, 16 (1931), 97. doi: 10.1016/S0092-8240(05)80011-4. Google Scholar

[103]

G. U. Yule, Mendel's laws and their probable relations to intra-racial heredity,, New Phytol., 1 (1902), 193. doi: 10.1111/j.1469-8137.1902.tb06590.x. Google Scholar

[104]

L. A. Zhivotovsky, M. W. Feldman and A. Bergman, On the evolution of phenotypic plasticity in a spatially heterogeneous environment,, Evolution, 50 (1996), 547. doi: 10.2307/2410830. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

M. G. Bulmer, Multiple niche polymorphism,, Amer. Natur., 106 (1972), 254. doi: 10.1086/282765. 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. Conley, Isolated invariant sets and the Morse index,, NSF CBMS Lecture Notes 38, (1978). Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

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

[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. Google Scholar

[50]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems,, University Press, (1988). Google Scholar

[51]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, University Press, (1998). Google Scholar

[52]

S. Karlin, Gene frequency patterns in the Levene subdivided population model,, Theor. Popul. Biol., 11 (1977), 356. doi: 10.1016/0040-5809(77)90018-1. Google Scholar

[53]

S. Karlin, Classification of selection-migration structures and conditions for a protected polymorphism,, Evol. Biol., 14 (1982), 61. Google Scholar

[54]

S. Karlin and R. B. Campbell, Selection-migration regimes characterized by a globally stable equilibrium,, Genetics, 94 (1980), 1065. Google Scholar

[55]

S. Karlin and M. W. Feldman, Simultaneous stability of $D=0$ and $D\ne0$ for multiplicative viabilities at two loci,, Genetics, 90 (1978), 813. Google Scholar

[56]

S. Karlin and J. McGregor, Application of method of small parameters to multi-niche population genetics models,, Theor. Popul. Biol., 3 (1972), 186. doi: 10.1016/0040-5809(72)90026-3. Google Scholar

[57]

S. Karlin and J. McGregor, Polymorphism for genetic and ecological systems with weak coupling,, Theor. Popul. Biol., 3 (1972), 210. doi: 10.1016/0040-5809(72)90027-5. Google Scholar

[58]

J. F. C. Kingman, An inequality in partial averages,, Quart. J. Math., 12 (1961), 78. doi: 10.1093/qmath/12.1.78. Google Scholar

[59]

A. Kolmogoroff, I. Pretrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantite de matiére et son application à un problème biologique,, Bull. Univ. Etat Moscou, 1 (1937), 1. Google Scholar

[60]

J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conf. Ser. Appl. Math. 25, (1976). Google Scholar

[61]

H. Levene, Genetic equilibrium when more than one ecological niche is available,, Amer. Natur., 87 (1953), 331. doi: 10.1086/281792. Google Scholar

[62]

S. Lessard, Fisher's fundamental theorem of natural selection revisited,, Theor. Pop. Biol., 52 (1997), 119. doi: 10.1006/tpbi.1997.1324. Google Scholar

[63]

R. C. Lewontin and K.-I. Kojima, The evolutionary dynamics of complex polymorphisms,, Evolution, 14 (1969), 458. doi: 10.2307/2405995. Google Scholar

[64]

C. C. Li, The stability of an equilibrium and the average fitness of a population,, Amer. Natur., 89 (1955), 281. doi: 10.1086/281893. Google Scholar

[65]

C. C. Li, Fundamental theorem of natural selection,, Nature, 214 (1967), 505. doi: 10.1038/214505a0. Google Scholar

[66]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics,, J. Diff. Eqs., 181 (2002), 388. doi: 10.1006/jdeq.2001.4086. Google Scholar

[67]

Y. Lou, T. Nagylaki and W.-M. Ni, An introduction to migration-selection PDE models,, Disc. Cont. Dyn. Syst. A, 33 (2013), 4349. doi: 10.3934/dcds.2013.33.4349. Google Scholar

[68]

Yu. I. Lyubich, Basic concepts and theorems of evolutionary genetics of free populations,, Russ. Math. Surv., 26 (1971), 51. doi: 10.1070/RM1971v026n05ABEH003829. Google Scholar

[69]

Yu. I. Lyubich, Mathematical Structures in Population Genetics,, Springer, (1992). doi: 10.1007/978-3-642-76211-6. Google Scholar

[70]

J. Maynard Smith, Genetic polymorphism in a varied environment,, Amer. Natur., 104 (1970), 487. doi: 10.1086/282683. Google Scholar

[71]

T. Nagylaki, Selection in One- and Two-Locus Systems,, Lect. Notes Biomath. 15, (1977). Google Scholar

[72]

T. Nagylaki, The diffusion model for migration and selection,, in Some Mathematical Questions in Biology (ed. A. Hastings), (1989), 55. Google Scholar

[73]

T. Nagylaki, Introduction to Theoretical Population Genetics,, Berlin, (1992). doi: 10.1007/978-3-642-76214-7. Google Scholar

[74]

T. Nagylaki, The evolution of multilocus systems under weak selection,, Genetics, 134 (1993), 627. Google Scholar

[75]

T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance,, Theor. Popul. Biol., 75 (2009), 239. doi: 10.1016/j.tpb.2009.01.004. Google Scholar

[76]

T. Nagylaki, Evolution under the multilocus Levene model without epistasis,, Theor. Popul. Biol., 76 (2009), 197. doi: 10.1016/j.tpb.2009.07.003. Google Scholar

[77]

T. Nagylaki, J. Hofbauer and P. Brunovský, Convergence of multilocus systems under weak epistasis or weak selection,, J. Math. Biol., 38 (1999), 103. doi: 10.1007/s002850050143. Google Scholar

[78]

T. Nagylaki and Y. Lou, Patterns of multiallelic poylmorphism maintained by migration and selection,, Theor. Popul. Biol., 59 (2001), 297. Google Scholar

[79]

T. Nagylaki and Y. Lou, Multiallelic selection polymorphism,, Theor. Popul. Biol., 69 (2006), 217. doi: 10.1016/j.tpb.2005.09.003. Google Scholar

[80]

T. Nagylaki and Y. Lou, Evolution under the multiallelic Levene model,, Theor. Popul. Biol., 70 (2006), 401. doi: 10.1016/j.tpb.2006.03.002. Google Scholar

[81]

T. Nagylaki and Y. Lou, Evolution under multiallelic migration-selection models,, Theor. Popul. Biol., 72 (2007), 21. doi: 10.1016/j.tpb.2007.02.005. Google Scholar

[82]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models,, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), (2008), 119. doi: 10.1007/978-3-540-74331-6_4. Google Scholar

[83]

A. Novak, The number of equilibria in the diallelic Levene model with multiple demes,, Theor. Popul. Biol., 79 (2011), 97. doi: 10.1016/j.tpb.2010.12.002. Google Scholar

[84]

S. Peischl, Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes,, Theor. Popul. Biol., 78 (2010), 12. doi: 10.1016/j.tpb.2010.03.006. Google Scholar

[85]

G.R. Price, Selection and covariance,, Nature, 227 (1970), 520. doi: 10.1038/227520a0. Google Scholar

[86]

T. Prout, Sufficient conditions for multiple niche polymorphism,, Amer. Natur., 102 (1968), 493. doi: 10.1086/282562. Google Scholar

[87]

W.B. Provine, The Origins of Theoretical Population Genetics,, Chicago Univ. Press, (1971). Google Scholar

[88]

D. Roze and F. Rousset, Multilocus models in the infinite island model of population structure,, Theor. Popul. Biol., 73 (): 529. doi: 10.1016/j.tpb.2008.03.002. Google Scholar

[89]

D. Rutschman, Dynamics of the two-locus haploid model,, Theor. Popul. Biol., 45 (1994), 167. doi: 10.1006/tpbi.1994.1009. Google Scholar

[90]

E. Seneta, Non-negative Matrices,, 2nd ed., (1981). doi: 10.1007/0-387-32792-4_6. Google Scholar

[91]

S. Shahshahani, A new mathematica framework for the study of linkage and selection,, Memoirs Amer. Math. Soc., 211 (1979). doi: 10.1090/memo/0211. Google Scholar

[92]

M. Spichtig and T. J. Kawecki, The maintenance (or not) of polygenic variation by soft selection in a heterogeneous environment,, Amer. Natur., 164 (2004), 70. doi: 10.1086/421335. Google Scholar

[93]

B. Star, R. J. Stoffels and H. G. Spencer, Single-locus polymorphism in a heterogeneous two-deme model,, Genetics, 176 (2007), 1625. doi: 10.1534/genetics.107.071639. Google Scholar

[94]

B. Star, R.J. Stoffels, and H.G. Spencer, Evolution of fitnesses and allele frequencies in a population with spatially heterogeneous selection pressures,, Genetics, 177 (2007), 1743. doi: 10.1534/genetics.107.079558. Google Scholar

[95]

C. Strobeck, Haploid selection with $n$ alleles in $m$ niches., Amer. Natur., 113 (1979), 439. doi: 10.1086/283401. Google Scholar

[96]

Yu. M. Svirezhev, Optimality principles in population genetics,, in Studies in Theoretical Genetics (in Russian), (1972), 86. Google Scholar

[97]

G. S. van Doorn and U. Dieckmann, The long-term evolution of multilocus traits under frequency-dependent disruptive selection,, Evolution, 60 (2006), 2226. doi: 10.1111/j.0014-3820.2006.tb01860.x. Google Scholar

[98]

J. Wakeley, Coalescent Theory: An Introduction,, Roberts & Company Publishers, (2008). Google Scholar

[99]

W. Weinberg, Über den Nachweis der Vererbung beim Menschen,, Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg, 64 (1908), 368. Google Scholar

[100]

W. Weinberg, Über Vererbungsgesetze beim Menschen,, Zeitschrift für induktive Abstammungs- und Vererbungslehre, 1 (1909), 377. doi: 10.1007/BF01975801. Google Scholar

[101]

T. Wiehe and M. Slatkin, Epistatic selection in a multi-locus Levene model and implications for linkage disequilibrium,, Theor. Popul. Biol., 53 (1998), 75. doi: 10.1006/tpbi.1997.1342. Google Scholar

[102]

S. Wright, Evolution in Mendelian populations,, Genetics, 16 (1931), 97. doi: 10.1016/S0092-8240(05)80011-4. Google Scholar

[103]

G. U. Yule, Mendel's laws and their probable relations to intra-racial heredity,, New Phytol., 1 (1902), 193. doi: 10.1111/j.1469-8137.1902.tb06590.x. Google Scholar

[104]

L. A. Zhivotovsky, M. W. Feldman and A. Bergman, On the evolution of phenotypic plasticity in a spatially heterogeneous environment,, Evolution, 50 (1996), 547. doi: 10.2307/2410830. Google Scholar

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