[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akin E (1979) The geometry of population genetics. Lecture Notes in Biomath, vol 31. Springer, Berlin

    Google Scholar 

  2. Akin E (1982) Cycling in simple genetic systems. J Math Biol 13: 305–24

    Article  MATH  MathSciNet  Google Scholar 

  3. Akin E (1993) The general topology of dynamical systems. American Mathematical Society, Providence

  4. Barton NH (1999) Clines in polygenic traits. Genet Res 74: 223–36

    Article  Google Scholar 

  5. Bennett JH (1954) On the theory of random mating. Ann Eugen 18: 311–17

    Google Scholar 

  6. Bulmer MG (1972) Multiple niche polymorphism. Am Nat 106: 254–57

    Article  Google Scholar 

  7. Bürger R (2000) The mathematical theory of selection, recombination, and mutation. Wiley, Chichester

    MATH  Google Scholar 

  8. Cannings C (1971) Natural selection at a multiallelic autosomal locus with multiple niches. J Genet 60: 255–59

    Article  Google Scholar 

  9. Christiansen FB (1975) Hard and soft selection in a subdivided population. Am Nat 109: 11–6

    Article  Google Scholar 

  10. Christiansen FB (1999) Population genetics of multiple loci. Wiley, Chichester

    MATH  Google Scholar 

  11. Christiansen FB, Feldman M (1975) Subdivided populations: a review of the one- and two-locus deterministic theory. Theor Popul Biol 7: 13–8

    Article  MathSciNet  Google Scholar 

  12. Conley C (1978) Isolated invariant sets and the Morse index. NSF CBMS Lecture Notes, vol 38. American Mathematical Society, Providence

  13. Deakin MAB (1966) Sufficient conditions for genetic polymorphism. Am Nat 100: 690–92

    Article  Google Scholar 

  14. Dempster ER (1955) Maintenance of genetic heterogeneity. Cold Spring Harbor Symp Quant Biol 20: 25–2

    Google Scholar 

  15. Deuflhard P, Bornemann F (1995) Numerical mathematics II. Integration of ordinary differential equations. de Gruyter, Berlin

    Google Scholar 

  16. Ewens WJ (1969) Mean fitness increases when fitnesses are additive. Nature 221: 1076

    Article  Google Scholar 

  17. Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York

    Google Scholar 

  18. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Ind Univ Math J 21: 193–26

    Article  MATH  MathSciNet  Google Scholar 

  19. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7: 355–69

    Google Scholar 

  20. Gantmacher FR (1959) The theory of matrices, vol II. Chelsea, New York

    Google Scholar 

  21. Garay BM (1993) Discretization and some qualitative properties of ordinary differential equations about equilibria. Acta Math Univ Comenianae 62: 249–75

    MATH  MathSciNet  Google Scholar 

  22. Garay BM, Hofbauer J (1997) Chain recurrence and discretization. Bull Austral Math Soc 55: 63–1

    Article  MATH  MathSciNet  Google Scholar 

  23. Geiringer H (1944) On the probability theory of linkage in Mendelian heredity. Ann Math Stat 15: 25–7

    Article  MATH  MathSciNet  Google Scholar 

  24. Haldane JBS (1930) A mathematical theory of natural and artificial selection Part VI. Isolation. Proc Camb Phil Soc 28: 224–48

    Google Scholar 

  25. Haldane JBS (1948) The theory of a cline. J Genet 48: 277–84

    Article  Google Scholar 

  26. Hastings A (1981) Stable cycling in discrete-time genetic models. Proc Natl Acad Sci USA 78: 7224–225

    Article  MATH  MathSciNet  Google Scholar 

  27. Hirsch MW, Pugh C, Shub M (1977) Invariant Manifolds, Lecture Notes in Mathematics, vol 583. Springer, Berlin

    Google Scholar 

  28. Hofbauer J, Iooss G (1984) A Hopf bifurcation theorem of difference equations approximating a differential equation. Monatsh Math 98: 99–13

    Article  MathSciNet  Google Scholar 

  29. Karlin S (1977) Gene frequency patterns in the Levene subdivided population model. Theor Popul Biol 11: 356–85

    Article  MathSciNet  Google Scholar 

  30. Karlin S (1982) Classification of selection-migration structures and conditions for a protected polymorphism. Evol Biol 14: 61–04

    Google Scholar 

  31. Karlin S, Campbell RB (1980) Selection-migration regimes characterized by a globally stable equilibrium. Genetics 94: 1065–084

    MathSciNet  Google Scholar 

  32. Karlin S, McGregor J (1972a) Application of method of small parameters to multi-niche population genetics models. Theor Popul Biol 3: 186–08

    Article  MathSciNet  Google Scholar 

  33. Karlin S, McGregor J (1972b) Polymorphism for genetic and ecological systems with weak coupling. Theor Popul Biol 3: 210–38

    Article  MathSciNet  Google Scholar 

  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

  35. Kun LA, Lyubich YuI (1980) Convergence to equilibrium in a polylocus polyallele population with additive selection. Probl Inform Transmiss 16: 152–61

    MATH  MathSciNet  Google Scholar 

  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

    Google Scholar 

  37. Levene H (1953) Genetic equilibrium when more than one ecological niche is available. Am Nat 87: 331–33

    Article  Google Scholar 

  38. Li CC (1955) The stability of an equilibrium and the average fitness of a population. Am Nat 89: 281–95

    Article  Google Scholar 

  39. Li W-H, Nei M (1974) Stable linkage disequilibrium without epistasis in subdivided populations. Theor Popul Biol 6: 173–83

    Article  MathSciNet  Google Scholar 

  40. Lyubich YuI (1971) Basic concepts and theorems of evolutionary genetics of free populations. Russ Math Surv 26: 51–23

    Article  MATH  Google Scholar 

  41. Lyubich YuI (1992) Mathematical structures in population genetics. Springer, Berlin

    MATH  Google Scholar 

  42. Nagylaki T (1992) Introduction to theoretical population genetics. Springer, Berlin

    MATH  Google Scholar 

  43. Nagylaki T (1993) The evolution of multilocus systems under weak selection. Genetics 134: 627–47

    Google Scholar 

  44. Nagylaki T (2008) Polymorphism in multiallelic migration-selection models with dominance (submitted)

  45. Nagylaki T, Hofbauer J, Brunovský P (1999) Convergence of multilocus systems under weak epistasis or weak selection. J Math Biol 38: 103–33

    Article  MATH  MathSciNet  Google Scholar 

  46. Nagylaki T, Lou Y (2001) Patterns of multiallelic poylmorphism maintained by migration and selection. Theor Popul Biol 59: 297–13

    Article  MATH  Google Scholar 

  47. Nagylaki T, Lou Y (2006) Evolution under the multiallelic Levene model. Theor Popul Biol 70: 401–11

    Article  MATH  Google Scholar 

  48. Nagylaki T, Lou Y (2007) Evolution under multiallelic migration-selection models. Theor Popul Biol 72: 21–0

    Article  MATH  Google Scholar 

  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

  50. Prout T (1968) Sufficient conditions for multiple niche polymorphism. Am Nat 102: 493–96

    Article  Google Scholar 

  51. Reiersøl O (1962) Genetic algebras studied recursively and by means of differential operators. Math Scand 10: 25–4

    MathSciNet  Google Scholar 

  52. Seneta E (1981) Non-negative matrices and Markov Chains. Springer, New York

    MATH  Google Scholar 

  53. Slatkin M (1975) Gene flow and selection in two-locus systems. Genetics 81: 787–02

    Google Scholar 

  54. Wright S (1931) Evolution in Mendelian populations. Genetics 16: 97–59

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Vienna, Austria

    Reinhard Bürger

Authors
  1. Reinhard Bürger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reinhard Bürger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bürger, R. Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration. J. Math. Biol. 58, 939–978 (2009). https://doi.org/10.1007/s00285-008-0236-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-008-0236-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation