Skip to main content
SpringerLink
Log in

Spatial Autocorrelation for Subdivided Populations with Invariant Migration Schemes

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

For populations with geographic substructure and selectively neutral genetic data, the short term dynamics is a balance between migration and genetic drift. Before fixation of any allele, the system enters into a quasi equilibrium (QE) state. Hössjer and Ryman (2012) developed a general QE methodology for computing approximations of spatial autocorrelations of allele frequencies between subpopulations, subpopulation differentiation (fixation indexes) and variance effective population sizes. In this paper we treat a class of models with translationally invariant migration and use Fourier transforms for computing these quantities. We show how the QE approach is related to other methods based on conditional kinship coefficients between subpopulations under mutation-migration-drift equilibrium. We also verify that QE autocorrelations of allele frequencies are closely related to the expected value of Moran’s autocorrelation function and treat limits of continuous spatial location (isolation by distance) and an infinite lattice of subpopulations. The theory is illustrated with several examples including island models, circular and torus stepping stone models, von Mises models, hierarchical island models and Gaussian models. It is well known that the fixation index contains information about the effective number of migrants. The spatial autocorrelations are complementary and typically reveal the type of migration (local or global).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions. With formulas, graphs, and mathematical tables. Tenth printing, U.S. Government Printing Office Washington, DC

  • Barbujani, G (1987) Autocorrelation of gene frequencies under isolation by distance. Genetics 117:777–782

    Google Scholar 

  • Barton NH, Depaulis F, Etheridge AM (2002) Neutral evolution in spatially continuous populations. Theor Popul Biol 61:31–48

    Article  MATH  Google Scholar 

  • Carmelli D, Cavalli-Sforza LL (1976) Some models of population structure and evolution. Theor Popul Biol 9:329–359

    Article  MathSciNet  MATH  Google Scholar 

  • Collet P, Martinez S, San Martin J (2013) Quasi stationary distributions, markov chains, diffusions and dynamical systems. Springer, Berlin

    Book  MATH  Google Scholar 

  • Cox DR, Miller HD (1965). The theory of stochastic processes. Methuen & Co Ltd, London

    MATH  Google Scholar 

  • Cox JT, Durrett R (2002) The stepping stone model: new formulas expose old myths. Ann Appl Probab 12(4):1348–1377

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett R (2008) Probability models for DNA sequence evolution, 2nd ed. Springer, New York

    Book  MATH  Google Scholar 

  • Durrett R, Restrepo M (2008) One-dimensional stepping stone modes, sardine genetics and Brownian local time. Ann Appl Probab 18(1):334–358

    Article  MathSciNet  MATH  Google Scholar 

  • Ewens WJ (2004) Mathematical population genetics I. Theoretical introduction, 2nd ed. Springer, New York

    Book  MATH  Google Scholar 

  • Felsenstein J (1971) Inbreeding and variance effective numbers in populations with overlapping generations. Genetics 68:581–597

    MathSciNet  Google Scholar 

  • Hare MP, Nunney L, Schwartz MK, Ruzzante DE, Burford M, Waples R, Ruegg K, Palstra F (2011) Understanding and estimating effective population size for practical applications in marine species management. Conserv. Biol. 25(3):438–449

    Article  Google Scholar 

  • Hardy OJ, Vekemans X (1999) Isolation by distance in a continuous population: reconciliation between spatial autocorrelation analysis and population genetics models. Heredity 83:145–154

    Article  Google Scholar 

  • Hardy OJ, Vekemans X (2002) SPAGeDI: a versatile computer program to analyse spatial genetic structure at the individual or population model. Mol Ecol Notes 2:618–620

    Article  Google Scholar 

  • Hössjer O, Jorde PE, Ryman N (2013) Quasi equilibrium approximations of the fixation index of the island model under neutrality. Theor Popul Biol 84:9–24

    Article  MATH  Google Scholar 

  • Hössjer O, Ryman N (2012) Quasi equilibrium, variance effective population size and fixation index for models with spatial structure. Report 2012:4, Mathematical Statistics, Stockholm University, Oct 2012

  • Kimura M (1953) Stepping stone model of population. Ann Rep Natl Inst Genet Japan 3:62–63

    Google Scholar 

  • Kimura M, Weiss GH (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 61:763–771

    Google Scholar 

  • Latter BDH (1973) The island model of population differentiation: a general solution. Genetics 73:147–157

    MathSciNet  Google Scholar 

  • Latter BDH, Sved JA (1981) Migration and mutation in stochastic models of gene frequency change. II. Stochastic migration with a finite number of islands. J Math Biol 13:95–104

    Article  MathSciNet  MATH  Google Scholar 

  • Malécot G (1948) Les Mathématiques de l’Hérédité. Masson et Cie, Paris

    MATH  Google Scholar 

  • Malécot G (1950) Quelques schémas probabilistes sur la variabilité des populations naturelles. Annales de l’Université de Lyon A 13:37–60

    Google Scholar 

  • Malécot G (1951) Un treatment stochastique des problemès linairés (mutation, linkage, migration) en géneétique de populations. Annales de l’Université de Lyon A 14:79–117

    Google Scholar 

  • Maruyama T (1970) Effective number of alleles in subdivided populations. Theor Popul Biol 1:273–306

    Article  MathSciNet  MATH  Google Scholar 

  • Maruyama T (1972) Rate of decrease of genetic variability in a two-dimensional continuous population of finite size. Genetics 70:639–651

    MathSciNet  Google Scholar 

  • Moran PAP (1950) Notes on continuous stochastic phenomena. Biometrika 37(1):17–23

    Article  MathSciNet  MATH  Google Scholar 

  • Morton NE (1973) Kinship and population structure. In: Morton NE (ed) Genetic structure of populations. University of Hawaii Press, Honolulu, pp 66–69

    Google Scholar 

  • Nagylaki T (1976) The decay of genetic variability in geographically structured populations. II. Theor Popul Biol 10:70–82

    Article  MathSciNet  MATH  Google Scholar 

  • Nei M (1973) Genetic distance between populations. Am Nat 106:283–292

    Article  Google Scholar 

  • Nei M (1977) Molecular evolution and population genetics. North-Holland Publishing Company, Amsterdam

    Google Scholar 

  • Nei M, Chakravarti A, Tateng Y (1977) Mean and variance of F ST in a finite number of incompletely isolated populations. Theor Popul Biol 11:291–306

    Article  MATH  Google Scholar 

  • Palstra FP, Ruzzante DE (2008) Genetic estimates of contemporary effective population size: what can they tell us about the importance of genetic stochasticity for wild populations persistence? Mol Ecol 17:3428–3447

    Article  Google Scholar 

  • Rohlf FJ, Schnell GD (1971) An investigation of the isolation-by-distance model. Am Nat 105:295–324

    Article  Google Scholar 

  • Rousset F (1997) Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance.

  • Rousset F (2000) Genetic differentiation between individuals. J Evol Biol 13:58–62

    Article  Google Scholar 

  • Rousset F (2001) Inferences from spatial population genetics. In: Balding BJ, Bishop M, Cannings C (eds) Handbook of statistical genetics, pp 239–269

  • Rousset F (2002) Inbreeding and relatedness coefficients: what do they measure? Heredity 88:371–380

    Article  Google Scholar 

  • Sawyer S (1976) Results for the stepping stone model for migration in population genetics. Ann Probab 4:699–728

    Article  MathSciNet  MATH  Google Scholar 

  • Sawyer S (1977) Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv Appl Probab 9:268–282

    Article  MathSciNet  MATH  Google Scholar 

  • Sawyer S, Felsenstein J (1983) Isolation by distance in a hierarchically clustered population. J Appl Probab 20:1–10

    Article  MathSciNet  MATH  Google Scholar 

  • Slatkin M (1991) Inbreeding coefficients and coalescent times. Genet Res 58(2):167–175

    Article  Google Scholar 

  • Slatkin M, Arter HE (1991) Spatial autocorrelation methods in population genetics. Am Nat 138(2):499–517

    Article  Google Scholar 

  • Slatkin M, Voelm L (1991) F ST in a hierarchical model. Genetics 127:627–629

    Google Scholar 

  • Smouse PE, Peakall R (1999) Spatial autocorrelation analysis of individual allele and multilocus genetic structure. Heredity 82:561–573

    Article  Google Scholar 

  • Sokal RR, Jacques MJ, Wooten MC (1989) Spatial autocorrelation analysis of migration and selection. Genetics 121:845–855

    Google Scholar 

  • Sokal RR, Oden NL (1978) Spatial autocorrelation in biology. Biol J Linn Soc 10:199–228

    Article  Google Scholar 

  • Sokal RR, Oden NL, Thomson BA (1997) A simulation study of microevolutionary inferences by spatial autocorrelation analysis. Biol J Linn Soc 60:73–93

    Article  Google Scholar 

  • Sokal RR, Oden NL, Thomson BA (1998) Local spatial autocorrelation in biological variables. Biol J Linn Soc 65:41–62

    Article  Google Scholar 

  • Sved JA, Latter BDH (1977) Migration and mutation in stochastic models of gene frequency change. J Math Biol 5:61–73

    Article  MathSciNet  MATH  Google Scholar 

  • Takahata N (1983) Gene identity and genetic differentiation of populations in the finite island model. Genetics 104:497–512

    Google Scholar 

  • Takahata N, Nei M (1984) F ST and G ST statistics in the finite island model. Genetics 107:501–504

    Google Scholar 

  • Vekemans X, Hardy OJ (2004) New insights from fine-scale genetic structure analyses in plant populations. Mol Ecol 13:921–935

    Article  Google Scholar 

  • Wang J, Caballero A (1999) Developments in predicting the effective size of subdivided populations. Heredity 82:212–226

    Article  Google Scholar 

  • Waples RS, Yokota M (2007) Temporal estimates of effective population size in species with overlapping generations. Genetics 175:219–233

    Article  Google Scholar 

  • Weiss GH, Kimura M (1965) A mathematical analysis of the stepping stone model of genetic correlation. J Appl Probab 2:129–149

    Article  MathSciNet  MATH  Google Scholar 

  • Wright S (1931) Evolution in Mendelian populations. Genetics 16:97–159

    Google Scholar 

  • Wright S (1938) Size of population and breeding structure in relation to evolution. Science 87:430–431

    Google Scholar 

  • Wright S (1943) Isolation by distance. Genetics 28:114–138

    Google Scholar 

  • Wright S (1946) Isolation by distance under diverse systems of mating. Genetics 31:39–59

    Google Scholar 

  • Wright S (1951) The genetical structure of populations. Ann Eugen 15:323–354

    Article  Google Scholar 

  • Zhao R, Xia H, Lu B-R (2009) Fine-scale genetic structure enhances biparental inbreeding by promoting mating events between more related individuals in wild soybean populations. Am J Bot 96(6):1138–1147

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Division of Mathematical Statistics, Stockholm University, SE 106 91, Stockholm, Sweden

    Ola Hössjer

Authors
  1. Ola Hössjer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ola Hössjer.

Additional information

Supported by the Swedish Research Council, contract nr. 621-2008-4946, and the Gustafsson Foundation for Research in Natural Sciences and Medicine.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hössjer, O. Spatial Autocorrelation for Subdivided Populations with Invariant Migration Schemes. Methodol Comput Appl Probab 16, 777–810 (2014). https://doi.org/10.1007/s11009-013-9321-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9321-3

Keywords

AMS 2000 Subject Classification

Search

Navigation