The strongmigration limit in geographically structured populations
 Thomas Nagylaki
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Summary
Some strongmigration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number N _{ e } (⩽ N_{ T }) appears instead of the actual total population number N _{ T }. 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers N _{ i } are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the N _{ i } are proportional to 1/u, as N _{ T }→∼8, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection coefficients proportional to 1/N _{ T }, let P _{ j } be the frequency of the allele A _{ j } in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As N _{ T } → ∼8, the deviations of the allelic frequencies in each of the subpopulations from P _{ j } converge to zero; the usual panmictic mutationselection diffusion is obtained for P _{ j }, with the selection intensities averaged with respect to the stationary distribution of M. In both models, N _{ e } = N _{ T } and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.
 Dempster, E. R.: Maintenance of genetic heterogeneity. Cold Spring Harbor Symp. Quant.Biol.20, 25–32 (1955)
 Ethier, S. N., Nagylaki, T.: Diffusion approximations of Markov chains with two time scales and applications to population genetics. Adv. Appl. Prob., in press (1980)
 Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edition, Vol. I. New York: Wiley, 1968
 Franklin, J. N.: Matrix Theory. Englewood Cliffs, N.J.: PrenticeHall, 1968
 Gantmacher, F. R.: The Theory of Matrices, 2 vol. New York: Chelsea, 1959
 Karlin, S., Taylor, H. M.: A First Course in Stochastic Processes, 2nd edition. New York:Academic Press, 1975
 Kimura, M.: Diffusion models in population genetics. J. Appl. Prob, 1, 177–232 (1964)
 Kimura, M., Crow, J. F.: The number of alleles that can be maintained in a finite populationGenetics 49, 725–738 (1964)
 Malécot, G.: Les mathématiques de l'hérédité. Paris: Masson 1948. Extended translation Malécot, G.: The Mathematics of Heredity. San Francisco: Freeman, 1969
 Malécot, G.: Un traitement stochastique des problèmes linéaires (mutation, linkage, migration)en Génétique de Population. Ann. Univ. Lyon, Sciences, Sec. A 14, 79–117 (1951)
 Nagylaki, T.: Selection in One and TwoLocus Systems. Berlin: SpringerVerlag, 1977
 Nagylaki, T.: Decay of genetic variability in geographically structured populations. Proc. Natl.Acad. Sci. USA 74, 2523–2525 (1977a)
 Nagylaki, T.: A diffusion model for geographically structured populations. J. Math. Biol. 6, 375–382 (1978)
 Sawyer, S.: Results for the steppingstone model for migration in population genetics. Ann.Prob. 4, 699–728 (1976)
 Wallace, B.: Topics in Population Genetics. New York: Norton 1968
 Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)
 Title
 The strongmigration limit in geographically structured populations
 Journal

Journal of Mathematical Biology
Volume 9, Issue 2 , pp 101114
 Cover Date
 19800401
 DOI
 10.1007/BF00275916
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Migration
 Random drift
 Geographical structure
 Markov chains
 Limit theorems
 Industry Sectors
 Authors

 Thomas Nagylaki ^{(1)}
 Author Affiliations

 1. Department of Biophysics and Theoretical Biology, University of Chicago, 920 East 58th Street, 60637, Chicago, IL, USA