Journal of Mathematical Biology
, Volume 9, Issue 2, pp 101114
First online:
The strongmigration limit in geographically structured populations
 Thomas NagylakiAffiliated withDepartment of Biophysics and Theoretical Biology, University of Chicago
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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.
Key words
Migration Random drift Geographical structure Markov chains Limit theorems Title
 The strongmigration limit in geographically structured populations
 Journal

Journal of Mathematical Biology
Volume 9, Issue 2 , pp 101114
 Cover Date
 198004
 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
 Authors

 Thomas Nagylaki ^{(1)}
 Author Affiliations

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