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Clines with partial panmixia

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  • Nagylaki, Thomas

Abstract

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ0 (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ0 is an increasing function of the panmictic rate and to deduce the effect on λ0 of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ0; bounds on λ0 are established; and λ0 is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.

Suggested Citation

  • Nagylaki, Thomas, 2012. "Clines with partial panmixia," Theoretical Population Biology, Elsevier, vol. 81(1), pages 45-68.
  • Handle: RePEc:eee:thpobi:v:81:y:2012:i:1:p:45-68
    DOI: 10.1016/j.tpb.2011.09.006
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    References listed on IDEAS

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    1. Nagylaki, Thomas, 2009. "Polymorphism in multiallelic migration–selection models with dominance," Theoretical Population Biology, Elsevier, vol. 75(4), pages 239-259.
    2. Peischl, Stephan, 2010. "Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes," Theoretical Population Biology, Elsevier, vol. 78(1), pages 12-25.
    3. Nagylaki, Thomas, 2011. "The influence of partial panmixia on neutral models of spatial variation," Theoretical Population Biology, Elsevier, vol. 79(1), pages 19-38.
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    Citations

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    Cited by:

    1. Nagylaki, Thomas, 2016. "Clines with partial panmixia across a geographical barrier," Theoretical Population Biology, Elsevier, vol. 109(C), pages 28-43.
    2. Nagylaki, Thomas & Zeng, Kai, 2014. "Clines with complete dominance and partial panmixia in an unbounded unidimensional habitat," Theoretical Population Biology, Elsevier, vol. 93(C), pages 63-74.
    3. Nagylaki, Thomas & Zeng, Kai, 2016. "Clines with partial panmixia across a geographical barrier in an environmental pocket," Theoretical Population Biology, Elsevier, vol. 110(C), pages 1-11.
    4. Nagylaki, Thomas, 2012. "Clines with partial panmixia in an unbounded unidimensional habitat," Theoretical Population Biology, Elsevier, vol. 82(1), pages 22-28.
    5. Nagylaki, Thomas & Su, Linlin & Dupont, Todd F., 2019. "Uniqueness and multiplicity of clines in an environmental pocket," Theoretical Population Biology, Elsevier, vol. 130(C), pages 106-131.
    6. Geroldinger, Ludwig & Bürger, Reinhard, 2015. "Clines in quantitative traits: The role of migration patterns and selection scenarios," Theoretical Population Biology, Elsevier, vol. 99(C), pages 43-66.
    7. Nagylaki, Thomas & Su, Linlin & Alevy, Ian & Dupont, Todd F., 2014. "Clines with partial panmixia in an environmental pocket," Theoretical Population Biology, Elsevier, vol. 95(C), pages 24-32.
    8. Forien, Raphaël, 2019. "Gene flow across geographical barriers — scaling limits of random walks with obstacles," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3748-3773.

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