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Clines with complete dominance and partial panmixia in an unbounded unidimensional habitat

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  • Nagylaki, Thomas
  • Zeng, Kai

Abstract

In spatially structured populations, global panmixia can be viewed as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection with complete dominance in an unbounded unidimensional habitat is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and symmetric; the latter is frequency independent. The spatial factor g̃(x) in the selection term, where x denotes position, is a single step at the origin: g̃(x)=−α<0 if x<0, and g̃(x)=1 if x>0. If α=1, there exists a globally asymptotically stable cline. For α<1, such a cline exists if and only if the scaled panmictic rate β is less than the critical value β∗∗=2α/(1−α). For α>1, a unique, asymptotically stable cline exists if and only if β is less than the critical value β∗; then a smaller, unique, unstable equilibrium also exists whenever β<β∗. Two coupled, nonlinear polynomial equations uniquely determine β∗. Explicit solutions are derived for each of the above equilibria. If β>0 and a cline exists, some polymorphism is maintained even at x=±∞. Both the preceding result and the existence of an unstable equilibrium when α>1 and 0<β<β∗ differ qualitatively from the classical case (β=0).

Suggested Citation

  • 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.
  • Handle: RePEc:eee:thpobi:v:93:y:2014:i:c:p:63-74
    DOI: 10.1016/j.tpb.2014.01.003
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    References listed on IDEAS

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    1. Nagylaki, Thomas, 2012. "Clines with partial panmixia," Theoretical Population Biology, Elsevier, vol. 81(1), pages 45-68.
    2. Nagylaki, Thomas, 2009. "Polymorphism in multiallelic migration–selection models with dominance," Theoretical Population Biology, Elsevier, vol. 75(4), pages 239-259.
    3. Nagylaki, Thomas, 2012. "Clines with partial panmixia in an unbounded unidimensional habitat," Theoretical Population Biology, Elsevier, vol. 82(1), pages 22-28.
    4. 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 & 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.
    2. Nagylaki, Thomas, 2015. "Dying on the way: The influence of migrational mortality on clines," Theoretical Population Biology, Elsevier, vol. 101(C), pages 54-60.
    3. Nagylaki, Thomas, 2016. "Clines with partial panmixia across a geographical barrier," Theoretical Population Biology, Elsevier, vol. 109(C), pages 28-43.
    4. 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.

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