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Stationary distribution of a 2-island 2-allele Wright–Fisher diffusion model with slow mutation and migration rates

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  • Burden, Conrad J.
  • Griffiths, Robert C.

Abstract

The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright–Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the forward Kolmogorov equation, the stationary distribution is obtained to leading order as a set of line densities on the edges of the sample space, corresponding to states for which one island is bi-allelic and the other island is non-segregating, and a set of point masses at the corners of the sample space, corresponding to states for which both islands are simultaneously non-segregating. Analytic results for the corner probabilities and line densities are verified independently using the backward generator and for the corner probabilities using the coalescent.

Suggested Citation

  • Burden, Conrad J. & Griffiths, Robert C., 2018. "Stationary distribution of a 2-island 2-allele Wright–Fisher diffusion model with slow mutation and migration rates," Theoretical Population Biology, Elsevier, vol. 124(C), pages 70-80.
    • Handle: RePEc:eee:thpobi:v:124:y:2018:i:c:p:70-80
    DOI: 10.1016/j.tpb.2018.09.004
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    References listed on IDEAS

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    1. Burden, Conrad J. & Tang, Yurong, 2017. "Rate matrix estimation from site frequency data," Theoretical Population Biology, Elsevier, vol. 113(C), pages 23-33.
    2. Burden, Conrad J. & Tang, Yurong, 2016. "An approximate stationary solution for multi-allele neutral diffusion with low mutation rates," Theoretical Population Biology, Elsevier, vol. 112(C), pages 22-32.
    3. Schrempf, Dominik & Hobolth, Asger, 2017. "An alternative derivation of the stationary distribution of the multivariate neutral Wright–Fisher model for low mutation rates with a view to mutation rate estimation from site frequency data," Theoretical Population Biology, Elsevier, vol. 114(C), pages 88-94.
    4. Vogl, Claus & Bergman, Juraj, 2015. "Inference of directional selection and mutation parameters assuming equilibrium," Theoretical Population Biology, Elsevier, vol. 106(C), pages 71-82.
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