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On the eigenvalue effective size of structured populations.

Ola Hössjer1

  • 1Divsion of Mathematical Statistics, Department of Mathematics, Stockholm University, Stockholm, Sweden, ola@math.su.se.

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Summary

We developed a general theory for the eigenvalue effective size (N(e)E) in structured populations. This theory provides novel expressions for N(e)E and unifies existing results in population genetics.

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Area of Science:

  • Population Genetics
  • Mathematical Biology
  • Evolutionary Theory

Background:

  • Understanding population structure is crucial for evolutionary studies.
  • Effective population size (N(e)) is a key parameter in population genetics.
  • Previous models often simplified population structures or genetic dynamics.

Purpose of the Study:

  • To develop a general theory for the eigenvalue effective size (N(e)E) in structured populations.
  • To generalize existing results on effective population size.
  • To derive new explicit expressions for N(e)E.

Main Methods:

  • Developed a general theory for N(e)E in discrete-time, two-allele models.
  • Utilized the largest non-unit eigenvalue of the allele frequency Markov chain transition matrix.
  • Applied Perron-Frobenius Theorem and coalescence theory.
  • Analyzed asymptotic behavior for small population sizes and migration rates.

Main Results:

  • Characterized N(e)E using a specific eigenvalue of the transition matrix.
  • Derived explicit, novel expressions for N(e)E.
  • Demonstrated that previously known results are special cases of the new theory.
  • Showed that coalescence effective size (N(e)C) is an asymptotic version of N(e)E for large populations.

Conclusions:

  • The developed theory provides a unified framework for N(e)E in structured populations.
  • The new expressions for N(e)E offer deeper insights into population genetic dynamics.
  • The relationship between N(e)E and N(e)C is clarified in the asymptotic limit.