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Journal of Mathematical Biology

, Volume 76, Issue 1–2, pp 1–35 | Cite as

Moran-type bounds for the fixation probability in a frequency-dependent Wright–Fisher model

  • Timothy Chumley
  • Ozgur Aydogmus
  • Anastasios Matzavinos
  • Alexander Roitershtein
Article

Abstract

We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright–Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite population with a positive probability. We also give asymptotically sharp bounds for the fixation time distribution.

Keywords

Evolutionary game dynamics Stochastic dynamics Finite populations Strong selection 

Mathematics Subject Classification

60J10 60J85 91A22 92D25 

Notes

Acknowledgements

The work of T.C. was partially supported by the Alliance for Diversity in Mathematical Sciences Postdoctoral Fellowship. O. A. thanks the Department of Mathematics at Iowa State University for its hospitality during a visit in which part of this work was carried out. A.M. would like to thank the Computational Science and Engineering Laboratory at ETH Zürich for the warm hospitality during a sabbatical semester. The research of A.M. is supported in part by the National Science Foundation under Grants NSF CDS&E-MSS 1521266 and NSF CAREER 1552903.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMount Holyoke CollegeSouth HadleyUSA
  2. 2.Department of EconomicsSocial Sciences University of AnkaraAnkaraTurkey
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA
  4. 4.Computational Science and Engineering LaboratoryETH ZürichZurichSwitzerland
  5. 5.Department of MathematicsIowa State UniversityAmesUSA

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