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Journal of Mathematical Biology

, Volume 78, Issue 4, pp 1147–1210 | Cite as

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

  • Benjamin AllenEmail author
  • Alex McAvoy
Article

Abstract

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Keywords

Evolution Population genetics Fixation probability Spatial structure Weak selection Markov chain 

Mathematics Subject Classification

91A22 92D15 

Notes

Acknowledgements

BA is supported by National Science Foundation Award #DMS-1715315. AM is supported by the Office of Naval Research, Grant N00014-16-1-2914. We thank Martin A. Nowak for helpful discussions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsEmmanuel CollegeBostonUSA
  2. 2.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA

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