Abstract
In evolutionary dynamics, a key measure of a mutant trait’s success is the probability that it takes over the population given some initial mutant-appearance distribution. This “fixation probability” is difficult to compute in general, as it depends on the mutation’s effect on the organism as well as the population’s spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation’s effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant’s fixation probability, from an arbitrary initial configuration of mutant and resident types. Our results apply to a broad class of stochastic evolutionary models, in which the size and spatial structure are arbitrary (but fixed). The problem of whether selection favors a given trait is thereby reduced from exponential to polynomial complexity in the population size, when selection is weak. We conclude by applying these methods to obtain new results for evolutionary dynamics on graphs.
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Acknowledgements
We thank Krishnendu Chatterjee, Joshua Plotkin, Qi Su, and John Wakeley for helpful discussions and comments on earlier drafts. We are also grateful to the anonymous referees for many valuable suggestions. This work was supported by the Army Research Laboratory (grant W911NF-18-2-0265), the John Templeton Foundation (grant 61443), the National Science Foundation (grant DMS-1715315), the Office of Naval Research (grant N00014-16-1-2914), and the Simons Foundation (Math+X Grant to the University of Pennsylvania).
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McAvoy, A., Allen, B. Fixation probabilities in evolutionary dynamics under weak selection. J. Math. Biol. 82, 14 (2021). https://doi.org/10.1007/s00285-021-01568-4
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DOI: https://doi.org/10.1007/s00285-021-01568-4