We consider genotypic convergence of populations and show that under fixed fitness asexual and haploid sexual populations attain monomorphic convergence (even under genetic linkage between loci) to basins of attraction with locally exponential convergence rates; the same convergence obtains in single locus diploid sexual reproduction but to polymorphic populations. Furthermore, we show that there is a unified theory underlying these convergences: all of them can be interpreted as instantiations of players in a potential game implementing a multiplicative weights updating algorithm to converge to equilibrium, making use of the Baum-Eagon Theorem. To analyse varying environments, we introduce the concept of 'virtual convergence', under which, even if fixation is not attained, the population nevertheless achieves the fitness growth rate it would have had under convergence to an optimal genotype. Virtual convergence is attained by asexual, haploid sexual and multi-locus diploid reproducing populations, even if environments vary arbitrarily. We also study conditions for true monomorphic convergence in asexually reproducing populations in varying environments.
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Data accessibility. We have supplied the code for the simulation generating figure 1. Authors’ contributions. All the authors contributed to the design of the model and the discussion of the results. O.E. and Z.H. conceived and proved the mathematical results. I.N. contributed to the simulations illustrating the mathematical results (see figure 1). Competing interests. We declare we have no competing interests. Funding. This work was partially supported by Israel Science Foundation grant no. 1626/18. Acknowledgements. The authors thank Daniel B. Weissman for suggesting corrections to an earlier version of this paper, as well as two anonymous reviewers.
© 2021 The Authors.
- Evolutionary dynamics
- Population genetics
- Potential games
- Sexual reproduction
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