2020/03/06 by Loren Coquille, Coquille, Loren, Anna Kraut +3
Biochemistry, Genetics and Molecular Biology · Social Sciences · Medicine · #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.2003.03452
We consider a stochastic individual-based model for the evolution of a\nhaploid, asexually reproducing population. The space of possible traits is\ngiven by the vertices of a (possibly directed) finite graph G=(V,E). The\nevolution of the population is driven by births, deaths, competition, and\nmutations along the edges of G. We are interested in the large population\nlimit under a mutation rate \μK given by a negative power of the carrying\ncapacity K of the system: \μK=K-1/\α,\α>0. This results in\nseveral mutant traits being present at the same time and competing for invading\nthe resident population. We describe the time evolution of the orders of\nmagnitude of each sub-population on the \log K time scale, as K tends to\ninfinity. Using techniques developed in [Champagnat, M 'el 'eard, Tran, 2019]\nwe show that these are piecewise affine continuous functions, whose slopes are\ngiven by an algorithm describing the changes in the fitness landscape due to\nthe succession of new resident or emergent types. This work generalises [Kraut,\nBovier, 2019] to the stochastic setting, and Theorem 3.2 of [Bovier, Coquille,\nSmadi, 2018] to any finite mutation graph. We illustrate our theorem by a\nseries of examples describing surprising phenomena arising from the geometry of\nthe graph and/or the rate of mutations.\n