We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph . The evolution of the population is driven by births, deaths, competition, and mutations along the edges of . We are interested in the large population limit under a mutation rate given by a negative power of the carrying capacity K of the system: . This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the time scale, as K tends to infinity. Using techniques developed in , we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work generalises  to the stochastic setting, and Theorem 3.2 of  to any finite mutation graph. We illustrate our theorem by a series of examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations.
This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy GZ 2047/1, Projekt-ID 390685813 and GZ 2151, Project-ID 390873048 and through the Priority Programme 1590 “Probabilistic Structures in Evolution”. This work was also partially funded by the Chair Modélisation Mathématique et Biodiversité of VEOLIA-Ecole Polytechnique-MNHN-F.X.
The authors thank Anton Bovier for stimulating discussions at the beginning of this work and comments, and L. Coquille and C. Smadi also thank him for his invitations and welcome at Bonn University. The authors would also like to thank Sylvain Billiard for pointing out some bibliographical references. Finally, the authors are very grateful to an anonymous referee for his/her numerous comments and questions.
"Stochastic individual-based models with power law mutation rate on a general finite trait space." Electron. J. Probab. 26 1 - 37, 2021. https://doi.org/10.1214/21-EJP693