Stochastic individual-based models with power law mutation rate on a general finite trait space

We consider a stochastic individual-based model for the evolution of a haploid, asex-ually reproducing population. The space of possible traits is given by the verticesof a (possibly directed) finite graph G = (V, E). The evolution of the population isdriven by births, deaths, competition, and mutations along the edges of G. We areinterested in the large population limit under a mutation rate μK given by a negativepower of the carrying capacity K of the system: μK = K−1/α, α > 0. This results inseveral mutant traits being present at the same time and competing for invading theresident population. We describe the time evolution of the orders of magnitude ofeach sub-population on the log K time scale, as K tends to infinity. Using techniquesdeveloped in [ 8], we show that these are piecewise affine continuous functions, whoseslopes are given by an algorithm describing the changes in the fitness landscape dueto the succession of new resident or emergent types. This work generalises [ 25 ] to thestochastic setting, and Theorem 3.2 of [6 ] to any finite mutation graph. We illustrateour theorem by a series of examples describing surprising phenomena arising fromthe geometry of the graph and/or the rate of mutations.