Stochastic individual-based models with power law mutation rate on a general finite trait space
Abstract
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 vertices
of a (possibly directed) finite graph G = (V, E). The evolution of the population is
driven by births, deaths, competition, and mutations along the edges of G. We are
interested in the large population limit under a mutation rate μK given by a negative
power of the carrying capacity K of the system: μK = K−1/α, α > 0. 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 log K time scale, as K tends to infinity. Using techniques
developed in [ 8], 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 [ 25 ] to the
stochastic setting, and Theorem 3.2 of [6 ] 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.
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