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Abstract

We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in
finite populations based on stochastic differential equations (SDEs). For large, but finite populations this allows
us to include demographic noise without requiring explicit simulations. Instead, the population size only rescales
the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types,
provided that mutation rates μ are not too small compared to the inverse population size 1/N. This ensures that
all types are almost always represented in the population and that the occasional extinction of one type does
not result in an extended absence of that type. For μN 1 this limits the use of SDEs, but in this case there
are well established alternative approximations based on time scale separation. We illustrate our approach by
a rock-scissors-paper game with mutations, where we demonstrate excellent agreement with simulation based
results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small
population sizes.