Stochastic differential equations for evolutionary dynamics with demographic noise and mutations.
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.
Evolutionary Theory Group, Max-Planck-Institute for Evolutionary Biology, August-Thienemann-Strasse 2, 24306 Plön, Germany.
Arne
A
Traulsen
Jens Christian
JC
Claussen
Christoph
C
Hauert
D000818:Animals; D005075:Biological Evolution; D003198:Computer Simulation; D005716:Game Theory; D006801:Humans; D008957:Models, Genetic; D015233:Models, Statistical; D011157:Population Dynamics; D013269:Stochastic Processes