Anomalous finite-size effects in the Battle of the Sexes

The Battle of the Sexes describes asymmetric conflicts in mating
behavior of males and females. Males can be philanderer or
faithful, while females are either fast or coy, leading to a cyclic
dynamics. The adjusted replicator equation predicts stable
coexistence of all four strategies. In this situation, we consider
the effects of fluctuations stemming from a finite population size.
We show that they unavoidably lead to extinction of two strategies
in the population. However, the typical time until extinction
occurs strongly prolongs with increasing system size. In the
emerging time window, a quasi-stationary probability distribution
forms that is anomalously flat in the vicinity of the coexistence
state. This behavior originates in a vanishing linear deterministic
drift near the fixed point. We provide numerical data as well as an
analytical approach to the mean extinction time and the
quasi-stationary probability distribution.