  Mating preferences of many biological species are not constant but
season-dependent. Within the framework of evolutionary game theory this can be
modeled with two finite opposite-sex populations playing against each other
following the rules that are periodically changing. By combining Floquet theory
and the concept of quasi-stationary distributions, we reveal existence of
metastable time-periodic states in the evolution of finite game-driven
populations. The evolutionary Floquet states correspond to time-periodic
probability flows in the strategy space which cannot be resolved within the
mean-field framework. The lifetime of metastable Floquet states increases with
the size $N$ of populations so that they become attractors in the limit $N
\rightarrow \infty$.
