Evolutionary game theory is a framework to formalize the evolution of
collectives ("populations") of competing agents that are playing a game and,
after every round, update their strategies to maximize individual payoffs.
There are two complementary approaches to modeling evolution of player
populations. The first addresses essentially finite populations by implementing
the apparatus of Markov chains. The second assumes that the populations are
infinite and operates with a system of mean-field deterministic differential
equations. By using a model of two antagonistic populations, which are playing
a game with stationary or periodically varying payoffs, we demonstrate that it
exhibits metastable dynamics that is reducible neither to an immediate
transition to a fixation (extinction of all but one strategy in a finite-size
population) nor to the mean-field picture. In the case of stationary payoffs,
this dynamics can be captured with a system of stochastic differential
equations and interpreted as a stochastic Hopf bifurcation. In the case of
varying payoffs, the metastable dynamics is much more complex than the dynamics
of the means