  In stochastic dynamical systems, different concepts of stability can be
obtained in different limits. A particularly interesting example is
evolutionary game theory, which is traditionally based on infinite populations,
where strict Nash equilibria correspond to stable fixed points that are always
evolutionarily stable. However, in finite populations stochastic effects can
drive the system away from strict Nash equilibria, which gives rise to a new
concept for evolutionary stability. The conventional and the new stability
concepts may apparently contradict each other leading to conflicting
predictions in large yet finite populations. We show that the two concepts can
be derived from the frequency dependent Moran process in different limits. Our
results help to determine the appropriate stability concept in large finite
populations. The general validity of our findings is demonstrated showing that
the same results are valid employing vastly different co-evolutionary
processes.
