  We study the evolutionary dynamics of games under environmental feedback
using replicator equations for two interacting populations. One key feature is
to consider jointly the co-evolution of the dynamic payoff matrices and the
state of the environment: the payoff matrix varies with the changing
environment and at the same time, the state of the environment is affected
indirectly by the changing payoff matrix through the evolving population
profiles. For such co-evolutionary dynamics, we investigate whether convergence
will take place, and if so, how. In particular, we identify the scenarios where
oscillation offers the best predictions of long-run behavior by using
reversible system theory. The obtained results are useful to describe the
evolution of multi-community societies in which individuals' payoffs and
societal feedback interact.
