  Recently, an evolutionary game dynamics model taking into account the
environmental feedback has been proposed to describe the co-evolution of
strategic actions of a population of individuals and the state of the
surrounding environment; correspondingly a range of interesting dynamic
behaviors have been reported. In this paper, we provide new theoretical insight
into such behaviors and discuss control options. Instead of the standard
replicator dynamics, we use a more realistic and comprehensive model of
replicator-mutator dynamics, to describe the strategic evolution of the
population. After integrating the environment feedback, we study the effect of
mutations on the resulting closed-loop system dynamics. We prove the conditions
for two types of bifurcations, Hopf bifurcation and Heteroclinic bifurcation,
both of which result in stable limit cycles. These limit cycles have not been
identified in existing works, and we further prove that such limit cycles are
in fact persistent in a large parameter space and are almost globally stable.
In the end, an intuitive control policy based on incentives is applied, and the
effectiveness of this control policy is examined by analysis and simulations.
