  The fast-slow dynamics of an eco-evolutionary system are studied, where we
consider the feedback actions of environmental resources that are classified
into those that are self-renewing and those externally supplied. We show
although these two types of resources are drastically different, the resulting
closed-loop systems bear close resemblances, which include the same equilibria
and their stability conditions on the boundary of the phase space, and the
similar appearances of equilibria in the interior. After closer examination of
specific choices of parameter values, we disclose that the global dynamical
behaviors of the two types of closed-loop systems can be fundamentally
different in terms of limit cycles: the system with self-renewing resources
undergoes a generalized Hopf bifurcation such that one stable limit cycle and
one unstable limit cycle can coexist; the system with externally supplied
resources can only have the stable limit cycle induced by a supercritical Hopf
bifurcation. Finally, the explorative analysis is carried out to show the
discovered dynamic behaviors are robust in even larger parameter space.
