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The fast-slow dynamics of eco-evolutionary systems 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 that 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 the self-renewing resource undergoes a generalized Hopf bifurcation, and thus one stable and one unstable limit cycle can coexist; the system with the externally supplied resource only has the stable limit cycle.

Different Environment Feedback in Fast-Slow Eco-Evolutionary Dynamics and Resulting Limit Cycles