The mechanisms underlying oscillations in a central pattern generator (CPG) may differ fundamentally in the intact organism versus an isolated CPG preparation. We investigate this aspect of rhythmogenesis in a computational model of closed-loop respiratory control, incorporating minimal models of the brainstem CPG, lung mechanics, oxygen handling, metabolism, and chemosensation. We analyze the model behavior using fast/slow dynamical systems analysis and open-loop/closed-loop control analysis. We show that eupnea-like bursting oscillations arise from a distinct mechanism in the intact (closed loop) versus isolated (open loop) systems. The closed-loop model exhibits bistability between eupnea-like bursting and tachypnea-like tonic spiking, and we demonstrate that imposed bouts of hypoxia can induce a dramatic transition from eupnea to tachypnea. For moderate bouts of hypoxia, the endogenous properties of the ionic conductances in the Butera-Rinzel-Smith CPG model can lead to spontaneous autoresuscitation and recovery to eupnea. We also find that chemosensory feedback gives the closed-loop control system greater robustness to changes in metabolic demand than the open-loop control system.