In the present study a mathematical model of the chemical control of respiration is described which attempts to simulate periodic breathing during sleep. The model is an extension of an earlier model which has been shown to successfully reproduce the transient effects of CO2 inhalation on breathing, controlled changes in ventilation on arterial gas tension, and Cheyne-Stokes breathing. Included in the extended model are the effects of chemical stimuli during sleep on both chest wall and upper airway muscle activity. Data is presented indicating that simulations from the model reproduce reasonably well the essential features of the results obtained in eight subjects with periodic respiration during sleep when breathing room air, O2, or low concentrations of CO2. Simulations from the model and the experimental data suggest that periodic breathing during sleep results from unstable operation in the respiratory control system analogous to that seen during instabilities in physical control systems. The model indicates that obstructive as well as central apneas can be produced by control system instability. Furthermore, central apneas increase the likelihood of obstructive apneas while obstructive apneas tend to aggravate the control instability. The model results predict that the characteristics of the periodic breathing seen during sleep, such as apnea length, will depend on circulation time and the sensitivity of both upper airway and chest wall muscles to hypercapnia and hypoxia.