We present a new model of the underlying dynamics of the oxygen uptake V̇O2(v,t) kinetics for various exercise intensities. This model is in the form of a set of nonlinear coupled vector fields for the V̇O 2(v,t) and v̇, the derivative of the exercise intensity with respect to time. We also present a new and novel means for calculating the oxygen demand, D(v,t), and hence also the oxygen deficit and debt, given the time series of the V̇O2(v,t). This enables us to give better predictions for these values especially for when exercising at or close to maximal exercise intensities. Our model also allows us to predict the oxygen uptake time series given the time series for the exercise intensity as well as to investigate the oxygen uptake response to nonlinear exercise intensities. Neither of these features is possible using the currently used three-phase model. We also present a review of both the underlying physiology and the three-phase model. This includes for the first time a complete set of the analytical solutions of the three-phase model for the oxygen deficit and debt. © 2005 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.