A new one-dimensional, zonally averaged high resolution spherical model of the accumulation and flow of continental ice sheets in response to variations in insolation forcing has been constructed. The model includes a detailed representation of the influence of the isostatic adjustment of the earth under the weight of the ice at a rate determined by the viscosity of the planetary mantle. With this model we address the issue as to whether the combination of physical processes that it includes is sufficient to allow us to reconcile the variations of continental ice volume that are inferred to have dominated the Pleistocene epoch on the basis of the analysis of δ18O data from deep-sea sedimentary cores. We use high resolution data from the newly analyzed ODP site 677 and from DSDP site 607 as principal observational constraints. These datasets reveal a marked transition in climate to have occurred at mid-Pleistocene time about 900 000 years BP. In order to understand this transition, we find that we are obliged to add an additional feedback process to the model that depends on the time rate of change of the ice volume itself and which might be understood as due to the rather sudden switching on of a marine-induced instability of continental ice volume, which could conceivably be associated with a reorganization of the general circulation of the oceans as postulated by Broecker and Denton. With this feedback active, the model predicts a climatic response to Milankovitch forcing that is dominated over the last 400 kyr by an intense 105 yr cycle in continental ice volume that matches late Pleistocene data from ODP site 677 and DSDP site 607. With the feedback turned off it also reconciles the spectral characteristics of the data from early Pleistocene time.