Within the general trend of post-Eocene cooling, the largest and oldest outlet of the East Antarctic Ice Sheet underwent a change from ice-cliff to ice-stream and/or ice-shelf dynamics, with an associated switch from line-source to fan sedimentation. Available geological data reveal little about the causes of these changes in ice dynamics during the Miocene Epoch, or the subsequent effects on Pliocene-Pleistocene ice-sheet history. Ice-sheet numerical modeling reveals that bed morphology was probably responsible for driving changes in both ice-sheet extent and dynamics in the Lambert-Amery system at Prydz Bay. The modeling shows how the topography and bathymetry of the Lambert graben and Prydz Bay control ice-sheet extent and flow. The changes in bathymetric volume required for shelf-edge glaciation correlate well with the Prydz Channel fan sedimentation history. This suggests a negative feedback between erosion and glaciation, whereby the current graben is overdeepened to such an extent that shelf-edge glaciation is now not possible, even if a Last Glacial Maximum environment recurs. We conclude that the erosional history of the Lambert graben and Prydz Bay in combination with the uplift histories of the surrounding mountains are responsible for the evolution of this section of the East Antarctic Ice Sheet, once the necessary initial climatic conditions for glaciation were achieved at the start of the Oligocene Epoch.