During the Auroral Probe/Alpha-3 experiment, the data about the dynamics of the H+ ion density deep in the plasmasphere were obtained before and in the course of the development of several geomagnetic storms in the postmidnight and noon plasmasphere sectors. It was found that, in the postmidnight sector, the development of weak and moderate geomagnetic storms is accompanied by a considerable decrease of the H+ ion density inside the plasmasphere. This seems to be explained by cold plasma escaping into the ionosphere. In the daytime sector, at the initial phase of a storm, a considerable decrease of the H+ ion density inside the plasmasphere was observed, which then was replaced by a considerable increase. Variations of the H+ ion density in the daytime plasmasphere seem to be a result of variations of cold plasma fluxes from the ionosphere into the plasmasphere during ionospheric storms. In contrast to the model of propagation of plasmasphere deformations during magnetic substorms from the night sector into the daytime sector with the corotation rate proposed by the authors [6], the practically simultaneous beginning of the plasmapause displacement to the Earth in the night and daytime sectors is found. It is also found that the plasmapause displacement to the Earth in the daytime sector lasts for 10-12 h, once the night plasmapause begins to move away from the Earth. The last fact seems to be explained by the arrival (from the nightside to the dayside as a result of corotation) of magnetic field tubes with a plasma of reduced density as a result of a magnetic substorm.