A substantial nightside ionosphere has been observed on Venus during most orbits of the Pioneer Venus Orbiter. However, on some orbits during which the solar wind dynamic pressure was large, the nightside ionosphere seems to have almost disappeared, existing only as irregular patches of low-density plasma. We interpret these observational results using a two-dimensional theoretical model of the ionospere of Venus in which empirical horizontal velocities are adopted. We show that the degree to which horizontal transport of ions from day to night can mmaintain the nightside ionosphere depends on two parameters: (1) the flow velocities and (2) the ionopause height at the terminator. We also investigate the role of electron precipitation in supporting a nightside ionosphere. Our model also provides indirect evidence for an enhanced deuterium to hydrogen ratio on Venus.