Unihemispheric sleep has been observed in numerous species, including birds and aquatic mammals. While knowledge of its functional role has been improved in recent years, the physiological mechanisms that generate this behavior remain poorly understood. Here, unihemispheric sleep is simulated using a physiologically based quantitative model of the mammalian ascending arousal system. The model includes mutual inhibition between wake-promoting monoaminergic nuclei (MA) and sleep-promoting ventrolateral preoptic nuclei (VLPO), driven by circadian and homeostatic drives as well as cholinergic and orexinergic input to MA. The model is extended here to incorporate two distinct hemispheres and their interconnections. It is postulated that inhibitory connections between VLPO nuclei in opposite hemispheres are responsible for unihemispheric sleep, and it is shown that contralateral inhibitory connections promote unihemispheric sleep while ipsilateral inhibitory connections promote bihemispheric sleep. The frequency of alternating unihemispheric sleep bouts is chiefly determined by sleep homeostasis and its corresponding time constant. It is shown that the model reproduces dolphin sleep, and that the sleep regimes of humans, cetaceans, and fur seals, the latter both terrestrially and in a marine environment, require only modest changes in contralateral connection strength and homeostatic time constant. It is further demonstrated that fur seals can potentially switch between their terrestrial bihemispheric and aquatic unihemispheric sleep patterns by varying just the contralateral connection strength. These results provide experimentally testable predictions regarding the differences between species that sleep bihemispherically and unihemispherically.