The evolution of sleep is inevitable in a periodic world.

There are two contrasting explanations of sleep: as a proximate, essential physiological function or as a behavioral, adaptive state of inactivity and these hypotheses remain widely debated. To investigate the adaptive significance of sleep, we develop an evolutionary argument formulated as a tractable partial differential equation model. We allow demographic parameters such as birth and mortality rates to vary through time in both safe and vulnerable sleeping environments. From this model we analytically calculate population growth rate (fitness) for sleeping and non-sleeping strategies. We find that, in a temporally heterogeneous environment, sleep behavior always achieves a higher fitness than non-sleeping behavior. As organisms do not exist in constant environments, we conclude that the evolution of sleep is inevitable. Further, we suggest that the two contrasting theories need not be mutually exclusive.