Humans, of course, can breathe while the conscious mind is asleep; our subconscious mechanisms have control of this involuntary system. But equipped with a voluntary respiratory system, whales and dolphins must keep part of the brain alert to trigger each breath.
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Other methods help marine mammals to hold their breath longer than other types of mammals can. Marine mammals can take in more air with each breath, as their lungs are proportionately larger than those in humans. In addition, they exchange more air with each inhalation and exhalation. Their red blood cells also carry more oxygen. And when diving, marine mammals' blood travels only to the parts of the body that need oxygen--the heart, the brain and the swimming muscles. Digestion and any other processes have to wait.
Finally, these animals have a higher tolerance for carbon dioxide (CO2). Their brains do not trigger a breathing response until the levels of CO2 are much higher than what humans can tolerate. These mechanisms, part of the marine mammal diving response, are adaptations to living in an aquatic environment and help during the process of sleeping. Cetaceans reduce the number of breaths they take during rest periods; a dolphin might average 8 to 12 breaths a minute when fairly active only to have their breathing rate drop to 3 to 7 per minute while resting.
It is actually rare for a marine mammal to "drown," as they won't inhale underwater; but they do suffocate from a lack of air. Being born underwater can cause problems for newborn whale and dolphin calves. It is the touch of air on the skin which triggers that first, crucial breath. And necropsies sometimes show that an animal never gets to the surface to take its first breath of air. The same problem can occur when an animal is caught in a fishing net. If unable to reach the surface, or if in a panic, the animal may dive deeper, where it will be unable to breathe and suffocate.
Obviously sleeping safely at sea can pose problems, but the marine mammal system has addressed them.