Foster, the male, was almost 14 months when the study started. He had a particular fondness for turning upside down and blowing bubbles in front of the one-way mirror in the aquarium wall through which the researchers observed and recorded what the dolphins were doing.

The animals also passed a test in which the researchers drew a mark on some part of the dolphin’s body it could not see without a mirror. In this so-called mark test, the animal must notice and pay attention to the mark. Animals with hands point at the mark and may touch it.

The dolphins passed that test at 24 months, which was the earliest researchers were allowed to draw on the young animals. Rules for animal care prohibited the test at an earlier age because of a desire to have the animals develop unimpeded. During testing, the young animals were always with the group of adults they live with, and only approached a one-way mirror in the aquarium wall when they felt like it.

Rules for drawing on human children are apparently less strict, and they pass the mark test at 18-24 months.

Frans de Waal, of Emory University, who studies cognition in apes and other animals and is the author of “Are We Smart Enough to Know How Smart Animals Are?” , said in an email, “Great study.”

Dr. de Waal worked with Dr. Reiss on an earlier study of self-recognition in elephants but was not involved in the dolphin research.

He said the study has value because science needs to go beyond asking whether species display mirror self-recognition (MSR) to ask “whether the emergence of MSR correlates, as it does in humans, with other milestones of development.” Connecting the ability to the rest of development can help researchers “begin to answer the question of what MSR means.”