The association requires zoos to keep at least three female elephants (or two males) and to have a certain amount of space in which they can roam. PETA is arguing that while the Bronx Zoo meets the letter of those regulations, it violates their intent by keeping Happy separate.

PETA and other advocates, as well as the former zookeeper, would like to see Happy released to a wildlife sanctuary, where she could live among other elephants in an environment closer to their natural habitat. There are two such facilities in the United States — the Elephant Sanctuary in Hohenwald, Tenn., and the Performing Animal Welfare Society (PAWS) in San Andreas, Calif. Each occupies more than 2,000 acres of woods and pasture, with lakes and ponds for bathing. Both sanctuaries have taken in elephants that once lived in zoos — including the zoos in Detroit, San Francisco, Anchorage, Toronto and Madison, Wis.

Mr. Breheny isn’t interested in following that path. Happy, he said, is “very well adjusted and, I would say, has deep bonds for the people who care for her.” During the day, Happy stays in a separate yard; at night in the barn, she has a separate stall. But the three elephants are able to see one another over the fence, and sometimes the barn door between them is kept ajar. “The keepers report that she’s totally content,” Mr. Breheny said. “Their reading of her is that she’s happy.”

Though Happy spends much of her time alone today, she was briefly a celebrity in the scientific world, the star of a study in animal cognition. Most animals show no interest in their own image, but when three scientists placed a mirror in the holding yard at the Bronx Zoo in the summer of 2005, Happy looked in the mirror and noticed a mark that had been painted on her head, and touched it repeatedly with her trunk. She seemed to understand that she was looking at herself.

Until then, only the great apes, dolphins and humans (starting around 18 to 24 months) had passed the test, which, the authors of the study wrote, “is thought to correlate with higher forms of empathy and altruistic behavior.” (More recently, magpies have joined the club.)

Diana Reiss, a professor of psychology at Hunter College and one of the authors of the study, said that all these species, from human babies to elephants, go through the same stages as they explore the mirror, and all show an interest in looking at parts of their bodies they cannot otherwise see, like their eyes. “To me, it links us all in a fundamental way,” Dr. Reiss said.