Brendan Burns, a guide and executive at the KUIU hunting equipment company, demonstrates a spring sheep count in the Bears Paw Mountains near St. PIerre, Montana. Though few outsiders know it, the high degree of difficulty and scarcity of opportunities to hunt bighorn sheep make them the ultimate pursuit of many a North American big game hunter.(Leah Nash photos/The New York Times)

ROCKY BOY’S RESERVATION, Mont. — For the herd of bighorn sheep, the rocky cliffs were a safe place, with 360-degree views and plenty of nooks to blend into the gray rocks. The ground was sprinkled with scat, and the air carried a scent like a barnyard. Thousands of feet below, the landscape unfurled into a smooth checkerboard of ranch land that stretched to the horizon. The only threat up here would be to newborn lambs, susceptible to being plucked away by eagles.

Crouched behind a stand of rocks last spring, Brendan Burns, a 38-year-old with a growing reputation as sheep hunter and guide, peered over the edge, careful not to be seen or heard. Wild sheep have acute senses, and when they spook, they bolt as one, like a flock of birds. But the sheep were not home. Amid the panorama below, Burns spotted a constellation of tiny dots in a faraway meadow. The horns gave them away.

“There aren’t a lot of circles in the wild,” Burns whispered. “When you see something curved — and they kind of shine, they have this kind of glow to them — you learn to pick them up. You just train your eye to it.”

He pulled a high-powered Swarovski scope from his pack and aimed it downhill. Eight years before, there were no sheep here. Then 21 ewes and five juvenile rams were transplanted to the Rocky Boy’s Reservation of the Chippewa Cree, which straddles part of the Bears Paw Mountains.

The herd quickly grew to 100, and 40 were relocated to South Dakota. It has again grown over 100, and another 40 are likely to be transplanted this spring, part of broad attempts to replant sheep populations that are a fraction of what they once were in the West.

“There’s obviously no coyotes around, for them to be that low and feel comfortable,” Burns whispered. “This is a nice day to be a sheep.”

A man from Michigan had paid $100,000 for the year’s only chance to hunt one sheep in the herd on the Rocky Boy’s Reservation. Burns brought him there in October, and the men traipsed through the steep and rocky terrain for days before getting themselves in position for a clean shot. The ram was 10 years old, with a scar on its forehead, a cloudy eye and several missing teeth.

Its massive horns and about 80 pounds of meat were hauled back to Michigan. In exchange, the Chippewa Cree tribe at Rocky Boy’s received the $100,000, which was used to fund two tribal game wardens overseeing wildlife on the reservation.

It is a paradox of hunting, rarely so conspicuous as with wild sheep: The hunters are often the primary conservationists. In 2013, a permit in Montana sold for $480,000, still a record. Burns assisted on that hunt, too, over 18 days in the Upper Missouri River Breaks. The result was a large ram, and hundreds of thousands of dollars that went into the budget of Montana Fish, Wildlife & Parks.

“As far as sheep-hunting being a rich man’s sport, that’s absolutely true,” said Vance Corrigan, 84, who lives along the Yellowstone River in Livingston, Montana, and is one of the most accomplished big-game hunters in the world. “But if it weren’t for the rich man, those sheep wouldn’t be there.”

Nonhunters often presume that the biggest prize in North America is something large and fierce — some kind of bear, perhaps, or an elk, a moose or a mountain lion. But the widespread belief among serious hunters is that rams are the ultimate pursuit.

That is for two reasons. One, opportunities to hunt sheep are scarce, and often prohibitively expensive. Two, the hunts are among the most difficult, often lasting weeks in some of the most remote regions on Earth.

“For 100 years, it’s been somewhat at the pinnacle of big-game hunting, especially in the United States,” said Bob Anderson, a hunter and author of books on sheep-hunting. “But there weren’t a lot of people or sheep on the mountains. Now it’s become a cocktail party of sorts. Some well-to-do people have gotten into it, and they’ve driven the market up.”

At the Wild Sheep Foundation’s convention each January, single hunting permits from various states, provinces and Indian reservations are auctioned off to the highest bidders. Most go for well more than $100,000.

“People who pay $300,000 for a tag, they just paid to recover 30 sheep to places that haven’t had sheep in 100 years,” Corrigan said. “Lewis and Clark saw sheep on every ridge. Those people buying tags are helping put sheep back where they were before we arrived.”

Globally, there are dozens of species and subspecies of wild sheep, many in Central Asia. The Wild Sheep Foundation, based in Bozeman, Montana, considers there to be four primary wild sheep in North America: the Rocky Mountain bighorn; the desert bighorn (in the U.S. Southwest and Mexico); Dall’s sheep (commonly called Dall sheep, in Alaska, British Columbia, the Northwest Territories and the Yukon); Stone’s sheep (or Stone sheep, in British Columbia and the Yukon).

Some have estimated that there were millions of wild sheep in North America 200 years ago. But by the 1950s, squeezed out by people and livestock and decimated by diseases (especially carried by domestic sheep), the wild sheep population dwindled into the tens of thousands.

Conservation efforts saved the sheep and have expanded their territory again, often by transplanting herds and greatly limiting hunting opportunities. It is estimated that there are now nearly 200,000 wild sheep in North America.

The privilege of killing one (or “harvesting” one, in a hunting euphemism) remains limited to the very few. It requires a lot of money or a lot of luck.

The U.S. Fish and Wildlife Service estimates that there are more than 10 million big-game hunters in the United States. But only about 2,500 wild sheep are hunted each year across North America, a fraction compared with nearly every other animal.

“They’re like the collector’s edition Ferrari,” said Lance Kronberger, who owns Freelance Outdoor Adventures in Alaska and guides big-game hunts of all kinds. “There’s 200 of them made, and you have to get lucky to get one.”

Generally, there are two ways to hunt wild sheep. One is hunting’s version of the lottery — pay a few dollars and apply for one of the limited number of licenses that are restricted to certain territories and raffled off. Odds of winning can be infinitesimal. In Montana, 19,439 applications were submitted by state residents in 2015; licenses went to 111, a success rate of about 1 in 200. Nonresidents have a tougher time.

The second way to secure a chance at a wild sheep is to spend a lot of money. While residents of Alaska and those in Canada generally can hunt sheep within their own state, province or territory, nonresidents are required to hire a registered outfitter. The laws of supply and demand push the price of hunting a Dall’s up to about $25,000 and a Stone’s sheep to about $50,000. Hunts in Mexico, through outfitters or private landowners, can reach $100,000.

A wealthy few go beyond that. They bid on exclusive permits that are auctioned off annually to raise money for states, provinces and Indian reservations, seeing their lavish spending as a charitable donation, a tax write-off and a chance to capture one of hunting’s premier trophies.

“Some rich people are into yachts or floor tickets to the Lakers,” Burns said. “Some sheep-hunt.”