Signs of bighorn sheep were scattered inches away from the cliff edge: Dirt packed down from where they slept, round droppings, and a still-pungent smell. The animals themselves were gone, melting away into distant layers of mountain and rock.

Steve Kilpatrick, a retired biologist now leading the Wyoming Wild Sheep Foundation, was in the mountains on the border of Grand Teton National Park in August surveying sheep as part of an annual effort to keep tabs on its dwindling population.

Fewer than 100 sheep now comprise the ancient Teton Range herd in northwest Wyoming—one of the most unique in the West. And biologists say it is on the brink of blinking out.

The threat list begins as a familiar mixture of habitat loss,