VIELHA, Spain—As the spring sun thaws the icy crags of the Pyrenees Mountains, the telltale signs begin to appear again. Tufts of fur are snagged on barbed wire. Paw prints the size of a ping-pong paddle materialize in the mud. Ant hills are violently clawed.

Pyros, the alpha bear of the Pyrenees, is on the move again.

In 1997, Pyros was brought...