Do animals which pose a threat to people and livestock have to be exterminated?

The Irish wolf fared better but it too went to the wall; the last one was hunted down and killed on Mount Leinster in 1786. Much the same happened to large carnivores throughout Europe but now, in more enlightened times, they are re-colonising former haunts. Wolves are breeding even in Denmark.
‘Apex’ predators are important; they provide beneficial ‘top-down control’ of ecosystems. Without wolves to limit their numbers, deer are eating themselves out of house and home in Scotland. Vulnerable plants and animals can’t survive the onslaught and bio-diversity is suffering.