I recall listening to a Radiolab podcast about this kind of thing but it was in the Galapagos. Feral goats were taking down the brush and hurting the native species -- a tortoise, in that particular case. Things got pretty darned crazy as they tried all sorts of things to eliminate the goats. They sure killed a bunch. But the goats are still around. Years later, the objective is still not achieved. Some bitter people are still dropping new goats off on the island just as a "&*%? YOU!" to the policy. In my very limited time working with a non-profit land trust, I learned that invasive species are almost impossible to eradicate. It's a doomed objective. Once established, control is usually as good as it gets.
Why nobody uses bounties anymore is something I don't understand. Tags have a cost so even a small fee will limit demand. Bounties, even small ones, can attract demand.
But I'll concede to McDave's point -- our attempts to control nature & society, even with good intentions, usually have unintended consequences. Good luck getting that point across to a politician or bureaucrat, though. They're doing the Lord's work as far as they know.