It seems to me that "continually kill large numbers of X" is an inherently broken solution to "I don't want X to live here".

If you have large numbers of X, it's because you're creating a very favorable habitat for it. If you don't address that issue and just keep killing them, you'll have both large numbers of X and large numbers of X corpses.

There are also arguably ethical issues with killing huge numbers of intelligent, social animals simply because we don't like their presence.