Problem here, as I see it, is that very few of the introduced species are considered game animals.
Couple this with the decline of hunting area, restrictions on firearms, and public disdain for the hunting fraternity.
I feel that the fox population in Tasmania could be wiped out within three years if there were youth hunting programs and incentives promoting such.

With a worldwide economic downturn, we may see hunting on the increase yet.
Even native species such as the Canada goose, which is way overpopulating here in Colorado, could be controlled with a few operations such as this

A little bow hunt to help out the local cemetery...