The problem is not sufficient habitat, the problem is the lack of predators to decimate the population of other predators, i.e. fox, badger and so on. lynx and eagle owl for example do prey on these smaller predators, so does wolf. and if you have even the littlest common knowledge about biology, you might know, that a population that grows beyond natural capacities will be most probably decimated by disease. You guys don't have the rabies over there as far as I know - we do. In the 1970s Foxhides (holes? what are they called?) were gassed to get a hold of rabies.
Another reason for shoot