D.I.E.D, yes, in the long-term, the generalist species will diversity and become specialists. In the short-term, we are loosing specialists species at an unbelievable rate.
A large number of species, especially rodents, are cyclical in their population. Without a doubt, your population will crash after they eat themselves out of house and home. It is almost impossible to blunt their "crash" population.
Our Coopers hawks raised two babies and they were so much fun to watch. The babies would sit on the railing on our deck. Turkeys and deer will also come onto our deck but these Coopers hawks were so much more fun to watch.