Simberloff studied mangrove islands off Florida. He killed off all insects and found an equilibrium of species numbers attained.
Diamond claimed turnover rates on the Channel Islands off California but he was attacked in print (Lynch and Johnson) who claimed extinction either not so (Diamond just didn't find species) or human-caused, so not natural.
Clearly you need a broad data base to work form. You have to be sure your first birds are really breeders. Also extinction is very hard to prove. Researchers in Peru recently rediscovered the White-winged Guan, a bird thought to be extinct for nearly 100 years, and Ivory-billed Woodpeckers may have been rediscovered in Arkansas.
On the other hand, there is a body of evidence against the equilibrium theory.