Species become locally rarer and rarer until they eventually disappear. Intensive grazing of this kind can wreck pastures for raising stock. The selective pressure of hordes of hungry little mouths taking the best plants as they emerge alters the balance of species in favour of those that are least palatable. Surprisingly, several authorities believe that rabbits only became pests in the latter part of the 19th century, after sheep had modified pastures in this very manner. From the opposite end of the scale, in much of the now rabbit-free regions following the calicivirus epidemic, it is abundantly clear that rabbits caused enormous environmental damage just through the emergence of seedling trees and shrubs that hadn't previously been able to establish for more than 50 years.
Rabbits are truly prolific and opportunistic breeders.
They use a reproductive strategy that might be thought of as 'eating hay while the sun shines'. Boom and crash responses to good and bad times create a strong selective pressure. In consequence, tough individuals emerge from the periodic biological 'bottleneck' events that bring the huge populations crashing down. Hunting though is too localised to create such an effect. A friend from one of the SSAA Hunting and Conservation teams told me that he and two mates had fired off three bricks of .22s and taken more than 1000 rabbits during three days in the Flinders Ranges just before the calicivirus escaped.