Here in Bay County, Florida this is reflected in the number of car-deer collisions.
When deer population is high, disease is more easily spread and a variety of other conditions set in.
The Greenies actually depend on a successful deer harvest by licensed hunters to take the pressure off them in the event of a deer population explosion.
They don't give birth to as many offspring as rabbits but a much higher percentage of them survive so the end result is probably deer >= rabbits, in successful offspring.
I remember an item in the Chicago Sun Times in 1980, about O'Hare Airport in suburban Des Plaines (I used to live there on Ballard Rd. close to O'Hare). They had to hire "professional hunters" to come in and "thin out" the deer herd on the airport grounds before their numbers could begin to pose a hazard to landings and take-offs.
Last winter there was a herd of 30+ on my dad's land starving (50 acres), this year there are probably close to 50, and I cant get a nuisance permit even though they are eating everything in sight.