White-tailed deer have only recently recovered from over-hunting and returned to their pre-colonization population levels.[1] Do you have a source that frames their growth as exponential?
Apologies, I'm based in Pennsylvania and it's definitely more noticeable here. I expect it will become more apparent elsewhere too.
We're at 3x the total population that existed when europeans started settling here, and without an appropriate way to cull the herd I don't see that changing. We have milder winters (so no starvation), less interest in hunting, and again a lack of natural predators.
The Hawaiian deer population follows boom and bust cycles. They were introduced for hunting purposes, have no natural predators, and tend to mess up the ecosystem so badly via overgrazing that they end up dying off and then recovering.
Deer population growth during that recovery period (of white tail deer at least) seems pretty exponential. [1] However in the absence of natural predators it seems that disease and human culling have pretty effectively kept their population to stay at about pre-colonial levels rather than continuing upward to the point of over population.
1. https://en.wikipedia.org/wiki/White-tailed_deer#Population_a...