Does the game put in place any limits on hunting? Is it possible to out-hunt a region, either temporarily or permanently? Or do animals just keep spawning?
While it's not possible to hunt an animal to total extinction, it's possible to out hunt them temporary, a good example for this is the place with the 2 bears just below the entrance to the frontier from the homestead. There are only 2 bears that spawn there, (1 in the winter). And if you go there and hunt them both, there won't be any bears there until they re spawn, which means that you have to leave the area and wait, or fast travel to the entrance and go to them from there.
It doesn't happen with beavers, elk, deer, raccoons, hare, bobcats, foxes and wolfs, only to the animals with few specific spawning places, which are bears and cougars. They will disappear from their spawning place if you hunt them faster than they re spawn, but will return once you fast travel or go and wait for some time.