More details: When you generate a world, every biome has a given population for every animal type. Animals that march onto your map and get killed are subtracted from this pool of available creatures. Unless the biome is tiny, you shouldn't run out of things like groundhogs, and I don't think land vermin EVER go extinct, but larger creatures will eventually be hunted to extinction, especially the fanciful beasts, like merpeople.
To combat this, you should work on capturing some of them, and having a breeding stock within your fortress - even better, this takes away the need to actually hunt them down and drag their corpses back, plus you ensure a regular supply, if you just keep your breeding bears in a pin, and your spawned eating bears in a cage, and occasionally mark a few to be slaughtered, they get killed right there in the butcher's shop.
Another option is to have automated slaughtering towers - tether up breeding animals, while letting their spawn roam free, and set up a retracting bridge connected to a pressure plate so that when the animal tries to leave the breeding pins, the bridge comes out from beneath them, and they fall several z levels to their death - right on the butcher's floor!
To capture the animals, you want to either dig a trench or build a wall. The animals in this game path by picking a target location at random, then choosing the best way to get there, instead of just randomly meandering one tile at a time. To take advantage of this, make a wall or a trench near the edges of the map where they like to loiter, seperating two large chunks of land, where there is a good chance of them wanting to randomly path there. If you have two whole continents seperated in the middle of your map, that's the easy solution - just make a bridge between the two on the surface (if you can), and they will want to migrate between the two continents, so long as you don't have an interposing fortress full of doors. Otherwise, just dig yourself a trench or build a wall that leads up to a line of trees so that there isn't a small gap on the edge of the map (trees block movement, just like a wall) then set up your cage traps on the corners of the area of the wall where you do have that little gap for the animals to get through.
I just caught 11 unicorns in a single season with a single trench.