You get like one piece of a bronze booster animal in every booster pack, right? And it takes 100 pieces to make the animal. Here's the history of that whole thing.
6/29/16: the first animal, a mhorr gazelle, which stuck around for 231 days
2/15/17 binturong - active for 280 days
11/22/2017 Albatross - 265 days
8/14/2018 Sloth Bear - 211 days
3/13/2019 Japanese Serow
Predicted next bronze animal: within a month of October 2019
If you visit every day but don't trade, you'll end up with around 130 pieces, which is nowhere near enough for a breeding pair; that means you will need to trade if you want one. When a new bronze animal appears, the old ones still appear in bronze boosters - but much more rarely, so if you find yourself 50 pieces short you're in a lot of trouble.