If you can get close enough to the animals, I think you could get good results with a fixed (prime) lens. However, for anything other than large animals, I think you'll really need something at least 200mm to get any good results.
That being said, if your camera is high enough resolution, you can probably get away with a shorter lens and cropping the photo - at least to a degree ... you'd still want something over 100mm though.
Personally, unless I'm shooting elephants, I'm typically right up there in the 300 - 400mm range when I'm shooting at a zoo - animals at least, enclosures you'd want something much wider (perhaps even as wide as 10-22mm sometimes).
When I was travelling overseas on my own a lot, I would often do two loops around a zoo, one with my 100-400mm lens on the camera taking animal shots, then another with my 10-22mm lens on, taking enclosure shots. If I was only going to get time for one lap, then I'd carry a compact camera for enclosure shots.