How I did it was I paid for a lot of marketing. I also decreased the ticket price a bit. You can also add animals before they ask for it. I think that scenario they eventually want you to buy Cheetahs, you can get them early to promote more visitors. I didn't realize this until later on. I ended up making a lion habitat which boosted the guest numbers.
Edit: Also, there is plenty of space in the map! So I wouldn't worry about running out of room.