How do I build a zoo in Minecraft that does not make the animals disappear?

What about the enclosures, the dolphins disappear even in small areas.

You can just buy the Minecraft Zoo, it's all there.

It's best to do your mob nametags (name tags) so that the animals do not despawnen. Name badges are indeed very hard to get (only in Mobspawner dungeons) but unfortunately only the only solution known to me.