This seems to be an issue that has cropped up from time to time. It seemed to be more common a year or so ago in game, but other bugs have re-surfaced since the most recent DLC and update. I remember one early zoo of mine where I basically had to fire my entire staff, scrap all my workzones, and then redo them all from scratch. That seemed to fix it, but I've no idea why. I also don't know why one zoo I make might have constant animal escapes and a different one won't. Obviously there's something about the way I build or set up habitats that is different, but I can never pinpoint what it is.
Hopefully the next update will help. The general advice is to go to the frontier issue tracking site and see if others are having the problem, but I've personally never gotten much help there. Even when others report the same issue, they always seem to come up as unconfirmed. I wish we could figure out how the play testers and developers have their games set up to never see the same issues many players encounter regularly.
Note as a last ditch save the animals thing, you can send them directly to quarantine, where they will be fed at least.