Have a look at this tutorial for a possible solution: How To Show/Hide WordPress Meta Boxes By Selecting Categories.
Basically, all of the taxonomies would be hidden via javascript, and a function attached to the click event of the Animals taxonomy, which inspects the selected item’s ID to show a corresponding child taxonomy. You’d also have to have a bit of logic on page load to show the taxonomies with terms that are already selected.
There are potential problems though – what happens if someone deselects “dog”, but not any child terms?