What are examples of animal species that there are a surplus number of in zoos in either side of the Atlantic? I'm curious because this is how some zoos get the iconic mega fauna animals or animals that people instantly recognize.
Hopefully I've understood your question correctly. In Australasia, we have a large population of Sumatran tigers. Females are often bred only once, and rarely at the age of three to four when they attain maturity, but more often at eight years of age. While there are enough zoos to accodomate the ones we have, several females (and males) are surplus to the breeding programme and have never been found mates/will be found mates. When a smaller zoo expresses an interest to hold this species, it is never hard to fill this request. If it were not for the tightly controlled breeding programme (a lot more could easily be bred) then it would overwhelm the system. Lions essentially fall into the same category, though we have an aging population of these in New Zealand which can easily be replaced by lions bred from the younger Australian population when needed. Other examples of species in a similar situation include red panda and to a lesser extent otter and meekat (though their reproduction tends to be more sporadic). Male giraffe have previously been considered a surplus, with zoos such as Hamilton holding bachelor herds and even a private owner taking on three that were surplus to the breeding programme. Zebra also fall into this category. White rhinos are following a similar trend with Auckland and the National Zoo in Canberra holding bachelor herds.
A lot of the smaller animals (reptiles, fish, invertebrates, etc) when there's no room for other institutions that can take the offspring in.