I have been on various japanese travel sites and read a document about Ōkunoshima or as its known by everyone lol Usagi Jima (rabbit island)
Theres no reports of killings to control the population. but reports of tourists asking the those working on the island to help the weaker, sickly buns...
like any large animal population there are buns who die of starvation, illness and disease... there will be some very gaunt looking bunnies who are dying ect... which obviously is upsetting to see.
The island has lots of stalls where you can purchase pellets to feed them. Amazingly due to vet recommendation they all sell the same pellets.
They also have posters asking you not to feed stuff that is bad for the buns and ones saying dont pick them up in case you scare and injure them and other important stuff like that
They dont interfere with the buns with regards to the population or saving ill buns