The answer to this can be found in Chapter 2 of Guns, Germs, and Steel. In that chapter, Jared Diamond is discussing the ways in which various Polynesian societies managed to subsist. He is talking about what sorts of plants and animals they had and what sorts of islands they lived on. On p. 60 in the paperback edition of the book, Diamond tells us that the Polynesians, at best, did not have very many animals at all. In fact, they only had three domesticated animals that they brought with them as they colonized the islands. They did not find any more animals that could be domesticated on those islands.
The three domesticated animals that the Polynesians had were pigs, chickens, and dogs. However, it is not as if each island had all three of these. One or more animals did not survive some of the voyages to various islands. For example, New Zealand only had dogs and Easter Island only had chickens.