There are various examples where breeders did not outcompete, and instead a population settled on a low birth rate. Any animal that is not predator-dominated, really, but the best-known examples are New Zealands Kakapo [1] and Panda bears [2].

[1] https://en.wikipedia.org/wiki/Kakapo#Breeding [2] https://en.wikipedia.org/wiki/Giant_panda#Reproduction