I don't know why you're seeing no results for those phrases. I see loads, as expected. However, "propagate" is not the correct verb when referring to animals reproducing.
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KaaJee"Breeding" really seems to be better, however, I don't know if it is used only when they are kept by people, or also outside human environment, I mean that the animals do so by their own will,Transitive "breed" is the human-directed kind of reproduction (e.g. "He breeds horses"); instransitive "breed" is something that animals do naturally.