I agree that lots of animals are killed by other animals, and we could explore some challenging questions about this topic. But I don't really understand why you think this means that animals don't or can't have rights. Lots of humans are killed by non-humans, too: by wild animals, by extreme weather and natural disasters, by disease. We don't take that to mean that humans can't have rights.

To be clear, I am not arguing that we should typically stop wild animals from killing each other. Wild animals are not moral agents, and so are unable to violate rights. (Equally, we don't arrest babies if they bite people, even if I'd expect to be arrested if I bit you.) If a gazelle has a right not to be killed, that means that we cannot shoot the gazelle. It doesn't mean that lions can't hunt gazelles.