So did tyrannosaurus have rights? On the savannah do lions have rights? If so, what would it have meant to violate the rights of these creatures?
Implicit in your questions there appears to be the notion of 'why are we any better than the rest of the animals and plants on Earth?'
By saying "better", you seem to be saying that the notion of rights is ultimately a value statement. So, rights requires the ability to form values? Make judgements? How then could mere predation be sufficient?
humanity has the axiom that we are above the rest of the species on Earth
What do you mean by above? Do you mean that humans value humans over all other species? As a matter of survivability, one might argue that bacteria, or insects would have the more hardy species.
if we stoop to preying upon our own species members, we will be more like the other species
So the characteristic that makes humans unique from other species is that humans (potentially) don't kill members of their own species? What about oak trees? Do they kill their own?
I'm not convinced we know what we are talking about when we use the word "rights". I wonder the same about the word "human".