IMHO, Cooperation or Competition tells more about the mindset of the one uttering the word than about the reality that is happening.
Of course, humans are capable of deliberately taking one stance or the other, like cooperating in P2P networks, or competing in sports. And of course, different strategy might well be at play at different levels (for P2P, it could be cooperation the work, but competing for customer contracts).
But when it comes to nature, apart from saying that one species is feeding on the other… what else can we say without blatantly exposing a bias? Not much I think.
Nature “just is” or does what nature does. It’s just living creature doing the stuff that makes creatures live (something Maturana & Varela have expressed much clearer (?) than myself). It’s permanent co-evolution. Sometimes in simple 1:1 systems like predator-prey, sometimes through more complex ones (Like in this video which you probably already know: https://www.youtube.com/watch?v=ysa5OBhXz-Q although the initial perspective probably only was the one of predator-prey — to me it’s just an equilibrium that was changed with unexpected consequences (for humans without requisite variety).