According to Genesis, God assigned the job to nature. It says that God commanded that the waters and the earth bring forth all the living creatures, and they did. That's what science says happened, too.
That is not quite right on either account at least as I see it. The next verse in the narrative after the one you are sorta quoting clearly states that God created the critters and told them to multiply AFTER THEIR KIND.
It is both true that 1) God assigned the task of creating all living creatures to nature and 2) that God created all living creatures, since he created nature to which he assigned the task.
