The following article is based on a subject that has not been officially named in any official Square Enix material; the current title is merely a placeholder.
The so-called "Circle of Life" in Final Fantasy XI is a relationship of family predatory relationships that exist between creatures in the game. It is more of a player-coined term, like SATA or Elemental Wheel.
Pictured easiest with the illustration on the right, there are multiple systems. The arrows show the predatory relationship between the different families of creatures. All creatures that have a predatory relationship versus the other, and have the equivalent of the player trait Killer. This means they have the chance to intimidate it.