I've got a book somewhere (small trade paperback, dull silver cover[1], title might be Life) which claims that no one has ever gotten those pretty predator-prey equations to cycle nicely in the real world, not even with two species of micro-organisms in a test tube.

With 3 species, the LK equation can become chaotic, so I wouldn't expect to be able to duplicate a real-world history even if the model were perfect.

Perhaps we could find a 2-species real-world LK case involving bacteria deep underground.