They are missing the third part of the rotation -- chickens.
Imagine a conversation between two chickens coming upon that pasture:
Henny Penny: "Hey look, there's a steaming pile of cowshit over there."
Chicken Little: "Wanna check it out? Maybe there's a bunch of juicy
maggots in it."
Henny Penny: "Yum! Watch and see how far I can fling this when I scratch."
Chickens have gross eating habits. But by searching out all the maggots they can find, they do control the fly population.