We use to notice this when we farmed. We would see rabbit populations grow substantially over two or three years and then they would decline as the fox population increased. Soon the foxes ran out of rabbits and they died or moved out and the rabbit population took off again. On one end of the cycle we were loosing crops to rabbits, on the other end we were loosing chickens to the foxes. No matter what, there never seemed to be a part of the cycle where the farmer won.
And professional wildlife management, with managed hunting seasons, can hunt more rabbits in one year and more foxes in another to balance out the cycles. If you ever want to see what happens when there are no predators and no hunting, go to Valley Forge National Park - it is infested with small deer that leave a browse line on trees like a weed eater has gone around at five feet off the ground and that eat just about everything in the understory of the forest - leaving a forest that will die off eventuall because no young trees can grow to replace those that die.
Yep, the prey/predator populations run in similar cycles but the predator cycle is a year or two out of phase with the prey cycle.