I heard about this just yesterday.
These cycles are pretty basic as habitat increases, the prey animals increase, as the prey animals increase the predators increase, as the predators increase the prey animals decrease when the prey animals decrease the predators decrease. There is some illnesses/parasites that can also decrease the population of prey species.
With that said, you could kill all the predators if the habitat can't support a given species they can't live there.
Predators on the other hand will change their focus to a different species.
Biologists can determine the amount of food per acre in pounds a given landscape provides. Increasing food, removing non-native plants, re-introducing native vegetation. This increased habitat will increase not only the species there already but will attract new species.
Dynamics of Predation: https://www.nature.com/scitable/know...tion-13229468/