Yes, from what I have seen, read, learned in school, etc about a 5-7 year cycle is correct. And yes, a host of other predators will move in to fill the niche when the alpha is removed or lessened. I do not believe it would be realistic to expect to remove ALL the coyotes from the area, statistically, you have to remove something like 75-80% of the breeding population for 7-10 years. I hunt em alot, ALOT, but not that much. I have spoken to ranchers and a host of trappers and none have been able to "remove" the presence of established coyotes from an area, however some have brought populations to a managable level.
A scalpel can be just as effective as a broadsword
Obviously, Occam was not a reloader