Ah, great topic!
This has been proven various times, Penn State did a study in the low country of SC on 200 sq miles (I believe) the findings that trapping predators, reduced populations for any duration of time were non-existent. As coyotes were removed, bobcats moved in, and visa/versa.
If you are wanting to kill coyotes in an effort to reduce populations - that is probably a lost cause. However, many great trappers have proven that you CAN positively impact fawn recruitment with coyote trapping, which I think is what most people want to do anyway.
This is a bit of a slippery slope, as you may trigger the compensatory reproduction response.
The way this works is simple. You trap coyotes, HEAVILY, right before/during the fawn drop.
You MUST continue to do this year after year.
Casey Shoopman of the MGT Advantage did a podcast on this and it was insightful. He was adamant about how intense the trapping must be (50-70, sometimes way more, coyotes a week being trapped on large swaths of land).
The end goal is not to reduce the coyote population, but simply put a gap in population density, relative to the area one hunts, during the most vulnerable time of year - fawn drop.