I know of no legal obligation on the part of any landowner toward any other person, regarding free-ranging wild animals on one's land. Deer or rabbit, coyote or cougar. A landowner is not obligated to do anything whatsoever.
State law enters into the equation when one considers initiating some action. Then, it's a matter of seasons, bag limits, and protections. Licensing and all that. Regulations concerning trapping, as well. (This is separate from an actual self-defense situation.)
As for cougars, a five-mile circle of hunting in a night is commonplace. They're great cross-country travellers, known to cover over 30 or 40 miles in a 24-hour period. (Example: A young male with a radio collar was released in Black Gap WMA, down by the Rio Grande. Two nights later he was killed in a rancher's sheep pen in the Glass Mountains north of Marathon, Texas, some 70 or 80 miles from the release point.)