Did you ever think that maybe that is why the previous owners didn't have a yard with grass? Maybe they had the same problems.
I have feral cats that have taken up residency here since they were kittens. They do their own hunting. It is nothing to see a bird, chipmunk, rabbit or a squirrel on my porch.
That story reminds me of the commercial with the two squirrels running in front of a car and when the car swerves and goes off the road the squirrels are giving each other the high 5...............maybe there are some intelligent ones out there...........and you have one coming into your yard.
Barb