Regarding the OP, it’s easy to overlook the re-population possible from a few fertile animals. In my rural area, it’s common to assume that if you have a barn, you attract cats. Very few barn owners bother to do anything to or for the cats – they are there, they perform a service to keep the mice down, and if they get too populous, you drown the new kittens in the creek. It’s just another chore like slopping the hogs and mucking out the stables.
Even if every pet/animal owner in the county neutered their own charges, one fertile barn in the area would quickly supply replacements.