Elementary stuff really, given the vast abundance of prey, the big cat population would have exploded, trouble is, we have no evidence of it.
We can't have our cake and eat it, either we go along with the idea of the odd escapee over the years or we go along with a healthy population of big cats. Given the scant evidence the more likely theory is the former, but the fact that we have sightings in the here and now is a contradiction, unless people are constantly releasing animals in to the wild to this day.
Predators and prey - How are populations affected by conditions in an ecosyst