I have a friend who use to own a pet store that specialized in bird's. He told me they were imported to be sold as pets (low dollar one's).
When they first started showing up here, I googled them and read on some websight that during one of the hurricanes a few years back, several survived and escaped from pet store's in Florida and have been spreading across the U.S. every since.
Here in NM there is no limit but they have to be hunted during the regular dove season. But, since you can't discharge a firearm in the city limit's, the only way to get them is with a pellet gun. I havent tried that yet (and dont own a pellet gun) but I am seriously thinking about it. I have about 3 dozen of the things that took up permanent residence on my propery last spring, eating the food for out outside pets and sh**ing all over the place. We have had 3 hard frosts in the past week and they are still here.