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Yes they are introduced species around the world. Originally from South America they are the only species of parrot who build their own communal nests rather than nesting in tree cavities. In fact these communal nests are often so large they actually can create homes for other species. Due to this and the fact that in my they have a small population size and live in urban areas almost exclusively they tend to be categorized as naturalized rather than invasive.
They stay right where they are when it's cold. Being from the foothills of the Andes they can handle cold weather. I've seen them flying around when it was below freezing. As to what they eat they are herbivorous, eating fruit seeds, flowers and leaves. More specifically in Brooklyn they have been observed eating sunflower seeds, dandelion flowers, crabapples, and grasses.