Wild pigs unfortunately are a threat to habitats and destroying more than just other animals ability to live, they are destroying farm land. I live in a state in the USA that has a major feral pig problem.
Deer hunting in every state is closely regulated with specific seasons, not indiscriminately killing of Deer and Deer are not Domesticated animals, never have been. Deer do carry diseases that (i.e. Lime Disease) that are dangerous to humans.
Here in Michigan, there are over 60,000 deer/auto accidents every year with most of them occurring here in S.E. Michigan where deer inhabit small forested areas among subdivisions where hunting is prohibited.
In fact, if you would like some fresh venison, there is a dead deer that was hit by a car at the entrance of my sub yesterday. Come and get it......
But I digress, what is your point? What is your solution to the rat infestation in New York? Trap and sterilize and release?
Deer are native to the country.