Yeah in Louisana they go out in trucks with guys in the bed with rifles and a big spotlight. They shine the canals for the nutria. They have to hunt them like that regularly to keep the populations down.
What about rabbits? They are destructive too. There is no feesable way to put a stop to any of it unless you can go back in time and prevent oceanic travel from ever happening. No boats, no immigrations, no settlers, nothing. Even then you have to consider all the countries that are connected. It just isnt preventable. Period.
Go back in time and tell the egyptions that they cant have ball pythons as pets. Tell Xerxes he cant take all those exotic animals from their homeland to show his wealth and power. It goes on and on.. Cant stop it.