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Generally not recommended to introduce a new predator to deal with the invasive species... the new predator often becomes a worse problem.
I am currently stationed at Guantanamo Bay, Cuba and dive here every week. The lion fish has indeed made a big appearance here. So much that the local dive clubs have a Lion Fish Round-Up every month to keep them under control.
As far as the lion fish not having a natural predator in these waters, that is not true. We have seen Green Moray Eels on numerous occasions attack, kill and eat the lion fish.
If that is true then wouldn't hunting lionfish be counterproductive? If lionfish are left alone and then they become numerous then wouldn't the moray eels have a large supply of food? The the eels would become numerous and the lionfish population reduced. This would allow for a natural balance to occur instead of an artificial one imposed by man.
There are far more lion fish than the eels could eat. We go diving with a group and kill 30 to 40 in an area no more than the size of a couple of football fields. We generally only see the same five or six eels along the entire area surrounding the base. You are talking ratio's of 1000's to 1.