CANÓVANAS, Puerto Rico—Some here have dubbed it the "green plague"—an infestation of iguanas that are wreaking havoc on the island, chewing up plants and crops and burrowing under roads and dikes.

The reptiles, which aren't native to Puerto Rico, have few natural predators and are highly fertile. They have proliferated so uncontrollably that their population, estimated at roughly four million, now outnumbers humans in this U.S. territory.

"This...