The fishermen, who are also cross about large cruise shops being banned from docking on the islands, have surrounded the Charles Darwin research station, known for breeding giant tortoises.
They are refusing to let food or supplies be brought into the compound.
Inside the compound, on the island of Santa Cruz, are a colony of the rare tortoises and 33 people, who are mainly scientists.
The fishermen's protest has been peaceful so far but there are big worries about the safety of the scientists and the tortoises.
Campaigners have told the government of Ecuador to stick with the bans to keep the local environment healthy.