— Judas 760 knew just where to swim last fall after federal trappers set him free: back to his home in the marshes of Blackwater National Wildlife Refuge, where other nutria lived.

But true to his name, Judas betrayed members of his colony by providing a virtual road map through dense cattails and inky inlets via a tiny GPS unit on his back. Trappers followed and, in a scene played out with other Judases, exterminated a handful of the destructive rodents that have been responsible for denuding thousands of acres on the Eastern Shore.

"They are social animals and we are exploiting their Achilles heel," said Steve Kendrot, a biologist for the U.S. Department of Agriculture and leader of the Chesapeake Bay Nutria Eradication Project, which has killed more than 13,000 nutria since 2002.

Acre by watery acre — 150,000 in all — trappers have reclaimed the wildlife refuge and private property in five counties, allowing the marshes to heal and once again become fertile sanctuaries for crabs, oysters and native wildlife. This fall, they have turned their attention to 350,000 acres of potential nutria habitat and places where nutria have been detected, such as the Wicomico River south of Salisbury and a peninsula in Somerset County.

While there is no way of knowing how many nutria remain, only how many are gone, the goal of the take-no-prisoners campaign is to make the Delmarva Peninsula nutria-free by the end of 2015. Now comes the really hard part.

"We're at a crossroads," said Jonathan McKnight, invasive species biologist for the Maryland Department of Natural Resources. "It turns out that it was easy to kill 13,000 of them. It's much harder to finish the job, to find the small pockets. And given how quickly they reproduce, you can't kill all the nutria but two. You have to kill all the nutria."

But looking for a single animal or a pair of them is like trying to find a black cat on a moonless night.

"The enormity of the task hits you when you step out of your boat and see how much real estate is around you," Kendrot acknowledged.

Having nearly exhausted the possibilities of the traditional tools of trapping and shooting, scientists are embracing more modern technology — like the Judas experiment — to target the remaining animals, and concocting new scents to lure them to traps.

Orange-toothed and web-footed, nutria are an invasive species brought to the United States from South America by the fur industry in the late 1880s. Newspaper ads taken out by companies selling nutria touted pelts "next to mink" in softness and costing less than 2 cents a day to feed. But when the market collapsed in the 1940s, ranchers opened cage doors, unleashing the full force of a creature that breeds at a frightening rate and devours marshlands, leaving mud flats behind.

Nutria established colonies in 16 states, from Oregon's rivers to the Mississippi Delta. In Louisiana, what started as 20 animals in 1938 became 20 million animals two decades later. Nutria that were released or escaped from fur farms where Blackwater stands took refuge in the marshes and then expanded their range throughout the lower Eastern Shore and into Delaware and Virginia.

"They are baby-making machines," Kendrot said. "The adult females are pregnant 80 percent of their lives. They mate for life, just not with the same animal."

Aerial photographs taken at Blackwater during the last three decades show just how destructive the beaver-like creatures are. Vast tracts of vegetation, more than 5,000 acres in all, were rototilled and eaten by nutria, creating open expanses of water. Without plant roots, erosion continued unchecked, increasing the loss of wetlands and allowing sediment and contaminants to flow into the Chesapeake Bay.

An economic report by Southwick Associates, a Florida-based wildlife consulting firm, concluded that Blackwater nutria cost Maryland $4 million annually in commercial and recreational shell fishing and fishing opportunities, as well as hunting and trapping activities. Left unchecked, economic losses caused by nutria damage on the Eastern Shore would exceed $30 million annually by 2050.

"The costs of nutria removal are potentially minor compared to the costs to the state if the loss of marsh habitats continue unabated," the report concluded.

About 30 federal, state and private agencies pooled their resources in 2000 to form a management team. Congress provided a $1.5 million annual budget. After conducting an assessment, the Agriculture Department deployed 19 trappers at Blackwater to destroy nutria, grid by grid.

They killed 5,000 in 2003, then saw the number drop to 250 the next year and fall further in each subsequent year. So far this year, Blackwater has been nutria-free.

"Had we stopped after the third year, we'd be right back where we started," Kendrot said.

In 2007 and 2008, trappers moved to the Choptank River and Ellis Bay, then to Deal Island, where they killed 1,000 nutria. It is painstaking labor, with swarms of mosquitoes and armies of chiggers in summer and freezing water and mud in the winter. Every ribbon of water must be checked for signs of burrowing and muddy footprints.