They scamper across the grass at Lafayette Square, peek out from behind flowers and bushes, dig for nuts buried in the ground, climb into trash cans and scramble up trees to get peanuts left on low branches by animals lovers.

They are not the bushy-tailed squirrels that by day overpopulate one of the nation's most historic squares. They are rats, the squirrels' bare-tailed, nocturnal cousins.

There are about 130 squirrels in the small square opposite the White House -- so many, the National Park Service says, that they are killing or stunting the park's trees by eating their small branches.

And there are perhaps hundreds of rats as well, though there is no scientific method of counting rats. All the experts will concede is that when you've seen one, you haven't seen them all.

The National Park Service admits that rodents -- squirrels by day and rats, usually by night -- are running rampant in Lafayette Square, and agrees that something should be done about it. The question is, what?

The Park Service now has a ranger working almost full time on the squirrel problem and recently hired a Pied Piper -- the John Muir Institute of California -- to devise an environmentally safe way of getting the rats and their round holes out of the square.

The institute is also studying environmentally safe ways of reducing the number of mosquitoes along the C&O Canal, Japanese beetles around the Tidal Basin, lace bugs that live on the azaleas and yellow jackets, which every summer sting more than 1,000 visitors to Great Falls Park.

But the insititute's fall project is Lafayette Square's rats, which can be seen brazenly running about the park from early evening until after dawn. Even in the middle of the day they occasionally come out to compete with squirrels and pigeons for handouts from the park's lunch-time crowds.

"I've seen them climb up trees to get peanuts people put in the tree crotches," says Park Service urban wild-life ranger David Manski, who spends his days at the square trying to solve the squirrel problem, which has defied the Park Service for more than 30 years.

At night, motorists, pedestrians and joggers can see the rats scurrying across park paths, across Pennsylvania Avenue to the White House grounds and across side streets to the Dolley Madison House and the New Executive Office Building. Their holes can be seen at the bases of the park's statues and trees, around its fountains and restrooms and even in the middle of its lawns.

Since the founding of the nation's capital, rats have burrowed with equal lack of respect under the Capitol, the Supreme Court, the nation's monuments, most downtown office buildings and Washington homes from Georgetown to Anacostia.

Not surprising actually since rats and cockroaches have been man's companions and household pests far longer than dogs and cats have been houshold pets.

Not all federal buildings here have rat problems, however. The White House has no rats inside, according to the Park Service, although there was a small invasion of rats at the White House grounds this summer, apparently caused by the demolition of buildings nearby on Pennsylvania Avenue.

Officials say the Kennedy Center has no rat problem at all inside the building -- just a mouse problem.

But the rats of Lafayette Square seem as firmly entrenched underground as the squirrels are above. Both thrive on the large quantities of peanuts, popcorn and snack foods generously sprinkled around the park by humans. Several squirrel lovers leave hundreds of pounds of peanuts in the park every month, says Kevin Hackett, head of the Muir Institute's project for the Park Service.

Rat poison alone -- which has been put down park rat holes regularly for years -- simply has not worked, Hackett said. "Our main effort will be to remove the rats' food and shelter," and in particular to end the freeloading on peanuts. Hackett also said the bases of statues, fountains and buildings will be rat-proofed. 25 million Europeans in the 14th century was spread by fleas from infected rats.

Only seven rat bites have been reported in the District so far this year. None was considered serious or caused any illnesses, according to city officials. A four-month-old Fairfax County child died of rat bites in 1972; it has been the only rat-related death in the Washington area in more than 20 years.

Since 1969, the federal government has been providing $13 to $15 million a year to more than 50 communities, including Washington, Baltimore and the Norfolk area, to help combat rat problems in residential neighborhoods.

"Washington's program has been one of the best," said John Gallagher, who heads the War on Rats program at the federal Center for Disease Control in Atlanta.

The war" began in the Shaw area, where surveys found more than 40 percent of the houses had rats. It later was extended to Anacostia, where more, than 70 percent of the buildings were rat-infested, then to Capitol Hill, Trinidad, LeDroit Park, Brightwood and, this summer, near North Capitol Street, where 11 percent of the homes reportedly have rats.

The overall infestation rate in the city now is only about 4 percent, says James E. Murphy. Most of the $520,000 in federal War on Rats money and $75,000 in city funds will pay for Murphy's staff of 54 to canvass rat-infested neighborhoods, distribute 23,000 pounds of rat poison and improve neighborhood trash disposal to remove sources of food.

"Garbage . . . food is one of the biggest problems," Murphy said. As for the Lafayette Square rats, he chuckled, "I don't know. That's a Park Service problem . . . but all those peanuts don't help."