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Bringing random walkers into new territory.

The introduction of muskrats into central Europe in 1905 afforded ecologists a unique opportunity to study the spread of a population. Five surveys taken during the succeeding 23 years revealed an intriguing pattern of expansion. As the territory occupied by the increasing population of muskrats grew larger, its initially smooth boundary became increasingly convoluted.

A team of researchers has now developed a mathematical model that provides a new avenue for studying the spread of populations. This model suggests that the roughening of a territorial boundary, as demonstrated by the rambling muskrats, occurs naturally in any situation in which groups of diffusing particles," whether muskrats or molecules, move at random from position to position.

The model's importance stems from the fact that researchers can express many different physical processes as random-walk problems. "The random motion of particles is at the root of much physics, chemistry and biology," says physicist and team member H. Eugene Stanley of Boston University.

The novel random-walk variation developed at Boston University arose out of a curriculum project designed to introduce concepts of randomness to high-school students. Going beyond the familiar, well-studied problem of depicting the territory covered by a single "walker" randomly wandering from square to square on a grid, project manager Paul Trunfio decided to look at the patterns formed by a swarm of random walkers, all starting simultaneously on the same square, as they independently explored the grid.

Trunfio's computer-generated patterns were sufficiently interesting to prompt further investigation. "We realized ... that this might be an unsolved problem in random walks," Stanley says. "We spent two or three months looking at computer images trying to discover the laws that seemed to govern the behavior we saw."

The computer experiments revealed that the set of visited sites initially has a relatively smooth boundary. But after the territory reaches a certain size, this edge grows increasingly jagged.

Graduate student Hernan Larralde, aided by Shlomo Havlin and George H. Weiss of the National Institutes of Health in Bethesda, Md., then worked out a precise, mathematical solution of the problem in one, two and three dimensions. "At first sight, it looks like a trivial problem to solve," Stanley says. "It was in fact a very difficult mathemtical problem."

The analysis unexpectedly revealed that this random-walk process goes through three distinct time regimes representing characteristic but different rates and patterns of growth. "We were surprised by the fact that a relatively simple question about random walks could be so rich," Stanley says. A paper detailing the findings appears in the Jan. 30 NATURE.

"The work of Larralde [and his co-workers] opens up a host of further possibilities -- of using interacting walkers, of working in fractal spaces ...and so on," Michael F. Shlesinger of the Office of Naval Research in Arlington, Va., comments in the same issue of NATURE. "These would find applications in fields as diverse as physics and ecology."

"What's nice about this particular problem is that it's sufficiently simple that you can get some feeling for the collective behavior of large numbers of particles," Weiss notes.
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Author:Peterson, Ivars
Publication:Science News
Date:Feb 8, 1992
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