Salt Lake Tribune

Published: 05/07/1998

Keywords: Research; Animals; Wildlife


Pack Math

Utah researcher discovers patterns in the way that carnivores mark territory; Researcher Finds Patterns in Pack Movements




Most people appreciate the visual splendor of the natural world. But nature also harbors a hidden beauty: living things organizing themselves according to mathematical rules.

            For example, the distance between ponderosa pines in a pure stand is proportional to the trees' size. Or consider how diversity of insects rises in a mathematically predictable way as you approach the boundary between a forest and a meadow -- a transition zone with more niches for insects.

            University of Utah mathematician Mark Lewis has found another natural mathematical pattern. He developed a set of equations to describe the way wolf and coyote packs establish territorial boundaries by scenting the ground with urine and moving away from scent marks of other packs.

            “There are a lot of patterns in nature,” Lewis said. “Mathematics can be used to understand the patterns and understand the mechanisms that give rise to the patterns.”

            Lewis discussed his findings during an interview and the U.'s Science at Breakfast lecture Wednesday in Salt Lake City.

            Federal officials and Lewis said the research does not have immediate use for predicting where endangered wolves might settle as they are reintroduced into such areas as central Idaho or Wyoming's Yellowstone National Park. For that, the equations must be improved to account not only for wolf movements and urination, but geographical barriers, the availability of prey and presence of humans.

            Nevertheless, “this is valuable research,” said Douglas Smith, Yellowstone wolf project leader for the National Park Service. “It helps you understand how wolves function behaviorally in the wild, how their territorial boundaries get set and change year to year, and how one wolf pack interacts with another wolf pack. It's behavioral ecology -- how animals space themselves.”

            Lewis has studied the mathematics of territoriality since his 1991 postdoctoral research days tracking gray wolves in Minnesota, the only place in the lower 48 states where they were not wiped out. Yellowstone reintroduced gray wolves in 1995.

            Lewis and colleagues started with estimates of how wolves move and leave urine scent marks. He said the animals can travel 5 mph, and when they encounter scent marks from another pack, they tend to move away from them.

            “The wolf sees the world through its nose like we see it through our eyes. Scent tells them where their prey is and where other wolves' scent marks are. Scents give wolves sort of an information superhighway to navigate around their territories.”

            While most members of a pack urinate by squatting anywhere in the territory, dominant wolves also perform “raised-leg urination” on the boundaries of their territory, particularly atop scents left by other packs. So the concentration of scent marks is heaviest along territory boundaries.

            Lewis converted such information into a set of formulas -- called a model -- describing how individual wolves behave “in terms of the speed they move, the direction they move and the rate at which they mark their territories.”

            Next, he took the equations describing behavior of individual wolves and translated them into equations describing “the expected density of wolves in an area -- the probability of encountering a wolf at a given point in space and time.”

            Those equations then were put in a computer to simulate or “model” how real wolves define their territorial boundaries. The results were compared with real observations of the density of real wolves or other carnivores.

            In a 1993 study in the journal Nature, Lewis and University of Washington mathematician James Murray showed the computer simulation accurately reflected how wolf packs in Minnesota established territories separated by mile-wide buffer zones.

The study showed that because wolves either stay out of buffer zones or fight each other when they meet, buffer zones are safe havens for deer. Field studies showed that is exactly what happens.

Over the years, Lewis refined his formulas. In a new study, equations describing how wolves and other carnivores set territory boundaries were tested by Lewis and biologists Paul Moorcroft of Princeton University and Robert Crabtree of Montana State University.

            They used Lewis' computer simulation to predict how six coyote packs define territories on flat ground near Hanford Nuclear Reservation in Washington state. The computer-generated map of the coyote territories closely matched real boundaries observed at the site when scientists tracked coyotes with radio collars.

Before Lewis' work, equations to predict where wolves establish boundary lines were statistical, meaning they were designed to match observations of where wolves were seen. They are more likely to be seen near a den than at the edge of their territory.

Lewis' method isn't just a set of formulas designed to make predicted territory lines match real ones. Instead, his equations result in maps of territory boundaries based on behavior, namely, how wolves urinate on other packs' scent marks and move away from those foreign scent marks.

Lewis said his ultimate goal “is to make predictions about where wolves move and where we can expect to find them over the long term.  People are concerned about the interactions with humans and livestock.  People are concerned about the impact on prey populations such as the

elk in Yellowstone. . . . The management of wolves requires that we know where they are.”

Ed Bangs, wolf recovery coordinator for the U.S. Fish and Wildlife Service in Helena, Mont., said he wanted to review Lewis' research but wasn't sure how it might help his efforts to reintroduce wolves in Idaho and Yellowstone.

Wolves once lived anywhere where prey was available in North America north of Mexico City, and still could live anywhere, Bangs said. “The reason they don't is people kill them. So if you try to predict where wolves live, you're talking about where people will let them live.”

Other mathematical equations show the density of wolves on a parcel is directly related to the number of prey animals, Bangs said. He said any computer model that predicts where wolves live also must include natural boundaries such as lakes, ridges and highways.

Nevertheless, Lewis' research reveals the mathematical simplicity of how wolves define their turf. “A significant feature of the study is that the seemingly complex formation of wolf territories can be reduced to a relatively simple formula involving scent marking,” said

Encyclopedia Britannica's 1995 annual supplement.

Lewis said his equations describing how wolves and coyotes set boundaries could be used for other mammals -- such as hyenas and badgers -- by modifying the math to reflect the behavior of those animals.

He said it would be intriguing to use the method to study human territoriality. “Human societies have buffer zones,” he said, showing a slide of the Berlin Wall before it was torn down.

Lewis cited a 1974 study that mapped gang territories in Philadelphia based on locations of graffiti and the street-corner hangouts.

Territory formation “is not restricted to wolves,” Lewis said, but to devise equations describing gang territories would require a new set of equations that incorporate rules of gang behavior.