# Math describes sheep herd fluctuations

## Equations quantify periodic spreading, clustering

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There’s something in the way sheep move.

In a herd, Merino sheep follow a predictable pattern of spreading out and clustering together. Now scientists have developed equations that can describe those movements. The sheep’s choreography may allow them to balance their needs for food and protection, researchers report September 28 in the Proceedings of the National Academy of Sciences.

“This is the first quantitative study of this kind of behavior,”says study coauthor Francesco Ginelli, a collective animal behavior and active-matter researcher at the University of Aberdeen in Scotland. Ginelli and his colleagues developed equations that describe how the sheep move and respond to their neighbors. The results suggest that a herd of sheep may exist in a delicate balance, close to a “tipping point” between dispersing and huddling together, says Andrea Cavagna, a statistical physicist at the Institute for Complex Systems of the National Research Council in Rome.

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An observed group of 100 ewes slowly drifted apart while grazing, only to suddenly clump back together roughly every 15 minutes, the researchers report. The behavior occurred unprompted, without any nearby predator to spook the herd. Sheep toward the outside of the herd seemed to initiate each woolly avalanche by running toward the center of the group, tailed by their neighbors.

 Sheep follow fluctuating patterns of spreading out and clumping together. Sheep on the outskirts of a herd initiate a clustering response by running towards the rest of the group.F. Ginelli et al/PNAS 2015

Ginelli and his coauthors believe that a herd’s oscillations balance two ovine interests: maximizing grazing space, and having safety in numbers should a predator approach. “The interpretation —balancing these two things — it’s simple and elegant,” Cavagna says. The mathematical model in the study supports this proposed behavior, he says.

Future experiments will examine larger groups of sheep, and may explore what happens if a single sheep is actually spooked by something, Ginelli says. He believes these results may help scientists understand the behavior of other groups of animals — for example, humans fleeing a burning building.

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