Ivars Peterson's MathTrek

October 1, 2001

In the sport of orienteering, a competitor uses a detailed map (and perhaps a compass) to navigate his or her way across varied terrain following a course drawn on the map. Selecting the best available route, each participant races from one marker to the next in the required sequence. The winner is the person who completes the course in the shortest time.

Even in recreational orienteering, participants typically start off at staggered times. In that way, each person (or group) navigates the course individually and obtains no help from others on the course. In practice, however, competitors sometimes come into sight of each other. Followers can then benefit from the map-reading and route-selection skills of the leaders.

Recently, my son and I encountered just such a clumping phenomenon. At the orienteering meets in which we participate, orienteers set off at 2-minute intervals on a given course. We happened to catch up with the pair that had started 2 minutes before us. Meanwhile, the pair that had gone off 2 minutes after us also came into view. From that point on, we could all see one another most of the time. It was difficult to avoid being influenced by whomever happened to be in the lead at any given moment. All three pairs tended to stay within sight the rest of the way. In some sense, the group dynamics masked the "true" abilities of the individual pairs.

Physicists Graeme J. Ackland and David Butler of the University of Edinburgh address this sort of "pack formation" in the Sept. 13 Nature. They created a mathematical model to predict when packs would form in orienteering and cycling competitions. "Our results may prove useful in helping to stage events so that pack formation can be avoided," they noted.

The physicists integrated the relevant equations of motion for interacting competitors moving in one dimension, with passing allowed, from start to finish. Each competitor races at a characteristic speed u, which would increase by u x b (where b is a "boost" factor) when another competitor is within a certain range ahead. In orienteering, shared map-reading and route-selection skills provide the boost. In cycling, cyclists in a pack move faster than those going alone because they can take turns in front and share the benefits of reduced air resistance.

Maximal bunching occurs for a mass start. If the interval between starts is longer than the typical time to run a course, no pack forms. With constant initial separation (as in time trials) and a random distribution of competitor speeds, "packs are ultimately formed, but only after a considerable time," Ackland and Butler discovered.

The researchers found a sharp shift from individual to pack behavior at a certain critical ratio of range to starting interval. Below this critical ratio, most competitors move individually and tend to separate. Above the critical ratio, "packs form that catch and absorb individuals," they observed.

Ackland and Butler reported that the threshold depends on a single criterion: the fraction of competitors in a given event who encounter others and can stay with them. Their simulations showed the critical threshold to be about 13 percent.

In time trials, the onset of pack formation coincides with the time required to reach the critical threshold. The boost factor and distribution of characteristic speeds determine the pack size. Using the physicists' model, it's possible to work out optimal start intervals for different events.

The results appear qualitatively similar to those related to the formation of shock waves or traffic jams. "An important difference is that our packs tend to be composed of the same competitors throughout," Ackland and Butler said.

The physicists' findings may be particularly relevant to triathlons--events in which competitors swim, then cycle and run. Typically, swimming precedes cycling, so start times in the cycling segment are determined by swim times. The length of the swimming leg could be adjusted so that competitors are properly spaced out by the time they reach the cycling stage.

References:

Ackland, G.J., and D. Butler. 2001. Pack formation in cycling and orienteering. Nature 413(Sept. 13):127.

Pearson, H. 2001. Cyclists sent packing. Nature Science Update (Sept. 13). Available at http://www.nature.com/nsu/010913/010913-13.html.

Peterson, I. 2001. Waves of congestion. MAA Online (Sept. 10).

Graeme Ackland has a Web page at http://www.cmse.ed.ac.uk/ackland.html.