Abstract
We examine the role that randomness or noise in individual motion may play in forming effective grazing strategies for herd members as they collectively move toward a destination. Through a model where animals are attracted to Voronoi neighbors as well as a destination endpoint, we show that including a significant random motion component can speed up the movement of a herd toward this destination, increase the efficiency that food is acquired during the travel, and facilitate a natural herd shape that mitigates predation risk. Specifically, if the influence of the Voronoi neighbors on individual motion is equal to the pull toward the destination, we find that optimal travel time and food consumption efficiency occurs for noise approximately twice as strong as the influence of herd members to each other, in a range of herd sizes from 10 to 100. We find that reducing the destination influence lowers this optimal noise only slightly, with random motion still exceeding the influence of neighbors. For a destination influence exceeding that of the Voronoi neighbors, an additional travel mode appears with minimal noise and aligned velocities in which the herd marches directly toward the endpoint. Our results are consistent with observational evidence of random motion in several animal groups, and motivate its generalization to traveling and grazing herds.