Appendix B. Details of the von Mises distribution of turning angles for Oreaster reticulatus.
We used a von Mises (or circular normal) distribution (Evans et al. 2000) to describe the turning angle distribution of Oreaster reticulatus in Model 1. The von Mises distribution describes the distribution of a random variate with period 2π. A high value of k results in a distribution of turning angles concentrated around a zero mean and a high degree of directional persistence (correlated random walk), whereas a low value of k results in a more uniform distribution of turning angles with no directional persistence (uncorrelated random walk) (Fig. B1). After simulating a correlated random walk in 1000 individuals for 4 moves (~4 h), we estimated a linearity index (distance between final and initial position divided by the total distance traveled) of 0.88. This is in close accord with empirical measures of linearity index (0.82 and 0.93, mean = 0.88) based on tracking movements of O. reticulatus over 4 h at two sand-bottom sites (Scheibling 1985).
FIG. B1. (A) Effect of the parameter k of the von Mises distribution on the distribution of turning angles. (B) Resulting foraging paths using probability distributions of turning angles in A. Five paths were simulated for each value of k. Each path is composed of 20 moves of length 1. Initial direction for the first move of each path is selected randomly from a uniform distribution of angles and the successive direction of each move are obtained by adding the direction of the previous move to a randomly selected turning angle from the appropriate von Mises distribution. |
LITERATURE CITED
Evans, M., N. Hastings, and B. Peacock. 2000. Statistical distributions. John Wiley and Sons, New York, New York, USA.
Scheibling, R. E. 1985. Directional movement in a sea star (Oreaster reticulatus): Adaptive significance and ecological consequences. Pages 244–256 in M. A. Rankin, editor. Animal migration: Mechanisms and adaptive significance. Contributions in Marine Science Supplement, Vol. 27.