Fundamental insights into the random movement of animals from a single distance-related statistic

Statistical theories of animal movement have often been based on models of random walks, where movements take place in discrete steps and occur at discrete times. The multiplicity of distributions required in these approaches to describe animal movement (i.e., the distributions of angles, discrete steps, and times) have effects that cannot be simply disentangled, and hence they cannot be unambiguously determined. Here we present a mathematical formulation of continuous animal movements. In this new framework, it is shown that a single time-dependent distance statistic, the mean square displacement, which may be directly measured or mathematically modeled, is a central determinant of such random walks and encapsulates key information about the statistical properties of animal movements. The model and methodology presented here not only allow the determination of what were previously viewed as independent aspects of animal movements, such as the distribution of angular changes in direction, but also, because of the new emphasis on the mean square displacement, they may open up a new set of questions concerning animal movement and related phenomena. The results established in this work are directly applied to the foraging behavior of Pharaoh's ants, and very close agreement is found between observation and theory.