Modelling animal movement as a persistent random walk in two dimensions: expected magnitude of net displacement

Abstract

We present semi-empirical model of persistent random walk for studying animal movements in two-dimensions. The model incorporates an arbitrary distribution for the angles between successive steps in the tracks. Inclusion of a turning angle distribution enables explicit computation of the effect of persistence in the direction of travel on the expected magnitude of net displacement of the animal over time. We employed a form-analogous approach to obtain expressions for the expected net displacement and derived root mean square of the expected displacement of an animal at the end of a multi-step random walk in which turning angles were drawn from the Lemicon of Pascal, the elliptical, the von Mises, and the wrapped Cauchy distributions. The accuracy of these expressions for the expected magnitude of net displacement was tested by comparison with simulated results of persistent random walks where turning angles were drawn form the wrapped Cauchy distribution. Our results should be useful in predicting two-dimensional distribution of moving animals for which frequency distributions of the turning angles can be measured.