Equivalence between short- and long-distance dispersal in individual animal movement

Abstract

Random walks (RW) provide a useful modelling framework for the movement of animals at an individual level. If the RW is uncorrelated and unbiased such that the direction of movement is completely random, the dispersal is characterised by the statistical properties of the probability distribution of step lengths, or the dispersal kernel. Whether an individual exhibits short-or long-distance dispersal can be distinguished by the rate of asymptotic decay in the end-tail of the distribution of step-lengths. If the decay is exponential or faster, referred to as a thin-tail, then the step length variance is finite -as occurs in Brownian motion. On the other hand, inverse power-law step length distributions have a heavy end-tail with slower decay, resulting in an infinite step length variance, which is the hallmark of a Lévy walk. Although different approaches to relate these different dispersal mechanisms have been used, they are ad hoc and sub-optimal. We provide a more robust method by ensuring that the survival probability, that is the probability of occurrence of steps longer than a fixed characteristic step length is the same for both distributions. Moreover, we derive an optimal value for the survival probability by minimising the L2-distance between the dispersal kernels. By computing the optimal probability for movement paths with commonly used thin- and heavy-tailed step length distributions, we form equivalence between short-and long-distance dispersal of animals in different spatial dimensions. We also demonstrate how our findings can be applied to ecological scenarios, to more accurately relate dispersal mechanisms within a modelling framework for spatio-temporal population dynamics.