Bridging short- and long-distance dispersal in individual animal movement

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. In theoretical studies of individual animal movement, various approaches have been employed to connect these dispersal mechanisms, yet they are often ad hoc. We provide a more robust method by ensuring that the survival probability, that is the probability of occurrence of steps longer than a certain threshold is the same for both distributions. Furthermore, the dispersal kernels are then standardised by adjusting the probability to minimise disparities between these distributions. By assuming the same survival probability for movement paths with commonly used thin- and heavy-tailed step length distributions, we form a relationship between the short- and long-distance dispersal of animals in different spatial dimensions. We also demonstrate how our findings can be applied in different ecological contexts, to relate dispersal kernels within theoretical models for boundary effects and spatio-temporal population dynamics. Moreover, we show that the relationship between these dispersal kernels can drastically affect the outcomes across various ecological scenarios.