Apparent power-law distributions in animal movements can arise from intraspecific interactions

Abstract

Lévy flights have gained prominence for analysis of animal movement. In a Lévy flight, step-lengths are drawn from a heavy-tailed distribution such as a power law (PL), and a large number of empirical demonstrations have been published. Others, however, have suggested that animal movement is ill fit by PL distributions or contend a state-switching process better explains apparent Lévy flight movement patterns. We used a mix of direct behavioural observations and GPS tracking to understand step-length patterns in females of two related butterflies. We initially found movement in one species (Euphydryas editha taylori) was best fit by a bounded PL, evidence of a Lévy flight, while the other (Euphydryas phaeton) was best fit by an exponential distribution. Subsequent analyses introduced additional candidate models and used behavioural observations to sort steps based on intraspecific interactions (interactions were rare in E. phaeton but common in E. e. taylori). These analyses showed a mixed-exponential is favoured over the bounded PL for E. e. taylori and that when step-lengths were sorted into states based on the influence of harassing conspecific males, both states were best fit by simple exponential distributions. The direct behavioural observations allowed us to infer the underlying behavioural mechanism is a state-switching process driven by intraspecific interactions rather than a Lévy flight.

Keywords: Lévy flight; animal movement; animal movement models; correlated random walk; diffusion; optimal search.

Figure 1.

Histograms (a,d,g) and fitted distributions in linear–linear (b,e,h) and log–log (c,f,i) space for the step-lengths of the three tracked butterfly populations.

Figure 2.

Raw tracking data from the single population of E. e. taylori flying in two adjacent fields (a,b) separated by 200 m. The panels show 40 pathways collected over the course of 5 days, with a total of 674 observed steps and a total of 9.07 h of observation time. Coloured points indicate sites where the butterflies alighted (cyan) or were courted/harassed by males (red). Blue arrows indicate steps where the butterflies left the field and could not be followed. Axes are latitude and longitude. Here, the tracks from the two fields have been separated so they can be more easily plotted; they are plotted on the same axes in electronic supplementary material, appendix B (figure B1) to show landscape orientation. Data from the two fields were combined. Individual pathways for a set of 16 example individuals are shown in the electronic supplementary material, appendix D figure D2.

Figure 3.

Raw tracking data from E. phaeton from two separate populations. (a) Movement pathways from the Bullitt population: 25 individuals observed over a span of 5 days, with a total observation time of 6.25 h during which 483 step were observed. (b) Movement pathways from the SCP population: 20 individuals observed over a span of 3 days, with a total observation time of 5 hours during which 573 step were observed. Coloured points indicate sites where the butterflies alighted (cyan) or were courted/harassed by males (red). Individual pathways for a set of 16 example individuals are show in the electronic supplementary material, appendix D, figure D1.

Figure 4.

Histograms showing (a) all step-lengths, (b) harassed step-lengths and (c) unharassed step-lengths from the E. e. taylori population. Steps longer than 40 m are not shown so that data can be compared on the same axes.