Movement patterns of Tenebrio beetles demonstrate empirically that correlated-random-walks have similitude with a Lévy walk

OPEN SUBJECT AREAS: BIOLOGICAL MODELS BEHAVIOURAL ECOLOGY Received 15 July 2013 Accepted 21 October 2013 Published 7 November 2013 Correspondence and requests for materials should be addressed to A.M.R. (andy. reynolds@rothamsted. ac.uk) Movement patterns of Tenebrio beetles demonstrate empirically that correlated-random-walks have similitude with a Lévy walk Andy M. Reynolds1, Lisa Leprêtre2,3 & David A. Bohan3 1 Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK, 2Université de Bourgogne, UMR CNRS 6282 Biogéosciences, F-21000 Dijon, France, 3INRA, UMR1347 Agroécologie, BP 86510, F-21000 Dijon, France. Correlated random walks are the dominant conceptual framework for modelling and interpreting organism movement patterns. Recent years have witnessed a stream of high profile publications reporting that many organisms perform Lévy walks; movement patterns that seemingly stand apart from the correlated random walk paradigm because they are discrete and scale-free rather than continuous and scale-finite. Our new study of the movement patterns of Tenebrio molitor beetles in unchanging, featureless arenas provides the first empirical support for a remarkable and deep theoretical synthesis that unites correlated random walks and Lévy walks. It demonstrates that the two models are complementary rather than competing descriptions of movement pattern data and shows that correlated random walks are a part of the Lévy walk family. It follows from this that vast numbers of Lévy walkers could be hiding in plain sight. T urchin1 argued that correlated random walks are the most influential and practicable approach to describing animal movement patterns. In these models, an individual’s trajectory through space is typically regarded as being made up of a sequence of distinct, independent, randomly oriented ‘steps’. It has long been recognized that the transformation of an animal’s continuous movement path into a broken line is necessarily arbitrary and that probability distributions of steps lengths and turning angles are model artifacts1. Nonetheless, for most researchers this shortcoming is not important on a practical level (smoothing out the simulation data will not really change anything) and that the ‘problem’ can be fixed using ‘continuous-time’ correlated random walk models in which velocities evolve as a Markovian process2–4. This position changed somewhat when it was realized that, as a result of auto-correlation of velocities, continuous-time correlated random walk models produce Lévy walk movement patterns5; an increasingly popular but controversial model of animal movement patterns6–8. Lévy walks comprise clusters of short step lengths with longer movements between them. This pattern is repeated across all scales with the resultant clusters creating fractal patterns that have no characteristic scale because the root-mean-square step-length is a divergent quantity. The distribution of step lengths has a powerlaw tail, pl(l) , l2m where 1 , m , 3. Lévy walks are controversial, in part, because many early studies had wrongly ascribed Lévy walks to some species through the use of inappropriate statistical techniques9,10. More recently, however, studies have provided compelling evidence that many organisms (a diverse range of marine predators, honeybees, mussels, Escherichia coli, T-cells) display movement patterns that can be approximated by Lévy walks, with m < 2, and these have been attributed to the execution of an innate, advantageous searching strategy11–18. Continuous-time correlated random walks do, however, present different Lévy walks characteristics. First, because they are present only over timescales shorter than the velocity autocorrelation timescale. Over longer timescales when there is complete de-correlation of velocities, motion is normally diffusive. Consequently continuous-time correlated random walks are like truncated Lévy walks. Second, the Lévy (power-law) exponent, m, is 4/3 and so different from the exponents found in many previous empirical studies which are typically around 2. Nonetheless, the theoretical ‘duality’ of continuous-time correlated random walks and Lévy walks with m 5 4/3 suggests that correlated random walks are also part of the Lévy family and that the binary arguments surrounding the ‘Levy flight foraging hypothesis’ are misguided as organisms move in ways that are well approximated by various types of Lévy walks, correlated random walks being but one5. It is also significant because correlated random walk modellers and state-space modellers, who rely heavily on the correlated random walk framework, SCIENTIFIC REPORTS | 3 : 3158 | DOI: 10.1038/srep03158 1 www.nature.com/scientificreports have not really embraced or understood the importance or usefulness of Lévy walks as models of animal movement patterns6,7. It remains to be seen, however, whether the ‘duality’ is realized in practice. Here we demonstrate that the production of Lévy walk movement patterns by continuous-time correlated random walk models, i.e., by biologically realistic correlated random walk models, is not just a theoretical, counter-intuitive curiosity resulting from an artifact of correlated random walk modelling. High-resolution movement pattern data for Tenebrio molitor L. beetles are shown to be explained both by simple continuous-time correlated random models and by distributions of distances travelled between consecutive turns that have heavy power-law tails – the hallmark of Lévy walks. We will show that the movement patterns of Tenebrio beetles are consistent with theoretical expectations for one of the simplest continuous-time correlated random walk models - the Langevin equation (an Ornstein-Uhlenbeck process for velocity), rffiffiffiffiffiffiffi u 2s2u djðt Þ du~{ dtz ð1Þ T T dx~udt where dj(t) is an incremental Wiener process with correlation property hdjðt ÞdjðtztÞi~dðtÞdt. According to this model, velocities, u, are Gaussian distributed with mean zero and variance s2u and are exponentially correlated on a timescale, T, i.e., RðtÞ:huðt ÞuðtztÞi =s2u ~e{t=T : Furthermore, conditional velocity increments are Gaussian with mean and variance u hdujui~{ dt T ð2Þ   2s2 s2duju : ðdu{hdujuiÞ2 ju ~ u dt T Mean-square displacements are described by h  i   xðt Þ2 ~2s2u tT{T 2 1{e{t=T ð3Þ   and are therefore ‘ballistic’ at short times (t , T) since xðt Þ2 ~s2u t 2   and ‘diffusive’ at long-times because xðt Þ2 ?2s2u Tt. The ballistic characteristic is typically associated with straight-line movements but can more generally be associated with Lévy walks19. We will also show that the distances travelled by the Tenebrio beetles between successive changes in the direction of travel, hereafter referred to as ‘steps’, have distributions with m 5 4/3 powerscaling across a broad range of scales, as predicted theoretically5. This is indicative of Lévy walk movements with Lévy exponent m 5 4/3. We will thereby demonstrate that th (...truncated)