Andy M. Reynolds

Free-Flight Odor Tracking in Drosophila Is Consistent with an Optimal Intermittent Scale-Free Search

During their trajectories in still air, fruit flies (Drosophila melanogaster) explore their landscape using a series of straight flight paths punctuated by rapid 90° body-saccades [1]. Some saccades are triggered by visual expansion associated with collision avoidance. Yet many saccades are not triggered by visual cues, but rather appear spontaneously. Our analysis reveals that the control of these visually independent saccades and the flight intervals between them constitute an optimal scale-free active searching strategy. Two characteristics of mathematical optimality that are apparent during free-flight in Drosophila are inter-saccade interval lengths distributed according to an inverse square law, which does not vary across landscape scale, and 90° saccade angles, which increase the likelihood that territory will be revisited and thereby reduce the likelihood that near-by targets will be missed. We also show that searching is intermittent, such that active searching phases randomly alternate with relocation phases. Behaviorally, this intermittency is reflected in frequently occurring short, slow speed inter-saccade intervals randomly alternating with rarer, longer, faster inter-saccade intervals. Searching patterns that scale similarly across orders of magnitude of length (i.e., scale-free) have been revealed in animals as diverse as microzooplankton, bumblebees, albatrosses, and spider monkeys, but these do not appear to be optimised with respect to turning angle, whereas Drosophila free-flight search does. Also, intermittent searching patterns, such as those reported here for Drosophila, have been observed in foragers such as planktivorous fish and ground foraging birds. Our results with freely flying Drosophila may constitute the first reported example of searching behaviour that is both scale-free and intermittent.

The Lévy flight paradigm: random search patterns and mechanisms

Over recent years there has been an accumulation of evidence from a variety of experimental, theoretical, and field studies that many organisms use a movement strategy approximated by Lévy flights when they are searching for resources. Lévy flights are random movements that can maximize the efficiency of resource searches in uncertain environments. This is a highly significant finding because it suggests that Lévy flights provide a rigorous mathematical basis for separating out evolved, innate behaviors from environmental influences. We discuss recent developments in random-search theory, as well as the many different experimental and data collection initiatives that have investigated search strategies. Methods for trajectory construction and robust data analysis procedures are presented. The key to prediction and understanding does, however, lie in the elucidation of mechanisms underlying the observed patterns. We discuss candidate neurological, olfactory, and learning mechanisms for the emergence of Lévy flight patterns in some organisms, and note that convergence of behaviors along such different evolutionary pathways is not surprising given the energetic efficiencies that Lévy flight movement patterns confer.

Oceanic Turbulence and Stochastic Models from Subsurface Lagrangian Data for the Northwest Atlantic Ocean

The historical dataset provided by 700-m acoustically tracked floats is analyzed in different regions of the northwestern Atlantic Ocean. The goal is to characterize the main properties of the mesoscale turbulence and to explore Lagrangian stochastic models capable of describing them. The data analysis is carried out mostly in terms of Lagrangian velocity autocovariance and cross-covariance functions. In the Gulf Stream recirculation and extension regions, the autocovariances and cross covariances exhibit significant oscillatory patterns on time scales comparable to the Lagrangian decorrelation time scale. They are indicative of sub- and superdiffusive behaviors in the mean spreading of water particles. The main result of the paper is that the properties of Lagrangian data can be considered as a superposition of two different regimes associated with looping and nonlooping trajectories and that both regimes can be parameterized using a simple first-order Lagrangian stochastic model with spin parameter V. The spin couples the zonal and meridional velocity components, reproducing the effects of rotating coherent structures such as vortices and mesoscale eddies. It is considered as a random parameter whose probability distribution is approximately bimodal, reflecting the distribution of loopers (finite V) and nonloopers (zero V). This simple model is found to be very effective in reproducing the statistical properties of the data.

Radar Tracking and Motion-Sensitive Cameras on Flowers Reveal the Development of Pollinator Multi-Destination Routes over Large Spatial Scales

Automated tracking of bumblebees and computer simulations reveal how bees locate a series of flowers and optimize their routes to visit them all.

HOW MANY ANIMALS REALLY DO THE LÉVY WALK? COMMENT

Two major challenges in spatial ecology are understanding the effects of landscape heterogeneity on movement, and translating observations taken at small spatial and temporal scales into expected patterns at greater scales (Morales and Ellner 2002). Correlated random walk (CRW) models emerged from the analysis of short-scaled movement data acquired in experiments usually lasting less than an hour and performed in arenas extending over no more than several meters (Kareiva and Shigesada 1983, Bovet and Benhamou 1988, Turchin 1991). Analysis of animal movements over much larger spatial scales and/or longer temporal scales has given rise to Lévy walk models (LW, often referred to as Lévy flight models; Viswanathan et al. 1996, 1999, Atkinson et al. 2002). Recently, however, Edwards et al. (2007) questioned the claim of Lévy flight behavior in the Wandering Albatross, first reported by Viswanathan et al. (1996). Edwards et al. (2007) also identified some other data sets (Viswanathan et al. 1999) in which the analysis was unable to definitively discriminate the presence of Lévy flights. Bartumeus et al. (2005) argued that CRW can be interpreted as being the by-product of local scanning mechanisms, whereas LW have fundamental properties (super-diffusivity and scale invariance) that allow for higher search efficiencies in random search scenarios. This prompted them to propose that some animals may have evolved the ability to perform LW when confronted with uncertainty. In a recent Ecology report, Benhamou (2007) enriched the debate. He stressed two points: (1) that LW can look like foraging patterns in patchy environments, but in such an environment LW are far from being an optimal searching strategy; and (2) with the usual methodology it is easy to find apparent LW in composite Brownian random walks movements (hereafter referred to as composite CRW movements in accordance with standard terminology). Here I point out that Benhamou’s (2007) composite CRW model can, in fact, be interpreted as being an adaptive LW model and that, as a consequence, he has demonstrated that adaptive LW are better than nonadaptive LW when searching in patchy environments. I then generalize Benhamou’s second point by showing that heavy-tailed distributions of move lengths, a hallmark of LW, are almost inevitable when collating movement data acquired at observational scales that encompass heterogeneity. This elucidates the notion that the scale of observation influences the description of the pattern (Levin 1992). The emergence of an inverse-square power-law tail does, however, require rather special combinations of local movements and patterns of heterogeneity. This is at variance with the prevalence of such scaling in a diverse range of animals moving within a diverse range of environments (Atkinson et al. 2002, Bartumeus et al. 2003, Reynolds 2007a, b, Reynolds and Frye 2007). Finally, I demonstrate that intrinsic LW characteristics are quite robust with respect to subsampling and that, as a consequence, reports on LW stemming from analyses based on moves between arbitrary location fixes remain secure.

Bridging the gulf between correlated random walks and Levy walks: autocorrelation as a source of Levy walk movement patterns

For many years, the dominant conceptual framework for describing non-oriented animal movement patterns has been the correlated random walk (CRW) model in which an individuals trajectory through space is represented by a sequence of distinct, independent randomly oriented ‘moves’. It has long been recognized that the transformation of an animals continuous movement path into a broken line is necessarily arbitrary and that probability distributions of move lengths and turning angles are model artefacts. Continuous-time analogues of CRWs that overcome this inherent shortcoming have appeared in the literature and are gaining prominence. In these models, velocities evolve as a Markovian process and have exponential autocorrelation. Integration of the velocity process gives the position process. Here, through a simple scaling argument and through an exact analytical analysis, it is shown that autocorrelation inevitably leads to Lévy walk (LW) movement patterns on timescales less than the autocorrelation timescale. This is significant because over recent years there has been an accumulation of evidence from a variety of experimental and theoretical studies that many organisms have movement patterns that can be approximated by LWs, and there is now intense debate about the relative merits of CRWs and LWs as representations of non-orientated animal movement patterns.

Hierarchical random walks in trace fossils and the origin of optimal search behavior

Significance How best to search for food in heterogeneous landscapes is a universal problem facing mobile organisms. Diverse modern animals use a random search strategy called a Lévy walk, composed of many small move steps interspersed by rare long steps, which theoretically is optimal for locating sparse resources. Here, we find the first evidence, to our knowledge, that extinct animals, in this case 50 My-old sea urchins, used a Lévy-like search strategy. Our results are important because they indicate Lévy walks likely have an ancient origin and may arise from simple behaviors observed in much older fossil trails. This foraging strategy may have adapted in response to decreased food availability after productivity collapse associated with past climate change and mass extinctions. Efficient searching is crucial for timely location of food and other resources. Recent studies show that diverse living animals use a theoretically optimal scale-free random search for sparse resources known as a Lévy walk, but little is known of the origins and evolution of foraging behavior and the search strategies of extinct organisms. Here, using simulations of self-avoiding trace fossil trails, we show that randomly introduced strophotaxis (U-turns)—initiated by obstructions such as self-trail avoidance or innate cueing—leads to random looping patterns with clustering across increasing scales that is consistent with the presence of Lévy walks. This predicts that optimal Lévy searches may emerge from simple behaviors observed in fossil trails. We then analyzed fossilized trails of benthic marine organisms by using a novel path analysis technique and find the first evidence, to our knowledge, of Lévy-like search strategies in extinct animals. Our results show that simple search behaviors of extinct animals in heterogeneous environments give rise to hierarchically nested Brownian walk clusters that converge to optimal Lévy patterns. Primary productivity collapse and large-scale food scarcity characterizing mass extinctions evident in the fossil record may have triggered adaptation of optimal Lévy-like searches. The findings suggest that Lévy-like behavior has been used by foragers since at least the Eocene but may have a more ancient origin, which might explain recent widespread observations of such patterns among modern taxa.

Optimal random Lévy-loop searching: New insights into the searching behaviours of central-place foragers

A random Levy-looping model of searching is devised and optimal random Levy-looping searching strategies are identified for the location of a single target whose position is uncertain. An inverse-square power law distribution of loop lengths is shown to be optimal when the distance between the centre of the search and the target is much shorter than the size of the longest possible loop in the searching pattern. Optimal random Levy-looping searching patterns have recently been observed in the flight patterns of honeybees (Apis mellifera) when attempting to locate their hive and when searching after a known food source becomes depleted. It is suggested that the searching patterns of desert ants (Cataglyphis) are consistent with the adoption of an optimal Levy-looping searching strategy.

Chemotaxis can take plant-parasitic nematodes to the source of a chemo-attractant via the shortest possible routes

It has long been recognized that chemotaxis is the primary means by which nematodes locate host plants. Nonetheless, chemotaxis has received scant attention. We show that chemotaxis is predicted to take nematodes to a source of a chemo-attractant via the shortest possible routes through the labyrinth of air-filled or water-filled channels within a soil through which the attractant diffuses. There are just two provisos: (i) all of the channels through which the attractant diffuses are accessible to the nematodes and (ii) nematodes can resolve all chemical gradients no matter how small. Previously, this remarkable consequence of chemotaxis had gone unnoticed. The predictions are supported by experimental studies of the movement patterns of the root-knot nematodes Meloidogyne incognita and Meloidogyne graminicola in modified Y-chamber olfactometers filled with Pluronic gel. By providing two routes to a source of the attractant, one long and one short, our experiments, the first to demonstrate the routes taken by nematodes to plant roots, serve to test our predictions. Our data show that nematodes take the most direct route to their preferred hosts (as predicted) but often take the longest route towards poor hosts. We hypothesize that a complex of repellent and attractant chemicals influences the interaction between nematodes and their hosts.

Ballooning dispersal in arthropod taxa: conditions at take-off

We have solved a long-standing and seemingly paradoxical set of questions that relate to the conditions which govern spider ballooning. We show that observations of spider ballooning excursions are best explained by meteorological conditions which maximize dispersal. Dispersal is predicted to be most effective in terms of distance when the stability of the atmosphere is non-ideally convective and is less effective during purely convective or neutrally stable conditions. Ballooners are most likely to travel a few hundred metres, but dispersal distances of several hundred kilometres are possible.