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Lévy walks are a mathematical construction useful for describing random patterns of movement with bizarre fractal properties that seem to have no place in biology. Nonetheless, movement patterns resembling Lévy walks have been observed at scales ranging from the microscopic to the ecological. They have been seen in the molecular machinery operating within cells during intracellular trafficking, in the movement patterns of T cells within the brain, in DNA, in some molluscs, insects, fish, birds and mammals, in the airborne flights of spores and seeds, and in the collective movements of some animal groups. Lévy walks are also evident in trace fossils (ichnofossils) – the preserved form of tracks made by organisms that occupied ancient sea beds about 252-66 million years ago. And they are utilised by algae that originated around two billion years ago, and still exist today. In September of 2017, leading researchers from across the life sciences, along with mathematicians and physicists, got together at a Company of Biologists' Workshop to discuss the origins and biological significance of these movement patterns. In this Review the essence of the technical and sometimes heated discussions is distilled and made accessible for all. In just a few pages, the reader is taken from a gentle introduction to the frontiers of a very active field of scientific enquiry. What emerges is a fascinating story of a truly inter-disciplinary scientific endeavour that is seeking to better understand movement patterns occurring across all biological scales.

Sinhuber et al. (Sci Rep 11:3773, 2021) formulated an equation of state for laboratory swarms of the non-biting midge Chironomus riparius that holds true when the swarms are driven through thermodynamic cycles by the application external perturbations. The findings are significant because they demonstrate the surprising efficacy of classical equilibrium thermodynamics for quantitatively characterizing and predicting collective behaviour in biology. Nonetheless, the equation of state obtained by Sinhuber et al. (2021) is anomalous, lacking a physical analogue, making its’ interpretation problematic. Moreover, the dynamical processes underlying the thermodynamic cycling were not identified. Here I show that insect swarms are equally well represented as van der Waals gases and I attribute the possibility of thermodynamic cycling to insect swarms consisting of several overlapping sublayers. This brings about a profound change in the understanding of laboratory swarms which until now have been regarded as consisting of non-interacting individuals and lacking any internal structure. I show how the effective interactions can be attributed to the swarms’ internal structure, the external perturbations and to the presence of intrinsic noise. I thereby show that intrinsic noise which is known to be crucial for the emergence of the macroscopic mechanical properties of insect swarms is also crucial for the emergence of their thermodynamic properties as encapsulated by their equation of state.

Composite correlated random walks (CCRW) have been posited as a potential replacement for Lévy walks and it has also been suggested that CCRWs have been mistaken for Lévy walks. Here I test an alternative, emerging hypothesis: namely that some organisms approximate Lévy walks as an innate CCRW. It is shown that the tri-modal CCRW found to describe accurately the movement patterns of mussels ( Mytilus edulis ) during spatial pattern formation in mussel beds can be regarded as being the first three levels in a hierarchy of nested movement patterns which if extended indefinitely would correspond to a Lévy walk whose characteristic (power-law) exponent is tuned to nearly minimize the time required to form patterned beds. The mussels realise this Lévy walk to good approximation across a biologically meaningful range of scales. This demonstrates that the CCRW not only describes mussel movement patterns, it explains them.

During their trajectories in still air, fruit flies (Drosophila melanogaster) explore their landscape using a series of straight flight paths punctuated by rapid 90 degrees 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 degrees 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.

In fine warm weather, the daytime convective atmosphere over land areas is full of small migrant insects, among them serious pests (e.g. some species of aphid), but also many beneficial species (e.g. natural enemies of pests). For many years intensive aerial trapping studies were the only way of determining the density profiles of these small insects, and for taxon-specific studies trapping is still necessary. However, if we wish to determine generic behavioural responses to air movements shown by small day-migrating insects as a whole , the combination of millimetre-wavelength ‘cloud radars’ and Doppler lidar now provides virtually ideal instrumentation. Here we examine the net vertical velocities of > 1 million insect targets, relative to the vertical motion of the air in which they are flying, as a succession of fair-weather convective cells pass over the recording site in Oklahoma, USA. The resulting velocity measurements are interpreted in terms of the flight behaviours of small insects. These behaviours are accounted for by a newly-developed Lagrangian stochastic model of weakly-flying insect movements in the convective boundary layer; a model which is consistent with classic characterisations of small insect aerial density profiles. We thereby link patterns to processes.

Understanding strategies used by animals to explore their landscape is essential to predict how they exploit patchy resources, and consequently how they are likely to respond to changes in resource distribution. Social bees provide a good model for this and, whilst there are published descriptions of their behaviour on initial learning flights close to the colony, it is still unclear how bees find floral resources over hundreds of metres and how these flights become directed foraging trips. We investigated the spatial ecology of exploration by radar tracking bumblebees, and comparing the flight trajectories of bees with differing experience. The bees left the colony within a day or two of eclosion and flew in complex loops of ever-increasing size around the colony, exhibiting Lévy-flight characteristics constituting an optimal searching strategy. This mathematical pattern can be used to predict how animals exploring individually might exploit a patchy landscape. The bees' groundspeed, maximum displacement from the nest and total distance travelled on a trip increased significantly with experience. More experienced bees flew direct paths, predominantly flying upwind on their outward trips although forage was available in all directions. The flights differed from those of naïve honeybees: they occurred at an earlier age, showed more complex looping, and resulted in earlier returns of pollen to the colony. In summary bumblebees learn to find home and food rapidly, though phases of orientation, learning and searching were not easily separable, suggesting some multi-tasking.

As the number or density of interacting individuals in a social group increases, a transition can develop from uncorrelated and disordered behavior of the individuals to a collective coherent pattern. We expand this observation by exploring the fine details of termite movement patterns to demonstrate that the value of the scaling exponent μ of a power law describing the Lévy walk of an individual is modified collectively as the density of animals in the group changes. This effect is absent when termites interact with inert obstacles. We also show that the network of encounters and interactions among specific individuals is selective, resembling a preferential attachment mechanism that is important for social networking. Our data strongly suggest that preferential attachments, a phenomenon not reported previously, and favorite interactions with a limited number of acquaintances are responsible for the generation of Lévy movement patterns in these social insects.

Most insect migrants fly at considerable altitudes (hundreds of meters above the ground) where they utilize fast-flowing winds to achieve rapid and comparatively long-distance transport. The nocturnal aerial migrant fauna has been well studied with entomological radars, and many studies have demonstrated that flight orientations are frequently grouped around a common direction in a range of nocturnal insect migrants. Common orientation typically occurs close to the downwind direction (thus ensuring that a large component of the insects' self-powered speed is directed downstream), and in nocturnal insects at least, the downwind headings are seemingly maintained by direct detection of wind-related turbulent cues. Despite being far more abundant and speciose, the day-flying windborne migrant fauna has been much less studied by radar; thus the frequency of wind-related common orientation patterns and the sensory mechanisms involved in their formation remain to be established. Here, we analyze a large dataset of >600,000 radar-detected \"medium-sized\" windborne insect migrants (body mass from 10 to 70 mg), flying hundreds of meters above southern UK, during the afternoon, in the period around sunset, and in the middle of the night. We found that wind-related common orientation was almost ubiquitous during the day (present in 97% of all \"migration events\" analyzed), and was also frequent at sunset (85%) and at night (81%). Headings were systematically offset to the right of the flow at night-time (as predicted from the use of turbulence cues for flow assessment), but there was no directional bias in the offsets during the day or at sunset. Orientation \"performance\" significantly increased with increasing flight altitude throughout the day and night. We conclude by discussing sensory mechanisms which most likely play a role in the selection and maintenance of wind-related flight headings.

Recently Cavagna et al. (Sci Rep 13(1): 8745, 2023) documented the swarming behaviors of laboratory-based Anopheles gambiae mosquitoes. Here key observations from this 3D-video tracking study are reproduced by a minimally structured (maximum entropy) stochastic trajectory model. The modelling shows that in contrast with midge swarms which are a form of collective behavior, unperturbed mosquito swarms are more like collections of individuals that independently circulate around a fixed location. The modelling predicts the observed response Anopheles gambiae mosquitoes in wild swarms to varying wind speeds (Butail et al. in J Med Entomol 50(3): 552–559, 2013). It is shown that this response can be attributed to shear hardening. This is because mosquitoes are found to be attracted to the centre of the swarm by an effective force that increases with increasing flight speed. Mosquitoes can therefore better resist the influence of environmental disturbances by increasing their flight speeds. This contrasts with other emergent mechanical-like properties of swarming which arise accidentally without a change in an individual’s behavior. The new results add to the growing realization that perturbations can drive swarms into more robust states.

One key objective of the emerging discipline of movement ecology is to link animal movement patterns to underlying biological processes, including those operating at the neurobiological level. Nonetheless, little is known about the physiological basis of animal movement patterns, and the underlying search behaviour. Here we demonstrate the hallmarks of chaotic dynamics in the movement patterns of mud snails ( Hydrobia ulvae ) moving in controlled experimental conditions, observed in the temporal dynamics of turning behaviour. Chaotic temporal dynamics are known to occur in pacemaker neurons in molluscs, but there have been no studies reporting on whether chaotic properties are manifest in the movement patterns of molluscs. Our results suggest that complex search patterns, like the Lévy walks made by mud snails, can have their mechanistic origins in chaotic neuronal processes. This possibility calls for new research on the coupling between neurobiology and motor properties.