Scale-free movement patterns in termites emerge from social interactions and preferential attachments.

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.

Stochasticity may generate coherent motion in bird flocks.

Murmurations along with other forms of flocking have come to epitomize collective animal movements. Most studies into these stunning aerial displays have aimed to understand how coherent motion may emerge from simple behavioral rules and behavioral correlations. These studies may now need revision because recently it has been shown that flocking birds, like swarming insects, behave on the average as if they are trapped in elastic potential wells. Here I show, somewhat paradoxically, how coherent motion can be generated by variations in the intensity of multiplicative noise which causes the shape of a potential well to change, thereby shifting the positions and strengths of centres of attraction. Each bird, irrespective of its position in the flock will respond in a similar way to such changes, giving the impression that the flock behaves as one, and typically resulting in scale-free correlations. I thereby show how correlations can be an emergent property of noisy, confining potential wells. I also show how such wells can lead to high density borders, a characteristic of flocks, and I show how they can account for the complex patterns of collective escape patterns of starling flocks under predation. I suggest swarming and flocking do not constitute two distinctly different kinds of collective behavior but rather that insects are residing in relatively stable potential wells whilst birds are residing in unstable potential wells. It is shown how, dependent upon individual perceptual capabilities, bird flocks can be poised at criticality.

Phase transitions in insect swarms.

In contrast with laboratory insect swarms, wild insect swarms display significant coordinated behaviour. It has been hypothesised that the presence of a fluctuating environment drives the formation of transient, local order (synchronized subgroups), and that this local order pushes the swarm into a new state that is robust to environmental perturbations. The hypothesis is supported by observations of swarming mosquitoes. Here I provide numerical evidence that the formation of transient, local order is an accidental by-product of the strengthening of short-range repulsion which is expected in the presence of environmental fluctuations. The results of the numerical simulations reveal that this strengthening of the short-range can drive swarms into a crystalline phase containing subgroups that participate in cooperative ring exchanges-a new putative form of collective animal movement lacking velocity correlation. I thereby demonstrate that the swarm state and structure may be tuneable with environmental noise as a control parameter. Predicted properties of the collective modes are consistent with observations of transient synchronized subgroups in wild mosquito swarms that contend with environmental disturbances. When mutual repulsion becomes sufficiently strong, swarms are, in accordance with observations, predicted to form near stationary crystalline states. The analysis suggests that the many different forms of swarming motions observed across insect species are not distinctly different phenomena but are instead different phases of a single phenomenon.

Swarm formation as backward diffusion.

Considerable progress has been made in understanding insect swarms-forms of collective animal behaviour that unlike bird flocks, fish schools and animal herds do not possess global order. Nonetheless, little is known about swarm formation. Here we posit a mechanism for the formation of insect swarms that is consistent with recent empirical observations reported by (Patel and Ouellette 2022). It correctly predicts new features of swarm formation that have not been reported on previously. Our simple analytically tractable model shows how harmonic potential wells, a characteristic feature of swarming, and so swarm cohesion, arise from diffusion and local fission-fusion dynamics and how, in accord with observations, these wells deepen over time. The overall form of these potential wells is predicted to depend on the number and spatial distribution of all individuals, making them manifestly a collective phenomenon. Finally, swarms are predicted to 'cool' (that is, condense) as they form.

Mosquito swarms shear harden.

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.

Comment on 'A physics perspective on collective animal behavior' 2022Phys. Biol.19 021004.

In his insightful and timely review Ouellette (2022Phys. Biol.19021004) noted three theoretical impediments to progress in understanding and modelling collective animal behavior. Here through novel analyses and by drawing on the latest research I show how these obstacles can be either overcome or negated. I suggest ways in which recent advances in the physics of collective behavior provide significant biological information.

Weierstrassian Lévy walks are a by-product of crawling.

Weierstrassian Lévy walks are one of the simplest random walks which do not satisfy the central limit theorem and have come to epitomize scale invariance even though they were initially regarded as being a mathematical abstraction. Here, I show how these Lévy walks can be generated intrinsically as a by-product of crawling, a common but ancient form of locomotion. This may explain why Weierstrassian Lévy walks provide accurate representations of the movement patterns of a diverse group of molluscs-certain mussels, mud snails and limpets. I show that such movements are not specific to molluscs as they are also evident in Drosophila larvae. The findings add to the growing realization that there are many idiosyncratic, seemingly accidental pathways to Lévy walking. And that the occurrence of Lévy walks need not be attributed to the execution of an advantageous searching strategy.

Intrinsic stochasticity and the emergence of collective behaviours in insect swarms.

Intrinsic stochasticity associated with finite population size is fundamental to the emergence of collective behaviours in insect swarms. It has been assumed that this intrinsic stochasticity is purely additive (position independent) in quiescent (unperturbed) swarms. Here, I identify the hallmarks of intrinsic multiplicative (position dependent) stochasticity and show that they are evident in quiescent laboratory swarms of the non-biting midge Chironomus riparius. In accordance with theoretical expectations, the smallest well-documented laboratory swarms (containing between 14 and 46 individuals) are found to have q-Gaussian density profiles with [Formula: see text] 1, whereas larger laboratory swarms have Gaussian ([Formula: see text]1) density profiles. I show that these newly identified states are analogous to interstellar clouds and thereby extend a long-standing analogy between insect swarms and self-gravitating systems. Smaller laboratory swarms have been observed and are predicted to be gas-like, filling the available space rather than occupying just a small proportion of it. The new results unify laboratory swarms with wild swarms. Unlike laboratory swarms, wild swarms must contend with environmental (extrinsic) noise and have density profiles that are accurately represented by q-Gaussians with [Formula: see text] 1. Finally, it is shown how intrinsic multiplicative noise allows for the nucleation of swarms away from prominent visual features (basins of attraction) known as swarm markers.

On the origin of the tensile strength of insect swarms.

Traditionally animal groups have been characterized by the macroscopic patterns that they form. It is now recognised that such patterns convey limited information about the nature of the aggregation as a whole. Aggregate properties cannot be determined by passive observations alone; instead one must interact with them. One of the first such dynamical tests revealed that swarms of flying insects have macroscopic mechanical properties similar to solids, including a finite Young's modulus and yield strength. Here I show, somewhat counterintuitively, that the emergence of these solid-like properties can be attributed to centre-of-mass movements (heat). This suggests that perturbations can drive phase transitions.

Does the Australian desert ant Melophorus bagoti approximate a Lévy search by an intrinsic bi-modal walk?

We suggest that the Australian desert ant Melophorus bagoti approximates a Lévy search pattern by using an intrinsic bi-exponential walk and does so when a Lévy search pattern is advantageous. When attempting to locate its nest, M. bagoti adopt a stereotypical search pattern. These searches begin at the location where the ant expects to find the nest, and comprise loops that start and end at this location, and are directed in different azimuthal directions. Loop lengths are exponentially distributed when searches are in visually familiar surroundings and are well described by a mixture of two exponentials when searches are in unfamiliar landscapes. The latter approximates a power-law distribution, the hallmark of a Lévy search. With the aid of a simple analytically tractable theory, we show that an exponential loop-length distribution is advantageous when the distance to the nest can be estimated with some certainty and that a bi-exponential distribution is advantageous when there is considerable uncertainty regarding the nest location. The best bi-exponential search patterns are shown to be those that come closest to approximating advantageous Lévy looping searches. The bi-exponential search patterns of M. bagoti are found to approximate advantageous Lévy search patterns.

Spatial correlations in laboratory insect swarms.

In contrast with flocks of birds, schools of fish and herds of animals, swarms of the non-biting midge Chironomus riparius do not possess global order and under quiescent conditions velocities are only weakly correlated at long distances. Without such order it is challenging to characterize the collective behaviours of the swarms which until now have only been evident in their coordinated responses to disturbances. Here I show that the positions of the midges in laboratory swarms are maximally anticorrelated. This novel form of long-range ordering has until now gone unnoticed in the literature on collective animal movements. Here, its occurrence is attributed to midges being, in nearly equal measure, attracted towards the centre of the swarm and repelled by one another. It is shown that the midge swarms are poised at the cusp of a stable-unstable phase transition.

Beating the odds in the aerial lottery: passive dispersers select conditions at takeoff that maximize their expected fitness on landing.

Passive airborne dispersal, in which the direction and distance of travel are determined by air movement, affects propagules and pollens, as well as mites, spiders, and small insect larvae. The takeoff or launch phase is, however, largely controlled, and many organisms become airborne only under particular weather conditions at takeoff, when the distribution of distances traveled will have a -3/2 power-law tail, a hallmark of Lévy flights. Here these movement patterns are shown to maximize the likelihood of dispersing to the nearest unoccupied site, thereby maximizing expected fitness on landing.

Stochastic modelling of bird flocks: accounting for the cohesiveness of collective motion.

Collective behaviour can be difficult to discern because it is not limited to animal aggregations such as flocks of birds and schools of fish wherein individuals spontaneously move in the same way despite the absence of leadership. Insect swarms are, for example, a form of collective behaviour, albeit one lacking the global order seen in bird flocks and fish schools. Their collective behaviour is evident in their emergent macroscopic properties. These properties are predicted by close relatives of Okubo's 1986 [Adv. Biophys.22, 1-94. (doi:10.1016/0065-227X(86)90003-1)] stochastic model. Here, we argue that Okubo's stochastic model also encapsulates the cohesiveness mechanism at play in bird flocks, namely the fact that birds within a flock behave on average as if they are trapped in an elastic potential well. That is, each bird effectively behaves as if it is bound to the flock by a force that on average increases linearly as the distance from the flock centre increases. We uncover this key, but until now overlooked, feature of flocking in empirical data. This gives us a means of identifying what makes a given system collective. We show how the model can be extended to account for intrinsic velocity correlations and differentiated social relationships.

Understanding the thermodynamic properties of insect swarms.

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.

A single wind-mediated mechanism explains high-altitude 'non-goal oriented' headings and layering of nocturnally migrating insects.

Studies made with both entomological and meteorological radars over the last 40 years have frequently reported the occurrence of insect layers, and that the individuals forming these layers often show a considerable degree of uniformity in their headings--behaviour known as 'common orientation'. The environmental cues used by nocturnal migrants to select and maintain common headings, while flying in low illumination levels at great heights above the ground, and the adaptive benefits of this behaviour have long remained a mystery. Here we show how a wind-mediated mechanism accounts for the common orientation patterns of 'medium-sized' nocturnal insects. Our theory posits a mechanism by which migrants are able to align themselves with the direction of the flow using a turbulence cue, thus adding their air speed to the wind speed and significantly increasing their migration distance. Our mechanism also predicts that insects flying in the Northern Hemisphere will typically be offset to the right of the mean wind line when the atmosphere is stably stratified, with the Ekman spiral in full effect. We report on the first evidence for such offsets, and show that they have significant implications for the accurate prediction of the flight trajectories of migrating nocturnal insects.

Linking Small-Scale Flight Manoeuvers and Density Profiles to the Vertical Movement of Insects in the Nocturnal Stable Boundary Layer.

Huge numbers of insects migrate over considerable distances in the stably-stratified night-time atmosphere with great consequences for ecological processes, biodiversity, ecosystem services and pest management. We used a combination of meteorological radar and lidar instrumentation at a site in Oklahoma, USA, to take a new look at the general assistance migrants receive from both vertical and horizontal airstreams during their long-distance flights. Movement in the nocturnal boundary layer (NBL) presents very different challenges for migrants compared to those prevailing in the daytime convective boundary layer, but we found that Lagrangian stochastic modelling is effective at predicting flight manoeuvers in both cases. A key feature for insect transport in the NBL is the frequent formation of a thin layer of fast-moving air - the low-level jet. Modelling suggests that insects can react rapidly to counteract vertical air movements and this mechanism explains how migrants are retained in the jet for long periods (e.g. overnight, and perhaps for several hours early in the morning). This results in movements over much longer distances than are likely in convective conditions, and is particularly significant for the reintroduction of pests to northern regions where they are seasonally absent due to low winter temperatures.

Aphid aerial density profiles are consistent with turbulent advection amplifying flight behaviours: abandoning the epithet 'passive'.

Seminal field studies led by C. G. Johnson in the 1940s and 1950s showed that aphid aerial density diminishes with height above the ground such that the linear regression coefficient, b, of log density on log height provides a single-parameter characterization of the vertical density profile. This coefficient decreases with increasing atmospheric stability, ranging from -0.27 for a fully convective boundary layer to -2.01 for a stable boundary layer. We combined a well-established Lagrangian stochastic model of atmospheric dispersal with simple models of aphid behaviour in order to account for the range of aerial density profiles. We show that these density distributions are consistent with the aphids producing just enough lift to become neutrally buoyant when they are in updraughts and ceasing to produce lift when they are in downdraughts. This active flight behaviour in a weak flier is thus distinctly different from the aerial dispersal of seeds and wingless arthropods, which is passive once these organisms have launched into the air. The novel findings from the model indicate that the epithet 'passive' often applied to the windborne migration of small winged insects is misleading and should be abandoned. The implications for the distances traversed by migrating aphids under various boundary-layer conditions are outlined.

On the emergence of gravitational-like forces in insect swarms.

Okubo (Okubo 1986 Adv. Biophys.22, 1-94. (doi:10.1016/0065-227X(86)90003-1)) was the first to propose that insect swarms are analogous to self-gravitating systems. In the intervening years, striking similarities between insect swarms and self-gravitating systems have been uncovered. Nonetheless, experimental observations of laboratory swarms provide no conclusive evidence of long-range forces acting between swarming insects. The insects appear somewhat paradoxically to be tightly bound to the swarm while at the same time weakly coupled inside it. Here, I show how resultant centrally attractive gravitational-like forces can emerge from the observed tendency of insects to continually switch between two distinct flight modes: one that consists of low-frequency manoeuvres and one that consists of higher-frequency nearly harmonic oscillations conducted in synchrony with another insect. The emergent dynamics are consistent with 'adaptive' gravity models of swarming and with variants of the stochastic models of Okubo and Reynolds for the trajectories of swarming insects: models that are in close accord with a plethora of observations of unperturbed and perturbed laboratory swarms. The results bring about a radical change of perspective as swarm properties can now be attributed to known biological behaviours rather than to elusive physical influences.

Incorporating terminal velocities into Lagrangian stochastic models of particle dispersal in the atmospheric boundary layer.

Lagrangian stochastic models for simulation of tracer-particle trajectories in turbulent flows can be adapted for simulation of particle trajectories. This is conventionally done by replacing the zero-mean fall speed of a tracer-particle with the terminal speed of the particle. Such models have been used widely to predict spore and pollen dispersal. Here I show that this modification predicts that particles become uniformly distributed throughout the air column, which is at variance with the seminal experimental studies of Hirst et al. (1967) that demonstrated spore concentrations (and pollen concentrations) declined exponentially with height in unstable air. This discrepancy arises because the terminal speed, which is a Lagrangian property of a particle, has always been treated as if it were an Eulerian property of an ensemble of particles. In this study models are formulated correctly. I show that the mean acceleration of a tracer-particle should be replaced by the mean acceleration of a particle. Model predictions for aerial density profiles then agreed with the observations of Hirst et al. (1967) and with observations of ground-level concentrations but differed significantly from predictions obtained using conventional models. In accordance with the results of numerical simulations, the models also predict that particles are moving downwind marginally more slowly than the wind itself. Finally, the new modelling approach can be extended to predict the dispersal of small insects with active flight behaviours.