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Insect pollinators such as bumblebees play a vital role in many ecosystems, so it is important to understand their foraging movements on a landscape scale. We used harmonic radar to record the natural foraging behaviour of Bombus terrestris audax workers over their entire foraging career. Every flight ever made outside the nest by four foragers was recorded. Our data reveal where the bees flew and how their behaviour changed with experience, at an unprecedented level of detail. We identified how each bee's flights fit into two categories-which we named exploration and exploitation flights-examining the differences between the two types of flight and how their occurrence changed over the course of the bees' foraging careers. Exploitation of learned resources takes place during efficient, straight trips, usually to a single foraging location, and is seldom combined with exploration of other areas. Exploration of the landscape typically occurs in the first few flights made by each bee, but our data show that further exploration flights can be made throughout the bee's foraging career. Bees showed striking levels of variation in how they explored their environment, their fidelity to particular patches, ratio of exploration to exploitation, duration and frequency of their foraging bouts. One bee developed a straight route to a forage patch within four flights and followed this route exclusively for six days before abandoning it entirely for a closer location; this second location had not been visited since her first exploratory flight nine days prior. Another bee made only rare exploitation flights and continued to explore widely throughout its life; two other bees showed more frequent switches between exploration and exploitation. Our data shed light on the way bumblebees balance exploration of the environment with exploitation of resources and reveal extreme levels of variation between individuals.

“Continuous-time correlated random walks” are now gaining traction as models of scale-finite animal movement patterns because they overcome inherent shortcomings with the prevailing paradigm - discrete random walk models. Continuous-time correlated random walk models are founded on the classic Langevin equation that is driven by purely additive noise. The Langevin equation is, however, changed fundamentally by the smallest of multiplicative noises. The inclusion of such noises gives rise to Lévy flights, a popular but controversial model of scale-free movement patterns. Multiplicative noises have not featured prominently in the literature on biological Lévy flights, being seen, perhaps, as no more than a mathematical contrivance. Here we show how Langevin equations driven by multiplicative noises and incumbent Lévy flights arise naturally in the modelling of swarms. Model predictions find some support in three-dimensional, time-resolved measurements of the positions of individual insects in laboratory swarms of the midge Chironomus riparius. We hereby provide a new window on Lévy flights as models of movement pattern data, linking patterns to generative processes.

Collective behaviour in flocks, crowds, and swarms occurs throughout the biological world. Animal groups are generally assumed to be evolutionarily adapted to robustly achieve particular functions, so there is widespread interest in exploiting collective behaviour for bio-inspired engineering. However, this requires understanding the precise properties and function of groups, which remains a challenge. Here, we demonstrate that collective groups can be described in a thermodynamic framework. We define an appropriate set of state variables and extract an equation of state for laboratory midge swarms. We then drive swarms through “thermodynamic” cycles via external stimuli, and show that our equation of state holds throughout. Our findings demonstrate a new way of precisely quantifying the nature of collective groups and provide a cornerstone for potential future engineering design.

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.

Central place foragers, such as pollinating bees, typically develop circuits (traplines) to visit multiple foraging sites in a manner that minimizes overall travel distance. Despite being taxonomically widespread, these routing behaviours remain poorly understood due to the difficulty of tracking the foraging history of animals in the wild. Here we examine how bumblebees (Bombus terrestris) develop and optimise traplines over large spatial scales by setting up an array of five artificial flowers arranged in a regular pentagon (50 m side length) and fitted with motion-sensitive video cameras to determine the sequence of visitation. Stable traplines that linked together all the flowers in an optimal sequence were typically established after a bee made 26 foraging bouts, during which time only about 20 of the 120 possible routes were tried. Radar tracking of selected flights revealed a dramatic decrease by 80% (ca. 1500 m) of the total travel distance between the first and the last foraging bout. When a flower was removed and replaced by a more distant one, bees engaged in localised search flights, a strategy that can facilitate the discovery of a new flower and its integration into a novel optimal trapline. Based on these observations, we developed and tested an iterative improvement heuristic to capture how bees could learn and refine their routes each time a shorter route is found. Our findings suggest that complex dynamic routing problems can be solved by small-brained animals using simple learning heuristics, without the need for a cognitive map.

Animals that visit multiple foraging sites face a problem, analogous to the Travelling Salesman Problem, of finding an efficient route. We explored bumblebees’ route development on an array of five artificial flowers in which minimising travel distances between individual feeders conflicted with minimising overall distance. No previous study of bee spatial navigation has been able to follow animals’ movement during learning; we tracked bumblebee foragers continuously, using harmonic radar, and examined the process of route formation in detail for a small number of selected individuals. On our array, bees did not settle on visit sequences that gave the shortest overall path, but prioritised movements to nearby feeders. Nonetheless, flight distance and duration reduced with experience. This increased efficiency was attributable mainly to experienced bees reducing exploration beyond the feeder array and flights becoming straighter with experience, rather than improvements in the sequence of feeder visits. Flight paths of all legs of a flight stabilised at similar rates, whereas the first few feeder visits became fixed early while bees continued to experiment with the order of later visits. Stabilising early sections of a route and prioritising travel between nearby destinations may reduce the search space, allowing rapid adoption of efficient routes.

Pollinating bees develop foraging circuits (traplines) to visit multiple flowers in a manner that minimizes overall travel distance, a task analogous to the travelling salesman problem. We report on an in-depth exploration of an iterative improvement heuristic model of bumblebee traplining previously found to accurately replicate the establishment of stable routes by bees between flowers distributed over several hectares. The critical test for a model is its predictive power for empirical data for which the model has not been specifically developed, and here the model is shown to be consistent with observations from different research groups made at several spatial scales and using multiple configurations of flowers. We refine the model to account for the spatial search strategy of bees exploring their environment, and test several previously unexplored predictions. We find that the model predicts accurately 1) the increasing propensity of bees to optimize their foraging routes with increasing spatial scale; 2) that bees cannot establish stable optimal traplines for all spatial configurations of rewarding flowers; 3) the observed trade-off between travel distance and prioritization of high-reward sites (with a slight modification of the model); 4) the temporal pattern with which bees acquire approximate solutions to travelling salesman-like problems over several dozen foraging bouts; 5) the instability of visitation schedules in some spatial configurations of flowers; 6) the observation that in some flower arrays, bees' visitation schedules are highly individually different; 7) the searching behaviour that leads to efficient location of flowers and routes between them. Our model constitutes a robust theoretical platform to generate novel hypotheses and refine our understanding about how small-brained insects develop a representation of space and use it to navigate in complex and dynamic environments.

Pesticide exposure has been cited as a key threat to insect pollinators. Notably, a diverse range of potential sublethal effects have been reported in bee species, with a particular focus on effects due to exposure to neonicotinoid insecticides. Here, a purpose-built thermal–visual arena was used in a series of pilot experiments to assess the potential impact of approximate sublethal concentrations of the next generation sulfoximine insecticide sulfoxaflor (5 and 50 ppb) and the neonicotinoid insecticides thiacloprid (500 ppb) and thiamethoxam (10 ppb), on the walking trajectory, navigation and learning abilities of the buff-tailed bumblebee (Bombus terrestris audax) when subjected to an aversive conditioning task. The results suggest that only thiamethoxam prevents forager bees from improving in key training parameters (speed and distanced travelled) within the thermal visual arena. Power law analyses further revealed that a speed–curvature power law, previously reported as being present in the walking trajectories of bumblebees, is potentially disrupted under thiamethoxam (10 ppb) exposure, but not under sulfoxaflor or thiacloprid exposure. The pilot assay described provides a novel tool with which to identify subtle sublethal pesticide impacts, and their potential causes, on forager bees, that current ecotoxicological tests are not designed to assess.

Background The flight patterns of albatrosses and shearwaters have become a touchstone for much of Lévy flight research, spawning an extensive field of enquiry. There is now compelling evidence that the flight patterns of these seabirds would have been appreciated by Paul Lévy, the mathematician after whom Lévy flights are named. Here we show that Lévy patterns (here taken to mean spatial or temporal patterns characterized by distributions with power-law tails) are, in fact, multifaceted in shearwaters being evident in both spatial and temporal patterns of activity. Results We tested for Lévy patterns in the at-sea behaviours of two species of shearwater breeding in the North Atlantic Ocean ( Calonectris borealis ) and the Mediterranean sea ( C. diomedea ) during their incubating and chick-provisioning periods. We found that distributions of flight durations, on/in water durations and inter-dive time-intervals have power-law tails and so bear the hallmarks of Lévy patterns. Conclusions The occurrence of these statistical laws is remarkable given that bird behaviours are strongly shaped by an individual’s motivational state and by complex environmental interactions. Our observations could take Lévy patterns as models of animal behaviour to a new level by going beyond the characterisation of spatial movements to characterise how different behaviours are interwoven throughout daily animal life.

Honey bees (Apis mellifera) are regularly faced with the task of navigating back to their hives from remote food sources. They have evolved several methods to do this, including compass-directed \"vector\" flights and the use of landmarks. If these hive-centered mechanisms are disrupted, bees revert to searching for the hive, but the nature and efficiency of their searching strategy have hitherto been unknown. We used harmonic radar to record the flight paths of honey bees that were searching for their hives. Our subsequent analysis of these paths revealed that they can be represented by a series of straight line segments that have a scale-free, Lévy distribution with an inverse-square-law tail. We show that these results, combined with the \"no preferred direction\" characteristic of the segments, demonstrate that the bees were flying an optimal search pattern. Lévy movements have already been identified in a number of other animals. Our results are the best reported example where the movements are mostly attributable to the adoption of an optimal, scale-free searching strategy.