Are Lévy flight patterns derived from the Weber–Fechner law in distance estimation?

Honeybees (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, using an optimal Lévy flight searching strategy. The same strategy is adopted when a food source at a known location ceases to be available. Here, we show that the programming for this Lévy strategy does not need to be very sophisticated or clever on the bee’s part, as Lévy flight patterns can be derived from the Weber–Fechner law in a bee’s odometer. Odometry errors of a different kind occur in desert ants (Cataglyphis spp., Melophorus bagoti). The searching behaviors of these ants are very similar in overall structure to that of honeybees but do not display any Lévy flight characteristics. We suggest that errors in the estimation of distance can be implicitly involved in shaping the structure of systematic search behavior and should not be regarded as merely deficiencies in the odometer.     

Detecting Lévy walks without turn designation

Many organisms have been reported to have movement patterns that are well approximated as Lévy walks. This is typically because distributions of straight line distances between consecutive significant turns in movement paths have heavy power law tails. This diagnostic tool has been called into question because there is currently no standard, unambiguous way to identify significant turns. Even if such a way could be found, statistical analyses based on significant turns cannot account for actual movements made between turns and as a consequence cannot distinguish between true Lévy walks and other fractal random walks such as Lévy modulated correlated random walks where organisms randomly meander rather than move in straight lines between consecutive reorientation events. Here, I show that structure functions (i.e. moments of net displacements made across fixed time intervals) can distinguish between different kinds of Lévy walks and between Lévy walks and random walks with a few scales such as composite correlated random walks and correlated random walks. Distinguishing between these processes will lead to a better understanding of how and why animals perform Lévy walks and help bridge the apparent divide between correlated random walks and Lévy walks. Structure functions do not require turn identification and instead take account of entire movement paths. Using this diagnostic tool, I bolster previous claims that honeybees use a movement strategy that can be approximated by Lévy walks when searching for their hive. I also show how structure functions can be used to establish the extent of self-similar behaviour in meandering Lévy walks.     

Honeybees use a Lévy flight search strategy and odour-mediated anemotaxis to relocate food sources

The availability of food resources changes over time and space, and foraging animals are constantly faced with choices about how to respond when a resource becomes depleted. We hypothesise that flying insects like bees discover new food sources using an optimal Lévy flight searching strategy and odour-mediated anemotaxis, as well as visual cues. To study these searching patterns, foraging honeybees were trained to a scented feeder which was then removed. Two new unrewarding feeders, or ‘targets’, were then positioned up- and downwind of the original location of the training feeder. The subsequent flight patterns of the bees were recorded over several hundred metres using harmonic radar. We show that the flight patterns constitute an optimal Lévy flight searching strategy for the location of the training feeder, a strategy that is also optimal for the location of alternative food sources when patchily distributed. Scented targets that were positioned upwind of the original training feeder were investigated most with the numbers of investigations declining with increasing distance from the original feeder. Scented targets in downwind locations were rarely investigated and unscented targets were largely ignored, despite having the same visual appearance as the rewarding training feeder.     

Lévy flight patterns are predicted to be an emergent property of a bumblebees’ foraging strategy

Bumblebees forage uninterrupted for long periods of time because they are not distracted by sex or territorial defense and have few predators. This has led to a long running debate about whether bumblebees forage optimally. This debate has been enriched by the possibility that bumblebees foraging within clover patches have flight patterns that can be approximated by Lévy flights. Such flight patterns optimise the success of random searches. Bumblebees foraging within a flower patch tend to approach the nearest flower but then often depart without landing or probing it if it has been visited previously; unvisited flowers are not rejected in this manner. Here, this foraging behaviour has been replicated in numerical simulations. Lévy flight patterns are found to be an inconsequential emergent property of a bumblebees’ foraging behaviour. Lévy flights are predicted to emerge when bees reject at least 99% of previously visited flowers. A foraging bumblebee can certainly empty a clover flower head of nectar in one visit, but lower rates of rejection are observed for many other flowers. These findings suggest that Lévy flight patterns in foraging bumblebees are rare and specific to a few flower species and that if they exist, then they are not part of an innate, evolved optimal searching strategy.     

Scale-invariant movements of fishermen: the same foraging strategy as natural predators.

We analyzed the movement of fishing vessels during fishing trips in order to understand how fishermen behave in space while searching for fish. For that purpose we used hourly geo-referenced positions of vessels, provided by a satellite vessel monitoring system, for the entire industrial fleet (809 vessels) of the world's largest single species fishery (Peruvian anchovy, Engraulis ringens) from December 1999 to March 2003. Observed trajectories of fishing vessels are well modeled by Lévy random walks, suggesting that fishermen use a stochastic search strategy which conforms to the same search statistics as non-human predators. We show that human skills (technology, communication, or others) do not result in the fishermen's spatial behavior being fundamentally different from that of animal predators. With respect to probability of prey encounter, our results suggest that fishermen, on average, evolved an optimal movement pattern (mu = 2.00) among the family of Lévy random walks. This Lagrangian approach opens several perspectives in terms of operational management of the pelagic fish stock.     

Measurement error causes scale-dependent threshold erosion of biological signals in animal movement data.

Recent advances in telemetry technology have created a wealth of tracking data available for many animal species moving over spatial scales from tens of meters to tens of thousands of kilometers. Increasingly, such data sets are being used for quantitative movement analyses aimed at extracting fundamental biological signals such as optimal searching behavior and scale-dependent foraging decisions. We show here that the location error inherent in various tracking technologies reduces the ability to detect patterns of behavior within movements. Our analyses endeavored to set out a series of initial ground rules for ecologists to help ensure that sampling noise is not misinterpreted as a real biological signal. We simulated animal movement tracks using specialized random walks known as Lévy flights at three spatial scales of investigation: 100-km, 10-km, and 1-km maximum daily step lengths. The locations generated in the simulations were then blurred using known error distributions associated with commonly applied tracking methods: the Global Positioning System (GPS), Argos polar-orbiting satellites, and light-level geolocation. Deviations from the idealized Lévy flight pattern were assessed for each track after incrementing levels of location error were applied at each spatial scale, with additional assessments of the effect of error on scale-dependent movement patterns measured using fractal mean dimension and first-passage time (FPT) analyses. The accuracy of parameter estimation (Lévy mu, fractal mean D, and variance in FPT) declined precipitously at threshold errors relative to each spatial scale. At 100-km maximum daily step lengths, error standard deviations of > or = 10 km seriously eroded the biological patterns evident in the simulated tracks, with analogous thresholds at the 10-km and 1-km scales (error SD > or = 1.3 km and 0.07 km, respectively). Temporal subsampling of the simulated tracks maintained some elements of the biological signals depending on error level and spatial scale. Failure to account for large errors relative to the scale of movement can produce substantial biases in the interpretation of movement patterns. This study provides researchers with a framework for understanding the limitations of their data and identifies how temporal subsampling can help to reduce the influence of spatial error on their conclusions.     

Distinguishing between Lévy walks and strong alternative models

Lévy walks are a widely used but contentious model of animal movement patterns. They are contentious because they have been wrongly ascribed to some animal species through use of incorrect statistical methods and because they have not been adequately compared against strong alternative models, such as composite correlated random walks. This lack of comparison has been partly because the strong alternative models do not have simple likelihood functions. Here I show that power-spectra and the distribution of the first significant digits (the leading non-zero digits) of the step lengths can distinguish between Lévy walks and composite correlated random walks. Using these diagnostic tools, I bolster previous claims that honey bees use a movement strategy that can be approximated by Lévy walks when searching for their hive or for a food source.     

How many animals really do the Lévy walk?

Lévy walks (LW) are superdiffusive and scale-free random walks that have recently emerged as a new conceptual tool for modeling animal search paths. They have been claimed to be more efficient than the "classical" random walks, and they also seem able to account for the actual search patterns of various species. This suggests that many animals may move using a LW process. LW patterns look like the actual search patterns displayed by animals foraging in a patchy environment, where extensive and intensive searching modes alternate, and which can be generated by a mixture of classical random walks. In this context, even elementary composite Brownian walks are more efficient than LW but may be confounded with them because they present apparent move-length-heavy tail distributions and superdiffusivity. The move-length "survival" distribution (i.e., the cumulative number of moves greater than any given threshold) appears to be a better means to highlight a LW pattern. Even once such a pattern has been clearly identified, it remains to determine how it was actually generated, because a LW pattern is not necessarily produced by a LW process but may emerge from the way the animal interacted with the environment structure through more classical movement processes. In any case, emergent movement patterns should not be confused with the processes that gave rise to them.     

Lévy walk patterns in the foraging movements of spider monkeys (<Emphasis Type="Italic">Ateles geoffroyi</Emphasis>)

Scale invariant patterns have been found in different biological systems, in many cases resembling what physicists have found in other, nonbiological systems. Here we describe the foraging patterns of free-ranging spider monkeys (Ateles geoffroyi) in the forest of the Yucatan Peninsula, Mexico and find that these patterns closely resemble what physicists know as Lévy walks. First, the length of a trajectorys constituent steps, or continuous moves in the same direction, is best described by a power-law distribution in which the frequency of ever larger steps decreases as a negative power function of their length. The rate of this decrease is very close to that predicted by a previous analytical Lévy walk model to be an optimal strategy to search for scarce resources distributed at random. Second, the frequency distribution of the duration of stops or waiting times also approximates to a power-law function. Finally, the mean square displacement during the monkeys first foraging trip increases more rapidly than would be expected from a random walk with constant step length, but within the range predicted for Lévy walks. In view of these results, we analyze the different exponents characterizing the trajectories described by females and males, and by monkeys on their own and when part of a subgroup. We discuss the origin of these patterns and their implications for the foraging ecology of spider monkeys.Communicated by D. Watts     

Does interaction between bumblebees (Bombus ignitus) reduce their foraging area?: bee-removal experiments in a net cage

To examine whether the interaction between bumblebees, Bombus ignitus, reduces their foraging area, we conducted bee-removal experiments in a net cage. In the cage, we set potted Salvia farinacea plants, allowed bumblebees to forage freely on those plants, and followed their plant-to-plant movements to identify a bee with a relatively small foraging area. We then removed all the other foraging bees, except for the bee with a small foraging area, and observed the change of the foraging area of the focal bee under conditions of no interaction with other bees. After the removal of the other bees, all five bees tested enlarged their foraging areas, suggesting that the interaction between bees is an important determinant of their foraging areas. The result also means that bumblebees are able to adjust their foraging areas in response to other foragers, indicating the necessity for future studies to clarify what cues bees use to interact with other bees. Moreover, after the removal treatments, all five bees showed temporary increases in the number of flower probes per plant. This can be explained by their optimal foraging according to the old average intake rate for the plant population and by the delayed changes in response to the new high average energy intake rate after the bee-removal treatments.Communicated by M. Giurfa     

Kin structure and the swarming behavior of the honey bee Apis mellifera

Summary Experimental hives obtained from cordovan queens that were instrumentally inseminated with semen from one cordovan and one Italian drone were set up and allowed to swarm. Cordovan provides a resessive genetic marker system (cuticle color) so that the workers from the cordovan and Italian male lines are distinguishable. Our results show that these patrilineal worker groups segregate non-randomly during colony fission and this segregation cannot be explained by observed age structure. Evidence of innate kin recognition in bees has been previously established. We argue that kin recognition could be responsible for the observed non-random grouping of kin during swarming.     

A mark-recapture study of male<Emphasis Type="Italic"> Colletes cunicularius</Emphasis> bees: implications for pollination by sexual deception

An unusual pollination strategy is pollination by sexual deception in which orchids sexually attract male insects as pollinators. One gap in knowledge concerns the pattern and extent of pollinator movement among these sexually deceptive flowers and how this translates to pollen and gene flow. Our aim was to use mark and recapture techniques to investigate the behavior and movement of male Colletes cunicularius, an important bee pollinator of Ophrys. Our study site was located in northern Switzerland where a large population of the bees was nesting. Within two plots, (10×40 m), we marked bees with different colors and numbered tags. Seventeen percent of the 577 marked bees were recaptured over a period of 1 to a maximum of 11 days. However, the number of recaptures dropped dramatically after 3–5 days, suggesting an average lifetime of less than 10 days. Mark-recapture distances varied from 0 to 50 m, with a mean of 5 m. Our findings show that individual male bees patrol a specific and restricted region of the nesting area in search of mates. This mark-recapture study provides the first clues about the potential movement of pollen within populations of Ophrys orchids. We predict that orchid-pollen movements mediated by bees will be similar to the mark-recapture distances in this study. Parallel studies within orchid populations, including direct studies of pollen movement, are now required to better understand how pollinator mate-searching behavior translates to pollination success and pollen movement within sexually deceptive orchid populations.Communicated by R.F.A. Moritz     

A comparison of three Eulerian numerical methods for fractional-order transport models

Tracer transport in complex systems like turbulent flows or heterogeneous porous media is now more and more regarded as a non-local process that can hardly be represented by second-order diffusion models. In this work, we consider diffusion models that assume that tracer particles follow a heavy-tail Lévy distribution, which allows for large displacements. We show that such an assumption leads to a fractional-order diffusion operator in the governing equation for tracer concentration. A comparison of three Eulerian numerical methods to discretize that equation is then performed. These consist of the finite difference, finite element and spectral element methods. We suggest that non-local methods, like the spectral element method, are better suited to transport models with fractional-order diffusion operators.     

Overturning conclusions of Lévy flight movement patterns by fishing boats and foraging animals

A surprisingly diverse variety of foragers have previously been concluded to exhibit movement patterns known as Lévy flights, a special type of random walk. These foragers range in size from microzooplankton in experiments to fishermen in the Pacific Ocean and the North Sea. The Lévy flight conclusion implies that all the foragers have similar scale-free movement patterns that can be described by a single dimensionless parameter, the exponent micro of a power-law (Pareto) distribution. However, the previous conclusions have been made using methods that have since been shown to be problematic: inaccurate techniques were used to estimate micro, and the power-law distribution was usually assumed to hold without testing any alternative hypotheses. Therefore, I address the open question of whether the previous data still support the Lévy flight hypothesis, and thus determine whether Lévy flights really are so ubiquitous in ecology. I present a comprehensive reanalysis of 17 data sets from seven previous studies for which Lévy flight behavior had been concluded, covering marine, terrestrial, and experimental systems from four continents. I use the modern likelihood and Akaike weights approach to test whether simple alternative models are more supported by the data than Lévy flights. The previously estimated values of the power-law exponent micro do not match those calculated here using the accurate likelihood approach, and almost all of them lie outside of the likelihood-based 95% confidence intervals. Furthermore, the original power-law Lévy flight model is overwhelmingly rejected for 16 out of the 17 data sets when tested against three other simple models. For one data set, the data are consistent with coming from a bounded power-law distribution (a truncated Lévy flight). For three other data sets, an exponential distribution corresponding to a simple Poisson process is suitable. Thus, Lévy flight movement patterns are not the common phenomena that was once thought, and are not suitable for use as ecosystem indicators for fisheries management, as has been proposed.     

Aging and demographic plasticity in response to experimental age structures in honeybees (<Emphasis Type="Italic">Apis mellifera</Emphasis> L)

Honeybee colonies are highly integrated functional units characterized by a pronounced division of labor. Division of labor among workers is mainly age-based, with younger individuals focusing on in-hive tasks and older workers performing the more hazardous foraging activities. Thus, experimental disruption of the age composition of the worker hive population is expected to have profound consequences for colony function. Adaptive demography theory predicts that the natural hive age composition represents a colony-level adaptation and thus results in optimal hive performance. Alternatively, the hive age composition may be an epiphenomenon, resulting from individual life history optimization. We addressed these predictions by comparing individual worker longevity and brood production in hives that were composed of a single-age cohort, two distinct age cohorts, and hives that had a continuous, natural age distribution. Four experimental replicates showed that colonies with a natural age composition did not consistently have a higher life expectancy and/or brood production than the single-cohort or double-cohort hives. Instead, a complex interplay of age structure, environmental conditions, colony size, brood production, and individual mortality emerged. A general tradeoff between worker life expectancy and colony productivity was apparent, and the transition from in-hive tasks to foraging was the most significant predictor of worker lifespan irrespective of the colony age structure. We conclude that the natural age structure of honeybee hives is not a colony-level adaptation. Furthermore, our results show that honeybees exhibit pronounced demographic plasticity in addition to behavioral plasticity to react to demographic disturbances of their societies.     

Comparisons of forager distributions from matched honey bee colonies in suburban environments

We conducted experiments designed to examine the distribution of foraging honey bees (Apis mellifera) in suburban environments with rich floras and to compare spatial patterns of foraging sites used by colonies located in the same environment. The patterns we observed in resource visitation suggest a reduced role of the recruitment system as part of the overall colony foraging strategy in habitats with abundant, small patches of flowers. We simultaneously sampled recruitment dances of bees inside observation hives in two colonies over 4 days in Miami, Florida (1989) and from two other colonies over five days in Riverside, California (1991). Information encoded in the dance was used to determine the distance and direction that bees flew from the hive for pollen and nectar and to construct foraging maps for each colony. The foraging maps showed that bees from the two colonies in each location usually foraged at different sites, but occasionally they visited the same patches of flowers. Each colony shifted foraging effort among sites on different days. In both locations, the mean flight distances differed between colonies and among days within colonies. The flight distances observed in our study are generally shorter than those reported in a similar study conducted in a temperate deciduous forest where resources were less dense and floral patches were smaller.     

Sex differences in the movement patterns of free-ranging chimpanzees (<Emphasis Type="Italic">Pan troglodytes schweinfurthii</Emphasis>): foraging and border checking

Most social primates live in cohesive groups, so travel paths inevitably reflect compromise: decision processes of individuals are obscured. The fission–fusion social organisation of the chimpanzee, however, allows an individual's movements to be investigated independently. We followed 15 chimpanzees (eight male and seven female) through the relatively flat forest of Budongo, Uganda, plotting the path of each individual over periods of 1–3 days. Chimpanzee movement was parsed into phases ending with halts of more than 20 min, during which individuals fed, rested or engaged in social activities. Males, lactating or pregnant females and sexually receptive females all travelled similar average distances between halts, at similar speeds and along similarly direct beeline paths. Compared to lactating or pregnant females, males did travel for a significantly longer time each day and halted more often, but the most striking sex differences appeared in the organisation of movement phases into a day's path. After a halt, males tended to continue in the same direction as before. Lactating or pregnant females showed no such strategy and often retraced the preceding phase, returning to previously visited food patches. We suggest that female chimpanzee movements approximate an optimal solution to feeding requirements, whereas the paths of males allow integration of foraging with territorial defence. The ‘continually moving forwards’ strategy of males enables them to monitor their territory boundaries—border checking—whilst foraging, generally avoiding the explicit boundary patrols observed at other chimpanzee study sites.     

Estimating the Density of Honeybee Colonies across Their Natural Range to Fill the Gap in Pollinator Decline Censuses

Abstract: Although pollinator declines are a global biodiversity threat, the demography of the western honeybee (Apis mellifera) has not been considered by conservationists because it is biased by the activity of beekeepers. To fill this gap in pollinator decline censuses and to provide a broad picture of the current status of honeybees across their natural range, we used microsatellite genetic markers to estimate colony densities and genetic diversity at different locations in Europe, Africa, and central Asia that had different patterns of land use. Genetic diversity and colony densities were highest in South Africa and lowest in Northern Europe and were correlated with mean annual temperature. Confounding factors not related to climate, however, are also likely to influence genetic diversity and colony densities in honeybee populations. Land use showed a significantly negative influence over genetic diversity and the density of honeybee colonies over all sampling locations. In Europe honeybees sampled in nature reserves had genetic diversity and colony densities similar to those sampled in agricultural landscapes, which suggests that the former are not wild but may have come from managed hives. Other results also support this idea: putative wild bees were rare in our European samples, and the mean estimated density of honeybee colonies on the continent closely resembled the reported mean number of managed hives. Current densities of European honeybee populations are in the same range as those found in the adverse climatic conditions of the Kalahari and Saharan deserts, which suggests that beekeeping activities do not compensate for the loss of wild colonies. Our findings highlight the importance of reconsidering the conservation status of honeybees in Europe and of regarding beekeeping not only as a profitable business for producing honey, but also as an essential component of biodiversity conservation.     

Propagation of olfactory information within the honeybee hive

Transfer of information about food source characteristics within insect societies is essential to colony-foraging success. The food odor communicated within honeybee hives has been shown to be important for food source exploitation. When successful foragers return to the nest and transfer the collected nectar to hive mates through mouth-to-mouth contacts (trophallaxis), potential recruits receiving these samples learn the food odor by associative learning. The food then becomes rapidly distributed among colony members, which is mainly a consequence of the numerous trophallaxes between hive-mates of all ages during food processing. We tested whether the distribution of food among hive mates causes a propagation of olfactory information within the hive. Using the proboscis extension response paradigm, we show that large proportions of bees of the age groups representing the main worker castes, 4 to 9-day-old bees (nurse-aged bees), 12 to 16-day-old bees (food processor-aged bees), and actual foragers (about 17+ day old bees) associatively learn the food odor in the course of processing food that has been collected by only a few foragers. Results further suggest that the information is shared more or less equally between bees of the three age groups. This shows that olfactory information about the flower species exploited by foragers is distributed within the entire colony and is acquired by bees of all age groups, which may influence many behaviors inside and outside the hive.