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Lévy flight is a type of random walk that characterizes the behaviour of many natural phenomena studied across a multiplicity of academic disciplines; within biology specifically, the behaviour of fish, birds, insects, mollusks, bacteria, plants, slime molds, t-cells, and human populations. The Lévy flight foraging hypothesis states that because Lévy flights can maximize an organism’s search efficiency, natural selection should result in Lévy-like behaviour. Empirical and theoretical research has provided ample evidence of Lévy walks in both extinct and extant species, and its efficiency across models with a diversity of resource distributions. However, no model has addressed the maintenance of Lévy flight foraging through evolutionary processes, and existing models lack ecological breadth. We use numerical simulations, including lineage-based models of evolution with a distribution of move lengths as a variable and heritable trait, to test the Lévy flight foraging hypothesis. We include biological and ecological contexts such as population size, searching costs, lifespan, resource distribution, speed, and consider both energy accumulated at the end of a lifespan and averaged over a lifespan. We demonstrate that selection often results in Lévy-like behaviour, although conditional; smaller populations, longer searches, and low searching costs increase the fitness of Lévy-like behaviour relative to Brownian behaviour. Interestingly, our results also evidence a bet-hedging strategy; Lévy-like behaviour reduces fitness variance, thus maximizing geometric mean fitness over multiple generations., Author summary In heterotrophs, incuding animals, survival depends on the net energy gained through foraging. The expectation, then, is that natural selection results in adaptations for efficient foraging that optimize the balance of searching costs and rewards. Lévy flight foraging has been proposed as an optimal foraging solution. The hypothesis states, if no information about resource locations are available, and the locations are re-visitable, then selection will result in adaptations for Lévy flight foraging, a type of random walk. It has been argued that Levy-like foraging behaviour may simply reflect how resources are distributed, but empirical and theoretical research suggests that this behaviour is intrinsic or innate. However, this research does not address evolutionary mechanisms, and lacks ecological breadth. We extend the current theoretical framework by including evolutionary ecological contexts. We treat an organism’s random walk as a heritable trait, and explore ecological contexts such as population size, lifespan, carrying capacity, searching costs, reproductive strategies, and different distributions of food. Our evolutionary simulations overwhelmingly resulted in selection for Lévy-like foraging, regardless of the distribution of food, and evidences Lévy flight foraging as a bet-hedging strategy. Thus, here we provide some of the first evidence for the evolutionary maintenance of Lévy flight foraging.

It is widely accepted that inverse square Levy walks are optimal search strategies because they maximize the encounter rate with sparse, randomly distributed, replenishable targets when the search restarts in the vicinity of the previously visited target, which becomes revisitable again with high probability, i.e., non-destructive foraging [Nature 401, 911 (1999)]. The precise conditions for the validity of this Levy flight foraging hypothesis (LFH) have been widely described in the literature [Phys. Life Rev. 14, 94 (2015)]. Nevertheless, three objecting claims to the LFH have been raised recently for $d \geq 2$: (i) the capture rate $\eta$ has linear dependence on the target density $\rho$ for all values of the Levy index $\alpha$; (ii) "the gain $\eta_{max}/\eta$ achieved by varying $\alpha $ is bounded even in the limit $\rho \to 0 $" so that "tuning $\alpha$ can only yield a marginal gain"; (iii) depending on the values of the radius of detection $a$, the restarting distance $l_c$ and the scale parameter $s$, the optimum is realized for a range of $\alpha$ [Phys. Rev. Lett. 124, 080601 (2020)]. Here we answer each of these three criticisms in detail and show that claims (i)-(iii) do not actually invalidate the LFH. Our results and analyses restore the original result of the LFH for non-destructive foraging.

We refute here the concernes raised in the Comment of our letter. This reply states clearly the validity range of our results and shows that the optimality of inverse-square Levy walks at the basis of the Levy flight foraging hypothesis is generically unfounded. We also give the precise set of conditions for which inverse-levy square Levy walks turn to be optimal, conditions which are unlikely to be verified biologically.

Autonomous aerial robots are commonly tasked with the problem of area exploration, surveillance and search for certain targets or objects of interest to be detected and tracked. Traditionally, the problem formulation considered is that of complete coverage and thus - ideally - identification of all targets of interest. However, an important problem which is not often addressed is that of memoryless time-efficient search under sparse rewards that may be worth visited any number of times. An indicative application scenario relates to surveillance for moving and possibly camouflaged targets thus making map coverage an inherently memoryless process. In this paper we specifically address the largely understudied problem of optimizing the “time-of-arrival” or “time-of-detection” to robotically search for sparsely distributed rewards (detect targets of interest) within large-scale environments and subject to memoryless exploration. At the core of the proposed solution is the fact that a search-based Levy walk consisting of a constant velocity search following a Levy flight path is optimal for searching and identifying distributed target regions in the lack of map memory. A set of results accompany the presentation of the method, demonstrate its properties and justify the purpose of its use towards large-scale area exploration autonomy using aerial robots.

© 2018 The Authors A theoretical and applied literature has suggested that foragers search using Lévy flights, since Lévy flights can maximize the efficiency of search in the absence of information on the location of randomly distributed prey. Foragers, however, often have available to them at least some information about the distribution of prey, gained either through evolved mechanisms, experience and memory, or social transmission of information. As such, we might expect selection for heuristics that make use of such information to further improve the efficiency of random search. Here we present a general model of random search behavior that includes as special cases: area-restricted search, correlated random walks, Brownian search, and Lévy flights. This generative model allows foragers to adjust search parameters based on encounter-conditional and other heuristics. Using a simulation model, we demonstrate the efficiency gains of these search heuristics, and illustrate the resulting differences in the distributions of step-size and heading angle change they imply, relative to Lévy flights. We conclude by presenting a statistical model that can be fit to empirical data and a set of testable, quantitative predictions that contrast our model of adaptive search with the Lévy flight foraging hypothesis.

We study through a reaction-diffusion algorithm the influence of landscape diversity on the efficiency of search dynamics. Remarkably, the identical optimal search strategy arises in a wide variety of environments, provided the target density is sparse and the searcher’s information is restricted to its close vicinity. Our results strongly impact the current debate on the emergentist vs. evolutionary origins of animal foraging. The inherent character of the optimal solution (i.e., independent on the landscape for the broad scenarios assumed here) suggests an interpretation favoring the evolutionary view, as originally implied by the Lévy flight foraging hypothesis. The latter states that, under conditions of scarcity of information and sparse resources, some organisms must have evolved to exploit optimal strategies characterized by heavy-tailed truncated power-law distributions of move lengths. These results strongly suggest that Lévy strategies—and hence the selection pressure for the relevant adaptations—are robust with respect to large changes in habitat. In contrast, the usual emergentist explanation seems not able to explain how very similar Lévy walks can emerge from all the distinct non-Lévy foraging strategies that are needed for the observed large variety of specific environments. We also report that deviations from Lévy can take place in plentiful ecosystems, where locomotion truncation is very frequent due to high encounter rates. So, in this case normal diffusion strategies—performing as effectively as the optimal one—can naturally emerge from Lévy. Our results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks., Author summary How organisms improve the search for food, mates, etc., is a key factor to their survival. Mathematically, the best strategy to look for randomly distributed re-visitable resources—under scarce information and sparse conditions—results from Lévy distributions of move lengths (the probability of taking a step ℓ is proportional to 1/ℓ2). Today it is well established that many animal species in different habitats do perform Lévy foraging. This fact has raised a heated debate, viz., the emergent versus evolutionary hypotheses. For the former, a Lévy foraging is an emergent property, a consequence of searcher-environment interactions: certain landscapes induce Lévy patterns, but others not. In this view, the optimal strategy depends on the particular habitat. The evolutionary explanation, in contrast, is that Lévy foraging strategies are adaptations that evolved via natural selection. In this article, through simulations we exhaustively analyze the influence of distinct environments on the foraging efficiency. We find that the optimal procedure is the same in all situations, provided density is low and landscape information is scarce. So, the best search strategy is remarkably independent of details. These results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

Weierstrassian Lévy walks are the archetypical form of random walk that do not satisfy the central limit theorem and are instead characterized by scale invariance. They were originally regarded as a mathematical abstraction but subsequent theoretical studies showed that they can, in principle, at least, be generated by chaos. Recently, Weierstrassian Lévy walks have been found to provide accurate representations of the movement patterns of mussels (Mytilus edulis) and mud snails (Hydrobia ulvae) recorded in the laboratory under controlled conditions. Here, we tested whether Weierstrassian Lévy walks and chaos are present under natural conditions in intertidal limpetsPatella vulgataandP. rustica,and found that both characteristics are pervasive. We thereby show that Weierstrassian Lévy walks may be fundamental to how molluscs experience and interact with the world across a wide range of ecological contexts. We also show in an easily accessible way how chaos can produce a wide variety of Weierstrassian Lévy walk movement patterns. Our findings support the Lévy flight foraging hypothesis that posits that because Lévy walks can optimize search efficiencies, natural selection should have led to adaptations for Lévy walks.

Summary Levy flight foraging represents an innovative paradigm for the analysis of animal random search by including models of heavy-tailed distribution of move length, which complements the correlated random walk paradigm that is founded on Brownian walks. Theory shows that the efficiency of the different foraging tactics is a function of prey abundance and dynamics with Levy flight being especially efficient in poor prey fields. Levy flights have been controversial in some quarters, because they previously have been wrongly ascribed to many species through the employment of inappropriate statistical techniques and by misunderstanding movement pattern data. More recent studies using state-of-the-art statistical tools have, however, provided seemingly compelling evidence for Levy flights. In this study, we employ these maximum-likelihood methods and their Bayesian equivalents by analysing both turning angles and move length distributions. We tested, for compliance with Levy flight foraging, a set of 77 independent foraging trajectories of Cory's shearwaters Calonectris diomedea diomedea. Birds were tagged with high-resolution GPS loggers in two Mediterranean colonies (Linosa and Tremiti) during both incubation and chick rearing. We found that the behaviour of six birds was fitted by a correlated random walk; the movement of 32 birds was better represented by adaptive correlated random walks by switching from intensive to extensive searches; and the trajectories of 36 birds were fitted by a Levy flight pattern of movement. The probability of performing Levy flights was higher for trips during chick provisioning when shearwaters were forced to forage in suboptimal areas. This study supports Levy flight foraging as an appropriate framework to analyse search tactics in this pelagic bird species and highlights that the adoption of a given search strategy is a function of biological and ecological constraints.

1. Search processes play an important role in physical, chemical and biological systems. In animal foraging, the search strategy predators should use to search optimally for prey is an enduring question. Some models demonstrate that when prey is sparsely distributed, an optimal search pattern is a specialised random walk known as a Levy flight, whereas when prey is abundant, simple Brownian motion is sufficiently efficient. These predictions form part of what has been termed the Levy flight foraging hypothesis (LFF) which states that as Levy flights optimise random searches, movements approximated by optimal Levy flights may have naturally evolved in organisms to enhance encounters with targets (e.g. prey) when knowledge of their locations is incomplete. 2. Whether free-ranging predators exhibit the movement patterns predicted in the LFF hypothesis in response to known prey types and distributions, however, has not been determined. We tested this using vertical and horizontal movement data from electronic tagging of an apex predator, the great white shark Carcharodon carcharias, across widely differing habitats reflecting different prey types. 3. Individual white sharks exhibited movement patterns that predicted well the prey types expected under the LFF hypothesis. Shark movements were best approximated by Brownian motion when hunting near abundant, predictable sources of prey (e.g. seal colonies, fish aggregations), whereas movements approximating truncated Levy flights were present when searching for sparsely distributed or potentially difficult-to-detect prey in oceanic or shelf environments, respectively. 4. That movement patterns approximated by truncated Levy flights and Brownian behaviour were present in the predicted prey fields indicates search strategies adopted by white sharks appear to be the most efficient ones for encountering prey in the habitats where such patterns are observed. This suggests that C. carcharias appears capable of exhibiting search patterns that are approximated as optimal in response to encountered changes in prey type and abundance, and across diverse marine habitats, from the surf zone to the deep ocean. 5. Our results provide some support for the LFF hypothesis. However, it is possible that the observed Levy patterns of white sharks may not arise from an adaptive behaviour but could be an emergent property arising from simple, straight-line movements between complex (e.g. fractal) distributions of prey. Experimental studies are needed in vertebrates to test for the presence of Levy behaviour patterns in the absence of complex prey distributions.

We review the general problem of random searches in the context of biological encounters. We analyze deterministic and stochastic aspects of searching in general and address the destructive and nondestructive cases specifically. We discuss the concepts of Levy walks as adaptive strategies and explore possible examples. We also review Levy searches in other media and spaces, including lattices and networks as opposed to continuous environments. We analyze empirical evidence supporting the Levy flight foraging hypothesis, as well as the more general idea of superdiffusive foraging. We compare these hypothesis with alternative theories of random searches. Finally, we comment on several issues relevant to the practical application of models of Levy and superdiffusive strategies to the general question of biological foraging.

The study of animal foraging behaviour is of practical ecological importance1, and exemplifies the wider scientific problem of optimizing search strategies2. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails3, 4, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical3, 4, 5 and chemical6 systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface7. This well known finding2, 4, 8, 9 was followed by similar inferences about the search strategies of deer10 and bumblebees10. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer13, microzooplankton14, grey seals15, spider monkeys16 and fishing boats17. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data7 using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood18 and Akaike weights19, 20. We apply this to the four original deer and bumblebee data sets10, finding that none exhibits evidence of Lévy flights, and that the original graphical approach10 is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms13, 14, 15, 16, 17, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool17. Our results question the strength of the empirical evidence for biological Lévy flights.

While evidence for optimal random search patterns, known as Lévy walks, in empirical movement data is mounting for a growing list of taxa spanning motile cells to humans, there is still much debate concerning the theoretical generality of Lévy walk optimisation. Here, using a new and robust simulation environment, we investigate in the most detailed study to date (24×10(6) simulations) the foraging and search efficiencies of 2-D Lévy walks with a range of exponents, target resource distributions and several competing models. We find strong and comprehensive support for the predictions of the Lévy flight foraging hypothesis and in particular for the optimality of inverse square distributions of move step-lengths across a much broader range of resource densities and distributions than previously realised. Further support for the evolutionary advantage of Lévy walk movement patterns is provided by an investigation into the 'feast and famine' effect, with Lévy foragers in heterogeneous environments experiencing fewer long 'famines' than other types of searchers. Therefore overall, optimal Lévy foraging results in more predictable resources in unpredictable environments.

We address the general question of what is the best statistical strategy to adapt in order to search efficiently for randomly located objects ('target sites'). It is often assumed in foraging theory that the flight lengths of a forager have a characteristic scale: from this assumption gaussian, Rayleigh and other classical distributions with well-defined variances have arisen. However, such theories cannot explain the long-tailed power-law distributions of flight lengths or flight times that are observed experimentally. Here we study how the search efficiency depends on the probability distribution of flight lengths taken by a forager that can detect target sites only in its limited vicinity. We show that, when the target sites are sparse and can be visited any number of times, an inverse square power-law distribution of flight lengths, corresponding to Lévy flight motion, is an optimal strategy. We test the theory by analysing experimental foraging data on selected insect, mammal and bird species, and find that they are consistent with the predicted inverse square power-law distributions.

Background The Lévy flight foraging hypothesis predicts a transition from scale-free Lévy walk (LW) to scale-specific Brownian motion (BM) as an animal moves from resource-poor towards resource-rich environment. However, the LW-BM continuum implies a premise of memory-less search, which contradicts the cognitive capacity of vertebrates. Results We describe methods to test if apparent support for LW-BM transitions may rather be a statistical artifact from movement under varying intensity of site fidelity. A higher frequency of returns to previously visited patches (stronger site fidelity) may erroneously be interpreted as a switch from LW towards BM. Simulations of scale-free, memory-enhanced space use illustrate how the ratio between return events and scale-free exploratory movement translates to varying strength of site fidelity. An expanded analysis of GPS data of 18 female red deer, Cervus elaphus, strengthens previous empirical support of memory-enhanced and scale-free space use in a northern forest ecosystem. Conclusion A statistical mechanical model architecture that describes foraging under environment-dependent variation of site fidelity may allow for higher realism of optimal search models and movement ecology in general, in particular for vertebrates with high cognitive capacity. Electronic supplementary material The online version of this article (doi:10.1186/2051-3933-1-9) contains supplementary material, which is available to authorized users.

International audience; Animal behaviour exhibits fractal structure in space and time. Fractal properties in animal space-use have been explored extensively under the Le'vy flight foraging hypothesis, but studies of behaviour change itself through time are rarer, have typically used shorter sequences generated in the laboratory, and generally lack critical assessment of their results. We thus performed an in-depth analysis of fractal time in binary dive sequences collected via bio-logging from free-ranging little penguins (Eudyptula minor) across full-day foraging trips (216 data points; 4 orders of temporal magnitude). Results from 4 fractal methods show that dive sequences are long-range dependent and persistent across ca. 2 orders of magnitude. This fractal structure correlated with trip length and time spent underwater, but individual traits had little effect. Fractal time is a fundamental characteristic of penguin foraging behaviour, and its investigation is thus a promising avenue for research on interactions between animals and their environments.

The organization of episodes of activity and inactivity of Drosophila melanogaster has a complex structure. Episodes of apparently continuous activity have shorter episodes of inactivity embedded within them. This pattern of activity has a self-similar structure; the activity record appears the same regardless of the time scale used. There is a power law dependence of the rate ‘constants’ that describes how the activity of flies turns on and off. As a consequence no natural time scale for the measurement of the activity level exists, and the amount of activity depends on the time scale of measurement. The observation of fractal time variability in animal behaviour may illuminate the processes that produce activity. Fractal time variation in movement activity can lead to Levy flight patterns of movement, which produce efficient searches.

It is an open question how animals find food in dynamic natural environments where they possess little or no knowledge of where resources are located. Foraging theory predicts that in environments with sparsely distributed target resources, where forager knowledge about resources’ locations is incomplete, Lévy flight movements optimize the success of random searches. However, the putative success of Lévy foraging has been demonstrated only in model simulations. Here, we use high-temporal-resolution Global Positioning System (GPS) tracking of wandering ( Diomedea exulans ) and black-browed albatrosses ( Thalassarche melanophrys ) with simultaneous recording of prey captures, to show that both species exhibit Lévy and Brownian movement patterns. We find that total prey masses captured by wandering albatrosses during Lévy movements exceed daily energy requirements by nearly fourfold, and approached yields by Brownian movements in other habitats. These results, together with our reanalysis of previously published albatross data, overturn the notion that albatrosses do not exhibit Lévy patterns during foraging, and demonstrate that Lévy flights of predators in dynamic natural environments present a beneficial alternative strategy to simple, spatially intensive behaviors. Our findings add support to the possibility that biological Lévy flight may have naturally evolved as a search strategy in response to sparse resources and scant information.

Tracks of moving animals have many short moves interspersed with longer moves, and frequencies of all moves together often indicate Levy flights (a type of power law)^1^. The Levy flight foraging hypothesis suggests that, 'since Levy flights and walks can optimize search efficiencies, therefore natural selection should have led to adaptations for Levy flight foraging^1^ and it has been apparently supported by several studies of large numbers of tracks of marine animals^234^. We show that Levy flights are caused by marine animals attempting to move in simple ways and they are unlikely to be evolved behavior. We do this by analyzing the tracks of autonomous underwater gliders which conform to Levy flights while they were programmed to perform simple directed movement. Gliders are autonomous underwater vehicles (AUV), propelled by a buoyancy engine^5^.The principle of operation is that by alternating positive and negative buoyancy, the winged AUV glides through the ocean in an undulating path, resurfacing after a pre-programmed number of undulations. During the periods at the surface, which typically last 15 minutes, GPS is used for positioning before and after transmission of data via a satellite link, whereas underwater the trajectory is deadreckoned from measured heading, pitch and pressure rate. Thus they are potentially a good analogy to a navigating animal which may also attempt to make directed movements using the best available (and often intermittent^6^) navigational information.

An optimal search theory, the so-called Levy-flight foraging hypothesis1, predicts that predators should adopt search strategies known as Levy flights where prey is sparse and distributed unpredictably, but that Brownian movement is sufficiently efficient for locating abundant prey2, 3, 4. Empirical studies have generated controversy because the accuracy of statistical methods that have been used to identify Levy behaviour has recently been questioned5, 6. Consequently, whether foragers exhibit Levy flights in the wild remains unclear. Crucially, moreover, it has not been tested whether observed movement patterns across natural landscapes having different expected resource distributions conform to the theory’s central predictions. Here we use maximum-likelihood methods to test for Levy patterns in relation to environmental gradients in the largest animal movement data set assembled for this purpose. Strong support was found for Levy search patterns across 14 species of open-ocean predatory fish (sharks, tuna, billfish and ocean sunfish), with some individuals switching between Levy and Brownian movement as they traversed different habitat types. We tested the spatial occurrence of these two principal patterns and found Levy behaviour to be associated with less productive waters (sparser prey) and Brownian movements to be associated with productive shelf or convergence-front habitats (abundant prey). These results are consistent with the Levy-flight foraging hypothesis1, 7, supporting the contention8, 9 that organism search strategies naturally evolved in such a way that they exploit optimal Levy patterns.

Many free-ranging predators have to make foraging decisions with little, if any, knowledge of present resource distribution and availability1. The optimal search strategy they should use to maximize encounter rates with prey in heterogeneous natural environments remains a largely unresolved issue in ecology1, 2, 3. Lévy walks4 are specialized random walks giving rise to fractal movement trajectories that may represent an optimal solution for searching complex landscapes5. However, the adaptive significance of this putative strategy in response to natural prey distributions remains untested6, 7. Here we analyse over a million movement displacements recorded from animal-attached electronic tags to show that diverse marine predators—sharks, bony fishes, sea turtles and penguins—exhibit Lévy-walk-like behaviour close to a theoretical optimum2. Prey density distributions also display Lévy-like fractal patterns, suggesting response movements by predators to prey distributions. Simulations show that predators have higher encounter rates when adopting Lévy-type foraging in natural-like prey fields compared with purely random landscapes. This is consistent with the hypothesis that observed search patterns are adapted to observed statistical patterns of the landscape. This may explain why Lévy-like behaviour seems to be widespread among diverse organisms3, from microbes8 to humans9, as a 'rule' that evolved in response to patchy resource distributions.

Levy 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.

The origin of fractal patterns is a fundamental problem in many areas of science. In ecological systems, fractal patterns show up in many subtle ways and have been interpreted as emergent phenomena related to some universal principles of complex systems. Recently, Lévy-type processes have been pointed out as relevant in large-scale animal movements. The existence of Lévy probability distributions in the behavior of relevant variables of movement, introduces new potential diffusive properties and optimization mechanisms in animal foraging processes. In particular, it has been shown that Lévy processes can optimize the success of random encounters in a wide range of search scenarios, representing robust solutions to the general search problem. These results set the scene for an evolutionary explanation for the widespread observed scale-invariant properties of animal movements. Here, it is suggested that scale-free reorientations of the movement could be the basis for a stochastic organization of the search whenever strongly reduced perceptual capacities come into play. Such a proposal represents two new evolutionary insights. First, adaptive mechanisms are explicitly proposed to work on the basis of stochastic laws. And second, though acting at the individual-level, these adaptive mechanisms could have straightforward effects at higher levels of ecosystem organization and dynamics (e.g. macroscopic diffusive properties of motion, population-level encounter rates). Thus, I suggest that for the case of animal movement, fractality may not be representing an emergent property but instead adaptive random search strategies. So far, in the context of animal movement, scale-invariance, intermittence, and chance have been studied in isolation but not synthesized into a coherent ecological and evolutionary framework. Further research is needed to track the possible evolutionary footprint of Lévy processes in animal movement.

10 páginas, 4 figuras., Recent advances in spatial ecology have improved our understanding of the role of large-scale animal movements. However, an unsolved problem concerns the inherent stochasticity involved in many animal search displacements and its possible adaptive value. When animals have no information about where targets (i.e., resource patches, mates, etc.) are located, different random search strategies may provide different chances to find them. Assuming random-walk models as a necessary tool to understand how animals face such environmental uncertainty, we analyze the statistical differences between two random-walk models commonly used to fit animal movement data, the Le´vy walks and the correlated random walks, and we quantify their efficiencies (i.e., the number of targets found in relation to total displacement) within a random search context. Correlated random-walk properties (i.e., scale-finite correlations) may be interpreted as the by-product of locally scanning mechanisms. Le´vy walks, instead, have fundamental properties (i.e., super-diffusivity and scale invariance) that allow a higher efficiency in random search scenarios. Specific biological mechanisms related to how animals punctuate their movement with sudden reorientations in a random search would be sufficient to sustain Le´vy walk properties. Furthermore, we investigate a new model (the Le´vy-modulated correlated random walk) that combines the properties of correlated and Le´vy walks. This model shows that Le´vy walk properties are robust to any behavioral mechanism providing short-range correlations in the walk. We propose that some animals may have evolved the ability of performing Le´vy walks as adaptive strategies in order to face search uncertainties., CNPq and FAPEAL (G. M. Viswanathan), and CNPq, CT-Infro, CT-Energ, Capes, and Fundac¸a˜o Arauca´ria (M. G. E. da Luz) are acknowledged for research grants.

© 2015 by The University of Chicago. Animal movements exhibit an almost universal pattern of fat-tailed step-size distributions, mixing short and very long steps. The Lévy flight foraging hypothesis (LFFH) suggests a single optimal food search strategy to explain this pattern, yet mixed movement distributions are biologically more plausible and often convincingly fit movement data. To confront alternative explanations for these patterns, we tracked vultures of three species in two very different ecosystems using high-resolution global positioning system/accelerometer tags accompanied by behavioral, genetic, and morphological data. The Lévy distribution fitted the data sets reasonably well, matching expectations based on their sparsely distributed food resources; yet the fit of mixed models was considerably better, suggesting distinct movement modes operating at three different scales. Specifically, long-range forays (LRFs)—rare, short-term, large-scale circular journeys that greatly exceed the typical foraging range and contribute to the tail-fatness of the movement distribution in all three species— do not match an optimal foraging strategy suggested by the LFFH. We also found no support for preferred weather conditions or population genetic structure as alternative explanations, so the hypothesis that LRFs represent failed breeding dispersal attempts to find mates remains our most plausible explanation at this time. We conclude that inference about the mechanisms underlying animal movements should be confronted with complementary data, and suggest that mixed behavioral modes likely explain commonly observed fat-tailed movement distributions.