Adaptive Lévy Walks in Foraging Fallow Deer

Background

Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy walks maximise search efficiency of mobile foragers. Recently, several studies raised some concerns about the reliability of the statistical analysis used in previous analyses. Further, it is unclear whether Lévy walks represent adaptive strategies or emergent properties determined by the interaction between foragers and resource distribution. Thus two fundamental questions still need to be addressed: the presence of Lévy walks in the wild and whether or not they represent a form of adaptive behaviour.

Methodology/Principal Findings

We studied 235 paths of solitary and clustered (i.e. foraging in group) fallow deer (Dama dama), exploiting the same pasture. We used maximum likelihood estimation for discriminating between a power-tailed distribution and the exponential alternative and rank/frequency plots to discriminate between Lévy walks and composite Brownian walks. We showed that solitary deer perform Lévy searches, while clustered animals did not adopt that strategy.

Conclusion/Significance

Our demonstration of the presence of Lévy walks is, at our knowledge, the first available which adopts up-to-date statistical methodologies in a terrestrial mammal. Comparing solitary and clustered deer, we concluded that the Lévy walks of solitary deer represent an adaptation maximising encounter rates with forage resources and not an epiphenomenon induced by a peculiar food distribution.     

Adaptive Lévy Processes and Area-Restricted Search in Human Foraging

A considerable amount of research has claimed that animals’ foraging behaviors display movement lengths with power-law distributed tails, characteristic of Lévy flights and Lévy walks. Though these claims have recently come into question, the proposal that many animals forage using Lévy processes nonetheless remains. A Lévy process does not consider when or where resources are encountered, and samples movement lengths independently of past experience. However, Lévy processes too have come into question based on the observation that in patchy resource environments resource-sensitive foraging strategies, like area-restricted search, perform better than Lévy flights yet can still generate heavy-tailed distributions of movement lengths. To investigate these questions further, we tracked humans as they searched for hidden resources in an open-field virtual environment, with either patchy or dispersed resource distributions. Supporting previous research, for both conditions logarithmic binning methods were consistent with Lévy flights and rank-frequency methods–comparing alternative distributions using maximum likelihood methods–showed the strongest support for bounded power-law distributions (truncated Lévy flights). However, goodness-of-fit tests found that even bounded power-law distributions only accurately characterized movement behavior for 4 (out of 32) participants. Moreover, paths in the patchy environment (but not the dispersed environment) showed a transition to intensive search following resource encounters, characteristic of area-restricted search. Transferring paths between environments revealed that paths generated in the patchy environment were adapted to that environment. Our results suggest that though power-law distributions do not accurately reflect human search, Lévy processes may still describe movement in dispersed environments, but not in patchy environments–where search was area-restricted. Furthermore, our results indicate that search strategies cannot be inferred without knowing how organisms respond to resources–as both patched and dispersed conditions led to similar Lévy-like movement distributions.     

Scavengers on the Move: Behavioural Changes in Foraging Search Patterns during the Annual Cycle

Background

Optimal foraging theory predicts that animals will tend to maximize foraging success by optimizing search strategies. However, how organisms detect sparsely distributed food resources remains an open question. When targets are sparse and unpredictably distributed, a Lévy strategy should maximize foraging success. By contrast, when resources are abundant and regularly distributed, simple Brownian random movement should be sufficient. Although very different groups of organisms exhibit Lévy motion, the shift from a Lévy to a Brownian search strategy has been suggested to depend on internal and external factors such as sex, prey density, or environmental context. However, animal response at the individual level has received little attention.

Methodology/Principal Findings

We used GPS satellite-telemetry data of Egyptian vultures Neophron percnopterus to examine movement patterns at the individual level during consecutive years, with particular interest in the variations in foraging search patterns during the different periods of the annual cycle (i.e. breeding vs. non-breeding). Our results show that vultures followed a Brownian search strategy in their wintering sojourn in Africa, whereas they exhibited a more complex foraging search pattern at breeding grounds in Europe, including Lévy motion. Interestingly, our results showed that individuals shifted between search strategies within the same period of the annual cycle in successive years.

Conclusions/Significance

Results could be primarily explained by the different environmental conditions in which foraging activities occur. However, the high degree of behavioural flexibility exhibited during the breeding period in contrast to the non-breeding period is challenging, suggesting that not only environmental conditions explain individuals'' behaviour but also individuals'' cognitive abilities (e.g., memory effects) could play an important role. Our results support the growing awareness about the role of behavioural flexibility at the individual level, adding new empirical evidence about how animals in general, and particularly scavengers, solve the problem of efficiently finding food resources.     

Experimental evidence for inherent Lévy search behaviour in foraging animals

Recently, Lévy walks have been put forward as a new paradigm for animal search and many cases have been made for its presence in nature. However, it remains debated whether Lévy walks are an inherent behavioural strategy or emerge from the animal reacting to its habitat. Here, we demonstrate signatures of Lévy behaviour in the search movement of mud snails (Hydrobia ulvae) based on a novel, direct assessment of movement properties in an experimental set-up using different food distributions. Our experimental data uncovered clusters of small movement steps alternating with long moves independent of food encounter and landscape complexity. Moreover, size distributions of these clusters followed truncated power laws. These two findings are characteristic signatures of mechanisms underlying inherent Lévy-like movement. Thus, our study provides clear experimental evidence that such multi-scale movement is an inherent behaviour rather than resulting from the animal interacting with its environment.     

Optimising the success of random destructive searches: Lévy walks can outperform ballistic motions

We re-evaluate the long standing and widely held belief that ballistic movements (i.e. straight-lines movements) outperform Lévy walks when searching for targets that once located are not revisited. The belief stems from the results of analyses of one-dimensional searches, analyses which have not accounted for the fact that target numbers can be continually depleted during the search process. This is a crucial oversight because continual depletion promotes the searching efficiencies of some Lévy walks above that of ballistic motion. The continual depletion effect is not so important for two- and three-dimensional searches. Nevertheless, we show that Lévy walks and ballistic movements can be equally or almost equally effective when searching within two- and three-dimensional environments for randomly and sparsely distributed targets or when searching for targets that are occasionally concealed. We also show that Lévy walks are advantageous when searching for targets that can occasionally evade capture. These situations represent common predator–prey interactions in which predators are involved in ‘imperfect destructive’ searches. Our model suggests that accounting for coevolutionary arms races at the predator–prey detection/reaction scales can explain to some extent Lévy walk searching patterns of predators at larger scales. This result provides new insights into the Lévy walk movement patterns of some destructive foragers.     

Anomalous diffusion of water in biological tissues.

This article deals with the characterization of biological tissues and their pathological alterations. For this purpose, diffusion is measured by NMR in the fringe field of a large superconductor with a field gradient of 50 T/m, which is rather homogenous and stable. It is due to the unprecedented properties of the gradient that we are able not only to determine the usual diffusion coefficient, but also to observe the pronounced Non-Debye feature of the relaxation function due to cellular structure. The dynamics of the probability density follow a stretched exponential or Kohlrausch-Williams-Watts function. In the long time limit the Fourier transform of the probability density follows a long-tail Lévy function, whose asymptotic is related to the fractal dimension of the underlying cellular structure. Some of the properties of Lévy walk statistics are discussed and its potential importance in understanding certain biophysical phenomena like diffusion processes in biological tissues are pointed out. We present and discuss for the first time NMR data giving evidence for Lévy processes that capture the essential features of the observed power law (scaling) dynamics of water diffusion in fresh tissue specimens: carcinomas, fibrous mastopathies, adipose and liver tissues.     

Lévy Flights and Self-Similar Exploratory Behaviour of Termite Workers: Beyond Model Fitting

Animal movements have been related to optimal foraging strategies where self-similar trajectories are central. Most of the experimental studies done so far have focused mainly on fitting statistical models to data in order to test for movement patterns described by power-laws. Here we show by analyzing over half a million movement displacements that isolated termite workers actually exhibit a range of very interesting dynamical properties –including Lévy flights– in their exploratory behaviour. Going beyond the current trend of statistical model fitting alone, our study analyses anomalous diffusion and structure functions to estimate values of the scaling exponents describing displacement statistics. We evince the fractal nature of the movement patterns and show how the scaling exponents describing termite space exploration intriguingly comply with mathematical relations found in the physics of transport phenomena. By doing this, we rescue a rich variety of physical and biological phenomenology that can be potentially important and meaningful for the study of complex animal behavior and, in particular, for the study of how patterns of exploratory behaviour of individual social insects may impact not only their feeding demands but also nestmate encounter patterns and, hence, their dynamics at the social scale.     

Incorrect Likelihood Methods Were Used to Infer Scaling Laws of Marine Predator Search Behaviour

Background

Ecologists are collecting extensive data concerning movements of animals in marine ecosystems. Such data need to be analysed with valid statistical methods to yield meaningful conclusions.

Principal Findings

We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (Nature 451∶1098; Science 332∶1551). The first study analysed vertical movement data to conclude that diverse marine predators (Atlantic cod, basking sharks, bigeye tuna, leatherback turtles and Magellanic penguins) exhibited “Lévy-walk-like behaviour”, close to a hypothesised optimal foraging strategy. By reproducing the original results for the bigeye tuna data, we show that the likelihood of tested models was calculated from residuals of regression fits (an incorrect method), rather than from the likelihood equations of the actual probability distributions being tested. This resulted in erroneous Akaike Information Criteria, and the testing of models that do not correspond to valid probability distributions. We demonstrate how this led to overwhelming support for a model that has no biological justification and that is statistically spurious because its probability density function goes negative. Re-analysis of the bigeye tuna data, using standard likelihood methods, overturns the original result and conclusion for that data set. The second study observed Lévy walk movement patterns by mussels. We demonstrate several issues concerning the likelihood calculations (including the aforementioned residuals issue). Re-analysis of the data rejects the original Lévy walk conclusion.

Conclusions

We consequently question the claimed existence of scaling laws of the search behaviour of marine predators and mussels, since such conclusions were reached using incorrect methods. We discourage the suggested potential use of “Lévy-like walks” when modelling consequences of fishing and climate change, and caution that any resulting advice to managers of marine ecosystems would be problematic. For reproducibility and future work we provide R source code for all calculations.     

Motile Male Gametes of the Araphid Diatom Tabularia fasciculata Search Randomly for Mates

Sexuality in the marine araphid diatom Tabularia involves an unusual type of gamete, not only among diatoms but possibly in all of nature. The non-flagellated male gamete is free and vigorously motile, propelled by pseudopodia. However, the cues (if any) in their search for compatible female gametes and the general search patterns to locate them are unknown. We tracked and compared male gamete movements in the presence and absence of receptive female gametes. Path linearity of male movement was not affected by presence of female gametes. Male gametes did not move towards female gametes regardless of their proximity to each other, suggesting that the detection range for a compatible mate is very small compared to known algal examples (mostly spermatozoids) and that mate recognition requires (near) contact with a female gamete. We therefore investigated how male gametes move to bring insight into their search strategy and found that it was consistent with the predictions of a random-walk model with changes in direction coming from an even distribution. We further investigated the type of random walk by determining the best-fit distribution on the tail of the move length distribution and found it to be consistent with a truncated power law distribution with an exponent of 2.34. Although consistent with a Lévy walk search pattern, the range of move lengths in the tail was too narrow for Lévy properties to emerge and so would be best described as Brownian motion. This is somewhat surprising because female gametes were often outnumbered by male gametes, thus contrary to the assumption that a Brownian search mode may be most optimal with an abundant target resource. This is also the first mathematically analysed search pattern of a non-flagellated protistan gamete, supporting the notion that principles of Brownian motion have wide application in biology.     

Computational modelling of cell motility modes emerging from cell-matrix adhesion dynamics

Lymphocytes have been described to perform different motility patterns such as Brownian random walks, persistent random walks, and Lévy walks. Depending on the conditions, such as confinement or the distribution of target cells, either Brownian or Lévy walks lead to more efficient interaction with the targets. The diversity of these motility patterns may be explained by an adaptive response to the surrounding extracellular matrix (ECM). Indeed, depending on the ECM composition, lymphocytes either display a floating motility without attaching to the ECM, or sliding and stepping motility with respectively continuous or discontinuous attachment to the ECM, or pivoting behaviour with sustained attachment to the ECM. Moreover, on the long term, lymphocytes either perform a persistent random walk or a Brownian-like movement depending on the ECM composition. How the ECM affects cell motility is still incompletely understood. Here, we integrate essential mechanistic details of the lymphocyte-matrix adhesions and lymphocyte intrinsic cytoskeletal induced cell propulsion into a Cellular Potts model (CPM). We show that the combination of de novo cell-matrix adhesion formation, adhesion growth and shrinkage, adhesion rupture, and feedback of adhesions onto cell propulsion recapitulates multiple lymphocyte behaviours, for different lymphocyte subsets and various substrates. With an increasing attachment area and increased adhesion strength, the cells’ speed and persistence decreases. Additionally, the model predicts random walks with short-term persistent but long-term subdiffusive properties resulting in a pivoting type of motility. For small adhesion areas, the spatial distribution of adhesions emerges as a key factor influencing cell motility. Small adhesions at the front allow for more persistent motility than larger clusters at the back, despite a similar total adhesion area. In conclusion, we present an integrated framework to simulate the effects of ECM proteins on cell-matrix adhesion dynamics. The model reveals a sufficient set of principles explaining the plasticity of lymphocyte motility.     

Motor Learning Characterized by Changing Lévy Distributions

The probability distributions for changes in transverse plane fingertip speed are Lévy distributed in human pole balancing. Six subjects learned to balance a pole on their index finger over three sessions while sitting and standing. The Lévy or decay exponent decreased as a function of learning, showing reduced decay in the probability for large speed steps and was significantly smaller in the sitting condition. However, the probability distribution for changes in fingertip speed was truncated so that the probability for large steps was reduced in this condition. These results show a learning-induced tolerance for large speed step sizes and demonstrate that motor learning in continuous tasks may be characterized by changing distributions that reflect sensorimotor skill acquisition.     

From Lévy to Brownian: a computational model based on biological fluctuation

Background

Theoretical studies predict that Lévy walks maximizes the chance of encountering randomly distributed targets with a low density, but Brownian walks is favorable inside a patch of targets with high density. Recently, experimental data reports that some animals indeed show a Lévy and Brownian walk movement patterns when forage for foods in areas with low and high density. This paper presents a simple, Gaussian-noise utilizing computational model that can realize such behavior.

Methodology/Principal Findings

We extend Lévy walks model of one of the simplest creature, Escherichia coli, based on biological fluctuation framework. We build a simulation of a simple, generic animal to observe whether Lévy or Brownian walks will be performed properly depends on the target density, and investigate the emergent behavior in a commonly faced patchy environment where the density alternates.

Conclusions/Significance

Based on the model, animal behavior of choosing Lévy or Brownian walk movement patterns based on the target density is able to be generated, without changing the essence of the stochastic property in Escherichia coli physiological mechanism as explained by related researches. The emergent behavior and its benefits in a patchy environment are also discussed. The model provides a framework for further investigation on the role of internal noise in realizing adaptive and efficient foraging behavior.     

Normal and Tumoral Melanocytes Exhibit q-Gaussian Random Search Patterns

In multicellular organisms, cell motility is central in all morphogenetic processes, tissue maintenance, wound healing and immune surveillance. Hence, failures in its regulation potentiates numerous diseases. Here, cell migration assays on plastic 2D surfaces were performed using normal (Melan A) and tumoral (B16F10) murine melanocytes in random motility conditions. The trajectories of the centroids of the cell perimeters were tracked through time-lapse microscopy. The statistics of these trajectories was analyzed by building velocity and turn angle distributions, as well as velocity autocorrelations and the scaling of mean-squared displacements. We find that these cells exhibit a crossover from a normal to a super-diffusive motion without angular persistence at long time scales. Moreover, these melanocytes move with non-Gaussian velocity distributions. This major finding indicates that amongst those animal cells supposedly migrating through Lévy walks, some of them can instead perform q-Gaussian walks. Furthermore, our results reveal that B16F10 cells infected by mycoplasmas exhibit essentially the same diffusivity than their healthy counterparts. Finally, a q-Gaussian random walk model was proposed to account for these melanocytic migratory traits. Simulations based on this model correctly describe the crossover to super-diffusivity in the cell migration tracks.     

Human Mammary Epithelial Cells Exhibit a Bimodal Correlated Random Walk Pattern

Background

Organisms, at scales ranging from unicellular to mammals, have been known to exhibit foraging behavior described by random walks whose segments confirm to Lévy or exponential distributions. For the first time, we present evidence that single cells (mammary epithelial cells) that exist in multi-cellular organisms (humans) follow a bimodal correlated random walk (BCRW).

Methodology/Principal Findings

Cellular tracks of MCF-10A pBabe, neuN and neuT random migration on 2-D plastic substrates, analyzed using bimodal analysis, were found to reveal the BCRW pattern. We find two types of exponentially distributed correlated flights (corresponding to what we refer to as the directional and re-orientation phases) each having its own correlation between move step-lengths within flights. The exponential distribution of flight lengths was confirmed using different analysis methods (logarithmic binning with normalization, survival frequency plots and maximum likelihood estimation).

Conclusions/Significance

Because of the presence of non-uniform turn angle distribution of move step-lengths within a flight and two different types of flights, we propose that the epithelial random walk is a BCRW comprising of two alternating modes with varying degree of correlations, rather than a simple persistent random walk. A BCRW model rather than a simple persistent random walk correctly matches the super-diffusivity in the cell migration paths as indicated by simulations based on the BCRW model.     

Space Use and Movement Patterns in a Semi-Free-Ranging Herd of European Bison (Bison bonasus)

The successful reintroduction and restocking of the European Bison demands a reliable knowledge of the biology of this species. Yet little is known to date about the European bison, and empirical data remains insufficient to set up a reliable plan ensuring the reintroduction, maintenance and survival of populations in habitats that have been largely modified by human activity. Studies of the ecology, social behaviour and management of bison are therefore crucial to the conservation of this species and its cohabitation with humans. To meet these challenges, we focused on movement patterns and space use in a semi-free-ranging herd of European bison living in the Réserve Biologique des Monts-d’Azur (France). Bison spend over 80% of their time foraging and resting; foraging mainly occurs around the artificial feeding sites (i.e., hay racks) or in meadows. The time of day and the presence of snow have no influence on the time budget allocated to each activity. Animals, however, spend more time at the food racks in winter. Bison also spend most of their time in small groups of individuals, confirming the occurrence of both fission-fusion dynamics and sexual segregation in this species. Bison seem to follow a Lévy walk pattern of movement, which is probably related to the geographical distribution and size of food patches in the reserve. The conclusions of this study provide a better understanding of the sociality, life habits and habitat use of bison, and also describe how the provision of hay affects all these behaviours. These results could be useful in the development of tools to select the most suitable habitats for the reintroduction, management and conservation of bison populations.     

Lévy flight and Brownian search patterns of a free-ranging predator reflect different prey field characteristics

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 Lévy flight, whereas when prey is abundant, simple Brownian motion is sufficiently efficient. These predictions form part of what has been termed the Lévy flight foraging hypothesis (LFF) which states that as Lévy flights optimise random searches, movements approximated by optimal Lévy 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 Lévy 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 Lévy 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 Lévy 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 Lévy behaviour patterns in the absence of complex prey distributions.     

Front dynamics in fractional-order epidemic models

A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic's front and a power-law decay of the front's leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics.     

How superdiffusion gets arrested: ecological encounters explain shift from Lévy to Brownian movement

Ecological theory uses Brownian motion as a default template for describing ecological movement, despite limited mechanistic underpinning. The generality of Brownian motion has recently been challenged by empirical studies that highlight alternative movement patterns of animals, especially when foraging in resource-poor environments. Yet, empirical studies reveal animals moving in a Brownian fashion when resources are abundant. We demonstrate that Einstein''s original theory of collision-induced Brownian motion in physics provides a parsimonious, mechanistic explanation for these observations. Here, Brownian motion results from frequent encounters between organisms in dense environments. In density-controlled experiments, movement patterns of mussels shifted from Lévy towards Brownian motion with increasing density. When the analysis was restricted to moves not truncated by encounters, this shift did not occur. Using a theoretical argument, we explain that any movement pattern approximates Brownian motion at high-resource densities, provided that movement is interrupted upon encounters. Hence, the observed shift to Brownian motion does not indicate a density-dependent change in movement strategy but rather results from frequent collisions. Our results emphasize the need for a more mechanistic use of Brownian motion in ecology, highlighting that especially in rich environments, Brownian motion emerges from ecological interactions, rather than being a default movement pattern.     

How landscape heterogeneity frames optimal diffusivity in searching processes

Theoretical and empirical investigations of search strategies typically have failed to distinguish the distinct roles played by density versus patchiness of resources. It is well known that motility and diffusivity of organisms often increase in environments with low density of resources, but thus far there has been little progress in understanding the specific role of landscape heterogeneity and disorder on random, non-oriented motility. Here we address the general question of how the landscape heterogeneity affects the efficiency of encounter interactions under global constant density of scarce resources. We unveil the key mechanism coupling the landscape structure with optimal search diffusivity. In particular, our main result leads to an empirically testable prediction: enhanced diffusivity (including superdiffusive searches), with shift in the diffusion exponent, favors the success of target encounters in heterogeneous landscapes.