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

As the number or density of interacting individuals in a social group increases, a transition can develop from uncorrelated and disordered behavior of the individuals to a collective coherent pattern. We expand this observation by exploring the fine details of termite movement patterns to demonstrate that the value of the scaling exponent µ of a power law describing the Lévy walk of an individual is modified collectively as the density of animals in the group changes. This effect is absent when termites interact with inert obstacles. We also show that the network of encounters and interactions among specific individuals is selective, resembling a preferential attachment mechanism that is important for social networking. Our data strongly suggest that preferential attachments, a phenomenon not reported previously, and favorite interactions with a limited number of acquaintances are responsible for the generation of Lévy movement patterns in these social insects.

Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity.

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson-Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower boundedness of the real part of the eigenvalue spectrum that corresponds to an upper limit of the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We also analyze Generalized Poisson-Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there is a critical value for the wave vector, above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case, and in fact, it represents a simple and clear example of a sub-quantum dynamics with hidden variables.

The topography of the environment alters the optimal search strategy for active particles.

In environments with scarce resources, adopting the right search strategy can make the difference between succeeding and failing, even between life and death. At different scales, this applies to molecular encounters in the cell cytoplasm, to animals looking for food or mates in natural landscapes, to rescuers during search and rescue operations in disaster zones, and to genetic computer algorithms exploring parameter spaces. When looking for sparse targets in a homogeneous environment, a combination of ballistic and diffusive steps is considered optimal; in particular, more ballistic Lévy flights with exponent [Formula: see text] are generally believed to optimize the search process. However, most search spaces present complex topographies. What is the best search strategy in these more realistic scenarios? Here, we show that the topography of the environment significantly alters the optimal search strategy toward less ballistic and more Brownian strategies. We consider an active particle performing a blind cruise search for nonregenerating sparse targets in a 2D space with steps drawn from a Lévy distribution with the exponent varying from [Formula: see text] to [Formula: see text] (Brownian). We show that, when boundaries, barriers, and obstacles are present, the optimal search strategy depends on the topography of the environment, with [Formula: see text] assuming intermediate values in the whole range under consideration. We interpret these findings using simple scaling arguments and discuss their robustness to varying searcher's size. Our results are relevant for search problems at different length scales from animal and human foraging to microswimmers' taxis to biochemical rates of reaction.

Rationalizing spatial exploration patterns of wild animals and humans through a temporal discounting framework.

Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that "Lévy random walks"-which can produce power law path length distributions-are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent's goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.

Anomalous diffusion and multifractality enhance mating encounters in the ocean.

For millimeter-scale aquatic crustaceans such as copepods, ensuring reproductive success is a challenge as potential mates are often separated by hundreds of body lengths in a 3D environment. At the evolutionary scale, this led to the development of remote sensing abilities and behavioral strategies to locate, to track, and to capture a mate. Chemoreception plays a crucial role in increasing mate encounter rates through pheromone clouds and pheromone trails that can be followed over many body lengths. Empirical evidence of trail following behavior is, however, limited to laboratory experiments conducted in still water. An important open question concerns what happens in the turbulent waters of the surface ocean. We propose that copepods experience, and hence react to, a bulk-phase water pheromone concentration. Here we investigate the mating behavior of two key copepod species, Temora longicornis and Eurytemora affinis, to assess the role of background pheromone concentration and the relative roles played by males and females in mating encounters. We find that both males and females react to background pheromone concentration and exhibit both innate and acquired components in their mating strategies. The emerging swimming behaviors have stochastic properties that depend on pheromone concentration, sex, and species, are related to the level of reproductive experience of the individual tested, and significantly diverge from both the Lévy and Brownian models identified in predators searching for low- and high-density prey. Our results are consistent with an adaptation to increase mate encounter rates and hence to optimize reproductive fitness and success.

Foraging success under uncertainty: search tradeoffs and optimal space use.

Understanding the structural complexity and the main drivers of animal search behaviour is pivotal to foraging ecology. Yet, the role of uncertainty as a generative mechanism of movement patterns is poorly understood. Novel insights from search theory suggest that organisms should collect and assess new information from the environment by producing complex exploratory strategies. Based on an extension of the first passage time theory, and using simple equations and simulations, we unveil the elementary heuristics behind search behaviour. In particular, we show that normal diffusion is not enough for determining optimal exploratory behaviour but anomalous diffusion is required. Searching organisms go through two critical sequential phases (approach and detection) and experience fundamental search tradeoffs that may limit their encounter rates. Using experimental data, we show that biological search includes elements not fully considered in contemporary physical search theory. In particular, the need to consider search movement as a non-stationary process that brings the organism from one informational state to another. For example, the transition from remaining in an area to departing from it may occur through an exploratory state where cognitive search is challenged. Therefore, a more comprehensive view of foraging ecology requires including current perspectives about movement under uncertainty.

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

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

Selection pressures give composite correlated random walks Lévy walk characteristics.

Composite correlated random walks have been posited as a strong alternative to Lévy walks as models of multi-scale forager movement patterns. Here it is shown that if plastic then intrinsic composite correlated random walks will, under selection pressures, evolve to resemble optimal Lévy walks when foraging is non-destructive. The fittest composite correlated random walkers are found to be those that come closest to being optimal Lévy walkers. This may explain why such a diverse range of foragers have movement patterns that can be approximated by optimal Lévy walks and shows that the 'Lévy-flight foraging' hypothesis has a broad hinterland. The new findings are consistent with recent observations of mussels Mytilus edulis and the Australian desert ant Melophorus bagoti which suggest that animals approximate a Lévy walk by adopting an intrinsic composite movement strategy with different modes.

Brain-localized CD4 and CD8 T cells perform correlated random walks and not Levy walks.

Background. For survival of the organism, T cells must efficiently control pathogens invading different peripheral tissues. Whether or not such control is achieved by utilizing different movement strategies in different tissues remains poorly understood. Liver-localized CD8 T cells perform correlated random walks  --- a type of a Brownian walk -- in liver sinusoids but in some condition these T cells may also perform Levy flights -- rapid and large displacements by floating with the blood flow. CD8 T cells in lymph nodes or skin also undergo Brownian walks. A recent study suggested that brain-localized CD8 T cells, specific to Toxoplasma gondii, perform generalized Levy walks -- a walk type in which T cells alternate pausing and displacing long distances --- which may indicate that brain is a unique organ where T cells exhibit movement strategies different from other tissues. Methods.  We quantified movement patterns of brain-localized Plasmodium berghei-specific CD4 and CD8 T cells by using well-established statistical and computational methods. Results.  We found that T cells change their movement pattern with time since infection and that CD4 T cells move faster and turn less than CD8 T cells. Importantly, both CD4 and CD8 T cells move in the brain by correlated random walks without long displacements challenging previous observations. We have also re-analyzed the movement data of brain-localized CD8 T cells in T. gondii-infected mice and found no evidence of Levy walks. We hypothesize that the previous conclusion of Levy walks of T. gondii-specific CD8 T cells in the brain was reached due to missing time-frames in the data that create an impression of large movement lengths between assumed-to-be-sequential movements.  Conclusion. Our results suggests that movement strategies of CD8 T cells are largely similar between LNs, liver, and the brain and consistent with correlated random walks and not Levy walks.

Individual exploration and selective social learning: balancing exploration-exploitation trade-offs in collective foraging.

Search requires balancing exploring for more options and exploiting the ones previously found. Individuals foraging in a group face another trade-off: whether to engage in social learning to exploit the solutions found by others or to solitarily search for unexplored solutions. Social learning can better exploit learned information and decrease the costs of finding new resources, but excessive social learning can lead to over-exploitation and too little exploration for new solutions. We study how these two trade-offs interact to influence search efficiency in a model of collective foraging under conditions of varying resource abundance, resource density and group size. We modelled individual search strategies as Lévy walks, where a power-law exponent (µ) controlled the trade-off between exploitative and explorative movements in individual search. We modulated the trade-off between individual search and social learning using a selectivity parameter that determined how agents responded to social cues in terms of distance and likely opportunity costs. Our results show that social learning is favoured in rich and clustered environments, but also that the benefits of exploiting social information are maximized by engaging in high levels of individual exploration. We show that selective use of social information can modulate the disadvantages of excessive social learning, especially in larger groups and when individual exploration is limited. Finally, we found that the optimal combination of individual exploration and social learning gave rise to trajectories with µ ≈ 2 and provide support for the general optimality of such patterns in search. Our work sheds light on the interplay between individual search and social learning, and has broader implications for collective search and problem-solving.

Efficient quantitative assessment of robot swarms: coverage and targeting Lévy strategies.

Biologically inspiredstrategieshave long been adapted to swarm robotic systems, including biased random walks, reaction to chemotactic cues and long-range coordination. In this paper we applyanalysis toolsdeveloped for modeling biological systems, such as continuum descriptions, to the efficient quantitative characterization of robot swarms. As an illustration, both Brownian and Lévy strategies with a characteristic long-range movement are discussed. As a result we obtain computationally fast methods for the optimization of robot movement laws to achieve a prescribed collective behavior. We show how to compute performance metrics like coverage and hitting times, and illustrate the accuracy and efficiency of our approach for area coverage and search problems. Comparisons between the continuum model and robotic simulations confirm the quantitative agreement and speed up by a factor of over 100 of our approach. Results confirm and quantify the advantage of Lévy strategies over Brownian motion for search and area coverage problems in swarm robotics.

Current status and future directions of Lévy walk research.

Lévy walks are a mathematical construction useful for describing random patterns of movement with bizarre fractal properties that seem to have no place in biology. Nonetheless, movement patterns resembling Lévy walks have been observed at scales ranging from the microscopic to the ecological. They have been seen in the molecular machinery operating within cells during intracellular trafficking, in the movement patterns of T cells within the brain, in DNA, in some molluscs, insects, fish, birds and mammals, in the airborne flights of spores and seeds, and in the collective movements of some animal groups. Lévy walks are also evident in trace fossils (ichnofossils) - the preserved form of tracks made by organisms that occupied ancient sea beds about 252-66 million years ago. And they are utilised by algae that originated around two billion years ago, and still exist today. In September of 2017, leading researchers from across the life sciences, along with mathematicians and physicists, got together at a Company of Biologists' Workshop to discuss the origins and biological significance of these movement patterns. In this Review the essence of the technical and sometimes heated discussions is distilled and made accessible for all. In just a few pages, the reader is taken from a gentle introduction to the frontiers of a very active field of scientific enquiry. What emerges is a fascinating story of a truly inter-disciplinary scientific endeavour that is seeking to better understand movement patterns occurring across all biological scales.

A Chemical Reaction Network to Generate Random, Power-Law-Distributed Time Intervals.

In Lévy walks (LWs), particles move with a fixed speed along straight line segments and turn in new directions after random time intervals that are distributed according to a power law. Such LWs are thought to be an advantageous foraging and search strategy for organisms. While complex nervous systems are certainly capable of producing such behavior, it is not clear at present how single-cell organisms can generate the long-term correlated control signals required for a LW. Here, we construct a biochemical reaction system that generates long-time correlated concentration fluctuations of a signaling substance, with a tunable fractional exponent of the autocorrelation function. The network is based on well-known modules, and its basic function is highly robust with respect to the parameter settings.

Mechanistic movement models to understand epidemic spread.

An overlooked aspect of disease ecology is considering how and why animals come into contact with one and other resulting in disease transmission. Mathematical models of disease spread frequently assume mass-action transmission, justified by stating that susceptible and infectious hosts mix readily, and foregoing any detailed description of host movement. Numerous recent studies have recorded, analysed and modelled animal movement. These movement models describe how animals move with respect to resources, conspecifics and previous movement directions and have been used to understand the conditions for the occurrence and the spread of infectious diseases when hosts perform a type of movement. Here, we summarize the effect of the different types of movement on the threshold conditions for disease spread. We identify gaps in the literature and suggest several promising directions for future research. The mechanistic inclusion of movement in epidemic models may be beneficial for the following two reasons. Firstly, the estimation of the transmission coefficient in an epidemic model is possible because animal movement data can be used to estimate the rate of contacts between conspecifics. Secondly, unsuccessful transmission events, where a susceptible host contacts an infectious host but does not become infected can be quantified. Following an outbreak, this enables disease ecologists to identify 'near misses' and to explore possible alternative epidemic outcomes given shifts in ecological or immunological parameters.This article is part of the themed issue 'Opening the black box: re-examining the ecology and evolution of parasite transmission'.

Multiple sources of slow activity fluctuations in a bacterial chemosensory network.

Cellular networks are intrinsically subject to stochastic fluctuations, but analysis of the resulting noise remained largely limited to gene expression. The pathway controlling chemotaxis of Escherichia coli provides one example where posttranslational signaling noise has been deduced from cellular behavior. This noise was proposed to result from stochasticity in chemoreceptor methylation, and it is believed to enhance environment exploration by bacteria. Here we combined single-cell FRET measurements with analysis based on the fluctuation-dissipation theorem (FDT) to characterize origins of activity fluctuations within the chemotaxis pathway. We observed surprisingly large methylation-independent thermal fluctuations of receptor activity, which contribute to noise comparably to the energy-consuming methylation dynamics. Interactions between clustered receptors involved in amplification of chemotactic signals are also necessary to produce the observed large activity fluctuations. Our work thus shows that the high response sensitivity of this cellular pathway also increases its susceptibility to noise, from thermal and out-of-equilibrium processes.

Extending Lévy search theory from one to higher dimensions: Lévy walking favours the blind.

A diverse range of organisms, including T cells, E. coli, honeybees, sharks, turtles, bony fish, jellyfish, wandering albatrosses and even human hunter-gatherers have movement patterns that can be approximated by Lévy walks (LW; sometimes called Lévy flights in the biological and ecological literature). These observations lend support to the 'Lévy flight foraging hypothesis' which asserts that natural selection should have led to adaptations for Lévy flight foraging, because Lévy flights can optimize search efficiencies. The hypothesis stems from a rigorous theory of one-dimensional searching and from simulation data for two-dimensional searching. The potential effectiveness of three-dimensional Lévy searches has not been examined but is central to a proper understanding of marine predators and T cells which have provided the most compelling empirical evidence for LW. Here I extend Lévy search theory from one to three dimensions. The new theory predicts that three-dimensional Lévy searching can be advantageous but only when targets are large compared with the perceptual range of the searchers, i.e. only when foragers are effectively blind and need to come into contact with a target to establish its presence. This may explain why effective blindness is a common factor among three-dimensional Lévy walkers.

Liberating Lévy walk research from the shackles of optimal foraging.

There is now compelling evidence that many organisms have movement patterns that can be described as Lévy walks, or Lévy flights. Lévy movement patterns have been identified in cells, microorganisms, molluscs, insects, reptiles, fish, birds and even human hunter-gatherers. Most research into Lévy walks as models of organism movement patterns has been shaped by the 'Lévy flight foraging hypothesis'. This states that, since Lévy walks can optimize search efficiencies, natural selection should lead to adaptations that select for Lévy walk foraging. However, a growing body of research on generative mechanisms suggests that Lévy walks can arise freely as by-products of otherwise innocuous behaviours; consequently their advantageous properties are purely coincidental. This suggests that the Lévy flight foraging hypothesis should be amended, or even replaced, by a simpler and more general hypothesis. This new hypothesis would state that 'Lévy walks emerge spontaneously and naturally from innate behaviours and innocuous responses to the environment but, if advantageous, then there could be selection against losing them'. The new hypothesis has the virtue of making fewer assumptions and being broader than the original hypothesis; it also encompasses the many examples of suboptimal Lévy patterns that challenge the prevailing paradigm. This does not detract from the Lévy flight foraging hypothesis, in fact, it adds to the theory by providing a stronger and more compelling case for the occurrence of Lévy walks. It dispenses with concerns about the theoretical arguments in support of the Lévy flight foraging hypothesis and so may lead to a wider acceptance of Lévy walks as models of movement pattern data. Furthermore, organisms can approximate Lévy walks by adapting intrinsic behaviour in simple ways; this occurs when Lévy movement patterns are advantageous, but come with an associated cost. These new developments represent a major change in perspective and provide the broadest picture yet of Lévy movement patterns. However, the process of understanding and identifying Lévy movement patterns still has a long way to go, and further reinterpretations and shifts in understanding will occur. In conclusion, Lévy walk research remains exciting precisely because so much remains to be understood, and because, even relatively small studies, are interesting discoveries in their own right.

Collective foraging in heterogeneous landscapes.

Animals foraging alone are hypothesized to optimize the encounter rates with resources through Lévy walks. However, the issue of how the interactions between multiple foragers influence their search efficiency is still not completely understood. To address this, we consider a model to study the optimal strategy for a group of foragers searching for targets distributed heterogeneously. In our model, foragers move on a square lattice containing immobile but regenerative targets. At any instant, a forager is able to detect only those targets that happen to be in the same site. However, we allow the foragers to have information about the state of other foragers. A forager who has not detected any target walks towards the nearest location, where another forager has detected a target, with a probability exp(-αd), where d is the distance between the foragers and α is a parameter characterizing the propensity of the foragers to aggregate. The model reveals that neither overcrowding (α → 0) nor independent searching (α → ∞) is beneficial for the foragers. For a patchy distribution of targets, the efficiency is maximum for intermediate values of α. In addition, in the limit α → 0, the length of the walks can become scale-free.

Signatures of active and passive optimized Lévy searching in jellyfish.

Some of the strongest empirical support for Lévy search theory has come from telemetry data for the dive patterns of marine predators (sharks, bony fishes, sea turtles and penguins). The dive patterns of the unusually large jellyfish Rhizostoma octopus do, however, sit outside of current Lévy search theory which predicts that a single search strategy is optimal. When searching the water column, the movement patterns of these jellyfish change over time. Movement bouts can be approximated by a variety of Lévy and Brownian (exponential) walks. The adaptive value of this variation is not known. On some occasions movement pattern data are consistent with the jellyfish prospecting away from a preferred depth, not finding an improvement in conditions elsewhere and so returning to their original depth. This 'bounce' behaviour also sits outside of current Lévy walk search theory. Here, it is shown that the jellyfish movement patterns are consistent with their using optimized 'fast simulated annealing'--a novel kind of Lévy walk search pattern--to locate the maximum prey concentration in the water column and/or to locate the strongest of many olfactory trails emanating from more distant prey. Fast simulated annealing is a powerful stochastic search algorithm for locating a global maximum that is hidden among many poorer local maxima in a large search space. This new finding shows that the notion of active optimized Lévy walk searching is not limited to the search for randomly and sparsely distributed resources, as previously thought, but can be extended to embrace other scenarios, including that of the jellyfish R. octopus. In the presence of convective currents, it could become energetically favourable to search the water column by riding the convective currents. Here, it is shown that these passive movements can be represented accurately by Lévy walks of the type occasionally seen in R. octopus. This result vividly illustrates that Lévy walks are not necessarily the result of selection pressures for advantageous searching behaviour but can instead arise freely and naturally from simple processes. It also shows that the family of Lévy walkers is vastly larger than previously thought and includes spores, pollens, seeds and minute wingless arthropods that on warm days disperse passively within the atmospheric boundary layer.