Lévy flights in human behavior and cognition

Lévy flights represent the best strategy to randomly search for a target in an unknown environment, and have been widely observed in many animal species. Here, we inspect and discuss recent results concerning human behavior and cognition. Different studies have shown that human mobility can be described in terms of Lévy flights, while fresh evidence indicates that the same pattern accounts for human mental searches in online gambling sites. Thus, Lévy flights emerge as a unifying concept with broad cross-disciplinary implications. We argue that the ubiquity of such a pattern, both in behavior and cognition, suggests that the brain regions responsible for this behavior are likely to be evolutionarily old (i.e. no frontal cortex is involved), and that fMRI techniques might help to confirm this hypothesis

Signatures of a globally optimal searching strategy in the three-dimensional foraging flights of bumblebees

Simulated annealing is a powerful stochastic search algorithm for locating a global maximum that is hidden among many poorer local maxima in a search space. It is frequently implemented in computers working on complex optimization problems but until now has not been directly observed in nature as a searching strategy adopted by foraging animals. We analysed high-speed video recordings of the three-dimensional searching flights of bumblebees (Bombus terrestris) made in the presence of large or small artificial flowers within a 0.5 m3 enclosed arena. Analyses of the three-dimensional flight patterns in both conditions reveal signatures of simulated annealing searches. After leaving a flower, bees tend to scan back-and forth past that flower before making prospecting flights (loops), whose length increases over time. The search pattern becomes gradually more expansive and culminates when another rewarding flower is found. Bees then scan back and forth in the vicinity of the newly discovered flower and the process repeats. This looping search pattern, in which flight step lengths are typically power-law distributed, provides a relatively simple yet highly efficient strategy for pollinators such as bees to find best quality resources in complex environments made of multiple ephemeral feeding sites with nutritionally variable rewards

Behavioural analysis of marine predator movements in relation to heterogeneous environments

An understanding of the spatio-temporal dynamics of marine predator populations is essential for the sustainable management of marine resources. Tagging studies are providing ever more information about the movements and migrations of marine predators and much has been learned about where these predators spend their time. However little is known about their underlying motivations, making it difficult to make predictions about how apex predators will respond to changing environments. While much progress has been made in behavioural ecology through the use of optimality models, in the marine environment the necessary costs and benefits are difficult to quantify making this approach less successful than with terrestrial studies. One aspect of foraging behaviour that has proved tractable however is the optimisation of random searches. Work by statistical physicists has shown that a specialised movement, known as Lévy flight, can optimise the rate of new prey patch encounters when new prey patches are beyond sensory range. The resulting Lévy flight foraging (LFF) hypothesis makes testable predictions about marine predator search behaviour that can be addressed with the theoretical and empirical studies that form the basis of this thesis. Results presented here resolve the controversy surrounding the hypothesis, demonstrating the optimality of Lévy searches under a broader set of conditions than previously considered, including whether observed Lévy patterns are innate or emergent. Empirical studies provide robust evidence for the prevalence of Lévy search patterns in the movements of diverse marine pelagic predators such as sharks, tunas and billfish as well as in the foraging patterns of albatrosses, overturning a previous study. Predictions from the LFF hypothesis concerning fast moving prey are confirmed leading to simulation studies of ambush predator’s activity patterns. Movement analysis is then applied to the assessment of by-catch mitigation efforts involving VMS data from long-liners and simulated sharks

Intermittent search strategies

This review examines intermittent target search strategies, which combine phases of slow motion, allowing the searcher to detect the target, and phases of fast motion during which targets cannot be detected. We first show that intermittent search strategies are actually widely observed at various scales. At the macroscopic scale, this is for example the case of animals looking for food ; at the microscopic scale, intermittent transport patterns are involved in reaction pathway of DNA binding proteins as well as in intracellular transport. Second, we introduce generic stochastic models, which show that intermittent strategies are efficient strategies, which enable to minimize the search time. This suggests that the intrinsic efficiency of intermittent search strategies could justify their frequent observation in nature. Last, beyond these modeling aspects, we propose that intermittent strategies could be used also in a broader context to design and accelerate search processes.Comment: 72 pages, review articl

First-passage properties of asymmetric Lévy flights

Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times

L\'evy walks

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The L\'{e}vy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to L\'{e}vy walks, surveys their existing applications, including latest advances, and outlines further perspectives.Comment: 50 page