Lévy flights and superdiffusion in the context of biological encounters and random searches

Environmental context explains Lévy and Brownian movement patterns of marine predators

“…Lévy flights describe a movement pattern characterized by many small steps connected by longer relocations, with this pattern having scale invariance under projection, such that the probability density function, P(l j ), has a power-law tail in the long-distance regime: P(l j ) < l j 2m , where l j is the flight length (step length of move j), and m, 1 , m # 3, is the power-law exponent. Lévy flights comprise instantaneous steps and hence involve infinite velocities, whereas a Lévy walk 10 refers to a finitevelocity walk such that displacement is determined after a time t, reflecting a dynamical process such as movement 1,10,11 . Lévy flights and walks are theorized to be the most efficient movement pattern for locating patchy prey in low concentrations on spatial scales beyond a searcher's sensory range, with an optimal search having a power-law exponent of m < 2 (refs 4, 13).…”

“…Hence, controversy remains over whether Lévy behaviour occurs in nature 6,9,17 , despite many empirical studies 1,18 . Furthermore, long time series of movements (over weeks to months) derived from animalattached electronic tags will very probably capture complex movement data resulting from different types of behaviour (for example searching, travelling and resting) as animals respond to various biotic and abiotic factors over time.…”

A Lévy-flight diffusion model to predict transgenic pollen dispersal

“…Lévy-type foraging patterns have been observed for a broad range of organisms, including honeybees [17,18], bumblebees [19], fruit flies [20], large marine predators [21,22], human T cells [23], jellyfishes [24], albatrosses [25] and even human hunter-gatherers [26]. All of these Lévy patterns have been identified through sound statistical techniques, but there is still some debate on the underlying driving mechanism: natural selection to optimize search efficiency [27] or spontaneous emergence from innate behaviours [28]. The goal of this paper is not to elaborate that discussion but to investigate how the beemediated spread of pollen is altered when it is assumed that the distribution of foraging honeybee populations is well described by a movement paradigm of Lévy type.…”

The Active Sampling of Gaze-Shifts

“…The basic assumption was the "foraging metaphor", namely that Lévy-like diffusive property of scanpath behavior mirrors Lévy-like patterns of foraging behavior in many animal species [24]. In this perspective, the Lévy flight, as opposed, for instance, to Gaussian walk, is assumed to be essential for optimal search, where optimality is related to efficiency, that is the ratio of the number of sites visited to the total distance traversed by forager [24]. An example depicting the difference between Gaussian and Lévy walks is provided in Fig.…”

“…Examples of Lévy flights and walks have been found in many instance of animal mouvement such as spider monkey [16], albatrosses [23], jackals [1] and in general such dynamics have been found to be be essential for optimal exploration in random searches [24].…”