Optimizing the Encounter Rate in Biological Interactions: Lévy versus Brownian Strategies

Bartumeus, Frederic; Catalán, Jordi; Fulco, U. L.; Lyra, M. L.; Viswanathan, G. M.

doi:10.1103/physrevlett.88.097901

Cited by 296 publications

(225 citation statements)

References 16 publications

(16 reference statements)

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“…These results agree with the prediction of the LFF hypothesis that Lévy behaviour should occur in environments where prey is sparsely distributed but that Brownian motion is theoretically optimal where prey is abundant 3 . To test the significance of this with our habitatmapped data, we compared the frequency of sections that conformed to this broad prediction.…”

supporting

confidence: 82%

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confidence: 99%

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Environmental context explains Lévy and Brownian movement patterns of marine predators

Humphries

Queiroz

Dyer

et al. 2010

Nature

785

810

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An optimal search theory, the so-called Lévy-flight foraging hypothesis 1 , predicts that predators should adopt search strategies known as Lévy flights where prey is sparse and distributed unpredictably, but that Brownian movement is sufficiently efficient for locating abundant prey [2][3][4] . Empirical studies have generated controversy because the accuracy of statistical methods that have been used to identify Lévy behaviour has recently been questioned 5,6 . Consequently, whether foragers exhibit Lévy 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 Lévy patterns in relation to environmental gradients in the largest animal movement data set assembled for this purpose. Strong support was found for Lévy search patterns across 14 species of open-ocean predatory fish (sharks, tuna, billfish and ocean sunfish), with some individuals switching between Lévy and Brownian movement as they traversed different habitat types. We tested the spatial occurrence of these two principal patterns and found Lévy 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 Lévy-flight foraging hypothesis 1,7 , supporting the contention 8,9 that organism search strategies naturally evolved in such a way that they exploit optimal Lévy patterns.Lévy flights are a special class of random walk with movement displacements (steps) drawn from a probability distribution with a power-law tail (the so-called Pareto-Lévy distribution) 1,10 , and give rise to stochastic processes closely linked to fractal geometry and anomalous diffusion phenomena 7,11 . 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). It is proposed that organisms have therefore naturally evolved search patterns that can be modelled as optimal Lévy flights 1,7,13 .However, burgeoning empirical support for this hypothesis recently foundered following studies sug...

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supporting

confidence: 82%

“…

…”

mentioning

confidence: 99%

Environmental context explains Lévy and Brownian movement patterns of marine predators

Humphries

Queiroz

Dyer

et al. 2010

Nature

785

810

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“…for 0 < α < 2 were shown to be advantageous (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16); occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling.…”

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confidence: 99%

Lévy strategies in intermittent search processes are advantageous

Lomholt

Koren

Metzler

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

239

229

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Intermittent search processes switch between local Brownian search events and ballistic relocation phases. We demonstrate analytically and numerically in one dimension that when relocation times are Lévy distributed, resulting in a Lévy walk dynamics, the search process significantly outperforms the previously investigated case of exponentially distributed relocation times: The resulting Lévy walks reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets. We also show that a searching agent that uses the Lévy strategy is much less sensitive to the target density, which would require considerably less adaptation by the searcher.random processes | optimization | Lévy walk | movement ecology R andom search processes occur in many areas, from chemical reactions of diffusing reactants (1) to the foraging behavior of bacteria and animals (2, 3). Of general importance is the search efficiency. Brownian search in one and two dimensions involves frequent returns to an area, leading to oversampling. Higher efficiency, can be achieved, for instance, by facilitated diffusion in gene regulation (4) or by controlled motion in foraging (2, 3). From theoretical and data analysis Lévy strategies, in which the searching agent performs excursions whose length is drawn from distributions with a heavy tailfor 0 < α < 2 were shown to be advantageous (5-16); occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling.As an alternative to Lévy search, intermittent strategies have been introduced to improve the efficiency of diffusive search (17-21). Intermittent search requires that the searcher occasionally shifts focus from the search and concentrates on fast relocation. The relocation phase implies that the searcher is wasting time in the short run because the target cannot be spotted during it. However, the overall search efficiency is improved by introducing the searcher to previously unexplored areas (17-21).In refs. 18 and 20 relocation events were assumed to occur in a random direction for exponentially distributed time spans, giving rise to a Markovian process. We show here analytically and numerically in one dimension that this is only a partial solution to oversampling, as eventually the central limit theorem (CLT) reduces the process to a Brownian random walk with jumps on the scale of vτ 2 , where τ 2 is the typical time spent in a relocation event. In practice, revisits can be reduced by adjusting the average time spent in search and relocation phases to the density of targets. Lévy strategies, on the other hand, fundamentally circumvent the CLT, and we here demonstrate a twofold advantage of them over the exponential distribution: Lévy walk intermittent processes find the target faster than exponential strategies in the critical case of rare targets, and their performance is much less dependent on adapting to the target density. Intermittent Search with Lévy RelocationsGeneralizing the model from ref. 20, w...

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“…1, the motion is ballistic, where the path of the animal is rather a straight line, when observed over a finite time. The most efficient searching, by a single forager looking for targets, is supposed to happen when m % 2 [1,12,13]. However, investigations [12,14 -16] have also probed the robustness of Lévy walks and have found these to be actually optimal or only marginally advantageous under conditions pertaining to the nature of the encounter with targets (destructive or non-destructive), the presence of memory, geometry of the foraging region, among others.…”

Section: Introductionmentioning

confidence: 99%

Collective foraging in heterogeneous landscapes

Bhattacharya

Vicsek

2014

J. R. Soc. Interface.

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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(2ad), where d is the distance between the foragers and a is a parameter characterizing the propensity of the foragers to aggregate. The model reveals that neither overcrowding (a ! 0) nor independent searching (a ! 1) is beneficial for the foragers. For a patchy distribution of targets, the efficiency is maximum for intermediate values of a. In addition, in the limit a ! 0, the length of the walks can become scale-free.

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Optimizing the Encounter Rate in Biological Interactions: Lévy versus Brownian Strategies

Cited by 296 publications

References 16 publications

Environmental context explains Lévy and Brownian movement patterns of marine predators

Environmental context explains Lévy and Brownian movement patterns of marine predators

Lévy strategies in intermittent search processes are advantageous

Collective foraging in heterogeneous landscapes

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