Lévy flights in random searches

Viswanathan, G. M.; Afanasyev, Vsevolod; Buldyrev, Sergey V.; Havlin, Shlomo; Luz, M. G. E. da; Raposo, Ernesto P.; Stanley, H. Eugene

doi:10.1016/s0378-4371(00)00071-6

Cited by 224 publications

(189 citation statements)

References 20 publications

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“…The strategy has been tested and applied to experimental foraging data of various species (insects, mammals and birds) with relevant consistency. The theoretical arguments and numerical simulations provided by [72] show that μ = 2 is the optimal value for a search in any dimension. With this biological relevance, our study comprises of Lévy walks with exponent μ = 2, in particular Cauchy distributed steps.…”

Section: Introductionmentioning

confidence: 93%

“…The central limit theorem [26] states that the sum of independent and identical distributions, with first three moments converges to a normal distribution when the number of steps tend to infinity. Viswanathan et al [72,73] demonstrated that μ = 2 is the optimal parameter for a random search in any dimension. This justifies our choice to characterize the step lengths according to an inverse square law.…”

Section: Cauchy Type Random Walkmentioning

confidence: 99%

“…This characterization introduces a natural parameter μ, such that we have a family of distributions [72]. Due to the central limit theorem, the Gaussian is the stable distribution for the special case μ ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

Ahmed

Petrovskii

2015

Math. Model. Nat. Phenom.

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Abstract. Insect trapping is commonly used in various pest insect monitoring programs as well as in many ecological field studies. An individual is said to be trapped if it falls within a well defined capturing zone, which it cannot escape. The accumulation of trapped individuals over time forms trap counts or alternatively, the flux of the population density into the trap. In this paper, we study the movement of insects whose dynamics are governed by time dependent diffusion and Lévy walks. We demonstrate that the diffusion model provides an alternative framework for the Cauchy type random walk (Lévy walk with Cauchy distributed steps). Furthermore, by calculating the trap counts using these two conceptually different movement models, we propose that trap counts for pests whose dynamics may be Lévy by nature can effectively be predicted by diffusive flux curves with time-dependent diffusivity.

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Section: Introductionmentioning

confidence: 93%

Section: Cauchy Type Random Walkmentioning

confidence: 99%

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Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

Ahmed

Petrovskii

2015

Math. Model. Nat. Phenom.

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“…To test the predictions of the LFF hypothesis, we focused our analysis on vertical movement data recorded over 5,700 days using electronic tags attached to open-ocean predators (sharks, tunas, billfish and ocean sunfish; Methods and Supplementary Table 1). These species may be among those most likely to exhibit Lévy behaviour because they occupy unpredictable and depauperate environments with highly patchy prey distributions 13 , where Lévy motion is proposed to increase new-patch encounter probability 19 .To allow for a more robust test of the LFF hypothesis, long and complex time series of vertical diving movements (hereafter tracks, or sections) undertaken as fish moved horizontally across their …”

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confidence: 99%

Environmental context explains Lévy and Brownian movement patterns of marine predators

Humphries

Queiroz

Dyer

et al. 2010

Nature

798

816

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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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“…In particular, the model implies that the forager will not interact with any prey it meets whilst moving between foraging locations. Adapting the model to account for such possibilities inevitably leads one to consider more cruise-like foraging strategies where the theories of [5,21,29] are more appropriate. We note that, compared to a saltatory forager, cruise predation is unlikely to be an effective way to leave regions of low prey density, i.e., to escape low quality habitats.…”

Section: Foraging Behaviourmentioning

confidence: 99%

Lévy Processes, Saltatory Foraging, and Superdiffusion

Burrow

Baxter

Pitchford

2008

Math. Model. Nat. Phenom.

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Abstract. It is well established that resource variability generated by spatial patchiness and turbulence is an important influence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulations indicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment can be formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly, if jumps are of a fixed size and occur as a Poisson process (embedded within a drift-diffusion), recruitment is effectively described by a diffusion process alone. Secondly, in the absence of diffusion, and for "patchy" jumps (of negative binomial size with Pareto inter-arrivals), the encounter process becomes superdiffusive. To synthesise these results we conduct a strategic simulation study where "patchy" jumps are embedded in a drift-diffusion process. We conclude that increasingly Lévy-like predator foraging strategies can have a significantly positive effect on recruitment at the population level.

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Lévy flights in random searches

Cited by 224 publications

References 20 publications

Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

Environmental context explains Lévy and Brownian movement patterns of marine predators

Lévy Processes, Saltatory Foraging, and Superdiffusion

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