Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios

Recently it has been found that composite Brownian walk searches are more efficient than any Lévy walk when searching is non-destructive and when the Lévy walks are not responsive to conditions found in the search. Here a new class of adaptive Lévy walk searches is presented that encompasses composite Brownian walks as a special case. In these new models, bouts of Lévy walk searching alternate with bouts of more intensive Brownian walk searching. Switching from extensive to intensive searching is prompted by the detection of a target. And here, switching back to extensive searching arises if a target is not located after travelling a distance equal to the ‘giving-up distance’. It is found that adaptive Lévy walks outperform composite Brownian walks when searching for sparsely distributed resources. Consequently there is stronger selection pressures for Lévy processes when resources are sparsely distributed within unpredictable environments. The findings reconcile Lévy walk search theory with the ubiquity of two modes of searching by predators and with their switching search mode immediately after finding a prey.     

Balancing the competing demands of harvesting and safety from predation: Lévy walk searches outperform composite Brownian walk searches but only when foraging under the risk of predation

Some foragers have movement patterns that can be approximated by Lévy walks whilst others may be better represented as composite Brownian walks. Many attempts have been made to interpret these movement patterns in terms of optimal searching strategies for the location of randomly and sparsely distributed targets. Here it is shown that the relative merits of Lévy walk and composite Brownian walk searches are sensitively dependent upon target encounter dynamics which set the initial conditions for an extensive search. It is suggested these initial conditions are determined, at least in part, by the competing demands of harvesting and safety from predation. In accordance with observations, it is shown that Lévy walks are expected in tritrophic systems and where intraguild predation operates. Composite Brownian walks, on the other hand, are found to be advantageous when the risk of predation is low. Despite having fundamentally different properties, Lévy walks and composite Brownian walks can therefore compete a priori as possible models of animal movements. Throughout, attention is focused on searching for randomly and sparsely distributed resources that are not depleted or rejected once located but instead remain targets for future searches. We re-evaluate and overturn the widely held belief that in numerical simulations this ‘non-destructive’ searching scenario can faithfully and consistently represent destructive searching for patchily distributed resources, i.e. for resources that tend to occur in clusters rather than in isolation.     

On the first passage time and leapover properties of Lévy motions

We investigate two coupled properties of Lévy stable random motions: the first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Lévy stable probability law λ_α,β(x), the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index α(0<α<2) and skewness parameter β=0, (ii) one-sided Lévy motions with 0<α<1, β=1, and (iii) two-sided skewed Lévy motions, the extreme case, 1<α<2, β=−1.     

The Renormalization Group and Optimization of Entropy

We illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scaling symmetry. We consider three examples: (1) Gaussian fixed-point criticality in a fluid or in the capillary-wave model of an interface; (2) Lévy-like random walks with self-similar cluster formation; and (3) long-ranged bond percolation. In all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distributions and order-parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long-tailed distributions.     

Reynolds stresses in a lattice gas

I use a previously proposed algorithm, based on Lévy walks, to calculate and discuss longitudinal and transverse velocity correlations in turbulent channel flow. The general approach is that of lattice gas hydrodynamics.     

Human memory retrieval as Lévy foraging

When people attempt to recall as many words as possible from a specific category (e.g., animal names) their retrievals occur sporadically over an extended temporal period. Retrievals decline as recall progresses, but short retrieval bursts can occur even after tens of minutes of performing the task. To date, efforts to gain insight into the nature of retrieval from this fundamental phenomenon of semantic memory have focused primarily upon the exponential growth rate of cumulative recall. Here we focus upon the time intervals between retrievals. We expected and found that, for each participant in our experiment, these intervals conformed to a Lévy distribution suggesting that the Lévy flight dynamics that characterize foraging behavior may also characterize retrieval from semantic memory. The closer the exponent on the inverse square power-law distribution of retrieval intervals approximated the optimal foraging value of 2, the more efficient was the retrieval. At an abstract dynamical level, foraging for particular foods in one's niche and searching for particular words in one's memory must be similar processes if particular foods and particular words are randomly and sparsely located in their respective spaces at sites that are not known a priori. We discuss whether Lévy dynamics imply that memory processes, like foraging, are optimized in an ecological way.     

Time evolution of the probability distribution in stochastic and chaotic systems with enhanced diffusion

We perform a detailed study of the time evolution of the probability distribution for two processes displaying enhanced diffusion: a stochastic process named the Lévy walk and a deterministic chaotic process, the amplified climbing-sine map. The time evolution of the probability distribution differs in the two cases and carries information which is peculiar to the investigated process.     

A geometric theory for Lévy distributions

Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT.     

Fluctuations in a lévy flight gas

We consider the density fluctuations of an ideal Brownian gas of particles performing Lévy flìghts characterized by the indexf. We find that the fluctuations scale as N(t) t^H, where the Hurst exponentH locks onto the universal value 1/4 for Lévy flights with a finite root-mean-square range (f>2). For Lévy flights with a finite mean range but infinite root-mean-square range (1相似文献   

Distribution of animal population fluctuations

We show that the fluctuations of a variety of animal (insect, mammal and fish) populations are well-modeled by stable (Lévy) distributions, a family of distributions which includes the Gaussian. Our findings together with the finding [A.P. Allen, B.-L. Li, E.L. Charnov, Ecol. Lett. 4 (2001) 1-3] that bird population fluctuations are well-modeled by Gaussian distributions suggest the hypothesis that, in general, animal population fluctuations are well-modeled by stable distributions. If the hypothesis is confirmed, the stable law for the distribution of animal population fluctuations would be useful for animal population modeling, prediction and management.     

Scaling,stability and distribution of the high-frequency returns of the Ibex35 index

In this paper we perform a statistical analysis of the high-frequency returns of the Ibex35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data using maximum likelihood estimation and different goodness-of-fit tests rejects the Lévy-stable law as a plausible underlying probabilistic model. The analysis shows that the Normal Inverse Gaussian distribution provides an overall fit for the data better than any of the other subclasses of the family of Generalized Hyperbolic distributions and certainly much better than the Lévy-stable laws. Furthermore, the right (resp. left) tail of the distribution seems to follow a power-law with exponent α≈4.60$\alpha \approx 4.60$ (resp. α≈4.28$\alpha \approx 4.28$). Finally, we present evidence that the observed stability is due to temporal correlations or non-stationarities of the data.     

Lévy processes and Schrödinger equation

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy-Schrödinger equation where the usual kinetic energy operator-the Laplacian-is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy-Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models-such as a form of the relativistic Schrödinger equation-that are in the domain of the non stable Lévy-Schrödinger equations.     

Lévy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place. To this end we employ two model patterns of dynamical behavior: Langevin-driven and (Lévy-Schrödinger) semigroup-driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Lévy oscillators), with respect to both dynamical mechanisms. We discuss the links of the Lévy semigroup motion scenario with that of random searches in spatially inhomogeneous media.     

Can spontaneous cell movements be modelled as Lévy walks?

Spontaneous cell movement is a random motion that takes place in the absence of external guiding stimuli. The spontaneous movements of HaCaT and NHDF cells (cells of the epidermis) are well represented as continuous Markovian processes driven by multiplicative noise [D. Selmeczi, S. Mosler, P.H. Hagedorn, N.B. Larsen, H. Flyvbjerg, Biophysical Journal 89 (2005) 912]. Model components are, however, ad hoc as they are inspired by fits to experimental data. As a consequence, model agreement with experimental data does not add much to our understanding of spontaneous movements of these cells beyond demonstrating that they can be modelled phenomenologically. Here it is noted that a slight re-parameterization and re-interpretation of the driving noise leads to the model of Lubashevsky et al. (2009) [I. Lubashevsky, R. Friedrich, A. Heuer, Physical Review E 79 (2009) 011110] that realises Lévy walks as Markovian stochastic processes. This brings forth new biological insight as Lévy walks are advantageous when searching in the absence of external stimuli and without knowledge of the target distribution, as may be the case with cells of the epidermis that form new tissue by locating and then attaching on to one another. The Hänggi-Klimontovich interpretation of the driving noise in the model of Lubashevsky et al. (2009) and Cauchy distributions of predicted velocities do, however, appear problematic, even unphysical. Here it is shown that these are perceived rather than actual difficulties. Intermittent stop-start motions of the kind displayed by some cells and protozoan are found to underlie the formulation of the model of Lubashevsky et al. (2009) and the velocities of starved Dictyostelium discoideum (a unicellular organism) are found to be Cauchy distributed to a good approximation. It is therefore suggested that the model of Lubashevsky et al. (2009) can describe the spontaneous movements of some cells, and that some cells have spontaneous movement patterns that can be approximated by Lévy walks, as first proposed by Schuster and Levandowsky (1996) [F.L. Schuster, M. Levandowsky, Journal of Eukaryotic Microbiology 43 (1996) 150].     

Random walks with infinite spatial and temporal moments

The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.     

Fluctuation-driven directed transport in the presence of Lévy flights

The role of Lévy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Lévy white noise with stability index α in the range 1<α<2. The second approach is based on the space fractional Fokker-Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Lévy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Lévy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the −1/α exponent corresponding to the force free case.     

Langevin Picture of Lévy Walks and Their Extensions

In this paper we derive Langevin picture of Lévy walks. Applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled Lévy walk. Next, we introduce extensions of Levy walks, in which jump sizes are some functions of waiting times. We prove that under proper scaling conditions, such generalized Lévy walks converge in distribution to the appropriate limiting processes. We also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes, showing that different coupling functions lead to various types of anomalous diffusion.     

An averaging principle for stochastic dynamical systems with Lévy noise

The purpose of this paper is to establish an averaging principle for stochastic differential equations with non-Gaussian Lévy noise. The solutions to stochastic systems with Lévy noise can be approximated by solutions to averaged stochastic differential equations in the sense of both convergence in mean square and convergence in probability. The convergence order is also estimated in terms of noise intensity. Two examples are presented to demonstrate the applications of the averaging principle, and a numerical simulation is carried out to establish the good agreement.