G. M. Viswanathan

Optimizing the success of random searches

We address the general question of what is the best statistical strategy to adapt in order to search efficiently for randomly located objects (‘target sites’). It is often assumed in foraging theory that the flight lengths of a forager have a characteristic scale: from this assumption gaussian, Rayleigh and other classical distributions with well-defined variances have arisen. However, such theories cannot explain the long-tailed power-law distributions of flight lengths or flight times that are observed experimentally. Here we study how the search efficiency depends on the probability distribution of flight lengths taken by a forager that can detect target sites only in its limited vicinity. We show that, when the target sites are sparse and can be visited any number of times, an inverse square power-law distribution of flight lengths, corresponding to Lévy flight motion, is an optimal strategy. We test the theory by analysing experimental foraging data on selected insect, mammal and bird species, and find that they are consistent with the predicted inverse square power-law distributions.

ANIMAL SEARCH STRATEGIES: A QUANTITATIVE RANDOM-WALK ANALYSIS

Recent advances in spatial ecology have improved our understanding of the role of large-scale animal movements. However, an unsolved problem concerns the inherent stochasticity involved in many animal search displacements and its possible adaptive value. When animals have no information about where targets (i.e., resource patches, mates, etc.) are located, different random search strategies may provide different chances to find them. Assuming random-walk models as a necessary tool to understand how animals face such environmental uncertainty, we analyze the statistical differences between two random-walk models commonly used to fit animal movement data, the Levy walks and the correlated random walks, and we quantify their efficiencies (i.e., the number of targets found in relation to total displacement) within a random search context. Correlated random-walk properties (i.e., scale-finite correlations) may be interpreted as the by-product of locally scanning mechanisms. Levy walks, instead, have fundamenta...

Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics

We discuss examples of complex systems composed of many interacting subsystems. We focus on those systems displaying nontrivial long-range correlations. These include the one-dimensional sequence of base pairs in DNA, the sequence of flight times of the large seabird Wandering Albatross, and the annual fluctuations in the growth rate of business firms. We review formal analogies in the models that describe the observed long-range correlations, and conclude by discussing the possibility that behavior of large numbers of humans (as measured, e.g., by economic indices) might conform to analogs of the scaling laws that have proved useful in describing systems composed of large numbers of inanimate objects.

Lévy flights in random searches

We review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager, and discuss its quantitative description using the theory of random walks. We illustrate Levy flight foraging by comparison to Brownian random walks and discuss experimental observations of Levy flights in biological foraging. We review recent findings suggesting that an inverse square probability density distribution P(l)∼l−2 of step lengths l can lead to optimal searches. Finally, we survey the explanations put forth to account for these unexpected findings.

Analysis of DNA sequences using methods of statistical physics

We review the present status of the studies of DNA sequences using methods of statistical physics. We present evidence, based on systematic studies of the entire GenBank database, supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range, i.e., base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the DNA. We discuss the mechanisms of molecular evolution that may lead to the presence of long-range power-law correlations in noncoding DNA and their absence in coding DNA. One such mechanism is the simple repeat expansion, which recently has attracted the attention of the biological community in conjunction with genetic diseases. We also review new tools – e.g., detrended fluctuation analysis – that are useful for studies of complex hierarchical DNA structure.

Lévy flight random searches in biological phenomena

There has been growing interest in the study of Levy flights observed in the movements of biological organisms performing random walks while searching for other organisms. Here, we approach the problem of what is the best statistical strategy for optimizing the encounter rate between “searcher” and “target” organisms—either of the same or of different species—in terms of a limiting generalized searcher–target model (e.g., predator-prey, mating partner, pollinator–flower). In this context, we discuss known results showing that for fixed targets an inverse square density distribution of step lengths can optimize the encounter rate. For moving targets, we review how the encounter rate depends on whether organisms move in Levy or Brownian random walks. We discuss recent findings indicating that Levy walks confer a significant advantage for increasing encounter rates only when the searcher is larger or moves rapidly relative to the target, and when the target density is low.

Average time spent by Levy flights and walks on an interval with absorbing boundaries

We consider a Lévy flyer of order alpha that starts from a point x(0) on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of alpha-->2, and the cases of x(0)-->0 and x(0)-->L. For alpha>2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x(0) is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

Variance fluctuations in nonstationary time series: a comparative study of music genres

An important problem in physics concerns the analysis of audio time series generated by transduced acoustic phenomena. Here, we develop a new method to quantify the scaling properties of the local variance of nonstationary time series. We apply this technique to analyze audio signals obtained from selected genres of music. We find quantitative differences in the correlation properties of high art music, popular music, and dance music. We discuss the relevance of these objective findings in relation to the subjective experience of music.

Scaling and universality in animate and inanimate systems

We illustrate the general principle that in biophysics, econophysics and possibly even city growth, the conceptual framework provided by scaling and universality may be of use in making sense of complex statistical data. Specifically, we discuss recent work on DNA sequences, heartbeat intervals, avalanche-like lung inflation, urban growth, and company growth. Although our main focus is on data, we also discuss statistical mechanical models.

Improvements in the statistical approach to random Lévy flight searches

Recently it has been shown that the most efficient strategy for searching randomly located objects, when the sites are randomly distributed and can be revisited any number of times, leads to a power law distribution P(l)=l−μ of the flights l, with μ=2. We show analytically that the incorporation of energy considerations limits the possible range for the Levy exponent μ, however, μ=2 still emerges as the optimal foraging condition. Furthermore, we show that the probability distribution of flight lengths for the short and intermediate flight length regimes depends on the details of the system.