Lévy flights and random searches

In this work we discuss some recent contributions to the random search problem. Our analysis includes superdiffusive Levy processes and correlated random walks in several regimes of target site density, mobility and revisitability. We present results in the context of mean-field-like and closed-form average calculations, as well as numerical simulations. We then consider random searches performed in regular lattices and lattices with defects, and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes. We invoke energy considerations in relation to critical survival states on the edge of extinction, and we analyze the emergence of Levy behavior in deterministic search walks. Finally, we comment on the random search problem in the context of biological foraging.

[1]  D'arcy W. Thompson.  On Growth and Form , 1945 .

[2]  J. W. Essam,et al.  Critical Percolation Probabilities by Series Methods , 1964 .

[3]  S. Goldhor.  Ecology , 1964, The Yale Journal of Biology and Medicine.

[4]  V. Smirnov,et al.  A course of higher mathematics , 1964 .

[5]  F. Dullien.  Porous Media: Fluid Transport and Pore Structure , 1979 .

[6]  H. Berg.  Random Walks in Biology , 2018 .

[7]  Shlesinger,et al.  Comment on "Accelerated diffusion in Josephson junctions and related chaotic systems" , 1985, Physical review letters.

[8]  M. Shlesinger,et al.  Lévy Walks Versus Lévy Flights , 1986 .

[9]  S. Benhamou,et al.  Spatial analysis of animals' movements using a correlated random walk model* , 1988 .

[10]  M. Levandowsky,et al.  Swimming behavior and chemosensory responses in the protistan microzooplankton as a function of the hydrodynamic regime , 1988 .

[11]  P. V. von Hippel,et al.  Facilitated Target Location in Biological Systems* , 2022 .

[12]  Peter Grassberger,et al.  The red queen's walk , 1992 .

[13]  G. Jackson,et al.  Viral dynamics: a model of the effects of size shape, motion and abundance of single-celled olanktonic organisms and other particles , 1992 .

[14]  John A. Wiens,et al.  Animal movement in heterogeneous landscapes : an experiment with Eleodes beetles in shortgrass prairie , 1992 .

[15]  Simon Benhamou,et al.  Efficiency of area-concentrated searching behaviour in a continuous patchy environment* , 1992 .

[16]  A L Goldberger,et al.  Generalized Lévy-walk model for DNA nucleotide sequences. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Shigetoshi Nara,et al.  Memory search using complex dynamics in a recurrent neural network model , 1993, Neural Networks.

[18]  H. Stanley,et al.  Fractals in Biology and Medicine: From DNA to the Heartbeat , 1994 .

[19]  C. Peng,et al.  Mosaic organization of DNA nucleotides. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Stanley,et al.  Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. , 1994, Physical review letters.

[21]  George M. Zaslavsky,et al.  Fractional kinetic equation for Hamiltonian chaos , 1994 .

[22]  David Gale,et al.  Mathematical entertainments , 1995 .

[23]  B. Cole.  Fractal time in animal behaviour: the movement activity of Drosophila , 1995, Animal Behaviour.

[24]  Ezequiel V. Albano.  Branching annihilating Lévy flights: Irreversible phase transitions with long-range exchanges , 1996 .

[25]  P. A. Prince,et al.  Lévy flight search patterns of wandering albatrosses , 1996, Nature.

[26]  M. Levandowsky,et al.  Chemosensory Responses of Acanthamoeba castellanii: Visual Analysis of Random Movement and Responses to Chemical Signals , 1996, The Journal of eukaryotic microbiology.

[27]  Cardy,et al.  Theory of Branching and Annihilating Random Walks. , 1996, Physical review letters.

[28]  B. L. Olla,et al.  Food deprivation affects vertical distribution and activity of a marine fish in a thermal gradient: potential energy-conserving mechanisms , 1996 .

[29]  J. Kärger,et al.  Flow and Transport in Porous Media and Fractured Rock , 1996 .

[30]  Albano.  Irreversible phase transitions in contact processes with Lévy exchanges and long-range interactions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  N. Hill,et al.  A biased random walk model for the trajectories of swimming micro-organisms. , 1997, Journal of theoretical biology.

[32]  P. Turchin.  Quantitative analysis of movement : measuring and modeling population redistribution in animals and plants , 1998 .

[33]  Hans C. Fogedby.  Lévy flights in quenched random force fields , 1998 .

[34]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[35]  Bastien Chopard,et al.  Cellular Automata Modeling of Physical Systems: Index , 1998 .

[36]  H. Stanley,et al.  Optimizing the success of random searches , 1999, Nature.

[37]  J. Catalán.  Small-Scale Hydrodynamics as a Framework for Plankton Evolution , 1999 .

[38]  R. Dickman,et al.  Nonequilibrium Phase Transitions in Lattice Models , 1999 .

[39]  M. Weigt,et al.  On the properties of small-world network models , 1999, cond-mat/9903411.

[40]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[41]  L. Amaral,et al.  Small-World Networks: Evidence for a Crossover Picture , 1999, cond-mat/9903108.

[42]  V. Zolotarev,et al.  Chance and Stability, Stable Distributions and Their Applications , 1999 .

[43]  H E Stanley,et al.  Classes of small-world networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[44]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[45]  H. Stanley,et al.  Lévy flights in random searches , 2000 .

[46]  D. de Cogan,et al.  Cellular Automata Modelling of Physical Systems, by Bastien Chopard & Michel Droz, Cambridge University Press (Aléa Saclay Collection), 336 pp., ISBN 0‐521‐46168‐5, hardback, £55 , 2000 .

[47]  H E Stanley,et al.  Average time spent by Lévy flights and walks on an interval with absorbing boundaries. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  Gandhi M. Viswanathan.  Improvements in the statistical approach to random Levy flight searches M.G.E. da Luz, S.V. Buldyrev, S. Havlin, E.P. Raposo, H.E. Stanley, , 2001 .

[49]  H. Stanley,et al.  Statistical physics of random searches , 2001 .

[50]  Lada A. Adamic,et al.  Search in Power-Law Networks , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Sergey V. Buldyrev,et al.  Properties of Lévy flights on an interval with absorbing boundaries , 2001 .

[52]  O Kinouchi,et al.  Deterministic walks in random media. , 2001, Physical review letters.

[53]  D. Macdonald,et al.  Scale‐free dynamics in the movement patterns of jackals , 2002 .

[54]  V. M. Kenkre,et al.  Simulations in the mathematical modeling of the spread of the Hantavirus. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  Frederic Bartumeus,et al.  Erratum: Optimizing the Encounter Rate in Biological Interactions: Lévy versus Brownian Strategies [Phys. Rev. Lett.88, 097901 (2002)] , 2002 .

[56]  A. Hofgaard,et al.  Foraging and movement paths of female reindeer: insights from fractal analysis, correlated random walks, and Lévy flights , 2002 .

[57]  F Bartumeus,et al.  Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies. , 2002, Physical review letters.

[58]  H. Stanley,et al.  Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series , 2002, physics/0202070.

[59]  R. Metzler,et al.  Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation , 2002 .

[60]  H. Stanley,et al.  Lévy flight random searches in biological phenomena , 2002 .

[61]  V. M. Kenkre,et al.  Spatiotemporal patterns in the Hantavirus infection. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  H. Larralde,et al.  Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi) , 2003, Behavioral Ecology and Sociobiology.

[63]  Osame Kinouchi,et al.  Escaping from cycles through a glass transition. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  Marcos C. Santos,et al.  Dynamical robustness of Lévy search strategies. , 2003, Physical review letters.

[65]  F. Bartumeus,et al.  Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[66]  H. Fritz,et al.  Scale–dependent hierarchical adjustments of movement patterns in a long–range foraging seabird , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[67]  Leonid A. Bunimovich,et al.  Deterministic walks in random environments , 2004 .

[68]  G. Viswanathan,et al.  Optimal random searches of revisitable targets: Crossover from superdiffusive to ballistic random walks , 2004 .

[69]  G. Viswanathan,et al.  CRITICAL BEHAVIOR OF AN EPIDEMIC MODEL OF DRUG RESISTANT DISEASES , 2003, cond-mat/0312002.

[70]  P. Kareiva,et al.  Analyzing insect movement as a correlated random walk , 1983, Oecologia.

[71]  T. Geisel,et al.  Forecast and control of epidemics in a globalized world. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[72]  C.A.J. Appelo,et al.  FLOW AND TRANSPORT , 2004 .

[73]  Juan M. Morales,et al.  EXTRACTING MORE OUT OF RELOCATION DATA: BUILDING MOVEMENT MODELS AS MIXTURES OF RANDOM WALKS , 2004 .

[74]  Deborah Austin,et al.  Intraspecific variation in movement patterns: modeling individual behaviour in a large marine predator , 2004 .

[75]  D. Boyer,et al.  Modeling the searching behavior of social monkeys , 2004 .

[76]  M. Sisterson,et al.  Costs and Benefits of Food Foraging for a Braconid Parasitoid , 2002, Journal of Insect Behavior.

[77]  Edward A. Codling,et al.  Random walk models for the movement and recruitment of reef fish larvae , 2004 .

[78]  G. Viswanathan,et al.  Necessary criterion for distinguishing true superdiffusion from correlated random walk processes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[79]  Alexandre Souto Martinez,et al.  Analytical results for the statistical distribution related to a memoryless deterministic walk: dimensionality effect and mean-field models. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[80]  Arthur Pewsey,et al.  A Family of Symmetric Distributions on the Circle , 2005 .

[81]  Denis Boyer,et al.  Looking for the right thing at the right place: Phase transition in an agent model with heterogeneous spatial resources , 2005, Complex..

[82]  Frederic Bartumeus,et al.  ANIMAL SEARCH STRATEGIES: A QUANTITATIVE RANDOM‐WALK ANALYSIS , 2005 .

[83]  O Bénichou,et al.  Optimal search strategies for hidden targets. , 2005, Physical review letters.

[84]  G M Viswanathan,et al.  Optimization of random searches on regular lattices. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[85]  H. Stanley,et al.  Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[86]  Sophie Bertrand,et al.  Lévy trajectories of Peruvian purse-seiners as an indicator of the spatial distribution of anchovy ( Engraulis ringens ) , 2005 .

[87]  M. Lambin,et al.  Effect of Starvation on the Searching Path of the Predatory Bug Deraeocoris Lutescens , 2005, BioControl.

[88]  J. Urry.  Complexity , 2006, Interpreting Art.

[89]  Germinal Cocho,et al.  Scale-free foraging by primates emerges from their interaction with a complex environment , 2006, Proceedings of the Royal Society B: Biological Sciences.

[90]  O Bénichou,et al.  Two-dimensional intermittent search processes: An alternative to Lévy flight strategies. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[91]  Andy M. Reynolds.  On the intermittent behaviour of foraging animals , 2006 .

[92]  Mathieu Coppey,et al.  Intermittent search strategies: When losing time becomes efficient , 2006 .

[93]  B. Barlaup,et al.  Feeding pattern and diet of first feeding brown trout fry under natural conditions , 2006 .

[94]  Anthony J Richardson,et al.  Encounter success of free-ranging marine predator movements across a dynamic prey landscape , 2006, Proceedings of the Royal Society B: Biological Sciences.

[95]  A. Bergman,et al.  Functional and evolutionary inference in gene networks: does topology matter? , 2006, Genetica.

[96]  T. Geisel,et al.  The scaling laws of human travel , 2006, Nature.

[97]  P. Brantingham,et al.  Measuring Forager Mobility , 2006 .

[98]  Marcin Magdziarz,et al.  First passage times of Lévy flights coexisting with subdiffusion. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[99]  J. Klafter,et al.  First-passage times in complex scale-invariant media , 2007, Nature.

[100]  A. Reynolds,et al.  Free-Flight Odor Tracking in Drosophila Is Consistent with an Optimal Intermittent Scale-Free Search , 2007, PloS one.

[101]  Norman L Carreck,et al.  Honeybees perform optimal scale-free searching flights when attempting to locate a food source , 2007, Journal of Experimental Biology.

[102]  Clifford T. Brown,et al.  Lévy Flights in Dobe Ju/’hoansi Foraging Patterns , 2007 .

[103]  S. Benhamou.  HOW MANY ANIMALS REALLY DO THE LÉVY WALK , 2007 .

[104]  D. Sims,et al.  Minimizing errors in identifying Lévy flight behaviour of organisms. , 2007, The Journal of animal ecology.

[105]  Ralf Metzler,et al.  Leapover lengths and first passage time statistics for Lévy flights. , 2007, Physical review letters.

[106]  A. M. Edwards,et al.  Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer , 2007, Nature.

[107]  Luca Giuggioli,et al.  Effects of finite probing windows on the interpretation of the multifractal properties of random walks , 2007 .

[108]  John Travis,et al.  Do Wandering Albatrosses Care About Math? , 2007, Science.

[109]  Sophie Bertrand,et al.  Scale-invariant movements of fishermen: the same foraging strategy as natural predators. , 2007, Ecological applications : a publication of the Ecological Society of America.

[110]  Michael F. Shlesinger.  Mathematical physics: First encounters , 2007, Nature.

[111]  G M Viswanathan,et al.  Origin of power-law distributions in deterministic walks: the influence of landscape geometry. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[112]  Andy M. Reynolds,et al.  Avoidance of conspecific odour trails results in scale-free movement patterns and the execution of an optimal searching strategy , 2007 .

[113]  O Mason,et al.  Graph theory and networks in Biology. , 2006, IET systems biology.

[114]  L. R. da Silva,et al.  Search dynamics at the edge of extinction: Anomalous diffusion as a critical survival state , 2007 .

[115]  F. J. Quirijns,et al.  Linkage between fishers' foraging, market and fish stocks density: Examples from some North Sea fisheries , 2007 .

[116]  R. Menzel,et al.  Displaced honey bees perform optimal scale-free search flights. , 2007, Ecology.

[117]  G. Viswanathan,et al.  Lévy flights and superdiffusion in the context of biological encounters and random searches , 2008 .

[118]  Nicolas E. Humphries,et al.  Scaling laws of marine predator search behaviour , 2008, Nature.

[119]  Ralf Metzler,et al.  Lévy strategies in intermittent search processes are advantageous , 2008, Proceedings of the National Academy of Sciences.

[120]  Frank Moss,et al.  Stochastic resonance and the evolution of Daphnia foraging strategy. , 2008, Physical biology.

[121]  David A. Bohan,et al.  Avoidance of conspecific odour by carabid beetles: a mechanism for the emergence of scale-free searching patterns , 2008, Animal Behaviour.

[122]  M. Plank,et al.  Optimal foraging: Lévy pattern or process? , 2008, Journal of The Royal Society Interface.

[123]  A. James,et al.  Optimizing the encounter rate in biological interactions: Ballistic versus Lévy versus Brownian strategies. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[124]  A M Reynolds,et al.  Deterministic walks with inverse-square power-law scaling are an emergent property of predators that use chemotaxis to locate randomly distributed prey. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[125]  Albert-László Barabási,et al.  Understanding individual human mobility patterns , 2008, Nature.

[126]  Ken Cheng,et al.  Search strategies of ants in landmark-rich habitats , 2008, Journal of Comparative Physiology A.

[127]  Edward A. Codling,et al.  Random walk models in biology , 2008, Journal of The Royal Society Interface.

[128]  G M Viswanathan,et al.  Optimization of random searches on defective lattice networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[129]  Andy M. Reynolds,et al.  Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios , 2009 .

[130]  F. Bartumeus.  Behavioral intermittence, Lévy patterns, and randomness in animal movement , 2009 .

[131]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[132]  A M Reynolds,et al.  The Lévy flight paradigm: random search patterns and mechanisms. , 2009, Ecology.

[133]  Bastien Chopard,et al.  Cellular Automata Modeling of Physical Systems , 1999, Encyclopedia of Complexity and Systems Science.