Journal of Modern Physics
Vol.4 No.3(2013), Article ID:28603,7 pages DOI:10.4236/jmp.2013.43046

Lévy Flights, 1/f Noise and Self Organized Criticality in a Traveling Agent Model

Oliver López Corona1, Pablo Padilla2, Oscar Escolero3, Alejandro Frank4, Ruben Fossion5

1Posgrado en Ciencias de la Tierra, Instituto de Geología, Universidad Nacional Autónoma de México, Ciudad de México, México

2IIMAS, Universidad Nacional Autónoma de México, Ciudad de México, México

3Instituto de Geología, Universidad Nacional Autónoma de México, Ciudad de México, México

4Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de México, México

5Instituto Nacional de Geratria, México

Email: oliverlc@geologí

Received October 29, 2012; revised December 1, 2012; accepted December 28, 2012

Keywords: Lévy Flights; 1/f Noise; Self Organized Criticality; Agents Modelling; Complexity


A new analysis of a previously studied traveling agent model, showed that there is a relation between the degree of homogeneity of the medium where the agents move, agent motion patterns, and the noise generated from their displacements. We proved that for a particular value of homogeneity, the system self organizes in a state where the agents carry out Lévy walks and the displacement signal corresponds to 1/f noise. Using probabilistic arguments, we conjectured that 1/f noise is a fingerprint of a statistical phase transition, from randomness (disorder) to predictability (order), and that it emerges from the contextuality nature of the system which generates it.

1. Introduction

Phenomena with Lévy like distributions have been widely reported, e.g. in experiments with CTAB mielles dissolved in salted water [1], chaotic transport in a laminar fluid flow of a water-glycerol mixture in a rapidly rotating annulus [2], subrecoil laser cooling [3], conservative motion in a two dimensional periodic poential [4], transport in heterogeneous catalysis [5], reactions and in polymer systems under conformational motion [6], in the description of behavior of dynamical systems [7], economic time series [8] and in the motion of gold nanoclusters in graphite [9].

The foraging patterns of a variety of animals [10-15] involve many spatio-temporal scales and are sometimes well described by Lévy walks. A good review on the subject may be found in [16]. This statistical behavior has also been reported in human movement patterns [17].

In this context, Boyer and López-Corona [18] introduced a model of traveling agents (e.g. frugivorous animals) who feed on randomly located vegetation patches and disperse their seeds. The foraging agents use a deterministic strategy with memory, that makes them visit the largest possible patches (most food content) accessible within minimal traveling distances. If the patches have a small initial size, the vegetation total mass increases with time and reaches a maximum corresponding to a self organized critical state with power-law distributed patch sizes and Lévy-like movement patterns for the foragers. They proved that the distribution of resources is not held fixed and spatial heterogeneities self organize spontaneously under the influence of positive feedback loops in the system dynamics. Particularly, at low plant competition, the power spectrum of biomass time series correspond to a 1/f noise.

Pink or 1/f noise (sometimes also called Flicker noise) is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency [19]. This statistical behavior appears in such diverse phenomena as Quantum Mechanics [20-23], Biology [18,24,25], Medicine [26] and Astronomy [27], among other fields. The frequent occurrence of pink noise in such a seemingly unrelated set of physical systems, has prompted an extensive search for common underlying physical principles [28].

It is clear that 1/f noise is part of a wider set of scale invariant signals defined by an inverse power law power spectrum of the form


where λ is the spectral density exponent, which classifies the signals depending of its value: λ = 0 for white noise, λ = 1 for pink and λ = 2 for brown noise. These tree types of noise exhibit quite different statistical characteristics.

Let’s define (following [29]) the autocorrelation function as the inverse Fourier transform of the power spectrum of the signal


If the power spectrum obeys the power law of Equation (1), and we apply a scale transformation in the time domain, then


and the general solution of Equation (3) is also a power law. In this way, one possible complexity measure is the memory of the time series, as expressed by the autocorrelation function which measures how quickly the time series looses similarity with a copy of itself displaced with some delay time τ. The correlations are zero for white noise, large for brown noise, and infinite for a periodic series. When corresponds to a very special case, as we show below. We shall see that the parameter λ gives a measure of the correlation strength and may be used as a control parameter for complexity.

A complex system [30] is a special type of dynamical composite system [31], where under critical circumstances new collective behavior emerges from the shortrange interactions between the constituent parts. Intuitively, complexity is inversely related to a simple behavior, i.e., the more we are able to provide a simple description of a phenomenon, the less complex it is. Uncorrelated random signals (white noise) may be highly unpredictable even in cases where the past history is fully known but, at a global level, they admit a very simple description and, therefore, are not really “complex”. From a complex-systems approach, random uncorrelated series are among the least complex signals, and those with longrange correlations are among the most complex [29].

The autocorrelation function of white noise follows a power law with; strongly correlated Brownian noise, although composed mostly of low-frequency and thus large-period waves, has only local, short-term correlations between neighboring points [32]; finally the autocorrelation function for 1/f noise is a logarithmic function, which decays more slowly than any power law. Thus 1/f noise has the largest possible memory for a scale invariant signal and therefore, under our definition, is the most complex signal too.

In recent works Eliazar and Klafter [33,34] showed that both Lévy walks and 1/f are the result of systems which superimpose the transmissions of infinite independent stochastic signals.

With this in mind, we proceeded to investigate under which conditions, if any, the power spectra of the agents motion (Lévy walks) follow a 1/f dynamics. We found a non trivial relationship between the media’s homogeneity coefficient β, the agents motion and the noise type observed. These results (Figure 1 and Table 1) are new and constitute our main result.

2. Method: A Traveling Agent Model

Let’s consider a two-dimensional square domain of unit area with N fixed, point-like food patches randomly and uniformly distributed. Each patch contains a k amount of food.

Initially, an agent is located on a patch chosen at random. Then the following deterministic foraging rules are iteratively applied at every time step:

1) The agent located at patch i feeds on that patch, the fruit content decreasing by one unit:.

2) When has reached the value 0, the agent chooses another patch, j, such that is maximal over all the allowed patches j = i in the system, where kj is the food content of patch j and the Euclidean distance between patches i and j. With this rule, the next visited patch (the “best” patch) has a large food content and/or is at a short distance from i. It is assumed that the travel from i to j takes one time unit.

3) The agent does not revisit previously visited patches.

The model exhibits some remarkable properties. Let’s define the agents displacement as with the agent position at time t. When averages are taken, different times t0 and independent realizations are considered. If the patch size k is taken froman inverse power-law distribution, where c is an arbitrary constant and β is a coefficient that represents the medium homogeneity. When β is high the medium is very homogeneous, meaning that all patches have similar food content values. On the contrary, when β is low () the medium is very heterogeneous, meaning that patches with high food content are numerous. The intermediate case (), corresponds to a complex medium where patches with high and low food contents are present.

This model produces complex trajectories that have been studied in detail in refs. [35,36] and discussed in connection with spider monkeys foraging patterns observed
Table 1. Relation between media homogeneity coefficient β, type of medium, agents motion, and the noise type observed.

Figure 1. Examples of agent walk (first column) and the corresponding power spectrum (second column) for three values of themedium homogeneity coefficient β = 2 (homogeneous medium), β = 3 (complex medium) and β = 5 (inhomogeneousmedium).

in the field [11]. The different media types produce distinctive motion patterns summarized in Figure 1.

3. Results

In Figure 1, three values of corresponding to disordered, complex and ordered media where used to construct time series for. The power spectrum defined by with the Fourier transformation of the agent displacement, was calculated for each independent run using the fast Fourier transform (fft) technique. In Table 1, 50 power spectra were averaged and fitted by an inverse power law . White noise corresponds to a, pink to a, and brown to a.

Figure 2 shows the spectral exponent λ as a function of the medium homogeneity coefficient β. Statistics were taken from the same 50 independent realization used for Table 1.

The analysis of time as a function of the step number (Figures 3 and 4), revels that the systems transit from a nonstationary to a stationary regimen. A second derivative parameter

was calculated in the third step of the walk in order to capture the transitory part of the signal.

• For β = 3.0 the system is highly non-stationary (ξ big) reaching the stationarity only for very big times

• For 3.0 ≤ β ≤ 4.0 the system is near the stationarity (ξ small).

• For β ≥ 4.0 the system is highly stationary (ξ tends to zero)

The system experiments a sudden transition from nonstationarity to stationarity in β = 3.0 which may be seen as a phase transition fingerprint.

These results suggests that the emergence of pink

Figure 2. For each realization the power spectrum was constructed and the spectral exponent calculated. Then averages were taken for 50 independent realization.

Figure 3. Discrete first derivative of time in function of the step number.

Figure 4. Discrete second derivative of time as a function of step number n, calculated in the third time step of the walk for capture the transitory part of the process.

noise for a traveling agent in a heterogeneous medium depends on the degree of heterogeneity of the medium. If the medium homogeneity coefficient is distributed as a power law, then a 1/f noise will be obtained when β = 3. Thus, this dynamical behavior may naturally arise from the motion of agents in a complex medium.

4. Conclusions

Our results suggest that 1/f noise may be a fingerprint of a statistical phase transition from randomness (low correlation associated with white noise), to predictability (high correlation associated to brown noise) an idea suggested in [37]. The authors proved that a transitional state in two different regimes implies the occurrence of 1/f time series and that this property is generic in both classical and quantum systems. They showed this by studying a classical system, the one-dimensional module-1 logistic map and a quantum one, the nuclear excitation spectra obtained with a schematic shell-model Hamiltonian.

Moreover, Relano et al. [38] proposed to interpret fluctuations in quantum excitation spectra as generalized time series and to study the corresponding power spectral density. While quantum excitation spectra (which is known to be chaotic) result in 1/f noise, the integrable excitation spectra follow instead a Brownian noise signal 1/f 2. In the same line of thought, autonomous (involuntary) physiological time series are found to correspond with power spectral density. Time series from healthy subjects are found to approach, and results have been reported for the fluctuations associated to heartbeat, gait, temperature, respiration [32,39- 41]. On the other hand, time series from aging subjects tend toward [42], while particular phenomena, such as heart fibrillation, tend to flatten out the power spectral density towards [43,44]. It has been suggested that the 1/f in healthy physiological systems indicates a critical state, and that deviations indicate aging or disease.

The main result of our work is the relationship between the media’s homogeneity of a traveling agent motion and the noise type observed. Of special interest is that 1/f noise is found when medium is complex. Additionally, we conjecture that 1/f noise is a fingerprint of a statistical phase (perhaps of second order) transition from randomness (low correlation associated with white noise), to predictability (high correlation associated to brown noise). Nevertheless a more detailed analysis should be carried out for prove this last point.

Finally, most interesting, but at this point somehow speculative, is the possible relation between 1/f and statistical contextuality. Meanwhile Eliazar and Klafter [33, 34] have proven that both Lévy walks and 1/f are the result of systems which superimpose the transmissions of infinite independent stochastic signals, it has been proposed that this would require a non-classical probabilistic theory, i.e. statistical contextuality [45-47]. Then we propose that 1/f ubiquity is a consequence of contextuality ubiquity in Nature.

5. Acknowledgements

This work was supported by CONACyT scholarship within the graduate program in Earth Sciences at Mexico’s National University (UNAM).


  1. A. Ott, J. P. Bouchaud, D. Langevin and W. Urbach, “Anomalous Diffusion in ‘Living Polymers’: A Genuine Levy Flight?” Physical Review Letters, Vol. 65, No. 17, 1990, pp. 2201-2204. doi:10.1103/PhysRevLett.65.2201
  2. T. H. Solomon, E. R. Weeks and H. L. Swinney, “Observation of Anomalous Diffusion and Lévy Flights in a Two-Dimensional Rotating Flow,” Physical Review Letters, Vol. 71, No. 24, 1993, pp. 3975-3978. doi:10.1103/PhysRevLett.71.3975
  3. F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect and C. Cohen-Tannoudji, “Subrecoil Laser Cooling and Lévy Flights,” Physical Review Letters, Vol. 72, No. 2, 1994, pp. 203-206. doi:10.1103/PhysRevLett.72.203
  4. J. Klafter and G. Zumofen, “Lévy Statistics in a Hamiltonian System,” Physical Review E, Vol. 49, No. 6, 1994, pp. 4873-4877. doi:10.1103/PhysRevE.49.4873
  5. O. V. Bychuk and B. O’Shaughnessy, “Anomalous Diffusion at Liquid Surfaces,” Physical Review Letters, Vol. 74, No. 10, 1995, pp. 1795-1798. doi:10.1103/PhysRevLett.74.1795
  6. I. M. Sokolov, J. Mai and A. Blumen, “Paradoxal Diffusion in Chemical Space for Nearest-Neighbor Walks over Polymer Chains,” Physical Review Letters, Vol. 79, No. 5, 1998, pp. 857-860.
  7. M. Bologna, P. Grigolini and J. Riccardi, “The Levy Diffusion as an Effect of Sporadic Randomness,” 1999.
  8. P. Santini, “Lévy Scaling in Random Walks with Fluctuating Variance,” Physical Review E, Vol. 61, No. 1, 2000, pp. 93-99. doi:10.1103/PhysRevE.61.93
  9. W. D. Luedtke and U. Landman, “Slip Diffusion and Lévy Flights of an Adsorbed Gold Nanocluster,” Physical Review Letters, Vol. 82, No. 19, 1999, pp. 3835-3838. doi:10.1103/PhysRevLett.82.3835
  10. G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. da Luz, E. Raposo and H. Stanley, “Optimizing the Success of Random Searches,” Nature, Vol. 401, No. 6756, 1999, pp. 911-914. doi:10.1038/44831
  11. G. Ramos-Fernández, J. L. Mateos, O. Miramontes, G. Cocho, H. Larralde and B. Ayala-Orozco, “Lévy Walk Patterns in the Foraging Movements of Spider Monkeys (Ateles Geoffroyi),” Behavioral Ecology and Sociobiology, Vol. 55, 2004, pp. 223-230. doi:10.1007/s00265-003-0700-6
  12. L. Seuront, A. Duponchel and C. Chapperon, “Statistical Mechanics and Its Applications,” Physica A, Vol. 385, No. 2, 2007, pp. 573-582. doi:10.1016/j.physa.2007.07.029
  13. R. Atkinson, C. Rhodes, D. MacDonald and R. Anderson, “Scale-Free Dynamics in the Movement Patterns of Jackals,” Oikos, Vol. 98, No. 1, 2002, pp. 134-140. doi:10.1034/j.1600-0706.2002.980114.x
  14. D. Austin, W. Bowen and J. McMillan, “Intraspecific Variation in Movement Patterns: Modeling Individual Behaviour in a Large Marine Predator,” Oikos, Vol. 105, No. 1, 2004, pp. 15-30. doi:10.1111/j.0030-1299.1999.12730.x
  15. D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, C. J. A. Bradshaw, J. W. Pitchford, A. James, M. Z. Ahmed, A. S. Brierley, M. A. Hindell, D. Morritt, M. K. Musyl, D. Righton, E. L. C. Shepard, V. J. Wearmouth, R. P. Wilson, M. J. Witt and J. D. Metcalfe, “Scaling Laws of Marine Predator Search Behaviour,” Nature, Vol. 451, No. 7182, 2008, pp. 1098-1102. doi:10.1038/nature06518
  16. O. Miramontes, D. Boyer and F. Bartumeus, “The Effects of Spatially Heterogeneous Prey Distributions on Detection Patterns in Foraging Seabirds,” PLoS One, Vol. 7, No. 4, 2012, p. e34317. doi:10.1371/journal.pone.0034317
  17. C. T. Brown, L. S. Liebovitch and R. Glendon, “Lévy Flights in Dobe Ju/’hoansi Foraging Patterns,” Human Ecology, Vol. 35, No. 1, 2007, pp. 129-138. doi:10.1007/s10745-006-9083-4
  18. D. Boyer and O. López-Corona, “Self-Organization Scaling and Collapse in a Coupled Automaton Model of Foragers,” Journal of Physics A, Vol. 42, No. 43, 2009, p. 4014.
  19. A. Downey, “Think Complexity,” O’Reilly Media, 2012, p. 79.
  20. O. Bohigas, M. Giannoni and C. Schmit, “Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws,” Physical Review Letters, Vol. 52, No. 1, 1984, pp. 1-4. doi:10.1103/PhysRevLett.52.1
  21. E. Faleiro, U. Kuhl, R. Molina, L. Muñoz, A. Relano and J. Retamosa, “Power Spectrum Analysis of Experimental Sinai Quantum Billiards,” Physics Letters A, Vol. 358, No. 4, 2006, pp. 251-255. doi:10.1016/j.physleta.2006.05.029
  22. R. Haq, A. Pandey and O. Bohigas, “Fluctuation Properties of Nuclear Energy Levels: Do Theory and Experiment Agree?” Physical Review Letters, Vol. 48, No. 16, 1982, pp. 1086-1089 doi:10.1103/PhysRevLett.48.1086
  23. J. M. Gómez, E. Faleiro, R. Molina, L. Muñoz, A. Relaño, and J. Retamosa, “Chaos and 1/f Noise in Nuclear Spectra,” AIP Conference Proceedings, Vol. 831, 2006, p. 80.
  24. A. Cavagna, A. Cimarelli, et al., “Scale-Free Correlations in Bird Flocks,” 2009. arXiv:0911.4393
  25. J. Buhl, D. Sumpter, et al., “From Disorder to Order in Marching Locusts,” Science, Vol. 312, No. 5778, 2006, pp. 1402-1406. doi:10.1126/science.1125142
  26. A. Goldberger, “Fractal Dynamics in Physiology: Alterations with Disease and Aging,” Proceedings of the National Academy of Sciences, Vol. 99, No. S1, 2002, pp. 2466-2472. doi:10.1073/pnas.012579499
  27. W. H. Press, “Flicker Noises in Astronomy and Elsewhere,” Comments on Astrophysics, Vol. 7, No. 4, 1978, pp. 103-119.
  28. S. L. Miller, W. M. Miller and P. J. McWhorter, “Extremal Dynamics: A Unifying and Activated Processes’ Physical Explanation of Fractals, 1/f Noise,” Journal of Applied Physics, Vol. 73, No. 6, 1993.
  29. R. Fossion, E. Landa, P. Stransky, V. Velazquez, J. C. Lopez Vieyra, I. Garduno, D. Garcia and A. Frank, “Scale Invariance as a Symmetry in Physical and Biological Systems: Listening to Photons, Bubbles and Heartbeats,” AIP Conference Proceedings, Vol. 1323, 2010, pp. 74-90. doi:10.1063/1.3537868
  30. P. Bak and M. Paczuski, “Why Nature Is Complex,” Physics World, Vol. 6, No. 12, 1993, pp. 39-43.
  31. E. R. Scheinerman, “Invitation to Dynamical Systems,” Dover Publications, Mineola, 2012.
  32. L. A. Lipsitz, “Dynamics of Stability the Physiologic Basis of Functional Health and Frailty,” The Journals of Gerontology Series A: Biological Sciences and Medical Sciences, Vol. 57, No. 3, 2002, pp. 115-125. doi:10.1093/gerona/57.3.B115
  33. I. Eliazar and J. Klafter, “A Unified and Universal Explanation for Lévy Laws and 1/f Noises,” Proceedings of the National Academy of Sciences, Vol. 106, No. 30, 2009, pp. 12251-12254.
  34. I. Eliazar and J. Klafter, “Universal Generation of Statistical Self-Similarity: A Randomized Central Limit Theorem,” Physical Review Letters, Vol. 103, No. 4, 2009, Article ID: 40602. doi:10.1103/PhysRevLett.103.040602
  35. D. Boyer, G. Ramos-Fernández, O. Miramontes, J. L. Mateos, G. Cocho, H. Larralde, H. Ramos and F. Rojas, “Scale-Free Foraging by Primates Emerges from Their Interaction with a Complex Environment,” Proceedings of the Royal Society B, Vol. 273, 2006, pp. 1743-1750
  36. D. Boyer, O. Miramontes and H. Larralde, “Lévy-Like Behaviour in Deterministic Models of Intelligent Agents Exploring Heterogeneous Environments,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 43, 2009.
  37. E. Landa, I. Morales, P. Stransky, R. Fossion, V. Velázquez, J. C. Vieyra and A. Frank, “Scale Invariance in Chaotic Time Series: Classical and Quantum Examples, Chaos Theory: Modeling, Simulation and Applications,” In: 3rd Chaotic Modeling and Simulation International Conference (CHAOS), World Scientific, London, 2010.
  38. A. Relano, J. M. Gómez, R. A. Molina, J. Retamosa and E. Faleiro, “Quantum Chaos and 1/f Noise,” Physical Review Letters, Vol. 89, No. 24, 2002, Article ID: 244102. doi:10.1103/PhysRevLett.89.244102
  39. A. L. Goldberger, “Non-Linear Dynamics for Clinicians: Chaos Theory, Fractals, and Complexity at the Bedside,” The Lancet, Vol. 347, No. 9011, 1996, pp. 1312-1314. doi:10.1016/S0140-6736(96)90948-4
  40. A. L. Goldberger, C. K. Peng and L. A. Lipsitz, “What Is Physiologic Complexity and How Does It Change with Aging and Disease?” Neurobiology of Aging, Vol. 23, No. 1, 2002, pp. 23-26. doi:10.1016/S0197-4580(01)00266-4
  41. A. L. Goldberger, et al., “Fractal Dynamics in Physiology: Alterations with Disease and Aging,” Proceedings of the National Academy of Sciences, Vol. 99, No. 1, 2002, pp. 2466-2472. doi:10.1073/pnas.012579499
  42. S. M. Pikkujämsä, T. H. Mäkikallio, L. B. Sourander, et al., “Cardiac Interbeat Dynamics from Childhood to Senescence: Comparison of Conventional and New Measures Based on Fractals and Chaos Theory,” Circulation, Vol. 100, No. 4, 1999, pp. 393-399. doi:10.1161/01.CIR.100.4.393
  43. J. Hayano, et al., Heart and Circulatory Physiology: American Journal of Physiology, Vol. 273, 1997, p. 2811.
  44. T. Hennig, P. Maass, J. Hayano and S. Heinrichs, “Exponential Distribution of Long Heart Beat Intervals during Atrial Fibrillation and Their Relevance for White Noise Behaviour in Power Spectrum,” Journal of Biological Physics, Vol. 32, No. 5, 2006, pp. 383-392. doi:10.1007/s10867-006-9022-z
  45. L. Hardy, “Quantum Theory from Five Reasonable Axioms,” 2001.
  46. A. Khrennikov, “Probabilistic Foundations of Quantum Mechanics and Quantum Information,” 2003.
  47. A. Khrennikov, “Växjio Interpretation-2003: Realism of Contexts,” 2004.