Abstract
In the previous Chapter we have reviewed different classical models of movement and have explored how they behave in the large time asymptotic regime. The emphasis has been put in showing the working methods and characteristics of the different levels of description (macroscopic, mesoscopic and microscopic). Next we shall show how this differentiation helps us when introducing more advanced concepts like that of memory in the transport processes. Memory has also been considered in the previous chapter, but only in a very elementary way through correlated (persistent) movement in which the direction of motion can be changed according to the current state (e.g. the current direction) of the individual. Nevertheless, the effects that the whole previous history of the trajectory may have on the motion process have not been considered yet.
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References
Alexander, S., Orbach, R.: Density of states on fractals – fractons. Journal de Physique Lettres 43(17), L625–L631 (1982)
Bartumeus, F., Levin, S.A.: Fractal reorientation clocks: linking animal behavior to statistical patterns of search. Proc. Natl. Acad. Sci. 105(49), 19072–19077 (2008). doi:10.1073/pnas. 0801926105
ben Avraham, D., Havlin, S.: Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge (2000)
Campos, D., Méndez, V., Fort, J.: Description of diffusive and propagative behavior on fractals. Phys. Rev. E 69(3), 031115 (5p.) (2004). http://dx.doi.org/10.1103/PhysRevE.69.031115
Czirok, A., Schlett, K., Madarasz, E., Vicsek, T.: Exponential distribution of locomotion activity in cell cultures. Phys. Rev. Lett. 81(14), 3038–3041 (1998). doi:10.1103/PhysRevLett. 81.3038
Deng, W., Barkai, E.: Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E 79(1, Part 1) (2009). doi:10.1103/PhysRevE.79.011112
Edwards, A.M., Phillips, R.A., Watkins, N.W., Freeman, M.P., Murphy, E.J., Afanasyev, V., Buldyrev, S.V., da Luz, M.G.E., Raposo, E.P., Stanley, H.E., Viswanathan, G.M.: Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449(7165), 1044–1048 (2007). doi:10.1038/nature06199. http://dx.doi.org/10.1038/nature06199
Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1971)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1990)
Gaveau, B., Schulman, l.: Anomalous diffusion in a random velocity-field. J. Stat. Phys. 66(1–2), 375–383 (1992). doi:10.1007/BF01060072
Giona, M., Roman, E.H.: Fractional diffusion equation for transport phenomena in random media. Physica A 185(1–4), 87–97 (1992). doi:10.1016/0378-4371(92)90441-R. http://dx.doi.org/10.1016/0378-4371(92)90441-R
Gómez Portillo, I., Campos, D., Méndez, V.: Intermittent random walks: transport regimes and implications on search strategies. J. Stat. Mech.-Theory Exp. (2011) doi:10.1088/1742-5468/2011/02/P02033
Halley, J.M., Hartley, S., Kallimanis, A.S., Kunin, W.E., Lennon, J.J., Sgardelis, S.P.: Uses and abuses of fractal methodology in ecology. Ecol. Lett. 7(3), 254–271 (2004). doi:10.1111/j.1461-0248.2004.00568.x. http://dx.doi.org/10.1111/j.1461-0248.2004.00568.x
Hapca, S., Crawford, J.W., Young, I.M.: Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level. J. R. Soc. Interface 6(30), 111–122 (2009). doi:10.1098/rsif.2008.0261
Hill, M., Caswell, H.: Habitat fragmentation and extinction thresholds on fractal landscapes. Ecol. Lett. 2(2), 121–127 (1999). doi:10.1046/j.1461-0248.1999.22061.x
Hill, N.A., Häder, D.P.: A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186(4), 503–526 (1997). url:http://dx.doi.org/10.1006/jtbi.1997.0421
Imre, A., Bogaert, J.: The fractal dimension as a measure of the quality of habitats. Acta Biotheor. 52(1), 41–56 (2004). doi:10.1023/B:ACBI.0000015911.56850.0f
Jansen, V.A.A., Mashanova, A., Petrovskii, S.: Comment on Lévy walks evolve through interaction between movement and environmental complexity. Science 335, 918 (2012)
Kaitala, V., Heino, M.: Complex non-unique dynamics in simple ecological interactions. Proc. R. Soc. Lond. B-Biol. Sci. 263(1373), 1011–1015 (1996). doi:10.1098/rspb.1996.0149
Klafter, J., Blumen, A., Shlesinger, M.F.: Stochastic pathway to anomalous diffusion. Phys. Rev. A 35(7), 3081–3085 (1987). http://dx.doi.org/10.1103/PhysRevA.35.3081
Kostylev, V., Erlandsson, J., Ming, M., Williams, G.: The relative importance of habitat complexity and surface area in assessing biodiversity: fractal application on rocky shores. Ecol. Complex. 2(3), 272–286 (2005). doi:10.1016/j.ecocom.2005.04.002
Levin, S., Muller-Landau, H., Nathan, R., Chave, J.: The ecology and evolution of seed dispersal: a theoretical perspective. Annu. Rev. Ecol. Evol. Syst. 34, 575–604 (2003). doi:10.1146/ annurev.ecolsys.34.011802.132428
Lévy, P.: Théorie de l’addition des variables aléatoires. Gauthiers-Villars, Paris (1937)
Lovejoy, S., Currie, W., Tessier, Y., Claereboudt, M., Bourget, E., Roff, J., Schertzer, D.: Universal multifractals and ocean patchiness: phytoplankton, physical fields and coastal heterogeneity. J. Plankton Res. 23(2), 117–141 (2001). doi:10.1093/plankt/23.2.117
Lovely, P.S., Dahlquist, F.W.: Statistical measures of bacterial motility and chemotaxis. J. Theor. Biol. 50(2), 477–496 (1975)
Lubelski, A., Sokolov, I.M., Klafter, J.: Nonergodicity mimics inhomogeneity in single particle tracking. Phys. Rev. Lett. 100(25), 250602 (2008). doi:10.1103/PhysRevLett.100.250602. http://link.aps.org/abstract/PRL/v100/e250602
Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman, New York (1983). http://books.google.es/books?id=0R2LkE3N7-oC
Mann, J., Ott, S., Pecseli, H., Trulsen, J.: Laboratory studies of predator-prey encounters in turbulent environments: effects of changes in orientation and field of view. J. Plankton Res. 28(5), 509–522 (2006). doi:10.1093/plankt/fbi136
Méndez, V., Campos, D., Fort, J.: Dynamical features of reaction-diffusion fronts in fractals. Phys. Rev. E 69(1), 016613 (7p.) (2004). http://dx.doi.org/10.1103/PhysRevE.69.016613
Meroz, Y., Sokolov, I.M., Klafter, J.: Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist. Phys. Rev. E 81(1, Part 1) (2010). doi:10.1103/PhysRevE.81.010101
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000). http://dx.doi.org/10.1016/S0370-1573(00)00070-3
Montroll, E.W., Weiss, G.H.: Random walks on lattices. ii. J. Math. Phys. 6(2), 167–181 (1965). http://dx.doi.org/10.1063/1.1704269
Okubo, A., Levin, S.A.: Diffusion and Ecological Problems. Springer, New York (2001)
O’Shaughnessy, B., Procaccia, I.: Analytical solutions for diffusion on fractal objects. Phys. Rev. Lett. 54(5), 455–458 (1985). doi:10.1103/PhysRevLett.54.455. http://link.aps.org/abstract/PRL/v54/p455
Patlak, C.: Random Walk with Persistence and External Bias: A Mathematical Contribution to the Study of Orientation of Organisms. University of Chicago, Committee on Mathematical Biology (1953). http://books.google.es/books?id=hWwvGwAACAAJ
Pecseli, H.L., Trulsen, J.K., Fiksen, O.: Predator-prey encounter rates in turbulent water: analytical models and numerical tests. Prog. Oceanogr. 85(3–4), 171–179 (2010). doi:10.1016/ j.pocean.2010.01.002
Petrovskii, S., Mashanova, A., Jansen, V.A.A.: Variation in individual walking behavior creates the impression of a Levy flight. Proc. Natl. Acad. Sci. U. S. A. 108(21), 8704–8707 (2011). doi:10.1073/pnas.1015208108
Ramos-Fernandez, G., Mateos, J., Miramontes, O., Cocho, G., Larralde, H., Ayala-Orozco, B.: Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi). Behav. Ecol. Sociobiol. 55(3), 223–230 (2004). doi:10.1007/s00265-003-0700-6
Richardson, L.: Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110(756), 709–737 (1926). doi:10.1098/rspa.1926.0043
Selmeczi, D., Mosler, S., Hagedorn, P.H., Larsen, N.B., Flyvbjerg, H.: Cell motility as persistent random motion: theories from experiments. Biophys. J. 89(2), 912–931 (2005). http://www.biophysj.org/cgi/content/abstract/89/2/912
Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987). http://dx.doi.org/10.1103/PhysRevLett.58.1100
Sims, D.W., Southall, E.J., Humphries, N.E., Hays, G.C., Bradshaw, C.J.A., Pitchford, J.W., James, A., Ahmed, M.Z., Brierley, A.S., Hindell, M.A., Morritt, D., Musyl, M.K., Righton, D., Shepard, E.L.C., Wearmouth, V.J., Wilson, R.P., Witt, M.J., Metcalfe, J.D.: Scaling laws of marine predator search behaviour. Nature 451(7182), 1098–1102 (2008). doi:10.1038/nature06518. http://dx.doi.org/10.1038/nature06518
Sokolov, I.M.: Statistics and the single molecule. Physics 1, 8 (2008). http://physics.aps.org/articles/v1/8
Sugihara, G., May, R.: Applications of fractals in ecology. Trends Ecol. Evol. 5(3), 79–86 (1990). doi:10.1016/0169-5347(90)90235-6
Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc. 2 20, 196–212 (1921)
Turchin, P.: Translating foraging movement in heterogeneous environments into the spatial-distribution of foragers. Ecology 72(4), 1253–1266 (1991). doi:10.2307/1941099
Visser, A., Stips, A.: Turbulence and zooplankton production: insights from PROVESS. J. Sea Res. 47(3–4), 317–329 (2002). doi:10.1016/S1385-1101(02)00120-X. In: 26th General Assembly of the European-Geophysical-Society, Nice (2001)
Visser, A.W., Mariani, P., Pigolotti, S.: Swimming in turbulence: zooplankton fitness in terms of foraging efficiency and predation risk. J. Plankton Res. 31(2), 121–133 (2009). doi:10.1093/plankt/fbn109
Viswanathan, G.M., Raposo, E.P., Bartumeus, F., Catalan, J., da Luz, M.G.E.: Necessary criterion for distinguishing true superdiffusion from correlated random walk processes. Phys. Rev. E 72(1), 011111 (2005). http://dx.doi.org/10.1103/PhysRevE.72.011111
Weron, A., Magdziarz, M.: Generalization of the Khinchin theorem to Levy flights. Phys. Rev. Lett. 105(26) (2010). doi:10.1103/PhysRevLett.105.260603
Wilson, R., Wilson, M.: Foraging behavior in 4 sympatric cormorants. J. Anim. Ecol. 57(3), 943–955 (1988). doi:10.2307/5103
With, K.: The landscape ecology of invasive spread. Conserv. Biol. 16(5), 1192–1203 (2002). doi:10.1046/j.1523-1739.2002.01064.x
Witten, T., Sander, L.: Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47(19), 1400–1403 (1981). doi:10.1103/PhysRevLett.47.1400
Zaburdaev, V.: Microscopic approach to random walks. J. Stat. Phys. 133(1), 159–167 (2008). http://dx.doi.org/10.1007/s10955-008-9598-8
Zaburdaev, V., Schmiedeberg, M., Stark, H.: Random walks with random velocities. Phys. Rev. E 78(1), 011119 (2008). doi:10.1103/PhysRevE.78.011119. http://link.aps.org/abstract/PRE/v78/e011119
Zumofen, G., Klafter, J.: Scale-invariant motion in intermittent chaotic systems. Phys. Rev. E 47(2), 851–863 (1993). doi:10.1103/PhysRevE.47.851
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Méndez, V., Campos, D., Bartumeus, F. (2014). Anomalous Diffusion and Continuous-Time Random Walks. In: Stochastic Foundations in Movement Ecology. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39010-4_4
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