Abstract
A standard assumption of continuous time random walk (CTRW) processes is that there are no interactions between the random walkers, such that we obtain the celebrated linear fractional equation either for the probability density function of the walker at a certain position and time, or the mean number of walkers. The question arises how one can extend this equation to the non-linear case, where the random walkers interact. The aim of this work is to take into account this interaction under a mean-field approximation where the statistical properties of the random walker depend on the mean number of walkers. The implementation of these non-linear effects within the CTRW integral equations or fractional equations poses difficulties, leading to the alternative methodology we present in this work. We are concerned with non-linear effects which may either inhibit anomalous effects or induce them where they otherwise would not arise. Inhibition of these effects corresponds to a decrease in the waiting times of the random walkers, be this due to overcrowding, competition between walkers or an inherent carrying capacity of the system. Conversely, induced anomalous effects present longer waiting times and are consistent with symbiotic, collaborative or social walkers, or indirect pinpointing of favourable regions by their attractiveness.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H.G. Othmer, T. Hillen, SIAM J. Appl. Math. 62, 1222 (2002)
T. Hillen, K.J. Painter, J. Math. Biol. 58, 183 (2008)
A. Stevens, SIAM J. Appl. Math. 61, 183 (2000)
J. Adler, Science 153, 708 (1966)
U. Alon, M.G. Surette, N. Barkai, S. Leibler, Nature 397, 168 (1999)
K. Oelschläger, Probab. Theory Rel. 82, 565 (1989)
P.C. Bressloff, Stochastic processes in cell biology (Springer, New York, 2014)
S. Fedotov V. Méndez, W. Horsthemke, Reaction-transport systems (Springer, Berlin Heidelberg, 2010)
J.D. Murray, Mathematical biology, 3rd ed. (Springer-Verlag, New York, 2002)
D.W. Macdonald, Nature 301, 379 (1983)
E. Matthysen, Ecography 28, 403 (2005)
V. Méndez, D. Campos, I. Pagonabarraga, S. Fedotov, J. Theor. Biol. 309, 113 (2012)
K.J. Painter, T. Hillen, Canad. Appl. Math. Quart. 10, 501 (2002)
M.J. Simpson, R.E. Baker, Phys. Rev. E 83, 051922 (2011)
A.E. Fernando, K.A. Landman, M.J. Simpson, Phys. Rev. E 81, 011903 (2010)
J. Klafter, A. Blumen, M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987)
J. Klafter, I.M. Sokolov, First steps in random walks: from tools to applications (Oxford University Press, London, 2011)
W. Feller, An introduction to probability theory and its applications, Vol. 2 (John Wiley & Sons, Inc., New York, 1966)
K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations (Wiley, New York, 1993)
W. Wyss, J. Math. Phys. 27, 2782 (1986)
S. Fedotov, N. Korabel, Phys. Rev. E 92, 062127 (2015)
S. Fedotov, N. Korabel, Phys. Rev. E 95, 030107 (2017)
P. Straka, S. Fedotov, J. Theor. Biol. 366, 71 (2015)
M. Bologna, C. Tsallis, P. Grigolini, Phys. Rev. E 62, 2213 (2000)
D. Schertzer, M. Larchevêque, J. Duan, V.V. Yanovsky, S. Lovejoy, J. Math. Phys. 42, 200 (2001)
C. Tsallis, E.K. Lenzi, Chem. Phys. 284, 341 (2002)
S. Fedotov, Phys. Rev. E 88, 032104 (2013)
H.G. Othmer, S.R. Dunbar, W. Alt, J. Math. Biol. 26, 263 (1988)
M.O. Vlad, J. Ross, Phys. Rev. E 66, 061908 (2002)
D.R. Cox, H.D. Miller, The theory of stochastic processes (CRC Press, Boca Raton, 1977)
A.G. Thompson, J. Tailleur, M.E. Cates, R.A. Blythe, J. Stat. Mech. Theory E 2011, P02029 (2011)
I.M. Sokolov, J. Klafter, Chaos Soliton. Fract. 34, 81 (2007)
A.V. Chechkin, R. Gorenflo, I.M. Sokolov, J. Phys. A Math. Gen. 38, L679 (2005)
A.I. Burshtein, A.A. Zharikov, S.I. Temkin, Theor. Math. Phys. 66, 166 (1986)
G. Krapivinsky, Y. Kirichok, D.E. Clapham, Nature 427, 360 (2004)
S. Boillée, C.V. Velde, D.W. Cleveland, Neuron 52, 39 (2006)
S. Fedotov, A. Iomin, L. Ryashko, Phys. Rev. E 84, 061131 (2011)
S. Fedotov, H. Al-Shamsi, A. Ivanov, A. Zubarev, Phys. Rev. E 82, 041103 (2010)
S. Fedotov, A. Tan, A. Zubarev, Phys. Rev. E 91, 042124 (2015)
E. Abad, S.B. Yuste, K. Lindenberg, Phys. Rev. E 81, 031115 (2010)
C.N. Angstmann, I.C. Donnelly, B.I. Henry, Math. Model. Nat. Phenom. 8, 17 (2013)
A.I. Shushin, Phys. Rev. E 64, 051108 (2001)
S. Fedotov, H. Stage, Phys. Rev. Lett. 118, 098301 (2017)
S. Fedotov, S. Falconer, Phys. Rev. E 89, 012107 (2014)
M.M. Meerschaert, Y. Zhang, B. Baeumer, Geophys. Res. Lett. 35, L17403 (2008)
B. Baeumer, M.M. Meerschaert, J. Comput. Appl. Math. 233, 2438 (2010)
J. Tailleur, M.E. Cates, Phys. Rev. Lett. 100, 218103 (2008)
J. Stenhammar, D. Marenduzzo, R.J. Allen, M.E. Cates, Soft Matter 10, 1489 (2014)
M.E. Cates, J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015)
S. Fedotov, S. Falconer, Phys. Rev. E 85, 031132 (2012)
N. Korabel, E. Barkai, Phys. Rev. Lett. 104, 170603 (2010)
B.A. Stickler, E. Schachinger, Phys. Rev. E 84, 021116 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Stage, H., Fedotov, S. Non-linear continuous time random walk models. Eur. Phys. J. B 90, 225 (2017). https://doi.org/10.1140/epjb/e2017-80400-5
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2017-80400-5