The European Physical Journal Special Topics

, Volume 226, Issue 3, pp 443–453 | Cite as

Fairy circles and their non-local stochastic instability

Regular Article
Part of the following topical collections:
  1. Nonlinearity, Nonequilibrium and Complexity: Questions and Perspectives in Statistical Physics

Abstract

We study analytically a non local stochastic partial differential equation describing a fundamental mechanism for patterns formation, as the one responsible for the so called fairy circles appearing in two different bio-physical scenarios; one on the African continent and another in Australia. Using a stochastic multiscale perturbation expansion, and a minimum coupling approximation we are able to describe the life-times associated to the stochastic evolution from an unstable uniform state to a patterned one. In this way we discuss how two different biological mechanisms can be collapsed in one analytical framework.

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References

  1. 1.
    S. Getzin, et al., Discovery of fairy circles in Australia supports self-organization theory, Proc. Natl. Acad. Sci. 113, 3551 (2016)ADSCrossRefGoogle Scholar
  2. 2.
    N. Juergens, The biological underpinnings of Namib Desert fairy circles, Science 339, 1618 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    W.R. Tschinkel, The life cycle and life span of Namibian fairy circles, PloS one 7, e38056 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    F. Carteni, A. Marasco, G. Bonanomi, S. Mazzoleni, M. Rietkerk, F. Giannino, Negative plant soil feedback explaining ring formation in clonal plants, J. Theor. Biol. 313, 153 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M.D. Cramer, N.N. Barger, Are Namibian fairy circles the consequence of self-organizing spatial vegetation patterning? PloS one 8, e70876 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    A.M. Turing, The chemical basis of morphogenesis, Philos. T. Roy. Soc. B 237, 37 (1952)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Deblauwe, N. Barbier, P. Couteron, O. Lejeune, J. Bogaert, The global bio-geography of semi-arid periodic vegetation patterns, Glob. Ecol. Biogeogr. 17, 715 (2008)CrossRefGoogle Scholar
  8. 8.
    J. von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett. 87, 198101 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    M. Rietkerk, J. van de Koppel, Regular pattern formation in real ecosystems, Trends. Ecol. Evol. 23, 169 (2008)CrossRefGoogle Scholar
  10. 10.
    M.A. Fuentes, M.N. Kuperman, V.M. Kenkre, Nonlocal interaction effects on pattern formation in population dynamics, Phys. Rev. Lett. 91, 158104 (2003)ADSCrossRefGoogle Scholar
  11. 11.
    M.O. Cáceres, Elementos de Estadistica de no Equilibrio y sus Aplicaciones al Transporte en Medios Desordenados (Reverte, Barcelona, 2003)Google Scholar
  12. 12.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)Google Scholar
  13. 13.
    V. Volterra, Theory of Functional and Integro Differential Equations (Dover, N.Y., 2005)Google Scholar
  14. 14.
    M.O. Cáceres, Passage Time Statistics in Exponential Distributed Time-Delay Models: Noisy Asymptotic Dynamics, J. Stat. Phys. 156, 94 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    M.A. Fuentes, M.N. Kuperman, V.M. Kenkre, Analytical considerations in the study of spatial patterns arising from nonlocal interaction effects, J. Phys. Chem. B 108, 10505 (2004)CrossRefGoogle Scholar
  16. 16.
    J.D. Murray, Mathematical Biology I: An Introduction, Vol. 17 of Interdisciplinary Applied Mathematics (Springer, New York, 2002)Google Scholar
  17. 17.
    M.O. Cáceres, M.A. Fuentes, First-passage time for pattern formation nonlocal partial differential equations, Phys. Rev. E 92, 042122 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    E.J. Gumbel, Les valeurs extrêmes des distributions statistiques, Annales de l’institut Henri Poincaré 5, 115 (1935)MathSciNetMATHGoogle Scholar
  19. 19.
    P. Colet, F. de Pasquale, M.O. Cáceres, M. San Miguel, Theory for relaxation at a subcritical pitchfork bifurcation, Phys. Rev. A 41, 1901 (1990)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico, 87501, USA IIF-Sadaf, Bulnes 642, 1428Buenos AiresArgentina
  2. 2.Facultad de Ingeniería y Tecnología, Universidad San SebastiánSantiago 7510157Chile
  3. 3.Centro Atómico Bariloche (CNEA), Instituto Balseiro (Uni. Nac. Cuyo), and CONICETBarilocheArgentina

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