Articles | Volume 12, issue 5
Nonlin. Processes Geophys., 12, 575–585, 2005
https://doi.org/10.5194/npg-12-575-2005
Nonlin. Processes Geophys., 12, 575–585, 2005
https://doi.org/10.5194/npg-12-575-2005

  09 Jun 2005

09 Jun 2005

Log-periodic behavior in a forest-fire model

B. D. Malamud1, G. Morein2, and D. L. Turcotte3 B. D. Malamud et al.
  • 1Environmental Monitoring and Modelling Research Group, Department of Geography, King’s College London, Strand, London, WC2R 2LS, UK
  • 2Center for Computational Science and Engineering, University of California, Davis, CA 95616, USA
  • 3Department of Geology, University of California, Davis, CA 95616, USA

Abstract. This paper explores log-periodicity in a forest-fire cellular-automata model. At each time step of this model a tree is dropped on a randomly chosen site; if the site is unoccupied, the tree is planted. Then, for a given sparking frequency, matches are dropped on a randomly chosen site; if the site is occupied by a tree, the tree ignites and an "instantaneous" model fire consumes that tree and all adjacent trees. The resultant frequency-area distribution for the small and medium model fires is a power-law. However, if we consider very small sparking frequencies, the large model fires that span the square grid are dominant, and we find that the peaks in the frequency-area distribution of these large fires satisfy log-periodic scaling to a good approximation. This behavior can be examined using a simple mean-field model, where in time, the density of trees on the grid exponentially approaches unity. This exponential behavior coupled with a periodic or near-periodic sparking frequency also generates a sequence of peaks in the frequency-area distribution of large fires that satisfy log-periodic scaling. We conclude that the forest-fire model might provide a relatively simple explanation for the log-periodic behavior often seen in nature.

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