Log-periodic behavior in a forest-fire model

• Published in: Nonlinear processes in geophysics Vol. 12; no. 5; pp. 575 - 585
• Main Authors: Malamud, B. D., Morein, G., Turcotte, D. L.
• Format: Journal Article
• Language: English
• Published: European Geosciences Union (EGU) 01.01.2005; Copernicus Publications
• Subjects: Astrophysics; Cosmology and Extra-Galactic Astrophysics; Earth Sciences; Physics; Sciences of the Universe
• ISSN: 1607-7946; 1023-5809; 1607-7946
• DOI: 10.5194/npg-12-575-2005

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.