Self-organized criticality of forest fire in China
Introduction
Self-organized criticality (SOC) was originally introduced as a general theory to understand fractals and 1/f noise as the natural coupled degrees of freedom (Bak et al., 1987, Bak et al., 1988, Bak and Chen, 1990). Irreversible dynamics would drive the system into a critical state without the fine tuning of parameters. When the system reaches the critical state, the ‘frequency-size’ distribution of energy dissipation events satisfies a power-law relation. The SOC idea is illustrated by computer models that have slow driving or energy input and rare, avalanche-like dissipation events that are instantaneous on the time scale of driving. These models include the sand-pile model (Bak and Chen, 1990), slide-block model (Carson and Langer, 1989) and forest-fire model (Drossel and Schwabl, 1992), etc. The ecosystem is a self-organized critical system (Jørgensen et al., 1998). The abundance of species versus the size of the species, the variations of changes in ecosystems versus the frequency, the sizes of the avalanches are plotted versus the frequencies all show power-law behaviors. SOC might ultimately explain the ubiquity of the fractal, which has been paid great attention in ecology (Li, 2000).
The forest-fire model proposed by Drossel and Schwabl (1992) is a stochastic cellular automaton. The forest is represented by a two-dimensional lattice, in which trees grow with a low probability, and fire occurs with less probability. A burning tree will ignite all its neighboring trees with the same probability, 1, so that a forest cluster will burn down in case it contains a burning tree. The forest-fire model will automatically reach a steady state characterized by the power-law relation of the ‘frequency-size’ distribution of forest fires. Drossel and Schwabl (1993) and Albano (1995) introduced tree immunity, g, to the forest-fire model and studied the change of forest-fire distribution versus g. In this generalized model, a tree will become a burning tree with probability 1−g if at least one of its nearest neighbors is burning. Schenk et al. (2000) studied finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and fire-size distribution. As the system size becomes smaller, the system contains fewer patches and finally becomes homogeneous, with large density fluctuations in time.
Compared with other models (Li et al., 1997, Miller and Urban, 1999, Peng and Apps, 1999, Mailly et al., 2000), the forest-fire model proposed by Drossel focuses on abstracting the basic characteristics of forest fires and investigating the global behaviors, especially SOC, of the forest and forest fires. Casagrandi and Rinaldi (1999) investigated fire regimes through a minimal model. Hargrove et al. (2000) developed a very similar model, with which they simulated fire patterns in different parameters. In these researches, the SOC characteristics were not studied.
Actual forest fires have SOC characteristics, and the forest-fire model can be used to describe actual forest fires (Malamud et al., 1998, Ricotta et al., 1999) and epidemics, etc. (Johansen, 1994, Rhodes and Anderson, 1996). ‘Frequency-size’ distributions of natural hazards provide important information on calculating risk and are used in hazard mitigation (Turcotte, 1989, Turcotte, 1999a).
Ricotta et al. (1999) analyzed a large series of wildfire records of the Regional Forest Service of Liguria from 1986 to 1993 and found a power-law relation between the frequency of occurrence and the size of the burned area. The idea of SOC applies well to explain wildfire occurrence on a regional basis. The records of burned area show a fractal distribution with a fractal dimension being about 1.446.
Malamud et al. (1998) analyzed some forest-fire data in USA and Australia and found that forest fires exhibit a power-law dependence of occurrence frequency on burn area over many orders of magnitude and that actual forest fires can be directly associated with the forest-fire model.
However, the occurrence frequency of very large fires shows obvious deviations from the power-law relation. A common reason of the deviations is because actual forest fires are affected by tree species, meteorological conditions and human fire-fighting efforts, etc. (Malamud et al., 1998, Ricotta et al., 1999).
This paper aims to explain these deviations through the forest-fire model and examine the SOC and fractal characteristics of actual forest fires in China. To explain the deviations between the occurrence frequency of large fires and the power-law relation, we make use of the forest-fire model with tree immunity (FFMTI) and generalize the meaning of tree immunity to involve external conditions such as tree species, meteorological conditions and human fire-fighting efforts. To examine the SOC characteristics of actual forest-fire data, we check SOC together with its time-invariant and scale-invariant attributes. The fractal characteristics of forest cluster and forest fires are also inspected. Based on these studies, we try to provide some suggestions to actual forest-fire protections.
Section snippets
Model and simulation
The forest-fire model is a cellular automata model combined with Monte Carlo simulation. The forest is denoted by a two-dimensional lattice. Each site is occupied by either a tree, a burning tree, or it is empty. The state of the system is updated in parallel by the following rules:
- 1.
A burning tree becomes an empty site.
- 2.
A tree becomes a burning tree if at least one of its nearest neighbors is burning.
- 3.
At an empty site, a tree grows with probability p.
- 4.
A tree without a burning nearest neighbor
Results and discussion
Simulation results of forest-fire model are compared with the actual forest-fire data in China from 1950 to 1989 (Chinese Academy of Forestry, 1999). Fire frequencies with different burned areas are calculated. As shown in Fig. 2, Fig. 3, Fig. 4, the occurrence frequency for the actual fire data is:Here, is the annual number of fires with a burned area being greater than a threshold area, s. In all figures of this paper, the burned area of fire is set to the quotient
Conclusions
In summary, in this paper, we have shown that actual forest-fire data in China have SOC behavior characterized by power-law relation of ‘frequency-size’ distribution of forest fires. The SOC characteristics of a forest are invariant with time and a broad range of forest size, and can be explained by the forest-fire model. The forest cluster and the forest-fire distribution are fractal, and the fractal dimensions are decided by SOC exponents. It is shown that forest-fire regimes are dominated by
Acknowledgements
We acknowledge the support from Special Fund for Major National Basic Research Projects in China (973 Project), the National Basic Research Climbing-up Project ‘Nonlinear Science’, and the National Natural Science Foundation in China under Project Grant No.59876039, 59936140 and 39970621.
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