Abstract

This report deals with the findings and results of our study on the Forest-Fire Model. The aim was to understand the general principles of Cellular Automata, the Forest-Fire Model itself and to write a sufficiently fast algorithm to create interesting results.
Our investigations encompass the formation, behaviour and lifetime of spiral-shaped fire fronts in the Bak Model. The anti-proportionality between the distance of the fire fronts and tree growth parameter p was confirmed. The percolation density in the Bak Model was determined and it was found a value of ρperc = 0.58 ... 0.60 which is in agreement with values in publications.
The major part of our investigations was carried out with the Drossel- Schwabl Forest-Fire Modelwhich contains the feature of lightning strikes realised by parameter f. The creation of clusters and their variation in size was studied. It turned out that the size of clusters increases with increasing θ = p · f-1. Furthermore, the analysis of the steady state density oscillation showed that they decline with increasing grid size. It was confirmed that the evolution of the system is quite independent of its initial state as expected for complex systems. By introducing the immunity parameter g it was shown that for increasing immunity the steady state density increases as well. Finally, the system was analysed in terms of Self-Organised Criticality. It was found that the system exhibits Self- Organised Criticality. However, not in the sense that the system can be described with power laws at all scales.
The model, similar to the Drossel-Schwabl algorithm, was implemented in Python. Furthermore, a more elaborate algorithm was programmed, which differs from the original rules but is equivalent in the limit of big systems.


Images from Simulations

In the top of this page four images from our simulations done with Python are presented.

The pictures 1-3 are snapshots of the simulated forest. A white pixel stands for an empty site, a black one for a tree, and a red one for a burning tree.

The fourth pictures shows the dependence of the system due to different initial tree densities (x-axis from 0 to 1: initial density; y-axis: time). The actual density is color coded, from 0 (blue) to 1 (red).


Code

The code was written in Python. Orientated on the Drossel-Schwabel Forest Fire Model [Self-organized critical forest-fire model, B. Drossel and F. Schwabl, Physik-Department der Technischen Universität München, 1992]

Hendrik Leusmann & Josef Lier in the framework of the Master Seminar: Chaotic, Complex & Evolving Systems (Summerterm 2015, Kurt Roth)

For further informations on this project please wrote an E-Mail.