The Drossel-Schwabl forest fire model (1992) is a paradigm for self-organized criticality (SOC) — systems that naturally evolve toward a critical state without tuning parameters.
Rules per step:
When p/f → 0 (slow growth, rare lightning), the system reaches a critical state: power-law distributed fire sizes. Small and large fires both occur; there is no characteristic scale.
The SOC state arises from the interplay of local spreading (fire) and global accumulation (trees). The system self-tunes — more trees → more connectivity → larger fires → more empty space → trees regrow. This feedback maintains near-criticality.
Similar SOC dynamics appear in earthquakes (Gutenberg-Richter law), avalanches, neural criticality, and stock market crashes.