A Study of the Drossel–Schwabl Forest-Fire Model: Local Connectivity and Fire-Size Statistics

Large wildfires impose substantial ecological and societal costs, motivating simplified models that isolate when small ignitions escalate into large burns. We study a two-dimensional stochastic forest-fire model with regrowth and lightning ignition, in an event-based formulation where each lightning event is preceded by θ random regrowth attempts, so θ controls pre-ignition density and connectivity. Beyond global density we evaluate a local predictor: tree density in the 9×9 neighbourhood around the strike. Across 72 independently seeded runs the median fire size versus local density curve shows an uptick threshold at ρ*_loc = 0.606 ± 0.006, resolving to one of two adjacent bins in every run and corresponding to roughly 48 of the 80 cells neighbouring the ignition site. The threshold is stable across regrowth rate, layout and system size, while its steepness grows 4.2-fold from θ = 250 to 3000 and large-fire probability per lightning event rises from 0.028 to 0.196. Roughly 91% of that rise comes from fires becoming more severe once started rather than from lightning striking fuel more often. Initial layout has no detectable effect on steady-state risk once each run has an independent random stream, with between-layout variation three to nine times smaller than seed-to-seed variation. We trace this to a memory-erasure timescale that collapses on φ = θ/N², falling from beyond 20,000 events at φ ≤ 0.031 to under 100 at φ = 0.183. The same variable governs finite-size comparison, where discrepancies up to 54.6× at fixed θ collapse to 1.00–1.11× at matched φ.