Why von Neumann (four-neighbor) fire instead of eight neighbors?

Both conventions exist; four-neighbor spread matches many textbook CA definitions and slightly anisotropic fire fronts. Eight-neighbor (Moore) coupling would burn faster and rounder; the qualitative p–f phase structure is similar for pedagogy.

Is this exactly the Drossel–Schwabl forest-fire model?

It is the standard three-state Drossel–Schwabl rule family on a finite torus: empty sites grow trees with probability p, trees ignite from burning neighbors or lightning f, and burning sites become empty. The caveats are finite size, synchronous updates, and no claim of true critical scaling unless one studies the usual separation-of-timescales limit with ensemble statistics.

What should I look for when tuning p and f?

Very large p with tiny f tends toward dense forests and rare megafires after long waits; very large f scatters small fires and keeps density low. Intermediate ratios on a finite torus produce bursty cycles that motivate the SOC narrative without claiming quantitative scaling exponents.

Forest Fire (Cellular Automaton)

Toroidal lattice: trees regrow with probability p, lightning ignites trees with probability f, fire spreads to four neighbors; explore intermittent bursts and SOC-like phenomenology.

About this model

This page implements a minimal forest-fire cellular automaton on a toroidal square lattice—three local states (empty, tree, burning) and synchronous parallel updates. Empty sites attempt to regrow into trees with probability p each timestep; each tree is struck by lightning and ignites independently with probability f; any tree with a burning von-Neumann neighbor catches fire; burning cells turn to ash (empty) on the next update. The competition between slow regrowth and rare sparks produces large, irregular fire avalanches: on finite systems you can explore parameter windows where activity looks intermittent and cluster statistics are reminiscent of self-organized criticality (strict SOC is a thermodynamic-limit story; here the goal is qualitative intuition). Use a small f and moderate p to watch long quiescent periods punctuated by spanning bursts, or raise f for a “noisy oven” of frequent small fires.

Who it's for: Intro courses in statistical physics, complex systems, or modeling where students already know percolation and want a second, dynamics-heavy lattice example.

Key terms

Forest-fire model
Cellular automaton
Self-organized criticality
Lightning probability
Regrowth probability
Toroidal lattice

How it works

Discrete-time forest-fire cellular automaton on a toroidal square lattice: trees regrow on empty sites with probability p, spontaneous lightning ignites each tree with probability f, and fire spreads to von-Neumann neighbors before burning sites turn to ash next step.

Key equations

States: empty 0, tree 1, burning 2. Synchronous update: burning → empty; tree → burning if any von-Neumann neighbor was burning or with prob f; empty → tree with prob p. The ratio p/f controls how “driven” the lattice is on finite grids.

Frequently asked questions

Why von Neumann (four-neighbor) fire instead of eight neighbors?: Both conventions exist; four-neighbor spread matches many textbook CA definitions and slightly anisotropic fire fronts. Eight-neighbor (Moore) coupling would burn faster and rounder; the qualitative p–f phase structure is similar for pedagogy.
Is this exactly the Drossel–Schwabl forest-fire model?: It is the standard three-state Drossel–Schwabl rule family on a finite torus: empty sites grow trees with probability p, trees ignite from burning neighbors or lightning f, and burning sites become empty. The caveats are finite size, synchronous updates, and no claim of true critical scaling unless one studies the usual separation-of-timescales limit with ensemble statistics.
What should I look for when tuning p and f?: Very large p with tiny f tends toward dense forests and rare megafires after long waits; very large f scatters small fires and keeps density low. Intermediate ratios on a finite torus produce bursty cycles that motivate the SOC narrative without claiming quantitative scaling exponents.

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About this model

How it works

Key equations

Frequently asked questions

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About this model

How it works

Key equations

Frequently asked questions