The Forest-Fire Model · The Atlas of Emergence

Green is forest, amber is the moving fire front, dark is bare ground. Trees grow back with probability p; each tree is struck by lightning with the much smaller probability f. Click or drag on the grid to start your own fire. Watch the readout's fire-size histogram fill across many orders of magnitude — the signature of self-organized criticality.

What you're seeing

A grid of cells, each empty, holding a tree, or actively burning. Time moves in discrete ticks, and every cell updates at once. Empty ground slowly greens over as trees grow; a tree catches fire the instant a neighbor is burning, so a flame races through any connected stand of forest and leaves bare ground behind. Very rarely a tree is struck by lightning and a new fire begins. The result is endless: forests fill in, a spark lands, a blaze sweeps out and clears a patch, and then it all regrows — forever, with no two fires quite the same size.

The rule

Each cell follows four transition rules, applied to every cell synchronously (a tree's "neighbors" are its four orthogonal — von Neumann — cells; the grid wraps at the edges):

A burning cell becomes empty (the fuel is spent).
A tree with at least one burning neighbor becomes burning (fire spreads).
A tree with no burning neighbor is struck by lightning and becomes burning with probability f.
An empty cell grows a tree with probability p.

The interesting regime is p ≫ f: trees grow far faster than lightning strikes. A fire then spreads through connected trees in exactly the way percolation spreads through occupied sites — a burning patch is just a connected cluster of trees ignited from one point. Because growth keeps refilling the forest while sparks keep clearing it, the tree density is driven, all on its own, to near the percolation threshold, where clusters of every size coexist.

Why it matters

This is a second face of self-organized criticality — compare the Abelian sandpile, the Atlas's other SOC entry. There, a slow drip of grains and sudden avalanches keep the pile poised at its critical slope. Here, slow growth and sudden fires keep the forest poised at its critical density. In both, the system tunes itself to the edge of a phase transition without any parameter being set to a special value, and the size distribution of events — avalanches there, fires here — follows an approximate power law: very many small events, a few enormous ones, and no characteristic scale in between. The forest-fire model is one proposed explanation for why power-law statistics show up so often in real cascading systems — wildfires, earthquakes, and the like — where a slow build-up of stress is released in bursts of every size.

In the wild

Real wildfire records do show approximate power-law size distributions over several orders of magnitude: Malamud, Morein, and Turcotte (1998) analyzed fires across several regions and found frequency–area statistics consistent with self-organized criticality, and the same cascading-failure framing has been applied to landslides, epidemics, and electrical-grid blackouts. But honesty matters here, as with every SOC claim in this Atlas: a power law is suggestive of criticality, not proof of it. Real forests have wind, terrain, fuel moisture, seasonality, and active human suppression — none of which this cartoon contains — and whether actual wildfire regimes are truly self-organized-critical (versus power laws arising from other mechanisms) remains genuinely debated. The model earns its place by making the idea vivid and checkable, not by being a faithful map of any real forest.

Try this

Keep the default SOC (p ≫ f) preset: lightning is tiny, so the forest grows dense and calm for long stretches, then a single spark touches off a fire that sweeps a large fraction of the grid. Watch max fire climb in rare jumps.
Switch to Frequent lightning (large f): now fires start everywhere, the forest never gets dense, and you see many small burns instead of rare giant ones — criticality is lost when f is no longer ≪ p.
Click and drag on a dense green region to start your own fire, and watch the front spread only as far as the trees are connected.

References

Drossel, B. & Schwabl, F. (1992). "Self-organized critical forest-fire model." Physical Review Letters 69(11), 1629–1632. doi:10.1103/PhysRevLett.69.1629
Bak, P., Chen, K., Tang, C. (1990). "A forest-fire model and some thoughts on turbulence." Physics Letters A 147(5–6), 297–300. doi:10.1016/0375-9601(90)90451-S
Malamud, B. D., Morein, G., Turcotte, D. L. (1998). "Forest Fires: An Example of Self-Organized Critical Behavior." Science 281(5384), 1840–1842. doi:10.1126/science.281.5384.1840