A spark ignites a single tree. Will the fire spread across the entire forest or burn out in a small patch? The answer depends critically on forest density — the simulation reveals the sharp percolation phase transition that governs real wildfire risk.
Each tree cell follows three states: Empty → Tree → Burning → Ash. Fire spreads to adjacent trees with probability p. At a critical density (≈59.3 % on a square lattice) a connected cluster spans the grid — fire roars through. Below it, fires are contained.
Adjust Tree density and Fire speed before clicking Ignite. Try density 0.55 and 0.65 to see how close to the percolation threshold changes outcomes. Enable Wind to favour directional spread.
The 2018 California Camp Fire destroyed Paradise in under two hours, consuming 18 000 structures. Models based on percolation theory are now used by fire agencies to identify high-risk landscapes before fire season.
The forest-fire model is a cellular automaton on a grid where each cell is empty, a living tree, or burning. Each step, burning cells turn to empty ground, trees adjacent to fire catch alight, trees may spontaneously ignite from a rare 'lightning' strike, and empty cells regrow trees with a small probability. Adjustable tree density, ignition probability and regrowth rate let you watch fires sweep through and the forest recover.
This model is a textbook example of self-organised criticality and percolation theory: as tree density rises past a critical threshold, fire suddenly gains the ability to span the whole grid, and the distribution of fire sizes follows a power law. The same percolation mathematics describes wildfire spread, epidemic outbreaks, the flow of fluids through porous rock and the robustness of connected networks.
What is the forest-fire cellular automaton?
It is a grid where every cell is empty, a tree, or burning. Simple rules update all cells together each step: fire spreads to neighbouring trees, burning cells become empty, empty cells slowly regrow trees, and lightning occasionally ignites a tree at random.
What is the percolation phase transition?
As tree density increases, there is a critical threshold at which clusters of connected trees suddenly span the entire grid. Below it fires stay local; above it a single ignition can burn across the whole forest. This abrupt change is the percolation phase transition.
How do the four states change over time?
A burning cell becomes empty next step. A tree becomes burning if a neighbour is on fire or if lightning strikes it. An empty cell becomes a tree with the regrowth probability. A tree with no burning neighbours and no strike stays a tree.
Density is the fraction of cells holding trees. It governs how connected the forest is: near the percolation threshold small changes in density cause dramatic changes in how far fires can spread.
With ongoing regrowth and rare lightning, the forest naturally hovers near the critical density without tuning. The result is fires of all sizes whose frequency follows a power law, a hallmark of self-organised criticality.
Near criticality, tree clusters exist at every scale, so an ignition may burn a tiny patch or, less often, a huge connected region. Many small fires and rare very large ones produce the characteristic power-law size distribution.
Lightning is the rare random ignition that starts new fires in an otherwise static forest. Its low probability relative to regrowth is what keeps the system cycling near the critical state rather than burning out or saturating.
Faster regrowth rebuilds connected forest sooner, allowing more frequent large fires; slower regrowth keeps the forest sparse and fires small. The balance between regrowth and ignition sets where the system self-organises.
It is an idealised model that captures the statistical character of spread and the critical-density effect, not exact fire physics. Real wildfire models add wind, terrain, moisture and fuel type, but this captures the essential percolation behaviour.
The same mathematics models epidemic spread through populations, fluid flow through porous rock, conduction in random materials, and the connectivity and resilience of networks, making the forest-fire model a window into many systems.