Each cell is empty, a tree, or burning. Empty cells grow trees with probability p; trees ignite from burning neighbours, or spontaneously by lightning with probability f. The Drossel-Schwabl model self-organizes to a critical state where fire sizes follow a power law.
How it works: Start runs the model forward one synchronous timestep at a time. Green cells are trees, orange cells are burning, dark cells are empty ground. Every burning cell becomes empty, every tree adjacent to a fire ignites, empty ground regrows a tree with probability p, and any tree is struck by lightning with probability f. The graph accumulates over all outbreaks and demonstrates convergence to a power law as the simulation progresses.
How it works
Start runs the model forward one synchronous timestep at a time.
Green cells are trees, orange cells are burning, dark cells are empty ground.
Each timestep every burning cell becomes empty, every tree adjacent to a fire
ignites, empty ground regrows a tree with probability p, and any
tree is struck by lightning with probability f.
When lightning strikes a tree, the connected cluster of trees containing it is
exactly what the fire will consume, so that cluster's size is recorded as one
outbreak of size s. The lattice is never tuned to a critical point; it reaches
one on its own from any starting density, which is what makes the model an
example of self-organized criticality[2].
Parameters:
p is the per-cell regrowth probability and f the
per-tree lightning probability. Only their ratio matters to the scaling. The
model requires a double separation of timescales — fires complete faster
than trees grow, and trees grow faster than lightning strikes — so the
scaling regime is $f/p \ll 1$ and the cleanest power law appears at the smallest
f/p the lattice size will support. Large f burns the forest down before clusters
can form; large p refills it faster than fire can clear it.
The apparent exponent is not universal. Measured over increasing system sizes it
drifts rather than converging, and the distribution is better described as a
crossover than as true critical scaling[3].
Graphs:
(a) fire-size distribution N(s), the log-binned
frequency of outbreaks of size s on log-log axes. A straight line means fires
have no characteristic size: small fires are common, system-spanning fires are
rare but never absent. The line straightens as the forest self-organizes, and
steepens at the largest s where the lattice is too small to hold a bigger fire.
