One of the classic agent-based and cellular automata models is so-called forest fire model. It is an excellent example of a modeling problem which could, in theory, be solved using standard mathematical modeling tools (such as, partial differential equations), but it would be extremely hard. On the other hand one could use the aforementioned discrete modeling frameworks. So, lets do that.
There are couple somewhat different variations of the model, but they all share the same core mechanism: fire spreads to nearby trees, and they spread the fire to their unburnt neighbors. The original rules of the model were defined in . And the original model was intended as a self-organized criticality model along the lines of the sandpile model. So, there could be multiple fires burning at the same time, while the trees were also replanted, and effectively long term temporal dynamics were observed.
Here we simplify the original model with a goal to see the size of the individual fires. We do so, as the model exhibits phase transition when consider the sizes of the fire "avalanches". If \( \rho \lesssim 0.55 \), then the fires usually will be small in comparison to the size of the forest. While with larger \( \rho \) fires would easily spread across the forest.
Still we keep the core rule as it was: a tree will burn if at least one neighbor is burning. In other words, during each time tick we (simultaneously) check if an unburnt tree has a burning neighbor. If so, then we set the unburnt tree on fire. We continue until no tree was set on fire during a time tick.
In the app below we use 10×10 forest with periodic boundary conditions. We consider four cardinal neighbors as neighbors for the purposes of spreading the fire.
Feel free to explore the model using the interactive app below. You can control the density of trees \( \rho \) (effectively this is a probability that a cell will contain a tree). You should be able to observe that around \( \rho \approx 0.55 \) the sizes of the fires change: being local for smaller values, and spanning whole forest with larger values.
- B. Drossel, F. Schwabl. Self-organized critical forest-fire model. Physical Review Letters 69: 1629 (1992). doi: 10.1103/PhysRevLett.69.1629.