Burning forests

I think I first encountered the term emergent behaviour in connection with fractals - in a book the title of I have regrettably forgotten.
The very first chapter discussed a phenomenon encountered in a simulation of a simple model of the forest fire.

Imagine the forest as a 2D grid of certain dimensions (let's say it's 100x100). Each of its rectangular tiles represents a small plot of land. On some of these tiles, a tree grows. Some are empty. And - in case of the fire - some host a merrily burning plant.

The rules of the model are as follows. On each turn:

• the empty plot remains empty, 
• a tree starts burning if it has at least one burning neighbour (we use von Neumann neighbourhood, so we only count the tiles which touch the current square on its sides and not the corners),
• otherwise the tree keeps living.

Variations of this model exist (see: https://en.wikipedia.org/wiki/Forest-fire_model). In these, an additional rule often applies: an empty square can in its own time spawn a new tree with a certain chance. Also, random fires can start (e.g. due to lightning strikes).

In our case we are concerned with a more short-term model - that is we do not account for the new growth. Moreover, our fire always starts at one side of the forest. Fig. 1. shows the burning forest.

Fig. 1. Burning forest model. Fire front is visible in red.

The question is: how does the fire front proceed? How many trees does the inferno claim? Surely it must depend on how dense the wood is. It does (see Fig. 2).

Fig. 2. The percentage of the remaining trees as a function of density.

This is where the emergent behaviour occurs. The forest is quite resilient: even with half the squares occupied by the living trees, most of them survive the fire. The things change quite quickly and unexpectedly very soon after the critical density is reached: just short of the 60% mark.

Once the density reaches 60-70% only a few trees ever remain unburned. The numbers of survivors dwindle further towards zero at 80% mark. Uniform front of fire advances on the 100% dense forest.

Where does the number that defines the critical density come from? It's hard to predict just basing on the model rules.

The forest-fire model does not only apply here. Similar approaches are used in the context of disease spread and resistance in populations. In such models, the squares are no longer the trees, but people. Disease replaces the fire. Surely, you can now appreciate the critical impact the vaccinations have on the outcome of epidemies.

I made an interactive tool for you to play with the matches (see Fig. 3). Check it out here: https://rawgit.com/dagothar/burning-forest/master/index.html

Fig. 3. Burning forest simulator.

You can check the code out here: https://github.com/dagothar/burning-forest