Forests are very important to us. They create oxygen, filter the air, provide shade and prevent the ground from drying out or getting washed away in the rain. It's important for us to understand how forests work and how sensitive they are to different conditions.

Fires are part of a forest’s life cycle. For example, some species of cypress trees release their seeds from their cones only after the tree is burned, so they need the heat of a fire to reproduce. Also, the conditions found in soils of recently burnt forests help seeds germinate. However, natural fires, like those caused by a stroke of lightning, make up only about five percent of forest fires globally. Most fires are actually caused by people (people also pose a threat to forests by chopping down trees and polluting the air, which causes acid rain and is a grave concern in rainy countries).

Mathematicians have developed models for understanding forest development. In fact, there are many such models with new ones are being developed all the time. One of the simpler, yet far from trivial ones, is the "cellular automaton" model.

A cellular automaton is based on a matrix of cells, where each cell can be in one of a small number of different states. For example, each cell could represent one spot large enough for a tree to grow in, and can be in one of the following three states: "Empty", "Occupied" (by a healthy tree) or "Burning". The state of a forest at any given moment is described by the states of all the cells. With every step in time, each cell "decides" what state it needs to be in based on a small set of simple, predetermined rules. Here's an example of a set of such rules:

- Any burning cell should turn into an empty cell.
- A healthy tree turns into a burning tree if at least one of its neighbors is burning.
- An empty cell turns into a healthy tree at probability r (r: tree reproduction rate).
- A healthy tree burns at probability f, even if none of its neighbors are burning.

Example of forest development simulation | Image: Claudio Rocchini, Wikipedia

These models provide a typical example of "self-organized criticality" in the complex systems theory. The system could move to a critical state, such as a huge forest fire, or otherwise change significantly, without any external factor. The system develops solely by the basic rules, defined identically for each element (cell) in the system.

Using these models, we can get a picture of how the forest might change as we change the parameters – in this case, r, the rate of reproduction, and f, the probability of forest fires.

Of course, this is a very simplified model. Mathematicians have also developed more complicated models, such as the "Cyclic Succession" model, which factors in the ages of trees, such that different aged areas of the same forest can exist at the same time.

Did you know? The type of model described here, the cellular automaton, became famous thanks to the "Game of Life" by John Conway.

**Dr. Sabine Stöcker-Segre**

Davidson Institute of Science Education

Weizmann Institute of Science

Article translated from Hebrew by Aviv J. Sharon, M.Sc. student at the Weizmann Institute of Science.

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