Forest Fires, Barnacles and Trickling Oil: Chance Plays a Key Role in Some Simple Mathematical Models That Suggest Natural Processes

By Peterson, Ivars | Science News, October 3, 1987 | Go to article overview

Forest Fires, Barnacles and Trickling Oil: Chance Plays a Key Role in Some Simple Mathematical Models That Suggest Natural Processes


Peterson, Ivars, Science News


Forest Fires, Barnacles and Trickling Oil

The spread of a raging forest fire is an awesome sight. Flames leap from tree to tree. The fire fans out in a blazing ring, consuming fresh timber at its outskirts and leaving a burnt residue within.

Mathematicians have their own, tamer version of this fiery ring. It's a simple mathematical procedure that on a computer screen seems to mimic the step-by-step growth of a forest fire. This particular procedure is one example of an "interacting particle system.'

The study of interacting particle systems is a relatively new activity for mathematicians. It began in the late 1960s as a branch of probability theory. During the two decades since then, says Thomas M. Liggett of the University of California at Los Angeles, "this area has grown and developed rapidly, establishing unexpected connections with a number of other fields.'

The first examples of interacting particle systems were suggested by research in statistical mechanics. Physicists wanted to understand how a collection of wandering or randomly scattered particles can suddenly organize itself, as happens when a liquid solidifies or a material is magnetized.

The mathematical models developed to simulate such phase transitions proved to be a rich source of inspiration. The idea was to look at what happens to particles scattered across a grid when each particle is allowed to interact with its neighbors according to certain rules. It became clear, says Liggett, that models with a very similar mathematical structure could also be useful for studying neural networks (SN: 8/1/87, p.76), tumor growth, ecological change and the spread of infections.

Moreover, the dynamic behavior of such models suggested intriguing mathematical questions. Mathematicians became interested in how certain models evolve over time and began searching for unusual types of behavior.

"Mostly, it's a mathematician's game of seeing what happens,' says Richard T. Durrett of Cornell University in Ithaca, N.Y. The mathematician defines the rules, sets up the game board and lets the game play itself out.

The rules for a mathematician's forest fire are simple. The playing field is a checkerboard grid, on which each cell represents a tree. The fire begins as a single, marked cell or a small cluster of cells at the grid's center. At each time step, a burning cell has a certain probability of spreading the fire to its four nearest neighbors, unless those neighbors have already been burnt.

The same rules also lead to a rough model for the spread of an infectious disease such as measles. In this case, each cell represents an individual who is healthy, ill or immune.

Of course, such a model isn't complicated enough to simulate a real measles epidemic, just as the forest-fire version doesn't take into account all the factors that influence a real forest fire. Nevertheless, the behavior of the model suggests some important features that these natural processes may possess. Eventually, researchers hope to develop more sophisticated variants that come closer to simulating the real thing.

The simplest set of rules mathematically worth investigating makes the unrealistic assumption that the fire (or infection) in a given cell lasts for only one unit of time. At each step, the toss of a coin or a similar randomizing procedure decides whether a certain burning cell spreads its flames to each of its neighbors. Hence, at any given time, a computer screen would show three types of cells: those that are burning, those already burnt up and those that have so far escaped unscathed. The burning cells usually form an irregular, broken ring that gradually expands as time goes on.

Mathematicians, particularly probabalists, are interested in how the process depends on the probability of transmission from one cell to another. One way to study this dependence is to assign a certain transmission probability to each affected cell and then see what the model does. …

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