1. One Dimensional Forest Fire CA–model
The 1d version of the forest fire CA of Drossal and Schwabl (Physica A, 212–229,
1994) employs a one–dimensional lattice with absorbing boundary conditions. Lattice sites may have a value of 0, 1 or 2, where 0 represents an empty site (or hole),
1 represents a healthy tree (or tree), and 2 represents a burning tree. During one time step, all of the sites are updated according to the following rule:
• If a tree has at least one burning tree as nearest neighbor, it becomes a burning tree with probability 1 − g (remains healthy with probability g ).
• If no nearest neighbor is a burning tree, a tree becomes a burning tree with probability f (1 − g ) where 0 ≤ f ≤ 1 (remains healthy with probability
1 − f (1 − g )).
• A burning tree becomes an empty site.
• An empty site becomes a tree with probability r (remains empty with probability 1 − r ).
The parameters 1 − g , f (1 − g ) and r represent the probability of the fire spreading out, the probability of spontaneous ignition of a tree, and the reforestation probability, respectively. In your simulations use a “natural” color code: 1: green, 2: red, and 0: blue or some other appropriate color. A reasonable lattice size is 200 sites and time may range from 0 to say 200.
(a) Investigate the time evolution of a single forest (all initial states 1) for some selected values of g, f, r to demonstrate the main effects of these parameters. Of particular interest is the average desert size as function of time. The average desert size is the total number of empty sites and burning trees together divided by the total number of deserts, where a single desert is defined as a connected sequence of empty sites or burning trees. For each chosen value of the parameters show the spatiotemporal evolution of the CA (color coded) and the average desert size as functions of time. You should see that after an initial phase the average desert size appears to fluctuate about a constant value. Compute this value by an additional time average (averaging over the last 100 time steps if the total time range extends to 200 should be sufficient). In this way the system is characterized by a single number. If you want to do more, compute also the standard deviation of the average desert size fluctuations.
(b) Assume there is no reforestation, i.e.
r = 0. Then, if g < 1 and f > 0, every initial configuration will evolve eventually into a single desert. Thus choose again a single forest as initial condition and call the time it takes until the forest has burned down completely the deforestation time , t d
. Of particular interest is the average
1

of t d
, denoted
Clearly, T d
T d
( g, f ), over several (ideally infinite) repetitions of the experiment.
( g, f ) is increasing when f and 1 − g are decreasing. Since g characterizes the spreading of the fire, we are mainly interested in as parameter.
T d as function of g and treat f
Choose f = 0 .
3 , 0 .
6 , 0 .
9 and compute an approximation of T d for, say 20 values of g in the range 0 ≤ g ≤ 0 .
8 Averaging over ≤ 20 repetitions should be sufficient.
For each value of f graph ln T d versus g (same plot for all three values of f ). The form of these graphs should suggest that T d an exponential function, T d
≈ τ e ag , where τ can be approximately represented by and a depend on f . Thus match τ and a to the computed data ( g, T d
) using the least squares method (same procedure as in the computation of Liapunov exponents). Graph the data (as points) and the approximating functions in one plot for all three values of f . Note, however, that the representation by an exponential function cannot be continued to g = 1 because
T d
→ ∞ when g → 1.
(c) Now consider the case f = 0 and r = 0. In this limit fire can only spread from an initial configuration with a certain number of burning trees because there is no spontaneous ignition. Start with a single burning tree and run some simulations for different values of g . When does the forest burn down within a time range of, say t ≤ 100?
Next consider a random initial distribution of burning and healthy trees with equal probabilities (no holes). For this kind of initial condition we quantify the global spreading of the fire by introducing the desert size, n d
, as follows. Starting from any random initial condition as introduced before, there is a certain time after which the CA states don’t change any longer (no burning trees left). The number n d is the average desert size as introduced in (a), i.e. the total number of holes divided by the number of deserts (number of connected sequences of 0s), but here this quantity is evaluated only for the final state.
From a statistical point of view the average of n d over, say 20 repetitions of the experiment is of particular interest. Denote this average by N d
( g ). Compute N d
( g ) for, say 20 values of g in the range 0 ≤ g ≤ 1. As in (b) try to match the data ( g, N d
) to a function. In this case you will, however, find that an exponential representation does not work very well (the data ( g, ln N d log–log plot of the data (ln g versus ln N d
) are not on a straight line). Instead, the
) suggests a power law behavior. Thus by matching the computed data (ln g, ln N d
) (with g = 0 excluded) to a straight line, determine a power law representation N d
≈ N
0 g − α . As in (b) graph the function representation and the data (as points) in one plot. If you were careful, you should see a very good fit.
Statistical Meaning.
The matching used in (b) is just a function modeling approach to represent the data. Other model functions such as splines could be used as well, the numbers a, τ in the representation T d
≈ τ e ag don’t have any statistical meaning.
On the other hand, the exponent α found in (c) has statistical meaning: it characterizes the divergence of N d for g → 0. Formally this divergence occurs only in
2

the limit M → ∞ , but you can “see” it already for finite, large values of M . When approaching g = 0 we encounter a phenomenon called percolation . Percolation refers to the emergence of a “large connected cluster”, in our case a cluster of holes. In statistical physics a power law divergence is always reminiscent of a phase transition combined with long range correlations and self similarity structures. In the 1d case there is no true percolation transition because the divergence occurs in the limit g → 0. In Problem 2(c) you investigate percolation for the 2d fire model. There you will find a true percolation transition at a positive value of g .
2. Two Dimensional Forest Fire CA and Percolation
The 2d version of the forest fire CA introduced in Problem 1 employs a two dimensional square or rectangular lattice with absorbing boundary conditions. The lattice site states and the update rule are as in Problem 1, but the neighborhood is now the von Neumann neighborhood (four nearest neighbors).
(a) To study the effect of r , set g = f = 0 and r = 0 , 0 .
01 , 0 .
5. Use a square lattice of size M × M . If you don’t work with a look up table, i.e. the update rule is directly included in the code in which you compute the evolution of the CA, a simulation with M = 100 and time range up to t = 100 takes about 3 min. Choose three different initial conditions: a random distribution of trees and burning trees, a single burning tree embedded in a forest, and, say four single burning trees at different locations embedded in a forest (no holes in each case). As in the case of the deterministic two dimensional CAs of Project 3 observe the time evolution of the forest fire CA in a movie, describe your observations, and document them by showing a selected set of frames in your report.
(b) Use again a square lattice, set f = 0 .
1, and choose as initial condition a single forest (all lattice states 1). Run simulations for g = 0 , 0 .
2 , 0 .
8 and r = 0 , 0 .
1. Describe and document your observations as in part (a).
(c) In these experiments you investigate the spreading of the fire in more detail. Set f = r = 0 and choose a rectangular lattice of size 50 × 100 say. As initial condition assume a forest with the leftmost layer of (50) trees burning, the other trees are healthy. The questions to be addressed are: how far does the fire spread out, when does it terminate, and how many trees will be eventually burned down.
Similarly as in Problem 1(b) let’s call the time it takes until the fire terminates
(no burning trees left) the fire duration time , and denote it by t d again. Run first some simulations for different values of g and describe your observations. In your report you only need to show the final state (after t d
). Note that t and may be relatively large (usually < 200) for certain values of g .
d varies with g
Start with g = 0. In this case you see a uniform fire front propagating to the right end and the fire lasts until this end is reached (100 time steps if the lattice size is 50 × 100). The opposite case is g = 1. Here the fire terminates immediately, leaving a healthy forest except for the initial layer (you don’t need to simulate
3

this). Between these extremes there must be a transition. To see this, compare first the CA evolution for g = 0 .
5 and g = 0 .
3. Your simulation should indicate that the transition is between these values, thus narrow the gap down by comparing g = 0 .
45 and g = 0 .
35. Again your simulation should suggest that the transition is between these values. If you would continue this way (you don’t need to do this, just pick some values of g ), you would locate the transition at a value g = g c
, where g c
≈ 0 .
41. For g < g c the forest will burn down nearly completely, whereas for g > g c still many trees survive. The “critical” value g c is called the percolation threshold.
For g close to g fire terminating.
c
, t d is very large. You may need up to ≈ 200 time steps to see the
T d
( g
One can show that for an infinite lattice the ensemble average of
), diverges with a power law when g → g c
+ 0, T b t
( g ) ∼ ( g − g c d
, denoted
) − β . The exponent β is a characteristic quantity of the percolation transition. Approximate
β similarly as in Problem 1(c) by approximating T d
( g ) for, say 10 values in the range 0 .
42 ≤ g ≤ 0 .
6, and matching the data (ln( g − 0 .
41) , ln T d
) to a straight line.
Approximating T d for a fixed value of g requires again to average t d over a certain number, K , of independent simulations. Since these computations are extensive, don’t choose K too large ( K = 5 is fine).
Challenging Computation (optional).
Another quantity which is even of more interest than T d is the average diameter, ξ , of the largest desert after the fire has terminated. This quantity is analogous to the average desert size introduced in
Problem 1(c).
ξ also diverges when g → g c
, with another critical exponent ν .
Compute an approximate value of ν .
4