VOLUME
PH YSICAL REVI
69, NUMBER 11
Self-Organized
Physik-Department
E% LETTERS
14 SEPTEIUIBER 1992
Critical Forest-Fire Model
B. Drossel and F. Schwabl
der Technischen Universitat Miinchen, D-8046 Garching, Germany
(Received 30 June 1992)
f.
This leads to a selfA forest-fire model is introduced which contains a lightning probability
0 provided that the time scales of tree growth and burning down
organized critical state in the limit
of forest clusters are separated. %e derive scaling laws and calculate all critical exponents. The values
of the critical exponents are confirmed by computer simulations. For a two-dimensional system, we show
that the forest density in the critical state assumes its minimum possible value, i.e. , that energy dissipation is maximum.
f
PACS numbers:
05.40.+j, 05.45.+b, 05.70.Jk
Recently, Bak, Chen, and Tang [1] introduced a
forest-fire model which they assumed to show selforganized critical behavior. Their forest-fire model is a
probabilistic cellular automaton defined on a d-dimensional hypercubic lattice with L sites. In the beginning
each site is occupied by either a tree or a burning tree or
it is empty. The state of the system is parallely updated
by the following rules: (i) A burning tree becomes an
empty site. (ii) A green tree becomes a burning tree if at
least one of its nearest neighbors is burning. (iii) At an
empty site a tree grows with probability p.
If the system size is larger than the correlation length
of the fire, the model assumes a steady state with finite
fire density. Bak, Chen, and Tang found that the fire-fire
correlation function obeys a power law in two and three
dimensions. They concluded that the system is critical in
0 where the fire correlation length diverges.
the limit p
Bak coined the expression "self-organized criticality" to
describe the behavior of extended dissipative systems
which assume a critical steady state independent of the
initial state and without tuning of parameters to a special
value. The most prominent example of a self-organized
critical system is the so-called sandpile model [2].
Grassberger and Kantz [3], as well as Mossner, Drossel, and Schwabl [41, performed computer simulations of
the forest-fire model with values of p smaller than those
used by Bak, Chen, and Tang. These simulations show
that the forest-fire model is not critical but instead it becomes more and more deterministic with decreasing p
and develops regular, spiral-shaped fire fronts. The size
of these spirals, as well as the distance between them, is
of the order 1/p. The temporal fire-fire correlation function oscillates regularly with a period proportional to 1/p.
By contrast, a critical system should contain fire fronts of
all sizes up to the correlation length, and their temporal
correlation function should show a power-law spectrum of
frequencies.
The reason that there are no fire fronts smaller than
1/p is the following: Trees that are next neighbors belong
to the same forest cluster. A tree only catches fire when
one of its neighbors burns. So a small forest cluster cannot be ignited; therefore it grows until it becomes part of
a burning cluster. Since the fire burns constantly in the
steady state, a burning forest cluster must be so large
that trees grow at one end while the fire burns the other
end; i.e., the diameter of a burning forest cluster is proportional to 1/p, and consequently the size of a fire front
is also proportional to 1/p.
The model becomes critical when a mechanism is included that allows for small forest clusters to burn also.
We therefore introduce a "lightning parameter"
and a
fourth rule: (iv) A tree without a burning nearest neighbor becomes a burning tree during one time step with
In order to understand
how criticality
probability
arises in this extended forest-fire model, let us first simplify dynamics and assume that a whole forest cluster is
burned down instantaneously,
i.e., during one time step
when one of its trees is struck by lightning.
In this case
dynamics are invariant (except for a change of the time
and p are multiplied by the same factor.
scale) when
Then there is only one relevant parameter f/p in the system. Let p be the mean overall forest density in the system in the steady state. The average number of lightning
strokes in the system during t time steps is tfpL . The
average number of trees growing in the system during t
time steps is tp(1 p)L . Conseque—
ntly, the average
number of trees destroyed by a lightning stroke is
f
f.
f
s = (f/p)
'(1 —p)/p.
In order to avoid finite-size eff'ects, the number of sites L
must be chosen much larger than the largest forest cluster. For any finite value of f/p, the value of s is then independent of L. Equation (1) represents a power law
six: (f/p)
for small values of f/p if limftp Qp( l.
This is the case for d
2 as we conclude from the following consideration: If the mean forest density p were near
1 for small values of f/p in d
2 dimensions, the largest
forest cluster would contain a finite percentage of all trees
in the system, and the average number of trees burned by
in
a lightening stroke would diverge in the limit L
contradiction to (1). We therefore expect a critical point
0. The special case d =1 will be treatin the limit f/p
ed later.
Let us now return to the real forest-fire model where a
forest cluster is not burned down instantaneously but dor-
1992 The American Physical Society
~
)
~
1629
PH YSICAL REVIEW
YOt UME 69, NUMBER 11
ing a finite time interval T(s) which depends on the number of trees s in the forest cluster and will be determined
below. Given a value of f/p, the forest dynamics of the
system are essentially the same as in the case of instantaneous burning down provided that p is so small that a
forest cluster is burned down before trees grow at its
edge, i.e. , if I/p» T(s) Eq.uation (I) remains valid, and
0 with
we expect critical behavior in the limit f/p
p((T '(s). Figure I shows the system in the steady
state with f/p =0. 1.
In order to show that the system really is critical in this
limit, we calculate the scaling relations and the critical
exponents which follow from the assumption of scale invariance. We then confirm the values of the critical exThe radius R(s) of a
ponents by computer simulations.
forest cluster is the square root of the mean quadratic distance of the cluster members from their center of mass.
We define the correlation length g by (=R(s), and the
exponents v and z by gee (f/p) " and T(s) eel . The
condition that forest clusters burn down rapidly then
reads p(&(f/p)". Let N(s)ds denote the number of
clusters consisting of s trees in a system of a given size.
For brevity we introduce the moments m„= 1" s"N(s)ds
of this non-normalized distribution function. Equation
(I) implies that N(s) does not decay faster than a power
law in the limit f/p
0. Since the density of trees in the
. The
system is finite, N(s) decays at least as fast as s
cluster distribution N(s) therefore obeys a power law in
the limit f/p
0. Under the scaling transformation
b' 'f/p, the normalized cluster distribux
x/b, f/p
tion N(s)ds/mo and the function R(s) must be invariant.
This implies the following scaling laws for N(s) and
f
R (s):
'X'
cx
C'(s/s, ,„), r
—aaaqm-qra-~
—
2,
'(s, „),
C'(s/s, „)In
R (s ) ee s '~" C'(s/s,
aaa~amr
Dn
0 0
&
r
=2,
„),
—
a
mr
"Jr" --'
0
—
'a a '-aaa
OO
&
'&
»oo
a»»nno
The logarithmic correction to N(s) for r =2
guarantees a finite mean forest density in the limit f/p
0. The exponent p is the fractal dimension of a forest
cluster. Equation (2) and the definition of the correlation
length lead to the following scaling relations:
X=vp,
d=p(r —I).
(3)
Since p & d, the exponent r must obey i ~ 2, as already
stated above. The mean number of trees destroyed by a
lightning stroke is
3&r &2f
„/ln(s, „), r =2.
[Eq. (I)] s eep/f. Since
sm,. „',
s,
(4)
We already derived
there is only
one diverging length scale in a critical system, one has
s
„. Together with Eqs. (3) and (4), this leads to
the following values of the critical exponents [5]:
ixs,
r =2, p =d, v= I/d.
s,
Since r =2, the power law for
„noted after Eq. (2)
acquires a logarithmic correction factor and now reads
s,,„ee (f/p)
'
ln(s, „)ee (p/f)
In(p/f)
.
Even for d = I, the scaling laws and critical exponents
derived are valid since the factor p/(I —
p) entering Eq.
(I) diverges only logarithmically, as we conclude from
the following considerations: The number of forest clusters in the system is constant in the steady state. It increases by 1 when an isolated tree grows, and it decreases
Lp(1
by I when lightning strikes a tree. Consequently
—p) —2pmo=fLp. In addition, we have the following
two relations that are valid in any dimension: L p=m]
and (p/f)L (I —p) =mz. From these equations we fiThe scaling law for
nally obtain p/(I —p) a:ln(sm,
N(s) again contains a logarithmic correction factor
. „, however, is exact in
'(sm, „); the power law for
ln
one dimension.
The distribution n(T) of the lifetime of a fire also
obeys a po~er law: The probability that lightning strikes
a cluster of size s is proportional to sN(s) ~s '. The
time steps
cluster burns down in T(s) ee R (s) ~ s '
with a= l. The mean lifetime
which implies n(T) ee T
of a fire is proportional to T(s) eel ee(f/p) ' which
leads to the dynamical critical exponent z = l.
Knowing that the number of burning trees in a forest
cluster t time steps after a lightning stroke is proportional
to t ', the Fourier transform of the fire-fire correlation
function can be shown to obey a power law ee
In
) for small frequencies
+O(f +', . . . ,
sandpile model where the number of active sites is assumed to be constant for the duration of an avalanche,
in one and two dithe corresponding power law is ~
mensions [6].
by circles, and burning
f
f
FIG. 1. Steady state of the forest-fire model for f/p =O. l
1630
x»1.
'
Oo»&oo
and d=2. Trees are represented
by crosses.
s,
„ix (f/p) . Here, k and r are new critical exwith
ponents, and the cutofT functions C (x) and C(x) decrease monotonically from the value 1 for x(&1 to 0 for
s,
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44
LETTERS
trees
f
f
the.
VOLUME
PH YSICAL REVI EW LETTERS
69, NUMBER 11
Our computer simulations for d=1, 2, 3 confirm the
of the critical exponents calculated above. For
d =2 and f/p =1/70, the cluster size distribution and the
cluster radius as functions of cluster size are shown in
I ig. 2. The slopes of the plots yield the exponents z =2
and v= 1/2.
We also determined the mean forest density p in the
steady state. For large values of f/p, the mean forest
density is small. It increases with decreasing f/p and ap0. We
proaches a constant value in the limit f/p
determined this value numerically and obtained p=0. 39
and p=0. 21 for 2 =2 and 3, respectively. In the following, we show that the critical value p=0. 39 in two dimensions is determined by an extremum principle: p assumes
its minimum possible value, i. e. , the fire destroys as much
forest as it can. Consider an area of the forest which is
very large compared to the lattice constant but very small
compared to the correlation length. The forest in this
area will grow until a fire passes through it which has
started somewhere in the forest around it (since the area
is small compared to the mean size of a forest cluster and
f/p is near 0, the lightning nearly never strikes it directly). Let hT denote the mean time interval between two
such fire fronts. Its value depends neither on the exact
size of the area nor on the precise value of f/p. The average forest density just before the fire enters the area is
pm, „, and is p;„just after the fire has passed. If the
value of hT were different from the actual value, the
values for
„and p;„and therefore the mean forest
density p would also be different. If hT were very large,
„would be near 1, and p;„near 0. With decreasing
„would decrease, and the value of
hT, the value of
p;„would therefore increase. For very small values of
hT, the difference between p~, „and p;„would also be
„would be just above the
very small, and the value of
percolation threshold 0. 59 for site percolation while the
value of p;„would be just below it. AT and p are related
to
„and p;„by the equations
values
p,
p,
p,
p,
p,
pAT
= 1n
p=l—Pmax
Pmin
Pmax
We still need a third relation between these four parame-
14 SEPTEMBER 1992
ters in order to determine all possible values of p and
therefore its minimum. Having in mind that the tree distribution just before the fire enters the area is stochastic
since trees grow stochastically, we determined p;„as a
function of
„numerically by randomly filling a lattice
with trees up to the density p~, „ for diAerent values of
0.59, and by counting the number of trees that belong to the largest forest cluster. This forest cluster connects the edges of the lattice and would be destroyed by a
fire sweeping the lattice. From the number of trees that
do not belong to this cluster we obtained the density p;„.
Using the above relations, we plot p as a function of
„
(Fig. 3). The minimum value of p is p=0. 39, and the
corresponding value of hT is AT=0. 91/p. The minimum
value of the mean forest density is just the value observed
in the critical state of our forest-fire model. We conclude
that the critical state is organized in such a way that the
number of growing trees and of burned trees is maximum, i.e. , that energy dissipation in the system is maximum. We expect that the extremum principle holds also
in higher dimensions.
Finally, we would like to comment on the physical significance of the two conditions f/p
0 and p
«(f/p)"', leading to the critical behavior of the forestfire model. Rewriting them in the form
p,
p, „)
p,
(f/p)
/
«p
«f
we see that
they describe a double separation of time
scales: The time in which a forest cluster burns down is
much shorter than the time in which a tree grows, which
again is much shorter than the time between two lightning occurrences at the same site. Separation of time
scales is quite frequent in nature, while the tuning of parameters to a certain finite value in nature only takes
place accidentally. Thus, the forest-fire model is critical
over a wide range of parameter values, and we expect
that its critical state should also be robust with respect to
slight modifications of the model rules, e.g. , another lattice symmetry or fire spreading to next-nearest neighbors.
Let us compare the critical behavior of the forest-fire
model to the sandpile model: Here too, a separation of
time scales is required: Sand must be added slowly com-
0.60
3.0
i
0.55—
0.50—
0.5
1.0
Z'.
33
CD
V)
-1.0
-30
0.0
-0.5
I
1.0
I
2.0
lksilii
3.0
log(s)
FIG. 2. Mean number of clusters and mean cluster radius as
functions of the cluster size for f/p = l/70 and d =2.
0.45—
0.40
0.35
0.55 0.60 0.65 0.?0 0.?5 0.80
I
p
s
I
max
FIG. 3. Mean forest density as a function of maximum
density for d =2.
forest
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VOLUME
69, NUMBER 11
PH
YSICAL REV I EW LETTERS
pared to the lifetime of an avalanche. This corresponds
to our condition that a forest cluster burns down rapidly.
The power-law distribution of the avalanches in the sandpile model is a consequence of the local conservation of
sand particles. In our forest-fire model, the power-law
distribution of forest clusters is a consequence of a second
', which guarantees
separation of time scales p
that a large amount of energy is deposited in the system
a
between two lightning occurrences and consequently
large number of trees is destroyed by a lightning stroke.
We thank W. Mossner and N. Knoblauch for writing
the simulation program.
'((f
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14 SEPTEMBER 1992
[1] P. Bak, K. Chen, and C. Tang, Phys. Lett. A 147, 297
(1990).
[2] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38,
364 (1988).
[3] P. Grassberger and H. Kantz, J. Stat. Phys. 63, 685
(1991).
B. Drossel, and F. Schwabl (to be published).
[5] These critical exponents can also be derived from an invariance of the dynamic equations of motion for the forest
density.
[6] H. J. Jensen, K. Christensen, and H. Fogedby, Phys. Rev.
B 40, 7425 (1989).
[4] W. Mossner,