The Self-Organized Critical Forest-Fire Model with Trees and Bushes

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The Self-Organized Critical Forest-Fire Model with Trees and Bushes

B. Strocka, J.A.M.S. Duarte, M. Schreckenberg

To cite this version:

B. Strocka, J.A.M.S. Duarte, M. Schreckenberg. The Self-Organized Critical Forest-Fire Model with Trees and Bushes. Journal de Physique I, EDP Sciences, 1995, 5 (9), pp.1233-1238.

�10.1051/jp1:1995193�. �jpa-00247132�

Classification Physics Abstracts

05.40+j 05.45+b 05.70Jk

The Self.Organized ^Critical Forest-Fire Model with Trees and Bushes

B. Strocka

(~),

^{J-A-M-S- Duarte}

(~)

^{and M.}

Schreckenberg (~)

(~) Institut für Theoretische Physik, Universitàt _zu KôIn, 50937 KôIn, Germany (~) Departamento Fisica, ^Faculdade de Ciencias, 4000 Porto, Portugal

(~) Theoretische Physik FB 10, Universitàt Duisburg, 47048 Duisburg, Germany

(Received

2 May 1995, accepted là May

1995)

Abstract. We present _a forest-tire model with trees and bushes. Additional to the usual

nearest-neighbour fire spreading bath populations _con igmte each other at the _sonne lattice site.

Simulations in two dimensions show that the bushes with the higher growth rate remain critical while the Self.organization of trie trees is destroyed by quick bush fires which ignite the tree clusters before they can become

large.

A mean-field treatment leads to coupled equations for the densities at the stationary point, depending _on the ratio k of the growth rates.

In recent years the

phenomenon

^{of self}

organized criticality (SOC)

^has been

attracting

much interest.

Many

mortels _are believed _to show

SOC,

but the best understood _so far is the

sandpile

model introduced

by Bak, Tang

^{and Wiesenfeld}

iii.

The forest-fire model _is of

special

interest because it has _no conservation laws. In its

original

form [2] ^it had

only

_one parameter _p, the

probability

that _a tree grows ^at an empty ^{site. This model} was

thought

to show SOC in the limit _{p -} 0, ^but in simulations with rather small _p

spiral-shaped

fire fronts occured with _a

typical length

scale of order

p~~

_{[3, 4].} It turned out to be _{necessary to} introduce

a

lightning

parameter

f,

_{a very} small

probability

that _{a tree} catches fire without

having

a

burning neighbour.

This mortel shows SOC under the condition of _a

separation

of time scales for the

burning

down of

dusters,

the

growing

of _new dusters and the time between two

lightnings [si.

It is diilicult to treat the model

analytically

and ii has been solved

exactly

for

one dimension

only

_[6].

Scaling theory

leads to relations between the critical exponents, ^which

can be

compared

to the result of computer simulations

[7-9].

It is also

possible

to introduce

an immunity _g so that the

probability

that _a tree gets

ignited by

_a

burning neighbour

^becomes

smaller than 1

[10, iii. Parallel,

and _even

substantially predating

these studies, the importance of two level systems ^{bushes and} small

vegetation

_near the

ground

and the tops ^{of the} trees on the _upper level has

long

been

recognized

in forest fire _science. A _mass of

mainly empirical

evidence has accumulated

steadily

and led to _some sort of basic

regimen

classification

[12,13].

But conclusions _are hindered

by

the

complexity

of the

phenomena

themselves and the lack of

adequate

treatment of both the

lower,

surface fuels

(subject

to _a wide

variety

of moisture and

© Les Editions de Physique 1995

1234 JOURNAL DE PHYSIQUE I N°9

combustion

parameters)

and the _crown canopy fuel

separately. Despite

these difliculties _some

consensus over the unlikelihood of

surface-independent

_crown fires exists

[12-14] although

the

principal

mechanism for their sustained

propagation

varies in

emphasis

between the various researchers. Van

Wagner's alternative,

for

example, hinges

_{on a} critical surface fire fine

intensity required

for the fire to _move to the _crown level and adds

propagation

rates

suitably

calculated.

Rothermel focusses _on fire

plume dynamics

and air entrainment at the surface

front,

while Albini in _a series of _papers

[là,16]

with

strong

wind conditions

emphasizes

the need for _a fire

spread

model

coupling

both surface and _crown fire eifects. And mention should also be made of the

independent

theories

developed by

Grishin

iii] through

_a detailed heat transfer balance system ^of diiferential

equations.

It is

precisely

such _{a consensus} that

brings

a somewhat realistic slant _to the model SOC _as described above and constitutes the motivation for _a

basic,

refinable two-level system. ^This cari be

improved

_upon

by considering culling

of the surface fuel under the tree canopy area, a

procedure

of great

ecological

importance

[18],

_or

by diiferentiating

between the

ignition

conditions of both fuel levels. It is also

appropriate

to mention that the

lightning probability

_{parameter, so}

fruitfully

introduced in Drossel and

Schwabl,

is indeed to be

kept

at _a

sufliciently

small _rate of occurrence,

according

to the system ^evolved

by Fuquay

et ai.

[19].

The forest-fire model is defined _{on a square or}

simple

cubic lattice

(more general

_{on a} d- dimensional

hypercubic lattice)

and each site _can be empty, contain _a tree _{or a}

burning

tree.

In the model _we discuss another

population,

the

bushes,

live _on the _saine lattice with the

analogous options

_so each site _can have 9 diiferent states instead of three. The

update

is

parallel

^for the whole lattice

according

to the

following

rules:

1. On _a site without _a tree, a tree grows with

probability

_p. On _a site without _a

bush,

_a

bush grows with

higher probability kp.

2. A

burning

tree _or bush vanishes.

3. A tree becomes _a

burning

tree if there is a

burning

bush _at the _saine site _{or a}

burning

tree at _one of the

neighbour

sites. A bush catches fire if there is _a

burning

tree at the

saine site _{or a}

burning

bush _at its

neighbour

sites.

4. A tree is

ignited by lightning

with

probability f.

These rules without those for the bushes _are

exactly

the Drossel-Schwabl model

[si.

^Our

model has _{a new} parameter ^{k > which determines the} ratio of the

growth speed

of bushes

and _trees. If the bushes _grow faster

they

_can build

larger

clusters

which,

when

burning down,

disturb the

self-organization

of the tree clusters.

We have dore computer simulations for _{a 1000 x} 1000 square lattice with

periodic

bound- ary conditions. The ratio 8

=

pff

of tree

growth

rate and

lightning probability

is

always

8 ₌ 10000. The

update

_was done step

by

_step

according

to the rules and not with the al-

gorithm

that burns down entire clusters _once

they caught

^fire _[9]. A cluster _means here _a connected number of trees _or

bushes, regardless

of whether _some of them _are

burning

_or not.

After _a

long equilibration

time, the sizes of bush and tree dusters _were

averaged

_over 2000 time steps.

In

Figures

and 2, normalized duster _size distributions

N(s)

_are shown _{as a} function of

s.

Figure

1 is the

special

_case k

= 1, in which trees and bushes _can be considered _as the

same

population. Although

the duster statistics _were done for trees and bushes

separately,

the exponent ^{remains the} same as in the Drossel-Schwabl model

N(s)

cc s~~ with

T =

2.15(1).

That is _trot too

surpnsing

because

snapshots

of the system ^{for k} ₌ 1

(trot shown)

indicate that tire fronts and _areas of dense forest fait

together

for both

populations.

In

Figure

2 the

bushes kees O.I

O.Oi

O,coi

O.oooi

i iO ioo iooo ioooo

s

Fig. l. Normabzed cluster size distribution of bushes and trees for k

= 1.

tree duster size distribution is

plotted

for diiferent values of k

compared

to the _one of the

bushes,

which

essentially

does not

depend

_on k. The numbers of

large

tree clusters _go down

more

rapidly

for

higher

values of k.

However,

the power law for the _trees still _seems to hold for _very small cluster

sizes,

but that cannot be

proved

from the simulations. Let

p(, pi

and

p[

denote the

density

of trees,

burning

trees and sites without trees at time t and

a[, ai, a(

for the

bushes, respectively.

The

changes

of the densities

during

_one time step due to the

update

rules then read:

PÎ~~

PI

"

~pPÎ

₊

_PI (i)

P(+~

PI

=

-PI

₊

_{(2d 1)PIPI}

_{+ PI}

aI (2)

PÎ~~ PI "

~12d

i

)PIPI

+

pPÎ PÎ~Î

13)

~Î~~ ~l

"

~kp~l

₊

~l

₁₄₎

a(+~ a(

=

-a(

₊

(2d 1)aba(

₊

abPÎ (5)

a)+~ a(

=

-(2d 1)a(ai

₊

pu( a( pi. (6)

For any initial state, ^after _a transition

penod

the system ^reaches a

stationary

state where the densities _are constant. Then the left hard sides of

equations il )-(6)

ail

equal

zero. In this _case

using

_pi + pi + pe = 1 and _{ab + ai + a~}

= 1, one obtains the

following coupled equations

^for

the densities _{pi ab:}

(2d -1)p( _[2d

₊

_{kil ab)]pt}

₊₁

= 0

ii)

(2d 1)ka( _[2dk

_{+ 1}

_pt]ab

₊ k

= 0.

(8)

1236 JOURNAL DE PHYSIQUE I N°9

k=1 k=2

0 ~~Î

kW k=10

~ ~~

~',",

k=12

'

'.', ',

O.OOI ' ',",

=, , ',

à

".,',,', _..,

(

0.0001 [ ',~, _",

~ Î ', Î ', ',

., '-'~, _{'- ).,}

le"05 ( ( _{j, ';}

, -, '.

Î; ~, ~l,,~ ~, _";

le"06 _Î~£; ^' '"..

'Î ", _'Î ..,,

".,

l e"07 ',

',, "-._

le"08

10 100 1000 10000

s

Fig. 2. Cluster _size distribution of the trees for dilferent values of k, normalized by the number of

single bushes

Nb(1).

»

a

~ m

*

'

~

a) b)

Fig. 3. Snapshot of _a 100 x100 section of the system at k

= 2, left hand: bushes, nght hand: trees.

Black dots indicate _a buming, _{grey ones a} non-bummg and white _{ones an} empty ^site.

The factor

(2d -1)

comes from the fact that _a

burning

site at the _{time t was}

ignited by

_a

burning neighbour

at t -1, so this

neighbour

site _is empty at time t and the tire _can

only spread

to the

(2d

₁₎ other sites. In

Figure

3, _a

snapshot

of _a 100 _x 100 area for both bushes and trees at the _same point ^of time is

given

for k

= 2. From this

picture

_{one can} see that tire

spreads typically

in

diagonal

tire fronts _so there _are two

burning

sites

igniting

the _same

site,

and both sites will be empty at the next

timestep.

Thus in _two

dimensions,

the mean-field solution should become better if the factor

(2d -1)

is

replaced by (2d 2)

in ail equations.

By inserting

equations

(7),(8)

into each

other,

_one obtains

equations

of fourth order in pi ab.

Due to the condition

f

< p, the

lightning

parameter

f

does trot enter the mean-field

equations il )-(6).

Hence the dense forest _pi

= ab " is _a trivial solution of

ii),(8)

and _con be eliminated

in the

equations

for _pi and _ab,

leading

to cubic

equations.

The relevant solution for the factor

(2d 1)

read

~~

2 ~~

~ ~~ ~

~~~~~~

_~ ~~~~ _~ ~~~ ~°~~

~

~~

~

~~~

,~~

125k~ ₊ 1995k~ ₊ 5559k ₊ 3689

~~~

~~ ~°~

~

(25k2

+ 266k ₊

241)3/2

while for _a factor

(2d 2)

_we gel

pt =

(k

^{+ 7}

~)

,

(ii) respectively.

^The solution

(10)

_is

simpler

because the trivial solution _pt

" is two-fold

degen-

erate in the

according equation.

The

update

rules _are

symmetric

in _trees and bushes _so the

bush densities _can be calculated from equations

(9),(10)

via

ablk)

=

Ptl)). ^l12)

In

Figure

4 the solutions

(9),(10)

_are shown

together

with simulation results for d

= 2. The

choice with the factor

(2d 2)

lits the bush data

better,

while for

high

values of k the

(2d -1)

o.5

0.45 o _~

Î,, «-

~~ j

~ o ^°

Î,-~""'"

04

~

~~,,,"'

035 o,""

0.3 ~

2~

,'~~''

i ₀₂₅ <'

+

', bushes o

# "._ ~' _,,~ ^{toees +}

o ~ "..,

+

'~ ÎÎÎÎ Î

'..

+ ~,

..._ +

~

~

015 ""...

~

'"~"---_

~""..

.±.,, '~"'

'""~-"-

0.1 "".+

.,,____

"

..~

o05

0

0 2 4 6 8 10 12

k

Fig. 4. Density of bushes and trees for dilferent _values of k, together with the mean-field solutions for

(2d

₂₎

(upper

pair of

fines)

and

(2d

1).

1238 JOURNAL DE PHYSIQUE I N°9

solution is better for the trees. This is due _to the fact that for

high

values of k the tree clusters become very small

(Fig. 2),

_so the above

assumption

_on

diagonal

tire fronts does _no

longer

holà for the trees.

Interestingly enough

the mean-field solution lits the data

quite well, although

d = 2 is _a very low dimension for

using

mean-field

theory.

In summary we have introduced _a forest-fire model with two

populations,

^{which ailler}

only

in their

growth

rate.

They

_are allowed to interact

by igniting

each other. Simulations show that the dominant bushes still retain the

properties

of the Drossel-Schwabl

model,

while the

self-organization

of the trees breaks down with

increasing

k. A mean-field

theory considering

the local

tire-spreading

leads to decent

results _even in two dimensions.

References

[1] Bak P., Tang C. and Wiesenfeld K., Phys. Re~. Lent. 59

(1987)

381.

[2] ^Bak P., Chen K. and Tang C., Phys. Lent. A 147

(1990)

297.

[3] Grassberger P. and Kantz H., J. Star. Phys. 63

(1991)

685.

[4] MofIner W., Drossel B. and Schwabl F., Physica A 190

(1992)

205.

[Si Drossel B. and Schwabl F., Phys. Re~. Lent. 69

(1992)

1629.

[6] ^Drossel B., Clor S. and Schwabl F., Phys. Re~. Lent. 71

(1993)

3739.

[7] Grassberger P., J. Phys. A 26

(1993)

2081.

[8] Christensen K., Flyvberg H. and Olami Z., Phys. Re~. Lent. 71

(1993)

2737.

[9] Henley Ci., Phys. Reu. Lent.

71(1993)

2741.

[loi

Drossel B. and Schwabl F., Physica A 199

(1993)

183.

[Il]

Drossel B. and Schwabl F., Physica A 204

(1994)

212.

[12] van

Wagner

C.E., ^{Ganad. J. Forest Res.} 7

(1977)

23.

[13] ^Williams F-A-, Prog. Energy ^{Gomb. Sci. 8}

(1982)

317.

[14] ^Rothermel R., Predicting the Behavior and Size of Crown Fires _m the Northem Rocky Mountains, USDA-FS, Ogden UT

(1990).

ils]

Albini F-A-, J. Appt. Metereol. 20

(1981)

1325.

[16] ^{Albini F-A- and Stocks} B-J-, Gomb. Soi. Technol. 48

(1986)

65.

[17] ^Grishin A.M., Sou. Phys. Dokl. 29

(1984)

917.

[18] ^Andre J., ^Gameiro Lopes A. and Viegas F-X-, A Broad Synthesis of Research

on Physical Aspects

of Forest Fires 3, University of Coimbra

(1992).

[19] Fuquay M., Baughman R-G- and Latham D.J., "A model for predicting hghtning fine igmtion _m

wildland fuels", USA-FS, Ogden UT, Res. Paper INT-217

(1979).