Model Forest Fires on a Grid

Simulate a forest fire as a probabilistic cellular automaton

The forest fire model is one of the simplest computational models to display self-organized criticality. The model is a probabilistic cellular automaton with the following rules. At each computation/time step:

◼ Trees on fire burn down, leaving wasteland.

◼ Trees catch fire if they are adjacent to at least one tree already on fire.

◼ Tree cells catch fire independently with probability

P(f)

◼ Trees grow in wasteland cells with probability

P(t)

We'll use the following values for the possible cell states:

In[1]:=

Dataset[0"Nothing",1"Tree",2"Wasteland",3"Fire"]

Out[1]=

Here is a visualization of the process:

Out[1]=

Generate some forests with varied amounts and patterns of clearings.

Forest with clearings and treeless zones:

In[2]:=

SeedRandom[0];lightForest=RandomChoice[{.6,.4}{1,0},{150,200}];

Dense forest with closely packed trees and no clearings:

In[3]:=

SeedRandom[0];fullForest=ConstantArray[1,{150,200}];

Clusters of forest with clearings and treeless zones, separated by continuous borders:

In[4]:=

SeedRandom[0];clusterForest=

;

Define helper functions to retrieve the positions of trees, actively burning fires and wasteland:

In[5]:=

trees[forest_]:=Position[forest,1]fire[forest_]:=Position[forest,3]wasteland[forest_]:=Position[forest,2]

Retrieve the positions of trees adjacent to actively burning fires:

In[6]:=

dangerZone[forest_]:=Module[{dims=Length[Dimensions[forest]],adjacent},adjacent=(Join[#1,-#1]&)[Permutations[Join[Table[0,dims-1],{1}],{dims}]];trees[forest]⋂Flatten[(adjacent+ConstantArray[#1,Length[adjacent]]&)/@fire[forest],1]]

Compute a forest fire model step, in which new fires ignite, new trees grow and existing fires spread:

In[7]:=

forestFireStepforest_,pf_:

1000

,pt_:

:=Module[{newForest=forest},(*Everystep:*)(*1)Newfiresignitewithprobabilitypf*)newForest=ReplacePart[newForest,Extract[trees[newForest],Position[RandomChoice[{pf,1-pf}{True,False},Length[trees[newForest]]],True]]3];(*2)Newtreesgrowinthewastelandwithprobabilitypt*)newForest=ReplacePart[newForest,Extract[wasteland[newForest],Position[RandomChoice[{pt,1-pt}{True,False},Length[wasteland[newForest]]],True]]1];(*3)Existingfirespreadstoneighbouringtrees,andalreadyburningtreesturntowasteland*)newForest=ReplacePart[newForest,{fire[newForest]2,dangerZone[newForest]3}]]

Compute and illustrate a single step of a forest fire cellular automaton:

In[8]:=

ModuletinyForest=ArrayPad[{{3}},2,1],newForest,forestPlot=

ArrayPlot[

]

&,(*Computethestateoftheforestafter1timestep:*)newForest=forestFireStep[tinyForest];(*Illustratetheresult:*)Rule@@Thread[Labeled[forestPlot/@{tinyForest,newForest},Text/@{"Initial forest","Forest after 1 step"},Top]]

Out[8]=

Initial forest



Forest after 1 step

Compute the state of light and full forests after 100 time steps with the default parameters:

WithforestPlot=

ArrayPlot[

]

&,legend=

SwatchLegend[

]

,LegendedRow

Thread[

]

,legend&@Map[forestPlot,(Nest[forestFireStep[#]&,#,100]&/@{lightForest,fullForest})]

Out[8]=

Make an animation of a forest fire trajectory exhibiting self-organized criticality:

In[9]:=

WithforestPlot=

ArrayPlot[

]

&,legend=

SwatchLegend[

]

,SeedRandom[0];ListAnimate@MapLegended[Labeled[forestPlot[#],Text["Full Forest"],Top],legend]&,NestListforestFireStep#,

10000

100

&,fullForest,200

Out[9]=

Show the time-series and phase-space plots corresponding to this trajectory:

In[10]:=

SeedRandom[0];Module{data=NestList[forestFireStep,fullForest,100],states,counts},states=Sort[DeleteDuplicates[Flatten[data]]];counts=Map[AssociationThread[states,Table[0,Length[states]]],Counts[Flatten[#]]&,data];counts=AssociationThread[statesTranspose[Values[counts]]];RowListLinePlotValues[counts],

,ListLinePlot3DTranspose[Values[KeyDrop[counts,0]]],AxesLabel{"trees","wasteland","active fire"},



Out[10]=

Animate a trajectory in which a fire ignites at the start of the simulation, but none ignite over its duration, and no new trees grow:

In[11]:=

WithforestPlot=

ArrayPlot[

]

&,legend=

SwatchLegend[

]

,SeedRandom[0];ListAnimate@Map[Legended[Labeled[forestPlot[#],Text["Light Forest"],Top],legend]&,FixedPointList[forestFireStep[#,0,0]&,ReplacePart[lightForest,{75,110}3],200]]

Out[11]=

Simulate a forest fire spreading in landscape containing isolated clusters of forest:

In[12]:=

ListAnimate

ArrayPlot[

]

&/@FixedPointListforestFireStep[#,0,0]&,



Out[12]=

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Contributed by: Phileas Dazeley-Gaist

Model Forest Fires on a Grid

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