Simple models of pattern and process

7 Pages·1995·2.7 MB·

by  M Bedward

Simple Models ofPattern and Process
MichaelBedward
(CommunicatedbyD.Keith)
Bedward, M. Simple models ofpattern and process. Proc. Linn. Soc. N.S.W., 115: 17-23
(1995).
Patternandprocessdynamicsinplantcommunitiescanbemodelledusingcellular
automata.Thesearesimuladonmodelsthatareeasytobuildandmodify,andthatareespe-
cially useful in visualizing the processes that are being modelled. To illustrate this
approach,Idescribeacellularautomatonthatsimulatespatternandprocessinasimple
bryophytecommunitygrowingonasteeprockface.
NSWNationalParksand WildlifeService, P.O. Box1967, HurstvilleN.S.W. 2220; manuscript
received26July1994,acceptedforpublication14December1994.
KEYWORDS:Cellularautomata,communitypopulation,lichen,moss.
Come-and-go pervades everything ofwhich we have knowledge, and though great
thingsgomore slowly, theytooarebuiltupofsmallonesandmustfareasthatwhich
makes them.
-SamuelButler
Introduction
Patternandprocessbehaviour,acontinualsequenceofdisturbanceorsenescence
andregrowth (Watt, 1947), occursin manytypesofvegetation butisparticularlyeasyto
observeinbryophytecommunitiesthatgrowonsteepsurfaces.Ashton (1986) givessome
interestingexamplesofpatternandprocessinbryophytecommunitiesonrockfacesand
tree trunks. Heavymossmatscanslumpfrom asurfacecreatingnewspace thatcan then
becolonizedbyalgae,liverworts,lichensandeventuallymosses.Mossmatsmayshadeout
and kill crustose and foliose lichens byovergrowing them. Alternatively, mosses can be
parasitizedandkilledbylichens (McWhorter, 1921:Ashton, 1986) and the germination
ofmoss spores can be inhibited by chemical compounds produced by lichens (Lawrey
1977). Such dynamics can produce a mosaic ofdifferentspecies and unoccupied space
that is always shifting over time. Both the distribution and the area occupied by each
speciesinthecommunitycanvarysubstantially.
One way ofviewing this shifting mosaic is long term observation. Another is to
attemptto modeltheprocessesinvolved. Modellingcan alsohelp to testthe logicofour
current understanding of these processes and help us to develop new hypotheses.
However,thereareproblemsintryingtoconstructmathematicalmodelsofthesetypesof
communities.Manyoftheinteractionsbetweenspeciesareverylocalizedandspatialcon-
figurationisimportant.Forexample,thepatternofgrowthofamossmatdependsonthe
arrangementofsuitablespacewhilethepatternofmossslumpingfromasteepsurfaceis
influencedbyhowconnectedthemossmatsare.Itisusuallydifficulttoincorporatethese
sortoffactorsintomathematicaltreatments (Hastings, 1991;DeAngelisandRose, 1992).
Analternativetomathematicalmodellingistousecomputersimulations. Onevery
simplefamilyofsimulationmodels,cellularautomata,havebeenusedforavarietyofeco-
logical studies including fire patterns in vegetation (Green, 1983), the fire ecology of
plant species (Green, 1985; Bradstock el al., in press), host-parasite dynamics in insects
(Hassell el al, 1991) and evolutionary theory (e.g. Nowak and May, 1992). A cellular
automaton isacollection ofcellsthatinteractinsimplewaysbutwherethewholecollec-
Proc.Linn.Soc.n.s.w.,i15, 1995
18 SIMPLEMODELSOFPATTERNANDPROCESS
tion can display very complex overall behaviour (Wolfram, 1984; Phipps 1992).
Simulationsusingcellularautomataproceedindiscretetimesteps.Eachcellassumesone
ofafinite number ofstates, and its state at the nexttime step depends upon its current
state,andthestatesofitsneighbours.Theneighbourhoodofacellcantakemanyforms,
e.g. theeightimmediatelyadjacentcellsorallcellswithinaspecifiedradius,andinsome
models the size and shape ofthe neighbourhood canvary over time. The rules used to
decidethestateofeachcellateachtimestepcanbepurelydeterministicorincludesome
elementofchance (Phipps, 1992).
In this paper, I describe a cellular automaton that simulates the dynamics of a
bryophytecommunitygrowingonasteeprockface.Althoughsimple,themodeldisplays
realistic pattern andprocessbehaviourand could easilybe extendedorrefined to study
specific communities. I have implemented the model as a computer program, MOBI
(ModelofBryophyteInteractions) thatdisplaystheshiftingmosaicofspeciesandvacant
,
spaceonacomputerscreen.
Description ofthe Model
Forsimplicity, themodelcommunitypresentedhere consistsofonlytwospecies: a
mossandafruticoselichenthatgrowsuponthemossandparasitizesit.Later,Iwilldiscuss
some possibilities for including further species as well as environmental variables. The
habitat, a steep rockface, is represented bya square gridwhere each cell can be either
vacant,occupiedbymoss,oroccupiedbymossandlichen.Ateachtimestepduringasim-
ulation,cellscanchangestateaccordingtorulesthatdescribethegrowthanddispersalof
themossandthelichen,andtheslumpingofmossmatsfromtherockface.Thesizeofthe
gridandthenumberoftimestepsinasimulationcanvary.Thetemporalandspatialscale
is flexible, but here I am assuming thateach cell represents an area ofabout 1 cm2 and
eachtimestepisabout1year.
Mossgrowthanddispersal
At the beginning ofa simulation, moss is assigned to agiven numberofrandomly
selectedcells.Themosscanthenspreadintoadjacentcellstosimulatethegrowthofmoss
clumps.Thegrowthmechanismincludesastochasticelementsuchthattheprobabilityof
mossspreadingintoavacantcellis:
l)=MAX[1.0,C./
Pmoss(t+ t]
where p moss(t+l) 1S the probabilityofthevacant cell being occupied at the next
time step; Cisaconstant;and/ isthefractionofneighboursofthecellthatalreadyhave
t
moss.Theneighbourhoodconsistsoftheeightimmediatelyadjacentcells.Figure 1 shows
thepatternofgrowththatisproducedusingC=2.
Mosssporelingestablishmentwasmodelledbyassumingthatarain ofmossspores
fallsequallyonallpartsoftherockface.Ateachtimestep,mosssporelingscouldestablish
insuitablevacantcellswithaspecifiedprobabilitythatwasconstantthroughoutthesimu-
lation. Conditions on the suitability of vacant cells for sporeling establishment are
explainedfurtherbelow.
Mossslumping
MOBI simulatestheslumpingofheavymossclumpsfrom therockwall.Todo this,
eachmosscellisassignedaweight.When thecellisfirstoccupiedbymoss,orreoccupied
afterbeingvacant, aweightof1 isassigned. Theweightisthen incrementedby 1 ateach
timestepuntilapresetmaximumvalueisreached.Theslumpingofmossclumpsissimu-
latedateach timestepbytestingeach mosscelltoseeifitwillinitiateaslump.Theproba-
ProcLinn.Soc.n.S.w.,115, 1995
M.BEDWARD 19
bilityofthisforeachmosscellisequaltotheweightofthemossmultipliedbyaconstant.
Cellswhoseweightsarebelowaspecifiedminimumcannotstartaslump.When afalling
cell isfounditbecomes the focusforaslump areathatspreads to contiguous mosscells
usingthesamestochasticmechanismdescribedaboveformossgrowth.Thegrowthofthe
slumpareacontinuesuntilnomoremosscellscan bereached, oruntilaspecifiedmaxi-
mumareaisattained,whicheverhappensfirst.Thenallofthemosscellsintheslumparea
Fig.1.Simulatedmossgrowthfromtenrandomlyplacedinitialcellsina100x100gridafter15timesteps.
areconvertedtovacantcells.Anumberofclumpsofmossmayslumpfromtherockfacein
asingletimestep.
Newlyvacantcells
In some bryophyte communities, newlycreated space mustbe colonized byother
species, such as algae, before moss spores can germinate and establish (e.g. Ashton,
1986). To simulate this, the number oftime steps that must elapse before newlyvacant
cells are suitable formoss sporelings can be specified as aconstant. Thisdoes notaffect
growthintothesecellsfromadjacentmossclumps.
Lichendispersalandgrowth
The lichen can onlyoccupycells that contain moss. Clumps oflichen can growin
the same stochastic mannerdescribed formossgrowth. The initial lichen population is
createdbyrandomlyassigninglichentoaspecifiednumberofmosscellseitheratthestart
ofthesimulationoratsomelatertime.Thedispersalofpropagules,andestablishmentin
mosscells,ismodelledusingaconstantprobabilityasformossdispersal.
Interactionofthelichenandmoss
Theeffectofthelichenonthemossissimplytopreventgrowthfrominfectedmoss
cells. Ifall cells in amoss clump become infected the clump ceases to expand. Once in-
fected, amoss cell can notfree itselfoflichen. The lichen doesnotaffect the pattern of
mossslumping.Thisisaveryconservativeandsimplifiedinteraction.
Proc.Linn. Soc.n.s.w.,ii5, 1995
20 SIMPLEMODELSOFPATTERNANDPROCESS
An ExampleApplication
IusedMOBI to see howtherate andpattern ofmossmatsslumpingfrom therock
facewouldaffectthesuccessofthelichen.Ivariedmossslumpingbysettingfourdifferent
valuesforthemaximumareaofanindividualslump.Ialsovariedmossandlichendisper-
sal to see what effect thiswould have on the behaviour ofthe model. Table 1 shows the
designusedwhilethecompletelistofMOBIvariablevaluesisgivenintheappendix.
Table 1
Numberofsimulationsforeachcombinationofmossslumpanddispersalvariables
Maximummossslumparea(cells)
100 250 500 1000
Dispersal
none 20 20 20 20
mossonly 20 20 20 20
mossandlichen 20 20 20 20
Figure 2showstheaveragemossandlichenpopulationsforeachsetof20replicate
simulations. As the maximum size of a slumping moss mat increased from 100 cells
(fig. 2a-c) to500cells (fig. 2g-i) theproportionofmossinfectedbylichendecreasedas
indicated by the gap between the lines in each graph. The presence ofmoss and lichen
dispersalmadelittledifference totheresults.Where themossfellinverylargeclumpsof
up to 1000 cells (fig. 2j-l) dispersalhadagreatereffect.Withneithermossorlichen dis-
persal (fig. 2j),bothspeciessoonbecameextinct.Wherethemossandlichencouldboth
disperse (fig. 21),bothpersistedthroughoutthesimulationperiodalthoughonlyasmall
proportionofthemosswasinfectedbylichen.
200
Fig. 2.Averagepopulationsofmossandlichenforeachsetof20replicatesimulations (seetext).Ineachgraph,
the upper line is the moss population and the lower line is the lichen population. Codeswithin each graph
indicatethesimulation settings:e.g. 100ismaxslumpareaof100cellswithnodispersal, 100+moissameslump
areawithmossdispersal, 100-mo-liissameslumpareawithmossandlichendispersal.
Proc.Linn. Soc.n.S.w.,i15, 1995
M.BEDWARD 21
Themechanismsbehindtheseresultsareeasytovisualizebywatchingthemossand
lichen mosaic asitunfoldsin each simulation. Figure 3shows the communityin asingle
simulationforeachlevelofmossslumpingwithnodispersal. Ineachsimulation,50time
stepshaveelapsed.Lowlevelsofmossslumpingledtosmallfragmentsofuninfectedmoss
inaseaoflichen.Whenwatchingoneofthese simulations, theuninfectedmossseemed
tobeconstantlychasedaroundtherockfacebythelichen. Increasingthelevelofslump-
ingledto the mossmatsbecomingmore discretewhich made itharderforthe lichen to
spread vegetatively. This also increased the chance of large patches of lichen being
removedfromtherockface.Atthehighestlevelofslumpingthemossitselfriskedextinc-
tionwhentherewasnodispersal.
c d
Fig.3.Thestateofthemodelinoneexamplesimulationforeachoffourmossslumpareas:a.100cells;b.250cells;
c.500cells;d.1000cells.Greydenotesuninfectedmoss;blackdenotesmossinfectedbylichen.
Discussion
The resultsshowthatMOBI is capable ofdisplayingcomplexand realistic pattern
and process behaviour. The model could be refined and extended in many ways for
specific applications to bryophytic communities. For example, the probabilities for
growth, dispersal and moss slumping could be linked to a historical set ofrainfall data.
Theinteractionbetween the mossandthelichencouldbe tailoredtodataforparticular
species.Asimulationcouldalsoincludevascularplantspeciesthatestablishinmossmats
and accelerate slumping ofthe moss when they grow large. With these sort ofelabora-
tions,MOBIcouldbeusedasatoolforpopulationviabilityanalysis.
Alternatively, it is possible to begin with a model such as MOBI that considers
ProcLinn.Soc.n.s.w.,115, 1995
.
22 SIMPLEMODELSOFPATTERNANDPROCESS
biologicalprocessesexplicitly,andthensimplifythemodeltoamoreabstractandgeneral
form.Hasselletal. (1991) tookthisapproachwiththeirworkoninsecthost-parasitepopu-
lations. Byobtainingverysimilarresultsfrommodelsthatwerebasedondetailedmathe-
matical formulations of species interactions, and alternative models that were purely
qualitativecellularautomata,theyshowedthatitwasthegeneralpatternofspeciesdisper-
salratherthanthefinedetailofthespeciesinteractionsthatdeterminedthebehaviourof
the system. This sort ofapproach seeks to identify common patterns that underlie the
behaviourofmanydifferentkindsofbiologicalsystems (e.g. Green, 1993) Theresultsof
.
MOBIsimulationscouldsuggestusefulhypothesesforothertypesofplantcommunities
thatdisplaypatternandprocessbehaviour.
Itis easy to design and build models that are based on cellular automata, such as
MOBI,toexplorepatternandprocessbehaviour.Perhapsthemostusefulfeatureofthese
models is thatyou can watch the progress ofeach simulation and notice patterns that
would not have been obvious from a mathematical treatment. This feature also makes
cellularautomataveryusefulteachingtools.
Acknowledgements
ThankstoMurrayEllisforhelpwiththefigures.
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Appendix
VariablesthatcanbesetinMOBIandvaluesusedforthesimulationsdiscussedinthetext.
Numberofrows 100
Numberofcols 100
Simulationperiod 200timesteps
Initialmosspopulation 100
Initiallichenpopulation 100
Timetoaddinitiallichencells 4
Lichengrowthrate(relativetomoss) 1
Probabilityofmosssporelingestablishmentinavacantcell or0.001 (seetext)
Probabilityoflichensporelingestablishmentinamosscell or0.001 (seetext)
Maximummossweight 10
Probabilityofanymossslippageateachtimestep 1.0
Minimumweightformosstostartslumping 5
SMlauxmipmcuomnsatraenatoCfs(lpurmopDisniugmmDos=sC.weight) 100to1000cells(seet0e.x0t1)
Periodthatnewlyvacantcellsareunsuitableformosssporelings 2
Mosswithlichencangrow? no
Proc.Linn.Soc.n.s.w.,i15,1995