Journalof Ecology1994, 82, 217-226

The influences oftreebiologyand firein thespatial oftheWestAfricansavannah structure MICHAEL JACQUES

E. HOCHBERG, GIGNOUX

JEAN CLAUDE

MENAUT

and

Ecole NormaleSuperieure, Laboratoired'Ecologie,CNRS - URA 258, 46, rued'Ulm,75230 Paris Cedex 05, France

Summary 1 Using a spatiallyexplicitcellularautomatonmodel we explorethe effects of tree demography,fire-induced mortality,and seed dispersalon the spatial spread of a singletreespeciesin a humidsavanna at Lamto in WestAfrica. 2 The modelsystemis describedbysixparametersand consistsofa grass-surrounded square gridof connectingcells,each beingeitherinhabitedbygrassalone or bygrass and an individualtree.In thebaselinenumericalsimulationsthetreecan onlyrecruit seedlingsin immediately adjacentcells.These seedlingsmayperishfromannual grass firesin theirfirstyearof lifeif theyare not protectedfromthe advancementof the fireby neighbouring reproductively maturetrees. 3 Based on preliminaryparameterestimatesfromdata collected at fieldsites at Lamto, we predictthatfireslows,but does not stop, the spread of the tree.In the absence of firethedoublingrateof thetreepopulationis about 6 years,whereaswe predictthatyearlyfiresprolongthisto at least 30 years. 4 The temporaldynamicsof thetreepopulationare fairlysmoothand predictableas long as thereare more than c. 100 cells in the system.As the numberof cells is decreasedbelow c. 100 the trajectoriesbecome increasinglyvariable fromyear to year. 5 Mortalitiesfromfireact in an inversespatiallydensity-dependent fashion,enhancing treeaggregation.The role of firein enhancingtreeaggregationis supportedby additionalsimulationsin whichdispersalof seeds to non-adjacentcells can occur. When a small amountof dispersalis possible the rate of treepopulationgrowthis greatlyacceleratedas comparedto whenno suchdispersaloccurs. 6 We presentseveralhypothesesto explainwhythe savanna at Lamto is not treedominatedas would be predictedby themodel,discusshow seed dispersaland fire influencetreedynamics,and make predictionsforfuturetesting. Keywords: cellularautomaton,dispersal,fire,populationdynamics,savannah,treegrassequilibrium

JournalofEcology(1994) 82,217-226

Introduction

217

The patchyoccurrenceof treesin what is otherwise an immensegrasslandcharacterizesthe large-scale vegetativestructureof the humidWest Africansavannas (Menaut 1983).At a 2500-hasiteat theLamto Reserve (5?02'W, 6?13'N; Ivory Coast) burning occursonce yearlyin thedryseason as a management practiceto decreasethegrasscoverforbetterhunting, to reducethe riskof accidentalfires,and to initiate grassregrowth fordomesticherbivores. Althoughthe

fireburnsmostor all of theabove groundbiomassof grass species,theirlarge undergroundroot systems enablethemto surviveeventhemostintensefiresand rapidlyto establishnewshootsbeforetheonsetofthe rainyseason. In contrastto grasses,individualsof woodyspecieswhichare lessthanc. 2min heightmay eithersuccumbto fireor have theirgrowthretarded. Mature treesand shrubsbeyondc. 2m are morefire resistantand only experiencedie-back (Menaut & Cesar 1979;Gillon 1983). Amongst the determinantscontrollingtree-grass

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218 Treebiology, fire and spatial population dynamics

coexistence(i.e. soil moisture,soil nutrients,herand firewouldappear bivoryand fire),onlyherbivory to variationin regimeand intensity to have sufficient induce rapid changes in vegetationstructureat the patchscale (Frost etal. 1986).The role of herbivory, alone or in interactionwithfire,has been givenconsiderableattentionin Eastern and SouthernAfrica 1979; Pellew 1983; Trollope 1984), (Norton-Griffiths but cannot be an importantfactorat Lamto where grazingand browsinganimals(domesticor wild) are absent.It is generallyagreedthatfire,per se, favours and maintenanceofa predominantly thedevelopment grasslandvegetation,sinceyoungtreeindividualsare likely to perish from firewhen surroundedby a large fuel-loadof dry grass. Tree-domisufficiently nated patchesare, however,maintainedsince reprowoody individuals ductivelymature, fire-resistant depress the growthof grass and hence reduce the effects of fireto youngertreeindividuals detrimental (Trollope 1984; Frost & Robertson 1987 and referencesin both). Very few simulationmodels have addressed the balance between grasses and trees in savannahs. Those that have, focused on soil water availability and differential absorptionby grassand treesystems eitheralone (Eagleson 1988) or in interactionwith herbivory(Walkeretal. 1981), and fewstudieshave (1979) modelled incorporatedfire.Norton-Griffiths of treepopulationsto the responseof size-structure fireand herbivoryin East Africansavannahs, but lacked data for the estimationof severalimportant injuveniletrees, suchas naturalmortality parameters, and and seedling establishment competitionwith simulated treedensityand only grasses.Thesemodels of account for the did not patchiness thevegetation. In the Lamto savannah, tree distributionsare of soil conditions,and highlyaggregatedirrespective the spread of firedepends on vegetationpatterning (Menaut etal. 1990). In the presentwork,we hypothesize that both the tree-grassbalance and vegetation patchinessare determinedby interactions betweenplantbiologicalattributes (e.g. treedemography,seed dispersal)and fire.We develop a simple, spatially explicit, cellular automaton simulation modelto explorethespatialaspectsoftreepopulation dynamicsand themajor mechanismswhichmay act to maintaina tree-grassequilibrium.We proposeseveral testablescenariosof thesavannahtreedynamics at Lamto.

The model In principle,the model developed here could apply boundary. to themovementsof the savannah-forest However,studyingthemechanismsthatunderliethe encroachmentof the rain foreston the savannah model structureand parwould requirea different ameterestimatesfromthosepresentedhere.Thus,we

onlyconsidervariationsin savannahtreedensityand althoughthesecould appear, in spatial distribution, thelongtermand undercertainciicumstances,as the firststagesofrainforestsuccessionintothesavannah.

BASIC

FRAMEWORK

The various modellingapproaches to plant communityecologyhave recentlybeen reviewedby Czacomplicated ran& Bartha(1992). We use a minimally thebasic processesacting algorithmso as to highlight in a singletreepoputo determinespatialpatterning lation.Our approach correspondscloselyto thecategoryof 'cellularautomaton'models,wherethehabitat is broken-upinto a grid of spatiallydistinct, interacting'cells'. These cells can be eitheruninhabitedor inhabitedby a singledynamicentity,such as an individualplantor a groupof plants. approach to modellingthe We take a reductionist systemby consideringonlya singletreepopulation. how demographymayinfluThis is done to highlight ence tree population dynamicsand to make preoffundamental processes dictionsabout theinfluence -canthenbe on thetreepopulation.These predictions and comparedto futuretheoretical testedempirically communities and grasspopustudieson multi-species lationdynamics. The systemis modelledas a n x n gridof square cells,the whole gridbeingsurroundedby grassland. The equations governingthe changes in cell occupation by treesare iteratedonce per calendar year, and at the beginningof a given iterationeach cell can be in one of threestates with respectto tree occupation: (i) unoccupied,(ii) occupied by reproductivelyimmaturetrees,or (iii) occupiedby mature trees. Demographiceventscan occur eitherpurely withina cell (i.e. nonfireinduced mortality),or in adjacentcells interactionwiththe eightimmediately or at longer (i.e. mortalityfromfire,recruitment), ofdispersedseeds).The size distances(i.e. recruitment of a givencell is 1m x 1m, whichis approximately thatpermitting thegrowthof,at most,a singlemaximally reproductiveindividual (P. Mordelet, pers. comm.). Intraspecificcompetitionis not explicitly consideredin the model; whencells are occupied by matureindividual,the morethanone reproductively largestis assumedto rapidlydisplaceitscompetitors (Antonovics& Fowler 1985). We assume that treetreecompetitionbetweencells is of the 'pre-emptive' type(sensu Schoener 1983), such that the firstindividual arrivingin a cell cannot be directlydisplaced by another.What littleis knownabout thedynamics of grassesat Lamto is thatinterspecific competition is highlyasymmetrical in favourof woodyspecies.In occurvia fire,suchthat effects ourmodel,interspecific the(constant)grasspopulationhas a negativeimpact on treeseedlings.Thus,theabsenceoftreesimplythe presenceof (fireconducting)grasses.

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219 M.E. Hochberg, J.C. Menaut& J. Gignoux

DESCRIPTION

OF PROCESSES

AND

PARAMETERS

The basic numericalalgorithmfunctionsby first apportioning maturetreesbetweencells.At thebeginning of year 1, thereis a probabilityInitial that a given cell will contain a newlymaturetree of age RepAge.Afterthisinitialgridofmaturetreesis established,the model is iteratedonce per year untilthe end of thesimulation.Withina givenyeara number of eventsmay occur: We takea firststeptowardsmodellingtheeffects of firebyconsidering onlymortalities inyoungseedlings. Fire is assumedto burnonce, at thebeginningof the year, and over the whole of the savannah. At the beginningof any givenyear t, seedlingsrecruitedin with yeart- 1 are susceptibleto fire-induced mortality probabilityBurn.Burnis a functionof thenumberof theseedling,such that maturetreessurrounding Burn =

1IS

1Np/BT

(1)

whereNpis thenumberof mature('protecting')trees in cells surroundingthe fire-prone plant (O < Np < 8), S is thesurvivaloftheplantfromfirewhenNp = 0, and BT is thenumberof protectingtreesat or above whichthejuvenilealwayssurvivesfire.IfBT> 8,then, even ifall eightcells surrounding a givenjuvenileare occupiedby maturetrees,thereis alwaysa riskof it perishingfromfire.If a seedlingdies fromfire,then thecell it occupiedbecomesopen to newrecruitment in yeart + 1. In anygivenemptycell thatis adjacentto a mature treethereis a probabilityColonizethata seed from that treegerminatesand goes on to surviveat least RepAgeyearsin theabsenceoffire.Ifan adjacentcell is alreadyoccupiedbya seedlingor a maturetreethen cannot occur in that cell. (We assume recruitment thatseeds germinateimmediately upon fallingto the ground).Colonizeis givenby Colonize = (1 - y)Recruit,

(2)

wherey is the probabilitythatthisseed is dispersed and 1 - y is therefore the fractionof seeds available in cells adjacent to the mothertree, forrecruitment and Recruitis the(constant)percell recruitment rate. Recruitment rateas usedhererefersto theprobability thata reproductively maturetreewillproducea seedlingin an adjacentemptycell and thatthatseedling willgo on to surviveat leastto theage ofreproductive maturity. seeds can be dispersed by mechAlternatively, anisms other than those which typicallyresult in in neighbouring cells such thatDisperse, recruitment in any unoccupiedcell theprobabilityof recruitment (once local recruitment via Colonizehas occurred),is a functionof maturetreedensityNm,

Disperse=

( 1 -Q)N",

(3)

whereQ is the probabilitythat a given cell is not

colonizedbya seedfromanygivenmature potentially tree [Q = 1 - (yRecruit/n2)]. Note that a particular mothertreeis not specifiedin Disperse,implyingthat a givenmaturetreecan producemore than one dispersedcolonisingseed per year. Mortalitiesin maturetreesoccurat theend of any givenyearwitha constantprobabilityDieOld thata maturetreedies (fromfactorsactingindependentof density).Otherwise,thematuretreeages one year. Finally,if a juvenile tree survivesRepAge years, thenit becomesan adult at age RepAge+ 1.

BASELINE

ESTIMATES

OF PARAMETER

CONSTANTS

There is littlequantitativeinformationthat can be used accuratelyto estimatethemodelparameters.To make firstapproximations,we employfieldstudies, not originallyconceivedforthepurposeofparameter estimation.We stressthat furtherexperimentsare neededto increasetheprecisionof theestimates.One of theobjectivesof thesensitivity analysis(see below) is to explorehow errorsin parameterestimationcan affectthemodel's behaviour. Four species account for c. 90% of the treesat Lamto. They are BrideliaferrugineaBenth., CrossopteryxfebrifugaBenth.,CussoniabarteriSeeman, and PiliostigmathoningiiMilne-Redhead (Menaut and Cesar 1979). What informationthereis on the populationbiologyof thesespeciesshows that(i) B. C. febrifuga and P. thoningii have thesame ferruginea, typesof growthcurve(Gignoux 1988), (ii) B. ferruginea, C. barteriand P. thoningii have highlyaggregatedspatialpatterns,whereasC. febrifugais almost random(Menaut & Cesar 1979; Guiblin 1992), and (iii) treeaggregationsare multi-specific and withno significantassociation between species (Guiblin 1992). Given thecoarse similarities betweenthesespecies, we do not distinguish betweenthemin makingmost of theestimates.The parametersare assignedthefollowingbaselinevalues (Table 1): n. We take n = 50, correspondingto a total system area of 0.25 ha. thereis on the age RepAge. What littleinformation ofthefourdominantspecies ofreproductive maturity suggestsa value on theorderof 5-10 years,although substantialvariationis likelyto existwithina given population. We employ the value of 5 years as a baselineestimate,meaningthata treebeginsto reproduce in itssixthyearof life. DieOld. The probabilitythata maturetreedies in a givenyearis estimatedfromyearto year recordsof individualsgreaterthan2 m in height(trees> 2 m are assumed to be mature) of the four dominanttree species fromtwo unburned0.25-ha sites (Gignoux 1988).This givesan estimateofDie Old = 0.015,indi-

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220 Treebiology, fire and spatial population dynamics

Table 1 Baselinevalues formodelparameters Processor parameter

Symbol

Value

Numberof cellsin system Initialprobabilityof occupationby maturetree Age of treeat firstreproduction Probabilityof mortality of maturetrees Fractionof potentialseedlingrecruitment dispersed Recruitment probabilityforadjacentcell Survivalof immaturetreesafterfire Numberof encircling maturetreesat or beyondwhichfiremortality is 0%

n2 Initial RepAge DieOld

2500 0.1 5 years 0.015 0 0.04 0.141 8

cating a 50% chance that a newlymaturetreewill survive46 years. y. We have no data on spatialaspectsofrecruitment, and thereforeinitiallyassume that all recruitment occurs adjacent to mother trees (y = 0 and Disperse = 0).

Recruit.We haveno directestimateoftheprobability thata maturetreewillgiveriseto matureprogenyin adjacent cells. We therefore estimatethisparameter indirectly by notingthat at a 0.25-ha site protected fromfirefrom1967 onwards,numbersof treesof all fourspecies greaterthan 2m in heightincreasedby 79% between1970 and 1975 (from132 to 236 trees, Menaut 1977). Note that this estimateshould be dependenton treedensityand age-structure, and the species compositionof the community.We iterated the model in the absence of firestartingwith 132 randomlydistributedmature trees and, employing thebaselinevaluesfortheotherparameters, adjusted Recruituntilthetreepopulationgrewby c. 80% over a 5 yearperiod.This resultedin Recruit= 0.04. S. The probabilityof a seedling(1 year old or less) survivingfromfireis estimatedfromdata collected on Bridelia.ferruginea fromfour0.25-hasites.In 1991 a totalof 239 out of 302 newseedlings(79.1%) and 5 out of 53 (9.4%) treeslessthan2 m tallperishedfrom fire.To keepthemodelas simpleas possiblewe group thesemortalitiesinto a singleeventoccurringin the first year following recruitment,giving 85.9% or S = 0.141. Simulationsof fire-induced mortality, mortalities distributed overeach ofthefirst5 yearsof a tree'slifedid not differappreciablyfromthe simplifiedmodel. BT. We are not able to estimateBT fromdata or

observations,and assume that when the seedlingis completelysurroundedby treesit is fullyprotected fromburning(i.e. BT = 8).

Recruit S BT

thenumberof new seedlingsproducedfroma single newseedlingin thesystem(i.e. in theabsenceofinitial competitors)over the life-timeof the latter.In the contextof our study,it is a contributing measureof the colonizationpotentialof unoccupiedgrasslands, and themaximumrateofspreadofthetreepopulation throughthegrasslandonce established.When Ro < 1 thetreepopulationdoes notpersistindefinitely. Based on 1000 simulations,we estimateRo = 3.17 (SD = 10.2). The large standarddeviationis indicativeof thefactthatin 85.7% of the simulations,the seedlingperishedfromfireand, in simulationswhere it survived,producedanywherefrom1 to 77 offspring duringits maturelife(Fig. la). EstimatingRo from

(a)

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E

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240

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18%

27%

36%

45%

InitialCell Occupation

Results BASIC

y

REPRODUCTIVE

RATE

AND

DOUBLING

TIME

An indexof centralimportancein characterising tree demographyis the basic reproductiverate,Ro. Ro is

Fig. 1 (a) Frequencydistributionof the 143 out of 1000 simulationsresultingin basic reproductive ratesof the tree populationgreaterthanzero. (b) Effectof initialcell occupancy by maturetreesand initialmaturetreedistribution (random,or aggregatedin the centreof the grid) on the doublingtimeof the maturetreepopulation.Baselineparametervalues are givenin Table 1.

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221 M.E. Hochberg, J.C. Menaut& J. Gignoux

fieldstudiesis therefore moreefficiently done byshifting the reference point of thecalculation,and countingthe numberof newlymaturetreesproducedby an initial,newlymaturetree. We predictthat,in thepresenceof yearlyfires,the doublingtimeof an initiallyrandomlydispersedtree populationis c. 30-40 years(Fig. lb). This estimation holds trueuntilapproximately 40% of the systemis initiallyoccupiedbytrees,beyondwhich(due to limits in thenumberofcellsin thesystem)thedoublingtime rapidlyapproaches infinity. When maturetreesare initiallyaggregatedin the centreof the system,the doublingtimeis substantially increased,therebeing a roughlylinearrelationship betweenitand initialcell occupancybymaturetreesat occupancyvaluesup to about 40% (Fig. Ib).

0.8 0 -0-0 C.) 0.2 0.6 C)

0.2

0 04 -0

0

80

Since our model is based on transitionprobabilities, the trajectoriesfrom one simulation to another foranygivensetofinitialconditions invariablydiffer and parametervalues. In general,these differences were veryminor,and we presentsingle,arbitrarily chosen, examples to expressthe typicaltrajectories observed. We note cases where simulationswere highlyvariablefromone to another. In accordance with the doubling time of 30-40 years,we foundthatthe treepopulationreachesan equilibriumof c. 90% cell occupancybymaturetrees in 1200 yearsfollowingthe introductionof a single maturetreeintothesystem(Fig. 2a). It can be shown thatan analyticalapproximationof the equilibrium N*, based solelyon thesurvivalof maturetrees,is -- 1

1(I-Die)RePAge

0010

0

DYNAMICS

N*

(a)

1

500

1000

1500

500

1000

1500

(b)

60

cm) c

40

-

a) 20

0 _

0

4

(4)

(C) C

resultingin N* = 0.927 forthe baselineparameters. Equation 4 becomesincreasingly accurateas NT -O

0, DieOld

-*

0, and RepAge -* 0.

2

The trajectoryof the mean age of maturetreesis somewhatoscillatoryfor the first500-1000 years, until the population reaches its equilibrium,after whichthemean age equilibratesto approximately 70 years(i.e. 1/DieOld)(Fig. 2b). Note thatthisdiffers fromthe medianage of the trees,whichis predicted to be 51 years(cf.estimationofbDieOld). To investigate theleveloftreeclumping,we employ two indices that calculate the average numberof 'unique couplings'per maturetree(numberof trees adjacentto one another,witha givencouplingcounted only once). For a saturatedsystemof n2mature trees,thereare on average 4 couplingsper tree as n oo x. (For n = 50 themaximumaverage number of couplingsis c. 3.9.) The index C is the number of couplingsper maturetree,or

x

X = ~~ Trees'

0)

0

500

1000

1500

Years Fig.2 Example of model dynamicsusingparametervalues presentedin Table 1. (a) Fractionofcellsoccupiedbyreproductivelymaturetrees. (b) Mean age in years of mature treesin system.(c) Aggregationofmaturetreesaccordingto indicesC and C. The simulationbeganwitha singlemature tree in cell 25,25. See text for definitionsof aggregation indices.

(5)

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222 Treebiology, fire and spatial population dynamics

whereX is the total numberof (adjacent) treecouplings,and Treesis thetotalnumberof occupiedcells. C increaseswith the population of maturetrees, reachingand stabilisingat a maximumvalue ofabout 2.7 mature neighboursper mature tree about 150 yearsaftertheintroduction of a singlematuretreein thesystem(Fig. 2c). C does not reachthetheoretical maximumof 3.9 due to the presenceof emptycells and of cellscontainingjuveniletrees. The index C discountsC by the expectednumber of couplingsifthetreesweredistributed at random, C=C-f

Trees-I

(6)

2

wheref is theaverageexpectedmaximumnumberof couplingspercell,or

f = [4(n- 2)2 + lO(n- 2) + 6]/n2

(7)

such thatf-* 4 as n -* oo, and in our 50 x 50 system (for example)f = 3.9.

The trajectories of C and C are thesame forabout 500 years,the formerdescendingthereafter towards 0 as the systemfillswithmaturetreesand the distributionapproachesrandomness.The highestlevels ofclumping(above thatexpectedbychance)therefore last forabout 300-400 yearsfollowinginitialcolonization. Figure3 shows a typicalexample of the spatial dynamicsof the systemfor the baseline parameter values and initial conditionspresentedin Table 1. Aggregationsof mature trees form around those maturetreesinitially inthesystem(a,b), and gradually coalesce (c), thetreepopulationcontinuingthereafter to spreadto theedgesof thesystem(d). It can further be seenfromtheexamplein Fig. 3 thatrelatively few of the juvenile treespresentin any given year will surviveto maturity. This is, to a considerableextent, due to fire-induced mortalities.

SENSITIVITY

The aim of thesensitivity analysisis to evaluatehow changes in initialconditionsand parametervalues affectthe dynamicsof the system.Here, we focus on the indices of tree population level (Nm/ln2)and clumping(C) forseveralof the more revealingparametersand processes. As expected,increasingthe initialtreepopulation resultsinthemorerapidattainment oftheequilibrium situation(Fig. 4a). The initial tree population can have an importantimpacton treeclumping,simply because the systemis more homogeneousfromthe start;aggregationattainsgreaterlevelsifinitialnumbersoftreesare smalland patchesareallowedto form (Fig. 4b). Tree population growthis initiallyfasterif the initialdistribution of treesis moredispersed(Fig. 4c). This is somewhatcounter-intuitive sincejuveniletrees

are more protectedwhennear large aggregationsof trees.However,thisprotectioneffectis outweighed by the much higherlevel of competitionforrecruitment sites in aggregatedsystems.Due to the probabilisticnature of the system,an initiallyhighly aggregatedpatch of treesbecomes increasingly tattered at the edges with time, thus decreasing C (Fig. 4d). The smallerthesystemsize,themoreproneit is to bothtemporalfluctuations and stochasticextinctions of trees,and the more idiosyncraticare the simulations.Beyonda systemof approximately100 cells (n = 10) the population dynamicsare highlypredictable(Fig. 4e,f). What happens if our estimateof BT is in error? Fig. 5a,b showsthatforsmallchangesin BT thismattersrelatively little;decreasingBT acceleratestherate ofpopulationgrowth(because fewermaturetreesare needed to offerprotectionto the seedlings)and the associated spatial clumpingpeaks quickly at low levels.As BT increases,aggregationpeaks higherand later(Fig. 5b) up to BT = 4, afterwhichthemaximum fallsalthoughthetimingofthepeak remainsconstant (60-70 yearsfromthestartof thesimulation). If we allow some seedlingsto be dispersedinto non-adjacentcellswe see thattreepopulationgrowth acceleratesvery rapidly,such that when 0.4% of potentialseedlingsare so dispersed,it takesless than 80 yearsfornear-complete occupationof the system froman initiallevel of 10% (Fig. 5c). We see that evenverysmallincreasesin dispersallead to therapid attainment of spatialhomogeneity and smalleraggregationpeaks (Fig. 5d) suggesting thatdispersaldiminishesspatialclumping. FIRE

INDUCED

EQUILIBRIUM?

Numericalsimulations(notpresentedhere)showthat firedoes not change the equilibriumlevel of trees, unless the value of S is less than about 0.02, or the mortalityof unprotectedseedlingsis greaterthan 98%, inwhichcase thetreepopulationalwayseventually goes extinct(correspondingto Ro < 1). This thresholdvalue of S would increaseif our estimate of BT weretoo high. ifRo < 1 is indeedtheonlycondition Furthermore, preventingthe eventual dominance of trees in the savannah,thenour estimationof Ro = 3.17 suggests thattreeswill eventuallygo extinctif the yearlyper cell recruitmentdrops below c. 1.26% (i.e. Recruit< 0.04/3.17= 0.0126). Numerical simulationsgivea value of c. 0.012, confirming theaccuracyof thisestimatebased on Ro.

Discussion Giventhatfireis insufficient to stopthenear-complete dominationby treesof the West Africansavannah, how can we explainareas inhabitedby grassesalone?

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223 M.E. Hochberg, J.C. Menaut& J. Gignoux

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Fig. 3 Developmentof thespatialdistributions of mature(solid squares) and juvenile(open squares)treesand corresponding 0. 0 0.39 10.2 O.O5.0 Year.1 mean age of(a)mature trees,fractionof cells0 occupiedby maturetrees,and aggregation indicesC and C. Parametervalues are 1*-* 0----53**23-018 20 Year *b indices. n fromTable 1,Yea except for See text for definitions of aggregation =25.28. 2 .15 0 ""905"0 "" 0 41." 50 (c

Year of simulation Mean age(years)

Cell occupation(%) C

C

(a) Year 1 (b) Year 2O (c) Year 50 (d) Year 100

10.2 23.5 41.0 65.8

0.011 0.532 0.905 0.628

5.0 18.3 28.0 38.3

One possibilityis thatthe basic reproductiverate of treesis less thanunityin certainareas and/orover thepersistenceof grasses certainperiods,permitting only. A mechanismwhichcould give rise to this is physicalor nutritiveconditionsfor tree insufficient growthand reproduction.Our resultsshow thattree dynamicsin patchesbelow about 100 cellsin size are less predictableand more prone to extinctionthan are largerareas (e.g. Fig. 4e,f).Systemscomposedof many small, internallyoscillatory,tree patches are indicativeof seed dispersaland/orseed dormancyas mechanismsresponsiblefor the persistenceof the whole treepopulation. The occurrenceof grassesover large areas could

0.39 1.14 2.15 2.76

also be explained by density-or age-dependent reductionsin the competitiveability,reproductive to offset outputor survivalof treesthatare sufficient inverse densitydependentmortalitiesdue to fire. Examples include the action of firein older stands (Sanfordet al. 1982),competitionforlimitingnutrients (Frost etal. 1986), or the impactof herbivores 1979). In previous and diseases (Norton-Griffiths modellingstudies,Prado (1988) and Menaut et al. (1990) foundthat,althoughfiredid notstoptheeventual dominationof thesavannahby treespecies,very strongcompetitiveinteractionscould do so. Below, we discuss brieflythe roles of seed dispersal and our model tree population and firein structuring

This content downloaded on Sat, 9 Mar 2013 12:52:33 PM All use subject to JSTOR Terms and Conditions

224 Treebiology, fire and spatial population dynamics

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parameterchangesand initialconditions.(a) and (b) Effectof initialfractionof cells occupied by maturetrees(Initial). (c) in a 16 x 16 all treesare initiallydistributed and (d) Effectof initialdistribution of maturetrees(foraggregateddistribution, cell block in thecentreof the grid).(e) and (f) Effectof systemsize (value of n). Unless otherwisespecifiedInitial =0.1I and See textforfurther treesare randomlydistributed. explanationsof parametersand indices.

present several testable predictionsbased on the results. THE

ROLE

OF SEED

DISPERSAL

IN TREE

DYNAMICS

We foundthat spatial spread of treesis more rapid in systemswhereseeds are dispersed,as opposed to

systemsin whichno dispersaloccurs.This is, at first sight,an unexpectedresultbecause seed dispersalin our modelentailsa cost to potentiallocal recruitment and dispersedseedlingshavea greaterchanceofburning than locally recruitedones. However,when tree low,dispersedseedlingshave an densityis sufficiently advantageovernondispersedones in thattheformer compeexperiencemuch lowerlevelsof pre-emptive

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1.2 - (b)

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225 M.E. Hochberg, J.C. Menaut& J. Gignoux

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of temporalchangesin thefractionof cellsoccupiedby maturetrees(Nm/fl2n) and theaggregationindex(C); Fig.5 Sensitivity (a) and (b) to changesin burnthreshold(BT), and (c) and (d) to changesin thefractionofseedlingsdispersed(y). Initial =0.1I and treesare randomlydistributed.Note thatin caption (b), C drops slightlybelow zero forBT1 0. See textforfurther explanationsof parametersand indices.

tition.Further,as maturetreedensityincreasesso too does theprobabilitythata dispersedseedlingwillfind itselfnear fire-protecting trees.The net resultis that even low levelsof seed dispersalgreatlyenhancetree spread. FIRE

MAINTAINS

PATCHINESS

In our model,firemaintainsclumpsof treesby preferentiallyburningthoseyoungplantswhichare isolated fromtheclumps.Protectionof a seedlingfrom oftrees annualfiresrequiresthatitbe in theproximity (i.e. seedlingmortality is inversely densitydependent) and thatthesetreesbe sufficiently largeto have competitivelydisplaced grass 'and thus reducedthe fuel to seedlings. available to cause firemortalities A previousstudybyGreen(1989) showedhow fire, of plants could seed dispersaland the distributions profoundlyaffectpopulationdynamics,spatial patA majordifference structure. terningand community betweenour model and Green's is in the dynamics of fire.Green assumed that firesigniteat random locationsin the habitat,witha Poisson distribution in frequency in and negativeexponentialdistribution

the area burnt.In his systemeach plant specieshas itsown levelof susceptibility to burningand post-fire timeintervalduringwhichgrowthand reproduction beginto occur. Fire contributesto thepatchinessof in areas thesystembyburningtheinferior competitor wheretwocompetingspeciesoverlap.Our mechanism and thatofGreenare twodifferent waysofpromoting patchinessthroughtheactionof fire. MODEL

PREDICTIONS

The simple modelling frameworkdeveloped here makes severaltestablepredictionsabout treepopulationdynamicsat Lamto. First, although clumpingis an inevitableconsequence of our assumptionthatmosttreerecruitment occurs in cells adjacent to the mother tree, fire reinforcesthe clumpeddistributionof treesand the 'smoothness'oftheseclumpsbyselectingout isolated and semi-isolated seedlings.We therefore predictthat, evenwhenno seed dispersaloccurs,highermaximum levelsof the aggregationindex C (eqn 6) should be attained in fire-treated plots than in fire-protected ones.

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226 Treebiology,fire and spatial population dynamics

Secondly,although it is not surprisingthat fire shouldslow down therateof spreadof some treesin humidsavannahs,itis notobviouson whattimescale thisshould occur. We predictthatin the absence of seed dispersal,fireshouldincreasethedoublingtime of savannah treesat Lamto from6 yearsto at least 30 years,and thatratherthan takingc. 350 yearsto attainan equilibriumleveloftreesina 0.25-hasystem, annual fireswillprolongthisto about 1200 years.A simplecalculation indicatesthat,in the absence of dispersal,treepatchesadvance at an averagerate of c. 1m every50 years.Of course,thislatterprediction willbe highlysensitiveto thescale ofthesystem,such as ours assumingat mostone maturetreeper square metre. Thirdly,our findingthatfireeitherhas no effect on theequilibriumtreedensityor drivestreesout of the systemis a consequenceof fireactingin an inversely fashionover all treedensities.We density-dependent predictthat at Lamto, mortalitiesfromfireshould have no effecton theeventualequilibriumdensityof trees,and thatat equilibriumapproximately 90% of 1-m2patchesshouldhave a maturetree.Further,the modelindicatesthatfireis mostlikelyto have a halting effecton the spread of treesin systemswithlow density,regulartreedistributions. Thereis currently littledata to testor corroborate these predictions.For instance,Dauget & Menaut (1992) founda c. 30% increasein treedensityovera 20-yearperiod (5 annuallyburntplots,each of 0.25 ha). Simulationsof our model withan initiallyrandom distributionof treespredicta largervalue of 44% (mean of 20 simulations).Possible reasons for this discrepancybased on the model include (i) an underestimationof mortalitiesdue to fire,(ii) an overestimationof recruitment rate, and/or(iii) the non-inclusionof directcompetitiveeffects. Untilfurther experimentaland modelling studies are conductedwe cannotweightherelativeinfluences ofthese and otherfactors.

Acknowledgements We thankIan Noble forhelpfuldiscussion,and Mark Rees and twoanonymousreviewers forcommentson an earlierdraftof thisstudy.

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