熱帯多雨林で樹木の種の多様性が高いのはなぜか?

(1)

A

Mathematical

Model for Latitudinal

Gradient

of

Forest Species Diversity

Yoh

Iwasa,

Takuya Kubo and Kazunori

Sato

Deparrment of Biology, Faculty ofScience,Kyushu University, Fukuoka812,Japan

熱帯多雨林で樹木の種の多様性が高いのはなぜか

?

九州大学理学部巖佐庸・久保拓弥・佐藤一憲

Thespeciesdiversity oftreesmaintained in tropical rain forests ismuchhigher than in

temperate,boreal,

_or

seasonally dry tropical forests. Inthis

paper,

we

analyze

a

mathematical model of tree-by-tree replacement. Withe

gap

formation ocuring throughout the

year,

a season

unfavorable for growth

causes a

peak ofregeneration opportunityin the beginning of the

growing

season.

Theresulting synchronization ofregeneration opportunityjeopardizes the coexistence of

many

similar species.

1.

Introduciton

An importantunsolved quesfion in ecology iS whatcontrolsffierichness ofspeciesofasimil と

lifeform living in the

same

habitat. The problem is illustratedmostclearly by the latitudinal gradient oftreespecies diversity,

as

tropicalrainforestsinclude by far

more

tree species than

temperate forests of the

same area.

Forexample, within

a

research

area

of2haof the tropical forestat Pasoh,Malaysia(I), there

are

1169

individualtrees withthe DBH(diameteratbreast height)larger than $10cm$,and they constitute

276

species. Eventhecommonestspecieshas

no

more

than

5

percentofthetotal, only

8

species have

more

than

18

individuals, and 114 species

are

represented by

a

singleindividual. This makes sharpcontrastwith

many

temperateand borealforests,in which

one

or a

few dominant species

occupy

a

large fraction of

area.

Among tropical forests where the temperatureis constantly high throughouttheyear, the species diversity clearly decreases withthe lengthof the dry

season

(1). Thespecies diversity oftrees isthe highest for tropical rain forests inBorneo, wheremonthlyprecipitationexceeds

(2)

have

a

few relatively drymonths,anditisstill lowerfortropical seasonal forests and

savanna

woodlands, where there is

a

clear dry

season

of severalmonths.

Gause’s principle ofcompetitiveexclusionstatesthatitis difficult for species similar in lifeform and

resource

utilizationtocoexiststably. However, trees apparentlyrequire

a

similar

setofresources, such

as

light, soilmoisture,_mineral nutrients,_andyet

many

species coexist

within

a

habitat(2). Whatprevents

one

or a

fewspeciesthat

are

themosteffective in

competitionfromeliminating others ? Numerousmechanisms havebeenspeculatedtoexplain thespecies diversity oftreesin tropicalrain forests(3). The following

are

some

examples(12).

2.

Hypotheses Explaining Latitudinal Gradient of Species Diversity

i)Specialization ofResource Use: Aclassical viewis thatcompetingspecies

are more

likelyto coexiststablywhen theydiffer inniche,

_or

_the

_resource

_use

pattem,and that thecommunity

can

maintain

a

larger numberofspeciesif eachspecies is

more

specialized(4). Unpredictable

or

fluctuating

resource

availability intemperateforests

may

inhibit theevolutionofniche

specialization, resulting in fewercoexisting speciesthan intropics(5). Although specialization oftree specieswithrespecttoregenerationisimportant(2, 6, 7),extremelyhighdiversity of tropicalrainforestsisunlikelytobe explained only by theobserved degree of niche

specialization(8).

ii) Mode of Disturbance: Randomdisturbance notonly delays the competitive exclusion betweenspecies,butalsoactively maintainsthespecies diversity(9). Mathematicalmodels that successfully explain the stablecoexistence of

a

largenumberofspecies with

very

similar life form often

assume

sedentaIy and long-lived adults and widely dispersing larvae

or

seeds

(”foundercontrol“ models ofcompetitionfor

space,

(9)),the examples including Hubbell’s

random drift model and Chesson and Wamer’slotterymodel$(10,11)$

.

_{A single disturbance}

eventsuch

as

a

fire

or a

bigstonn

may

killtrees

over a

large area,and

cause

spatially clumped and temporally synchronizedtreeregeneration. Iflarge-scaleddisturbances

are more

pronounced intemperateandborealregions than intropics, then this

may

possibly explain the latitudinal difference ofspeciesdiversity (12).

iii) Smaller Opportunity forCompetition: Accordingto the

survey

offield observations and

experimentsin

a

varietyof plantcommunities,including freshwateralgalcommunities, species

diversity isoften the highest in habitats of relativelylow

resource

supply(13). _{This trend}

can

be explained intuitively

as

that

a

slowgrowthratereduces theopportunityforcompetitive

exclusion. Theidea

can

bemade

more

rigorous(14). _{Negative correlation between soil}

(3)

tropicalrain foreststheavailability ofnutrients in the soil wouldbeconstantlylow,

as

most nutrient

resources

are

likelytobecapturedbytrees (15). Incontrast,intemperateand boreal forests,the availability ofsoil nutrient

may

have

a

seasonal peak, due for exampletothe synchronized defoliationoftrees

or

tosnow-thawing, causingtemporarily eutrophic

environmentprovidingopportunity for

a

few fastgrowing speciestodominate thecommunity.

iv)Productivity: Species-energy hypothesis postulates that the

energy

availability

may

constrain the number ofcoexisting species (16). Recentcomparative studiesof forest species

diversity

over a very

large scale havedemonsrrated

a

strongcorrelation between thespecies

diversity and theannual actualevapotranspiration,

a very

good predictor ofprimary

productivity (17). However the analysis of finer scaled comparisonssometimesreveals

negativecorrelation between plantspeciesdiversity and thenetprimaryproductivity (13). In

addition,

_no

convincing theory is currently available that explains why speciesdiversity should be higher in productive habitats.

v) Specific Herbivores and Pathogens: Alarge effectof

a

generalist predator

on

species

diversity of its

prey

species has often been demonstrated(18). Probably

more

effective in enhancingspecies diversity

are

theparasites,pathogens, and predators that

are

specifictohost

species. These

cause

greaterdamagewhenthesusceptivehostplantis

more

abundant, producingstrongfrequency dependencefavoring

rare

species and enhancing the host guild diversity (19). This

can

explain the latitudinaldiversity gradientifpathogens and herbivores

are

somehow

more

activein the tropicsthan intemperate

zones.

Janzen’s prediction that specific predators inthe tropics should

suppress

therecruitmentoftreespecies

near

conspecific adulttreesis sometimessupported (20),but the observedeffectis notstrongenoughtoexplain theextremediversity of tropical forests (21).

vi)EvolutionarylEcological History: Lowtreediversity of temperate and boreal forestsis sometimesconsidered

as

a

resultof shorttime sincetheretreatof the last glacier. However the lineages oftrees intemperateand borealregions andthesehabitats themselves

are

old in evolutionary history(13). Therateofspeciation

may

be higher inthetropics than intemperate

regions. For example,animal-pollinated tropical trees

may

experience faster speciation than wind-pollinated boreal foresttrees. Habitatfragmentation of tropicalrain forestsduring glaciationisalso suggestedto havecaused genetic differentiation and produced

a

largenumber of tropical species.

All the six hypotheses

seem

tobeplausible and

are

likelytobeimportantin

some

(4)

latitudinal gradient oftree speciesdiversity. One effective approach istomodeleach hypothesis

or process

andtoexamine theoreticallythecondition in which that mechanism works.

In thefollowing,

we

study

a

mathematical model describing the dynamical changes in

a

communityby replacement oftrees. We analyze inparticular howspeciesrichness decreases withthe length of the cold

or

dry

seasons

based

on a

hypothesis that lower species diversity in

temperateregions is

a consequence

ofthe greatersynchronization of regenerationopportunities

thaninthetropics. Weconclude(1) the

mere

existence of unfavorable

season

can

reduce significantly the diversity of coexisting species. (2) Diversity intheequilibrium community

can

be high when niche width of eachspecies is broad and

resource use

is strongly overlapped.

(3) _{Equilibrium community includes several distinct}

groups

ofspeciesdiffering in phenology ofregeneration.

3.

Model

Chesson and Wamer analyzed the lottery model and demonstrated that the temporal fluctuation ofrecruitmentabilityuncorrelated between species isable tomaintain

a

high species diversity(I1). Theconditionthisrequires (calledstorage effect)_is thatsedentaryadults

once

successfullysettled

can

survive

over

time sufficiently longer than the intervals between

intermittentfavorable periods that give

a

rremendous

success.

Runklepostulated thatthe storageeffectisthebasic mechanism for

many

similartree speciestocoexist in

a

forest(22). Only

a

small fraction of sites

are

disturbed each

year.

Even tropicaltrees havestrong seasonality in fruit production, andthis subsequently

causes

a

higher

regenerationability of thespecies for subsequent months.

Runklethen notedthatthetemporalpattemof

gap

formation and the

gap

size

are

similar between tropical andtemperateforests (23). In seasonalenvironments, however,

gaps

created during the unfavorable

season

(eithercold

or

dry)remainunfilled andincrease in number until the beginning of the following favorable

season.

This produces

a

synchronizedregeneration opportunityfortree species andgives competitiveadvantagetothespecies having the peak

regenerationabilityatthe startof thefavorablegrowing season,resulting in

a

lowerspecies

diversity. As the number of cold

or

drymonths

per year

increases, the peakrateof supply of

gaps

in the beginning of the favorable

season

becomes

more

importantand speciesdiversity decreases.

Here

we

studythis hypothesis. Theforestis composed of

a

largenumberofsites, each

of which occupied by

a canopy

tree. Each

year,

only

a

small fraction ofsitesreceive disturbance andthe

gaps

thuscreated

are

filled bythespecies randomly chosenin the

(5)

community,considering seasonality of regeneration ability. Let$X_{j}$ bethefraction ofsites

occupied by the ith species $(i=1,2,.., n)$. The change of$X_{j}$

per year

is:

$\Delta X_{i}=\lambda\{- X_{i}+\int_{0^{T}}d(\frac{M\iota)X_{i}}{\sum_{j--1}^{n}\beta 4t)\kappa_{j}}\{$

(1)

where$\lambda$is the annualrateofdisturbance. Theinverse $1/\lambda$isequaltothe

average

tumover time,

andisof the order of

100

to

200 years.

$T$is the length of

a year

($T=3\omega$days) and$t$indicates

day within

a year.

Regenerationopportunity$p(t)$is the distribution of thedateatwhich

gaps

are

available forregeneration. Let$b$bethe length of the unfavorable

season

(Fig. 1). _$p(t)$ is

zero

for$0\leq t$

$\leq b$,and itis large forthefirsttwoweeksof the growingseason,indicating that all the

gaps

that

are

accumulated in the preceding unfavorable

season

thenbecome available forregeneration. Regeneration opportunity$p(t)$is normalized

so

that its integral is equaltounity.

Theregeneration ability of the ith species$(i=1,2,.., n)$

on

day$t(0\leq t\leq T)$is:

$\beta_{i}\{t)=\{$

$1+\cos((t- iT/n)\pi/w)$, $|t- iT/n|<w$

$0$, otherwise

(2)

whichis larger than 1, half-peak height, forthetime period of length $w$. Note that multiplying

regenerationability by

a

constantdoesnotchangethedynamics, Eq. (1). Function$\beta_{i}\{t$) has

a

bell shapedcuIveandthedateatwhich eachspecies has the peakis spaced regularly

over

the

yeと.

Thepresentmodeldescribescompetitionof plants differing in the phenological aspectof theregenerationniche (2), where speciescompetefor newlyformed

gaps.

Accordingto this

interpretation,regenerationopportunity$p(t)$isthe

resource

supplyrateand regeneration ability

$\beta_{i}(t)$

isthephenological niche for the ith species. Theparameter$w$henceindicatestheniche

width.

4.

Equilibrium Species Diversity

Thedynamicalsystem Eq. (1) has

a

uniqueequilibrium that is globally stable. This

can

beproved byconsidering the following function:

(6)

$V(X_{1},.., X_{n})=- \int_{0}^{\tau_{At)\log}}5\sum_{=1}^{n}\beta_{J}\{t$)_{$X_{j}]dt$}

(3)

whichdecreases withtime

as

$X_{1},$

$,$

$X_{n}$ change

withtime$t$

as

changingfollowing Eq. (1) and

attains the minimumatthe equilibrium,it is

a

Lyapunov function(26). Eq. (3)is

a measure

of thedistancenamed”Kullback’s divergence” between thetwodistributions

over

$t,$$\mu_{t)}$and $\sum_{j=1}^{n}\beta X^{t)\chi_{j}}(24)$

. Eq. (3)_therefore statesthat the

sum

of regeneration ability ofthespecies

becomes closer with timetotheregeneration opportunity$\iota Xr$

),

and inthe equilibrium, the

community species composition gives the bestapproximation of$\iota Xt$) in the

environmentby

a

linear combination ofregenerationability

curves

of$n$ species

$\beta_{i}(t)$

.

We hereexaminehow much species diversity

can

bemaintained in the equilibrium of Eq. (1),the diversity being measured simply by the total number ofspecieshaving

a

positive

abundance.

The model analyzed exrremely speciesrichcommunitieswith

many sparse

speciesthat differ

very

little from each other inregenerationniche. As the total number ofspecies$n$

increases,

more

specieswithsimilar phenology become included, and eachspecies decreases its abundance. The number ofspeciespresentin the equilibriumcommunity$S$increases almost in

proportion tothe total number ofspecies $n$.

$S_{-}D$ecies diversitv

versus

the len th of unfavorable

season

Fig. $2A$illustrates the relationships

forthe number ofspecies$S$andthe length of unfavorable

season

$b$

.

The total numberof

speciesis $n=80$. The diversity decreases with the length of unfavorableseason,

as

postulated

(22). Howeverthe

way

it decreases greatly depends

on

niche width$w$

.

$S_{-}oecies$diversitv

versus

niche width Fig. $2B$illustrates the relationship ofthenumber of exisiting species$S$and the niche width _$w$. The diversityisnotmonotonically decreasing with

nichewidth $w$. Itdecreases with$w$for small$w$, takes

a

minimumfor

an

intermediate$w$, and

thenincreases again forlarge$w$.

Atraditionalconceptofspecies packingsuggeststhat

a

larger number ofspecies

can

coexistifthespecies

are more

specialized, andhenceitpredicts the decrease of thespecies

diversitywithniche width$w$. However,in thepresentmodel this holds only for small niche

(7)

ofspecies increases(ratherthandecreases)with the niche width and with the degree of niche overlapping. This seeminglycounter-intuitiveresult

can

be understood by considering

a

limiting

case

of

very

flat$\beta_{i}\{t$).

in which the species

are

similar in regeneration ability, and allthe

species

can

bemaintained inthe system. Competitive exclusion is themosteffective when$\beta_{\iota}\{r$)

has

an

intermediatewidth,

as one or a

few specieswiththe peakregenerationdate coinciding with the peakregeneration opportunityin theenvironmentdominate andexcludeothers.

Phenolomofcoexisting_species _Fig.

₃

_{illustrates the phenological}_pattems_of_{the equilibrium}

community. The abundance of each species isindicated by

a

symbol

on

the dateatwhich its regenerationabilityisatmaximum.

Thespecieshavingitspeak

near

the beginning ofthegrowing

season

suppresses

other

species withsimilar peak regeneration dates butnotthose with sufficientlydifferent peak dates.

As

a consequence,

there

appears

a

wave-likepattemof the abundanoeof species. For example,

Fgi.$3A$illustratesthe

case

with unfavorable

season

of five months ($b=150$ days)andniche width $w=60$days. There

are

three

groups

of species withpositive abundance. Thespeciesin the first

group

have the peakregenerationabout 170; the second

group,

between

240

and 260; the third

group,

between

320

and

325.

The distance between adjacent

groups

is about

70

to 80

days. As the niche width increases,thewavelength ofthepattemincreasesand the number of

groups

decreases, and finally thereremainsonly

a

single peak located in the middle(ratherthan

in thebeginning) of thegrowing

season

(e.g.Fig. $3B$ is for$w=165$ days).

Fig. $3C$and$3D$

are

the resultswhenthe lengthof unfavorable

season

is

one

month, shorter than in Fig. $3A$and $3B$. Interestingly, for short unfavorable

season

and wide niche

(Fig. $3D$),there

are

many

specieshaving peakregenerationduring theunfavorableperiod. This isbecause

a

dominantspecieswith peakregeneration in themiddle of

a

growing

season

suppresses

au

thespecies having peakregenerationclosetothem.

We alsostudiedthe

case

in which the recruirment ability decreases with the abundance ofthespecies and the

case

in whichthereisrecurrentreinvasionofspecies from outside

source.

Theresults

are

qualitatively the

same

as

thebasic

case

studied here.

6.

ConcIusion

Thepresentstudy shows the importanceof temporalpattem,especiaUyseasonalpattem,

of the opportunityforregenerationin understanding thespeciesdiversity maintained in the forests (22).

(8)

In tropicalforests, seasonalvariation

occurs

in arrival of seeds at

a

site,

as

thereis

a

clear seasonal rhythm of both fruitfall and seedgermination(26). Althoughmosttropicalrain

foresttreespecies

are

presentin the understory before

gaps

are

produced, theturnoverofstems

among

the seedling poolishigh. For example,Augspurgerreported thatmostseedlings die

within three months ofgermination(27). Hence

among

treespecies there is

a

seasonal

variationof the germination advantage ofregenerationrates,aUowing theircoexistencein

a

habitat through thestorageeffect mechanism (11).

Onthe otherhand, the length of period during which eachtreespecies

are

ableto

regenerate,probably extends

over

several months instead of

a

week

or

two. Since the niche widthin the model,denotedby$w$, should bemuch longer than the period for high regeneration

oppormnity, coexisting speciesmusthaveconsiderable niche overlap with each other.

One of

our

findings in this

paper

isthat

a

broader niche of eachspecies

may

result in

a

larger number ofcoexisting species with

an

extremeniche overlap. As illustrated by Fig. $2B$_, $4B$ and$5B$_,the relation of the speciesdiversity and the niche width is notmonotonical. The

previous understanding about the relation of diversity and niche width

was

that

narrow

niche

(orspecialized

resource

use)should enhancethediversity of coexisting species$(4,5)$. Thisis

the

case

when the nichewidth of each species isshort compared with the width of regeneration

opportunityin theenvironment. When the nichewidthissufficientlybroad,the number of

coexisting species increase,ratherthandecreases,with the niche width. Thisis because

many

species

can

coexistifthey

are

very

similar. Huston (14)andHubbell and Foster(8) stated this

as

the basic mechanism for

numerous

tree species tocoexist in tropicalforests without

sufficient degree of specialization.

Compared with otherhypotheses explaining latitudinal gradient of diversity, thepresent

model identifies

a

directlogicalconnection betweenspecies diversity difference andthe

existenceofwinter

or

dry

season.

Amerit ofthehypothesisis its simplicity. At thismoment,

however,

we

cannottell which of

many

possible

processes

is the dominant factorexplaining the observedlatitudinalgradient of forest diversity, allsixclasses ofhypotheses listed

up

before look plausibleto

us.

Tomodeleachaspectof the hypotheses andtospecify theconditionsin which eachproposed mechanismworks,

as we

havedone in this

paper,

isprobably themost effectiveapproach understanding thebasicand generalmechanism maintaining biodiversityin naturalecosystems.

(9)

Referencesand Notes

(1)Kira$T$(1983)“Ecology

of

tropical

forests.

“Jinbun-Shoin,Kyoto(in Japanese)

WhitmoreTC(1984)“Tropical rain_forests

_of

the

_far

east.“ _{(2nd ed.)}_{Oxford University}_Press

(2)Grubb$P$(1977) BiolRev52: 107-145

(3)Begon$M$,HarperJL,TownsendCR(1990)”Ecology:individuals,populations andcommunities“(2nd ed.).

BlackwellSci,London

Diamond$J$,CaseTJ eds. (1986)“Community ecology“ Harper andRow,N.Y.

Ricklefs RE (1990)“Ecology“ (3rd ed.)FreemanandCo.,N.Y.

(4)MacArthur RH(1972)“Geographical ecology:patternsinthe disrributionofspecies“. Harper andRow,N.Y

(5) Pianka ER (1966) Am$Nat1\alpha$)$;33- 46$

Pianka ER(1978) “Evolutionary ecology.“ (2nd ed.)Harper andRow,N.Y.

(6)Ricklefs RE(1977) Am$Nat111:376- 381$

(7)Shmida$A$,Ellner$S$ (1985) Vegetatio58: 29-55

(8)HubbellSP,FosterRB (1986) In“Community ecology“ (J.Diamond andT.J. Caseeds.) _{pp. 314-329.}

Harper&Row,New York

(9)Connell JH(1978)Science 199: 1302-1310

Chesson PL (1986) In “Community ecology”(J.DiamondandTJ.Caseeds.) _{pp 204-256} Harper&

Row,New York

Yodzis$P$(1986) In “Community ecology” (J.Diamond and T.J.Caseeds.) pp. 480-491. HaIper&

Row,New York

(10)Hubbell SP (1979) Science203: 1299-1309

(11) ChessonPL,WamerRR(1981)Am$Nat117:923- 943$

WamerRR,ChessonPL(1985) Am$Nat125:769- 787$

(12) Iwasa,Y.,K.Sato,M.Kakita and T. Kubo.(1992) In “Ecosystemfunction of biodiversity“(E.-D.

Schulze,H. Mooneyeds.) Springer-Verlag.(inpress)

(13)AshtonPS(1977) AnnalMissouri$Bot$Gard64: 694-705

Huston $M$(1980) J Biogeogr7: 147-157

(14)HustonM(l979) AmNat113:81-101

Tilman$D$(1982) “Resourcecompetitionandthecommunitystructure“. Princeton UnivPress,Princeton

(15)Odum EP(1969) Science 164: 262-270

(16)HutchinsonGE(1959) Am$Nat93:145- 159$

(17)CurrieDJ,PaquinV(1987) Nature329: 326-327

AdamsJM,WoodwardFI(1989) Nature339: 699-701

(18) e.g. PaineRT(1966) Am Nat 100:65-75

(19)BremermannHJ,Fiedler$B$ (1985)$J$theor Biol117:621-631

(20)ClarkDA,Clark DB (1984) _{Aus J}Zool 10: 362-380

HoweHF, SchuppEW,WestleyLC (1985)Ecology66: 781-791

Janzen$D$(1970) Am$Nat104:501- 528$

(21) HubbellSP, Condit$R$,FosterRB(1990) Phil Trans R SocLond$B330:269- 281$

(22)RunkleJR(1989) Ecology70: 546-547.

(23)DenslowJS (1987) $Ann$RevEcol Syst18: 431-451

(24)Kullback,S. andR.A.Leibler(1951)Ann. Math. Statist.22,79.

(25)Simpson EH(1949) Nature 163: 688

(26)Garwood NC(1983) EcolMonogr53: 159-181

(27)AugspurgerCK(1983) Oikos40: 189-196

$\sum^{n}X_{i}- 1$

(28) Toprovethe stability of the dynamics.weconsider the Eq. (3)plus$i=1$ whichiszerobecasethe summationof$X_{i}$ is unityin Eq. _(1).

Bynoting annualchange in$X$; _is_small,

theone-yearchangeof$V$ following Eq.(1)isalwayspositive:

$\Delta V(X_{1},.., X_{n})=\sum_{j--1}^{n}\frac{\partial V}{\partial X_{i}}\Delta X_{i}=-\sum_{i=1}^{n}\lambda X\{\int_{0^{T}}dt\frac{\beta(t)}{\sum_{j--1}^{n}Mt)X_{j}}- 1)^{2}\leq 0$

This together with theconvexityof function$V$(i.e.Hessianmatrix is negativedefinite) leadstothe conclusion

(10)

Figure 1 AnilluStrationof themodel Gapsareformedataconstant ratethroughouttheyear During山$e$

seasonunfavorable fortreegrowthwiththe length$b$,gapsareaccumulated withoutbeing filled,andbecome

available for regenerationinthe beginningof the growingseason. (Top)Regeneration opportuniry$p(t)$iszero

duringtheseasonunfavorable fortree growthwithlength$b$. It hasasharppeakinthebriefperiod at the

beginningof growingseasonandisalow constantafterwards. (Bottom)Thepeakregenerationopportunityis

taller$wi$_由山$e$lengthof unfaVorableSeaSon

Figure2 The numberof species$S$withpositive abundance intheequilibrium. (A) Horizontalaxisis thelength

of unfavorableseason$b$.Thenumber ofspecies$S$dccreasessharplywith thelengthof unfavorableseason$b$,

differentlines correspondingtodifferent niche width$w$. $(B)$ Horizontalaxis isthe niche width$w$. The number

ofspeciestendtobe large both forverynarrowandverybroadniche,butisthelowest forintermediateniche

width$w$. Thelengthofayear is$T=360$days,and thetotal number ofspeciesis$n=80$. The numerals in the

figureisthe lengthofbadseason$b$.

Figure3 Thephenological patternof speciesattheequilibrium. The abundance of each speciesisindicated bya

circleonthe dayatwhichits regencration abilityisatmaximum. Solidcirclcsareforspecieswithpositive

abundance$(^{X_{i}>0})$andopcnones areforspeciesabsent$(^{X_{i}=0})$in the equilibrium. Shadedareasarefor

regeneration opportunitycurve$p(t)$. $(A)$Unfavorableseasonextcnds five months($b=150$days),andnichewidth

is$w=60$days. The communityincludes threegroupsofspeciesdiffering in the date of peak regeneration. Peak

datesareseparatedabout70to80days(alittlelonger than$w$). $(B)$ Nichewidthisbroader$(w=165)$. Thereisa

single peak of dominantspeciesthatsuppressesallthe otherspecies. (C)and(D)areforthecaseswithashorter

unfavorableseason($b=30$days). Interestingly, thecommunity mayincludemany specieswhosethe peak

regenerationoccurinthe middleofunfavorableseason.

Fig. 1 lwasa

et

al.

regeneration opportunity

niche

$\beta$

(11)

Fig.2 $\Phi$ $u^{\Phi_{)}} \alpha\frac{4y_{)}}{o}$ $\vee 0$ $\overline{DZ\in\Phi\supset}$ $0$ 60 120 180 240

The Lengthof unfavorableseason $b$ _(days)

$0$ 60 120 180

(12)

Fig.3

$\chi_{j}$

$\chi_{T}$