Journal of Ecology

2004

92

, 97–109

© 2004 British Ecological Society

Blackwell Publishing, Ltd.

Seed mass and the competition/colonization trade-off: competitive interactions and spatial patterns in a guild of annual plants

LINDSAY A. TURNBULL*†, DAVID COOMES†‡, ANDY HECTOR* and MARK REES†

*

Institute of Environmental Sciences, University of Zürich, Winterthurerstrasse 190, Zürich, CH-8057, Switzerland,

†

Department of Biological Sciences and NERC Centre for Population Biology, Imperial College, Silwood Park, Ascot, Berkshire, SL5 7PY, UK, and

‡

Department of Plant Sciences, University of Cambridge,Cambridge, CB2 3EA, UK

Summary

1

We used neighbourhood modelling to estimate individual-level competition coeffi-cients for seven annuals growing in limestone grassland over 2 years. We calculated therelative strength of intra- and interspecific competition and related this to differences inseed size and plant size between targets and neighbours.

2

Significant differences in the impact of neighbours on each target species wereobserved in half the models fitted, allowing us to reject a null hypothesis of competitiveequivalence.

3

In one year we found that as the seed size or plant size of neighbours increased relativeto targets, so did their competitive effect. Although this is consistent with the competi-tion/colonization trade-off model the competitive interactions were not sufficientlyasymmetric to allow coexistence. In a second year we found only weak interspecificcompetition and no relationship with plant or seed size.

4

We found no overall relationship between competition coefficients and the degree ofsegregation, contradicting the spatial segregation hypothesis for coexistence. However,segregation was linked to differences in plant traits: when targets were smaller thanneighbours the degree of segregation increased with relative neighbour size.

5

Most species were positively associated with each other due to a shared preference forotherwise unvegetated patches. The degree of association was negatively correlated withdifferences in plant and seed size, particularly when interspecific competition was weak.This might reflect (i) decreasing overlap in microhabitat use with increasing trait diver-gence or (ii) density-dependent mortality.

6

Seed size is a key trait within this group of species, determining both competitive andcolonizing ability. The presence of such a competition/colonization trade-off undoubtedlystabilizes community dynamics although other mechanisms may also be at work.

Key-words

: seed size, competitive interactions, neighbourhood modelling, spatial patterns,spatial segregation, models of coexistence

Journal of Ecology

(2004)

92

, 97–109

Introduction

A recent review of spatial dynamics by Bolker

et al

.(2003) is the latest to lament the current lack of know-ledge about competitive interactions and spatial distri-butions in natural communities. It goes on to suggestthat such studies would be best undertaken with annual

plants, given their small size and relatively simple life-history. Annual plants often persist within a perennialmatrix by taking advantage of a fugitive niche (Levins &Culver 1971), i.e. by exploiting gaps from which they willeventually be excluded by perennial species (Skellam1951; Crawley & May 1987). Within a particular habitat,however, it is common to find a range of species usingthis strategy, which begs the question of how do thesespecies carve up the fugitive niche. There are severalpossibilities: for example, it has been suggested that

Correspondence: Lindsay A. Turnbull (tel. +41 16356120; fax+41 16355711; e-mail lindsayt@uwinst.unizh.ch).

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annuals in grasslands are only present at low den-sities and experience relatively little intraguild com-petition, so that niche differentiation is not required(Grubb 1986). However, if the outcome of competitionis deterministic, this would only slow down the rates ofcompetitive exclusion (Chesson 1991). Alternatively,the species might be competitively equivalent and thecommunity exhibit neutral dynamics (Aarssen 1983;Hubbell 2001). However, the short site tenure of annuals,and the documented differences in plant size, seedsize, fecundity and competitive ability (Mack & Harper1977) makes this unlikely. It has also been suggestedthat spatial processes alone can allow the persistence ofan inferior competitor provided that dispersal is local(the spatial segregation hypothesis, Weiner & Conte1981; Bolker & Pacala 1999), although the conditionsunder which this is true are limited (Takenaka

et al

.1997; Neuhauser & Pacala 1999; Neuhauser 2001).

Another possible way of dividing the fugitive niche isvia a competition/colonization trade-off. This modeldemands that a single, fully transitive competitivedominance hierarchy exists across the entire habitat(Tilman 1994) and that competition is completelyasymmetric (Geritz

et al

. 1999; Adler & Mosquera2000; Levine & Rees 2002). We have previously sug-gested that seed size variation within annuals mightrepresent different points on a competition/coloniza-tion trade-off axis, with large-seeded species beingsuperior competitors but inferior colonisers (Rees1995; Turnbull

et al

. 1999). We have shown that, withina community of limestone grassland annuals, thelarger-seeded species produce fewer seeds on an indi-vidual basis and have used a sowing experiment todemonstrate that they are much more strongly recruit-ment limited (Turnbull

et al

. 1999). Large-seeded spe-cies also dominated sown plots, suggesting that theywere competitively superior; however, our experimentcould not measure competitive interactions directly. Ingeneral, large-seeded species have been shown to havean establishment advantage (Gross & Werner 1982;Gross 1984; McConnaughay & Bazzaz 1987; Eriksson& Eriksson 1997; Freckleton & Watkinson 2001b;Dalling & Hubbell 2002; Leishman 2001), although fewerstudies have directly quantified a relationship betweenseed size and competitive advantage (Freckleton &Watkinson 2001b; Rodriguez-Girones

et al

. 2003).In order to test alternative theories of competi-

tion and coexistence we need to measure interactionstrengths directly; however, there are two fundamentalproblems. First, the number of parameters increaseswith the square of the number of species (64 interactioncoefficients are required in a guild of 8 competingspecies). Secondly, it is essential that this task is carriedout under field conditions (Freckleton & Watkinson2001a). One recently advocated approach is to quantifythe effect of natural changes in density on the popu-lation growth rates of target species (Freckleton &Watkinson 2000). This method has successfully providedfield-based estimates of population–level competitive

interactions for several groups of species (Emlen

et al

.1989; Rees

et al

. 1996; Law

et al

. 1997; Freckleton

et al

.2000; Watkinson

et al

. 2000). However, such studies donot reveal the pattern of individual-level competitionon which the competition/colonization trade-off is based.For example, a study of sand-dune annuals revealedthat, at the population level, interspecific interactionswere weak (Rees

et al

. 1996), although this apparently con-flicted with earlier experimental work at the individuallevel (Mack & Harper 1977).

Discrepancies between individual- and population-level interactions occur when there is systematic bias inthe frequency of interactions. For example, spatial seg-regation generated by local dispersal will reduce thefrequency of inter- relative to intraspecific interactions,therefore lessening the impact of interspecific com-petition at the population level (Pacala & Levin 1997;Bolker & Pacala 1999; Stoll & Prati 2001). However,spatial segregation might also arise from specializationon different microhabitats, making the interpretationof spatial pattern problematic. One could argue thatstrong competition results in spatial segregation, orthat spatial segregation due to some other cause resultsin species only rarely interacting and thus suggestingthat competition is weak and unimportant. Only bymeasuring individual-level interaction strengths dir-ectly can we determine whether spatial patterns arecorrelated with competition coefficients as predicted bythe spatial segregation hypothesis, or are more likely toresult from microhabitat preferences.

In this paper we show how a neighbourhood model-ling technique can be used to derive individual-levelcompetitive interactions from fully mapped data underfield conditions (see also Coomes

et al

. 2002b). Thistechnique was developed to reveal the impact of dif-ferent neighbour species on plant performance and hasmainly been used in the analysis of greenhouse experi-ments (Pacala & Silander 1985, 1987; Purves & Law2002a). We show that the estimates are unbiased byspatial structure, and that information on perennials,including grasses and clump-forming species, caneasily be incorporated. We measure the extent ofcompetition in the field and test the hypothesis ofcompetitive equivalence by assessing the significanceof species-specific interactions. We then specifically testthe competition/colonization trade-off hypothesis byrelating the magnitude of competitive interactions todifferences in plant traits such as seed size and plantsize. If there is a competition/colonization trade-offbased on seed-size, the strength of competitive inter-actions should be related to differences in seed size. Inaddition, we calculate association and segregationindices to explore the relationship between competi-tion coefficients and spatial patterns, particularly todetermine whether the extent of segregation is relatedto the relative magnitude of inter- and intraspecificcompetition coefficients. We also assess whether theprimary component of spatial structure is the tend-ency of species to form aggregates generated by local

99

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dispersal (Mahdi & Law 1987) or whether there arehigher-order terms generated by interactions betweenspecies (Purves & Law 2002b).

Materials and methods

The study site is within coastal calcareous grassland inSouth Wales, UK (National Grid Ref. SS/423872). Thearea consists of Carboniferous limestone cliffs coveredby shallow soils with exposed bedrock and is heavilysheep-grazed for much of the year. A total of 57 specieswere recorded (nomenclature follows Stace, 1997).Eight species of dicotyledonous annuals (

Aphanesarvensis

,

Arenaria serpyllifolia

,

Cerastium diffusum

,

Geranium molle

,

Myosotis ramosissima

,

Saxifraga tri-dactylites

,

Sherardia arvensis

and

Veronica arvensis

)were common at the site and are suspected to competestrongly for the fugitive niche: these form the focus ofthis study. All species are small: 0.5–10 cm tall with anerect growth form and rarely exceed 1–2 cm in dia-meter. We set up twenty-eight 60

×

40 cm permanentquadrats and recorded the positions of all plants forthree consecutive years. The size and fecundity of allannuals were estimated at the end of the flowering sea-son to provide a measure of relative plant performance.For fecundity measures we counted the number ofseedpods and dissected a sample to estimate the totalseed output of each individual. For size measureswe used either rosette diameter or total stem lengthdepending on the species. We then harvested 30 indi-viduals to span the range of observed sizes fromaround the site, estimated their sizes and weighed themto establish calibration curves. This enabled us to con-vert our size estimates into biomass estimates.

Although our study focused on these annuals, werequired information on the remaining species to in-clude as covariates in our neighbourhood modelling.All species with a single identifiable rooting point, bothannual and perennial, were recorded. These mapswere later digitized. Species belonging to other groups(bryophytes, grasses and cushion-forming species suchas

Armeria maritima

) were sketched as irregular poly-gons, and later converted to pixel maps. This lattercomponent is referred to as the background matrix.The local environment of each annual plant couldtherefore be described in terms of the number andidentity of neighbours (both annual and perennial) andin terms of the background matrix in which the plantwas rooted. Species with a single identifiable rootingpoint belonging to non-focal species were separatedinto six categories based on life-form characteristics(Appendix S1a in Supplementary Material). The back-ground matrix was initially divided into 16 mainlysingle-species categories (see Appendix S1b), whichcould be further simplified to five broader categories(bare earth, perennial grasses, annual grasses, bryophytesand perennial forbs).

The average seed mass of each annual species wasdetermined by weighing ripe seeds from at least 30 dif-ferent individuals in the population. We determinedthe average plant mass by averaging over the wholepopulation once the biomass of each individual hadbeen estimated using the calibration curves. Populationsizes of all focal species crashed in the second yearof the study, which was probably due to drought, andno analysis was therefore performed in this year. Theplant size hierarchy changed between 1994 and 1996,although the two were correlated (

r

S

=

0.762,

n

=

8,

P

<

0.05). The relevant size hierarchy for the year inquestion was always used as an explanatory variable.Plant size was not significantly correlated with seedsize within individual years (1994:

r

S

=

0.524,

n

=

8,

P

>

0.05; 1996:

r

S

=

0.571,

n

=

8,

P

>

0.05) althoughthere was a significant correlation when plant size wasaveraged over the two years (

r

S

=

0.71,

n

=

8,

P

<

0.05).

Neighbourhood modelling was used to quantify theimpact of individuals of each neighbour species on theperformance of each target species. A more detaileddescription of neighbourhood modelling has beenprovided elsewhere (Pacala & Silander 1987, 1990;Coomes

et al

. 2002b), but briefly the approach relies onthe fact that plant competition is spatially local. Thefinal size or fecundity of each plant is therefore influ-enced by the number and identity of close neighboursrather than by the average or mean-field density.Regressing plant performance against the number ofneighbours of each type provides regression coeffi-cients which can be used to assess the relative impactof different neighbour species. The distance at whichplants are considered to be neighbours must be selectedas part of the modelling process and may vary betweentargets and neighbours. However, when the number ofspecies is large it becomes impossible to estimate inde-pendently both the effective interaction distance andthe strength of interaction for each neighbour species,as these parameters co-vary too strongly.

For each target species we began by constraining allneighbours to interact over the same radius, and fittedthis model for all radii from 5 mm to 50 mm in 5 mmsteps (as described in Pacala & Silander 1987). Thisfailed to reveal a clear best distance at which the like-lihood peaked, but instead the likelihood was anasymptotically increasing function of radius (Turnbull1998; Coomes

et al

. 2002b). We then tried an altern-ative weighted-distance approach where we used themaximum radius of 50 mm and modelled the impact ofneighbours within this area as an exponential decayfunction. The rate of this decay determines both theimpact of a neighbour at a particular distance and thedistance at which the effect of neighbours of that typeeffectively falls to zero, although the two are intrins-ically conflated. This model usually performed betterthan a single fixed-radius model (Fig. 1a) and has the

100

L. A. Turnbull

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© 2004 British Ecological Society,

Journal of Ecology

,

92

, 97–109

advantage that only one model needs to be fitted foreach target species. We also tried fitting models wherethe influence of neighbours decayed as the squareof distance, but these never performed better. It isencouraging that the parameter values from a fixedradius model and a weighted distance model are highlycorrelated (Fig. 1b;

r

=

0.906,

n

=

6). Two other annualneighbour species were estimated to have no effect onthe target species by both models (fixed radius

=

0;weighted distance

=

−

5) and are not shown in Fig. 1bfor purposes of clarity. Parameter values do not there-fore appear to be dependant on the exact model choice.

The weighted distance model takes the followingform and is fitted using maximum likelihood tech-niques with a Poisson error structure:

. eqn 1

Here,

f

is the biomass or fecundity of the target plant,

F

MAX

is the average performance of an individual withno neighbours (and must be estimated from the data),

n

is the number of species in the community,

N

j

is thenumber of neighbours of the

j

th species, and

d

k

is thedistance to the

k

th individual of the

j

th species.

M

b

represents the background matrix in which the plant is

rooted, and separate parameters are estimated for eachof the

b

categories. Both annuals and point-rooted peren-nials can be included in the summation term withunique

α

values for each species or category. Note, thatin this model, species which have a large impact whenclose to the target are constrained to affect the targetover long distances. Although this provided a good fitto our data, in other communities, or with differentspecies, a fixed-radius model might be a better solution.

The estimation of competition coefficients can bebiased by the spatial structure present in most com-munities (Freckleton & Watkinson 2000). In particular,it is difficult to separate the frequency of interactionsfrom the strength of interactions. To assess this pos-sibility we examined the relationship between meanlocal density (number of neighbours within a radiusof 30 mm) and the estimated interaction strengths. Wealso determined how well the parameters had been esti-mated by examining the sensitivity of the likelihood tochanges in the value of the parameter

α

ij

.

We began by fitting a maximal model which containedseparate parameters for each species of annual, sixcategories of non-guild neighbours and 16 categoriesof background matrix (see Appendix S1). To determinethe degree of competitive equivalence within theannual guild we assessed the significance of combiningall guild members into a single category. We alsoexamined the performance of a model with no species-specific interactions. Differences in competition coeffi-cients between years were assessed by pooling the datafrom both years into a single model with one set ofcompetition coefficients. Two alternatives were con-sidered: one with different

F

MAX

values for each year, andone with a single value of

F

MAX

. The significance of theabove simplifications was assessed using likelihoodratio tests and analysis of deviance with approximate

F

-tests to avoid problems of overdispersion (Williams1982; McCullagh & Nelder 1989).

To examine coexistence criteria, we calculated a scaledcompetition coefficient which compares the relativemagnitude of inter- and intraspecific competitiveinteractions:

. eqn 2

If the effect of an interspecific neighbour is greater thanthat of a conspecific neighbour this value is positive,while negative values indicate the opposite (low valuesof the decay coefficients indicate strong competitiveinteractions, and vice versa). To determine whetherthe relative magnitude of intra- and interspecific

Fig. 1 Model comparisons for Arenaria in 1994. (a) Thepercentage deviance explained by a fixed-radius model for arange of radii between 5 mm and 50 mm (�) and a weighteddistance model using a maximum radius of 50 mm (�). (b)The correlation between the parameters from the weighteddistance model and the fixed radius model at 20 mm.

fF

e Mij k

jd

bk

k N

j

n

=+ +−

=

=

=∑∑

MAX

111

α

scaledCC ii

ij

log=

αα

101

Competition and spatial patterns in annuals

© 2004 British Ecological Society,

Journal of Ecology

,

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, 97–109

competition is related to the difference in plant or seedsize between target and neighbour, we regressed scaledcompetition coefficients against both average plant sizeand seed size ratios for each neighbour/target com-bination in each year. For those species where fecunditymeasures were available we also looked at the effect ofcompetition on population seed output. We calculatedthe proportional reduction in seed output using thereal seed output and the estimated seed output in theabsence of competition (when all individuals havefecundity equal to

F

MAX

).

We calculated two aspects of the pattern between spe-cies pairs: association and segregation (Fig. 2; Coomes

et al

. 2002a). Association is an absolute quantity: itcompares the actual number of species

j

neighbours tothat expected if species

j

was distributed at randomwith respect to the target species. The expected numberof species

j

neighbours in radius

r

around the targetplant is therefore the population size multiplied by thefraction of the total area sampled (

π

r

2

N

j

/

A

; where

N

j

isthe population size of species j, and A is the total area).The strength of association is the degree to which indi-viduals of species j are more or less likely to be foundaround a target species than around a random point.Positive associations might arise because different

species are typically found in the same environmentalpatch types (Fig. 2), and negative associations can ariseeither from preferences for different patch types orfrom interspecies interactions such as unequal com-petition. Our association index is based on Ripley’s K12(t)statistic (Ripley 1981). The index is calculated using:

. eqn 3

Here, H is the habitable area (in this case the total areaminus the area of bare rock) nii (t) and nij (t) are, respect-ively, the number of species i and j neighbours inradius t around individuals of species i, and Nj and Ni

are the population sizes of species j and i. If there is noassociation between the two species, the expected valueof the index is 0. Association indices are symmetricalfor a particular species pair: that is the strength of theassociation between species i and j will always be thesame as that between j and i for a particular radius.

Second, we calculated the degree of segregationbetween pairs of species. Segregation is a relative meas-ure: it quantifies the degree to which the local ratio ofconspecific to heterospecific neighbours is different tothat expected from the population sizes regardless ofthe absolute pattern of association (see Fig. 2). It is per-fectly possible for species to be associated and not seg-regated (Fig. 2a), yet strong segregation can also occurbetween two associated species (Fig. 2b). Segregationis the relevant measure for testing the spatial segrega-tion hypothesis, as the strength of segregation shouldreflect the relative strengths of intra- and interspeci-fic competitive interactions, regardless of any sharedpreference for the same environmental patch types(Bolker & Pacala 1999). Following Dixon (1994), oursegregation index is calculated using

. eqn 4

For consistency this index is based on all neighbourswithin a given radius and not just the nearest neigh-bour (as is the case for Dixon’s index). Positive values ofthe segregation index indicate that a species has arelative excess of conspecific neighbours, while a valueof 0 indicates that the neighbourhood is an unbiasedsample of the local habitat patch. Positive values of thesegregation index arise when single-species distribu-tions are aggregated or through negative associationsbetween species pairs. Unlike association, segregationindices are not symmetrical for a particular speciespair: that is the segregation of i from j might exceedthe segregation of j from i, because i is more highlyaggregated. If segregation results from unequal com-petition, the weaker competitor, which experiences

Fig. 2 The difference between association and segregation. In(a) and (b) the two species (� and �) are positively asso-ciated; however, in (b) the species are also segregated whereasin (a) they are not.

A tn t

t N N Hij

iji

N

i j

i

( ) log ( )

/=

=∑

12π

S tn t

n t

N

Nij

iii

N

iji

Nj

i

i

i( ) log

( )

( )

= ×−

=

=

∑

∑1

1

1

102L. A. Turnbull et al.

© 2004 British Ecological Society, Journal of Ecology, 92, 97–109

weak intra-relative to interspecific competition, isexpected to be more strongly segregated from itsstronger neighbour than vice versa. Both indices werecalculated for a range of radii between 10 mm and30 mm.

Results

In 1994, population sizes of all annual species werevery high: 11 243 plants were mapped, 6500 of whichbelonged to the eight focal species. In 1996, populationsizes were lower: 6078 plants were mapped, 3500 plantsof which belonged to the eight focal species. Geraniumwas the only species to have a higher population size in1996 than in 1994. The discrepancy between years isprobably due to the extreme drought of 1995 whichreduced the population sizes of most species.

Neighbourhood competition was detectable in allthe focal species in both years. Models with no species-specific interactions explained up to 8% of the variance,and models incorporating species-specific interactionsexplained between 10% and 30%. Myosotis was notincluded in the species-specific analyses because of lowpopulation sizes in both years (see Table 1). The matrixof competition coefficients for both years is given inTable 2 and the sensitivity of the likelihood to changesin the estimates in Appendix S2. Overall there wasno significant relationship between mean neighbourdensity and the magnitude of the competition coeffi-cient in either 1994 (F1,47 = 0.5298, P = 0.4703) or 1996(F1,47 = 0.1008, P = 0.7523). There were also no signi-ficant relationships between mean neighbour density

Table 1 Number of target individuals of each species used in the neighbourhood fecundity analysis. This is lower than thepopulation sizes recorded because individuals near quadrat edges with incomplete neighbourhoods had to be excluded. Thechange in scaled deviance associated with fitting a model with separate competition coefficients for the guild members rather thana single coefficient for all species is also given. The significance of this change is assessed with a chi-squared distribution with 7degrees of freedom (***P < 0.001, **P < 0.01, *P < 0.05, (*) P < 0.1)

Seed mass (g)

Average plant mass (g)

Number of target individuals in analysis

Significance of aggregating the guild

1994 1996 1994 1996

Myosotis 0.159 16.012 103 66 – –Veronica 0.0545 4.8015 992 210 9.408 17.737*Saxifraga 0.0126 3.188 565 110 13.933(*) 1.059Cerastium 0.0414 21.7635 1023 402 17.299* 18.537**Arenaria 0.0759 15.988 295 190 42.273*** 6.837Aphanes 0.134 9.065 184 176 14.356* 5.800Sherardia 2.497 49.903 281 269 18.228* 15.293*Geranium 0.701 61.020 175 806 7.838 33.499***

Table 2 The matrix of decay coefficients in (a) 1994 and (b) 1996. From the shape of the likelihood surface, parameters in boldtype are considered to be well estimated. (See also Appendix S2). For Arenaria, Aphanes and Veronica separate coefficients wererequired for the different years. For Saxifraga, Cerastium, Sherardia and Geranium coefficients did not differ significantly betweenyears and the coefficients from the combined-years model are presented

Target species

Neighbour species

Veronica Saxifraga Cerastium Arenaria Aphanes Sherardia Geranium

(a) 1994Veronica −−−−0.284 −−−−0.617 −−−−1.00 −−−−0.835 −−−−0.394 −−−−0.097 −−−−0.747Saxifraga −−−−0.236 −−−−0.278 −−−−1.427 −−−−0.132 −−−−0.124 −−−−0.393 −0.153Cerastium −−−−0.237 −−−−0.352 −−−−0.114 −−−−0.066 −−−−0.236 −−−−0.931 −−−−0.133Arenaria −0.140 −−−−0.877 −−−−0.182 −−−−0.040 −0.182 −0.387 −0.103Aphanes −3.005 −−−−0.017 −2.047 −0.360 −−−−0.024 −2.174 −0.026Sherardia −−−−0.226 −−−−0.254 −−−−4.972 −−−−1.396 −−−−0.315 −−−−0.114 −−−−0.060Geranium −−−−0.082 −−−−0.694 −−−−0.129 −−−−0.064 −−−−0.191 −−−−4.707 −−−−0.071

(b) 1996Veronica −0.255 −0.227 −−−−0.005 −0.092 −0.055 −−−−0.012 −−−−0.038Saxifraga −−−−0.236 −−−−0.278 −−−−1.427 −−−−0.132 −−−−0.124 −−−−0.393 −0.153Cerastium −−−−0.237 −−−−0.352 −−−−0.114 −−−−0.066 −−−−0.236 −−−−0.931 −−−−0.133Arenaria −−−−0.053 −0.678 −0.973 −−−−0.484 −−−−1.411 −0.281 −0.111Aphanes −0.104 −0.047 −0.333 −0.182 −−−−0.078 −−−−0.060 −0.115Sherardia −−−−0.226 −−−−0.254 −−−−4.972 −−−−1.396 −−−−0.315 −−−−0.114 −−−−0.060Geranium −−−−0.082 −−−−0.694 −−−−0.129 −−−−0.064 −−−−0.191 −−−−4.707 −−−−0.071

103Competition and spatial patterns in annuals

© 2004 British Ecological Society, Journal of Ecology, 92, 97–109

and the magnitude of the competition coefficient wheneach target species in each year was analysed separately(14 tests; F1,5-values ranged between 0.0001088 and1.977; P > 0.05 in all cases). This gives us confidencethat our estimates are not biased by spatial structure.

There were significant differences in the magnitudeof within-guild competition coefficients for six of theremaining seven target species in at least one of theyears, and Saxifraga was marginal (Table 1) suggestingthat guild members are not competitive equals. Forfour of the species, the competition coefficients were notsignificantly different between the 2 years, althoughseparate values of FMAX for the 2 years were requiredfor all species except Saxifraga (Turnbull 1998). Thecompetition coefficients for Veronica, Arenaria andAphanes were significantly different in 1994 and 1996(Table 2).

In 1994, interspecific competition was weak relativeto intraspecific competition and the mean value of thescaled competition coefficients (−1.199) was signi-ficantly less than zero (t = 5.065, P < 0.001). There wasno relationship between scaled competition coeffi-cients and the log ratio of plant sizes (F1,41 = 0.04117,P = 0.8402: Fig. 3c) or seed sizes (F1,41 = 0.7981,P = 0.3769: Fig. 3a). By contrast, in 1996, the inter-specific competition coefficient was greater than theintraspecific coefficient in a much higher proportion ofcases (18 vs. 9 out of 42). There was also a significant

positive relationship between scaled competitioncoefficients and the log ratio of neighbour/target plantsizes (F1,41 = 8.100, P < 0.007: Fig. 3d) and seed sizes(F1,40 = 5.512, P = 0.024: Fig. 3b) with log plant sizeratio being the slightly better explanatory variable. Inaddition, we tested Goldberg & Werner’s (1983) sug-gestion, that all species might be equal on a per-grambasis, by testing whether the slope of the relationshipsin 1996 was significantly different from 1: it was not forseed size where the slope was 0.5592 (t = 1.851, n = 41,P = 0.120), but it was for plant size where the slope was0.4723 (t = 3.181, n = 41, P = 0.0175).

In these analyses all points are weighted equally.However, because some interactions are better esti-mated than others we could, in principle, weight pointsby the precision of the estimates, as determined fromthe shape of the likelihood surface (the change in like-lihood which occurs when α is changed, see AppendixS2) or by the frequency of the interaction on which theestimate is based. Because frequent interactions areinevitably better estimated these weightings are closelyrelated. Using either weighting led to marginal (P <0.06) or non-significant relationships between planttraits and competition coefficients.

There was a significant positive relationship betweenseed size and the proportional reduction in fecundity(averaged over 2 years) that can be attributed to com-petition (F1,4 = 47.53, P = 0.00232). This means that

Fig. 3 The relationship between scaled competition coefficients and the log ratio of neighbour/target seed size or plant size in twoyears. A positive value of the scaled competition coefficients means that the per-capita effect of an interspecific neighbour isgreater than that of an intraspecific neighbour at the same distance and vice versa.

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the total impact of competition on the seed output oflarge-seeded species is greater than on small-seededspecies.

Segregation

All values of the segregation index were greater thanzero at all the spatial scales examined and in bothyears (Table 3). The local ratio of conspecific toheterospecific interactions is therefore always higherthan expected from the population sizes. For ease ofpresentation we only present analyses for a radius of30 mm, with other radii giving quantitatively similarresults. There were significant main effects of targetspecies (F6,29 = 70.612, P < 0.0001), neighbour species(F6,29 = 16.168, P < 0.0001) and year (F1,29 = 37.381,P < 0.0001) on the segregation indices and a significantneighbour–target interaction term (F29,29 = 3.3068, P <0.001). This interaction term means that the observedsegregation cannot solely be explained by intraspecificaggregation, but that some species-pairs must haveparticularly strong or weak associations. Segregationindices were higher in 1994 than 1996 although therewas a significant interaction between target and year(F6,29 = 13.3614, P < 0.0001).

If the relative magnitudes of intra- and interspecificcompetition determine the strength of segregation wewould expect a correlation between the segregationindices and the scaled competition coefficients. How-ever, there was no relationship between these twovariables in either 1994 (F1,40 = 0.688, P = 0.412) or

1996 (F1,40 = 0.0172, P = 0.896). There was, however,some evidence for a relationship between plant traitsand segregation indices. The lack of symmetry in thesegregation indices slightly complicates the analysis asfor each species-pair there are two ways of calculatingthe index: either with species 1 as the target and species2 as the neighbour, or with their positions reversed.Therefore, separate regressions were carried out; firstincluding only the value from each species pair wherethe bigger plant was the target, and, second, includingonly those values where the smaller plant was thetarget.

In 1994, the segregation index increased with the logneighbour/target (n/t) plant size ratio when neighbourswere larger than targets, n > t (F1,19 = 8.989, P < 0.008;Fig. 4c) but not when neighbours were smaller thantargets (F1,19 = 0.3467, P = 0.563; Fig. 4a). The rela-tionship was not significant when seed size rather thanplant size was used, although it was marginal in onecase (n > t, F1,19 = 3.887, P = 0.0634: n < t, F1,19 = 2.83,P = 0.109). In 1996, when the neighbour’s seed sizewas larger than the target’s, the segregation indexincreased with the log neighbour/target seed size ratio(F1,19 = 5.283, P = 0.033; Fig. 4d). This relationshipwas not significant when the neighbour’s seed size wassmaller than the target’s (F1,19 = 0.6042, P = 0.447;Fig. 4b). In 1996, neither relationship was significant ifplant size rather than seed size was used (n > t, F1,19 =2.947, P = 0.1023: n < t, F1,19 = 2.36, P = 0.141). There-fore, there is a tendency in both years for smaller plantsor those with smaller seeds to be more strongly segre-gated from their larger or larger-seeded neighbours,but not vice versa.

Table 3 The matrix of segregation indices in (a) 1994 and (b) 1996 calculated at a radius of 30 mm (the values given here were log-transformed for the analyses). The segregation index quantifies the degree to which the local ratio of conspecific to heterospecificneighbours is different to that expected from the population sizes. Segregation indices are not symmetrical for a given species-pair:for example, in 1996 the local ratio of Saxifraga to Geranium around Saxifraga target plants was about 55 times higher thanexpected from their respective population sizes, but the ratio of Geranium : Saxifraga around Geranium target plants was onlyabout 9 times higher than expected

Target species

Neighbour species

Veronica Saxifraga Cerastium Arenaria Aphanes Sherardia Geranium

(a) 1994Veronica 1 5.312 4.779 5.491 3.284 5.033 6.946Saxifraga 13.289 1 11.820 5.696 3.800 14.963 39.554Cerastium 2.054 1.200 1 1.515 2.091 2.650 1.986Arenaria 6.872 2.728 4.383 1 6.444 5.121 10.870Aphanes 9.695 4.449 14.318 15.108 1 8.142 15.810Sherardia 14.862 18.038 18.273 12.908 8.473 1 18.633Geranium 4.121 8.919 2.711 5.296 3.224 3.646 1

(b) 1996Veronica 1 6.729 4.848 5.134 4.131 5.416 7.990Saxifraga 10.944 1 7.245 9.580 7.464 14.436 54.051Cerastium 2.426 2.021 1 1.739 1.687 2.178 5.070Arenaria 2.235 2.371 1.537 1 2.306 2.552 4.639Aphanes 5.978 5.735 5.132 7.080 1 8.647 10.136Sherardia 3.916 5.013 3.251 4.401 4.470 1 6.539Geranium 2.383 9.023 3.065 3.158 2.160 2.892 1

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Association

There was a predominance of positive associationsbetween species pairs in both years rising from around50% at 10 mm to 75% at 30 mm. This was due to con-sistent associations of most species with bare earth(results not presented). Once again, only results from aradius of 30 mm are presented (Table 4) as results from

other radii are quantitatively similar. There weresignificant differences between target species (F6,30 =19.9472, P < 0.0001), neighbour species (F6,30 = 2.7716,P = 0.0289) and years (F1,30 = 22.215, P < 0.0001) in theassociation indices. Association indices were lower(less positive) in 1994 than in 1996. As the relative plantsizes of target and neighbour became more disparatethe value of the association index decreased in both

Fig. 4 The relationship between segregation indices and the log ratio of neighbour/target plant or seed sizes for all pairwisecombinations. The relationship was only significant when targets are smaller than their neighbours.

Table 4 The matrix of association indices in (a) 1994 and (b) 1996 calculated at a radius of 30 mm. Association indices aresymmetrical for the two members of a given species pair

Species 2

Species 1

Veronica Saxifraga Cerastium Arenaria Aphanes Sherardia Geranium

1994Veronica 1.665Saxifraga 0.00339 2.587Cerastium 0.103 0.130 0.821Arenaria −0.0297 0.861 0.414 1.865Aphanes 0.478 1.255 0.0852 0.0382 2.738Sherardia 0.053 −0.115 −0.152 0.231 0.650 2.744Geranium −0.271 −1.083 0.137 −0.521 −0.0140 −0.173 1.097

1996Veronica 4.829Saxifraga 3.219 5.845Cerastium 2.629 2.698 2.701Arenaria 1.835 2.774 1.633 2.441Aphanes 2.112 2.941 1.075 0.367 2.836Sherardia 1.576 1.462 0.727 0.460 0.639 2.624Geranium 0.142 −0.623 −0.0967 −1.406 −1.137 −1.406 −1.254

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1994 (F1,26 = 25.24, P < 0.0001; Fig. 5a) and 1996(F1,26 = 6.594, P = 0.0163; Fig. 5b). The same relation-ship occurred with seed size in 1994 (F1,26 = 14.9, P =0.0006; Fig. 5c) and 1996 (F1,26 = 5.701, P = 0.0245;Fig. 5d). There is a tendency for the variance toincrease with the mean but the relationships remainsignificant when tested with Spearman rank correla-tion (plant size: 1994 rs = − 0.719, P < 0.0001; 1996rs = −0.419, P < 0.0273; seed size: 1994 rs = −0.659,P = 0.0001; 1996 rs = −0.452, P < 0.0165).

Discussion

The neighbourhood modelling provided us withwell-defined estimates of competition coefficients forapproximately 75% of the interaction matrix and alsoappeared to be unbiased by the considerable spatialstructure in the community: there were no significantrelationships between mean local density and com-petition coefficients or between scaled competition coef-ficients and the degree of segregation. This is importantas spatial structure can potentially bias the measure-ment of competitive interactions (Freckleton &Watkinson 2000). The method is particularly usefulas it allows field-based estimation of individual-levelcompetitive interactions and these coefficients under-pin different models of coexistence.

One problem with the method is that the confidencein the estimates varies with the degree of spatial segre-

gation. This could lead to the incorrect grouping ofbiologically dissimilar species due to a lack of statist-ical power to distinguish them (type II errors). Forexample, Geranium was often a strong competitoragainst other species in the guild. However, associ-ations with Geranium were often negative and so inter-actions involving Geranium as a neighbour tend to bepoorly estimated (Appendix S2). A similar problemarises with the weighted regressions. Here the con-fidence in the estimates is largely determined by the fre-quency of interactions which is itself related to planttraits: species pairs with more disparate traits weremore strongly segregated (Fig. 5). This introduces asystematic bias between the weighting and plant traitswhich will inevitably flatten the regression slope aspoints at the ends (plants with different traits) willreceive lower weighting than points close to the origin(plants with similar traits). More generally, if stronginteractions result in spatial separation these inter-actions will be less well estimated and less stronglyweighted.

As expected, there were clear competitive differenceswithin the guild which rule out a null model of coexistencebased on species equivalence (Aarssen 1983; Hubbell2001). If species are not competitive equivalents thenthere must be some other mechanism to explain theircontinued coexistence (Chesson 1991). In 1996, scaledcompetition coefficients were positively correlated withthe neighbour/target seed size and plant size ratios.Large or large-seeded species therefore not only havean establishment advantage (Turnbull et al. 1999) but

Fig. 5 The relationship between association indices and the log ratio of neighbour/target seed size or plant size in two years.

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a true competitive advantage, and the greater thesize differential the greater the competitive differentialbetween species. To our knowledge, this is the first timesuch a result has been obtained for a multispecies com-munity in the field. Our results are also in agreementwith other studies of sand dune annuals which showcompetitive asymmetries linked to seed size (Mack& Harper 1977; Rees 1995; Coomes et al. 2002b) andclosely matches those for a group of perennialsgrown in pairwise competition trials in a greenhouse(Goldberg & Landa 1991; Freckleton & Watkinson2001b). In 1996, the existence of a single fully transitivedominance hierarchy was consistent with the assump-tions of the competition/colonization trade-off model(Tilman 1994). We also know from our seed sowingexperiment that large-seeded species are the moststrongly recruitment-limited, thus fulfilling the otherhalf of the trade-off (Turnbull et al. 1999). However,competitive interactions were not completely asym-metric and smaller neighbours had some impact onlarger targets. Indeed, the slope of the relationshipbetween the scaled competition coefficients andrelative plant sizes was less than 1, indicating that thecompetitive advantage of larger species is less than pre-dicted on a per-gram basis (Goldberg & Werner 1983).In addition, it is the large-seeded species which suffermost from competition overall, presumably becausethey achieve greater maximum size in the absence ofcompetitors by exploiting a larger area for resources.The cumulative effect of many small neighbours istherefore proportionally greater. Without absoluteasymmetry in competitive interactions the com-petition/colonization trade-off alone is insufficientto explain coexistence (Geritz et al. 1999; Adler &Mosquera 2000; Levine & Rees 2002). In this case, agroup of species would not evolve complementary seedsizes as a result of a seed size/seed number trade-off(Rees & Westoby 1997), but instead a single optimalseed size would always be selected (Levine & Rees 2002).Note, that the seed size/seed number trade-off can stillprovide a stabilizing mechanism and be a critical deter-minant of relative abundance patterns, provided someother mechanism maintains diversity (Levine & Rees2002). Other authors have concluded that a seed size/seed number trade-off is unlikely to be the maindeterminant of community structure in other annualsystems (Leishman 2001; Coomes et al. 2002a; Coomes& Grubb 2003).

In 1994, there was a very different pattern of com-petition: most interspecific coefficients were weakand the average scaled competition coefficient (whichexpresses the strength of inter- relative to intraspeci-fic competition) was significantly less than 0. Therewas also no relationship between the impact of a neigh-bour and its size relative to the target. This pattern ofindividual-level coefficients is very similar to the patternof population-level coefficients found by Rees et al.(1996) in a study of similar species in sand dunes. Theauthors thought that this might be due to spatial

segregation which resulted in weak interspecificinteractions. However, weak individual-level coeffi-cients cannot be explained in this way. One possibleexplanation for this pattern is that species specialize ondifferent patches in the environment for which they arethe best competitor. For example, sites might differ inmycorrhizas, water, or soil nutrient availability (Van derHeijden et al. 1998; Silvertown et al. 1999; McKaneet al. 2002). If species are largely confined to thepatches in which they are the best competitor, intra-specific coefficients will generally be stronger thaninterspecific coefficients at both the individual andpopulation levels. However, it is not clear why specieswould be confined to such patches in only one year ofour study.

The target species tend to show positive associationsbecause of their shared preference for patches free fromcompetition with perennials. The exception was Gera-nium, which showed negative associations with mostother target species (see also Coomes et al. 2002a). Theobserved spatial patterns support the hypothesis thatcommunities show fine-scale structure above andbeyond that generated by local dispersal (Mahdi &Law 1987; Purves & Law 2002b). Certain species pairsare more strongly associated than others and some ofthis variation can be attributed to plant traits: in bothyears species with more disparate seed or plant sizeshad more negative associations. Despite the lack of adirect link between spatial patterns and the strengthof competition, it is possible that these patterns are aresult of unmeasured density-dependent mortalityacting at the seedling stage. For example, seedlingsof different species might initially be well mixed butdifferences in plant traits could lead to unequal com-petition, with bigger species outcompeting smallerones (e.g. Coomes et al. 2002b). This would eventuallycause pairs of species with the most disparate planttraits to show the strongest negative associations.Alternatively, these spatial patterns could arise if spe-cies with different seed sizes specialize on differentmicrohabitat types. In this case, species which are moredisparate in their plant traits would also be moredisparate in terms of microhabitat use, and so havestronger negative associations.

All target species were segregated from each other,although to widely differing degrees. This occursdespite the predominance of positive associations. Thiswas also the case for an overlapping group of annualsgrowing in sand-dunes (Coomes et al. 2002a). Thedegree of segregation was occasionally extreme; thelocal ratio of Saxifraga : Geranium neighbours aroundSaxifraga plants was 55 times higher than expectedfrom the respective population sizes. Once again, planttraits were useful in predicting the degree of segrega-tion, but only when considering the patterns of largerneighbours around smaller targets. In this case, the

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more disparate the plant or seed sizes the higher thedegree of segregation. This would seem to benefit weakercompetitors because they experience a high relativefrequency of weak intraspecific interactions (Weiner &Conte 1981; Pacala 1997; Bolker & Pacala 1999). Thishas been shown greatly to increase the performance ofweaker competitors in experimental annual plant com-munities (Stoll & Prati 2001).

If the competition/colonization trade-off within our spe-cies is insufficiently asymmetric, then the coexistenceproblem remains. For sand-dune annuals, Coomeset al. (2002a) suggested that differential responses totemporal variability in the environment might beimportant, although this would require a much longertime series to confirm (Coomes & Grubb 2003). Inthe case of these annuals it would be interesting toinvestigate further the different patterns of competi-tion observed in the different years, in particularthe potential role of variability in the physical environ-ment. For example, seed size might be an adaptationto a particular type of microsite, with similarities inseed size representing similarities in microsite use. Thiswould explain why the association indices are related toseed size differences, with species becoming increas-ingly spatially separated as their seed size differentialincreases. Further investigation of competitive inter-actions under a range of controlled environmentalconditions would be required to confirm or refute thesehypotheses.

Conclusions

The competition/colonization trade-off can only workas a sole mechanism of coexistence when competitiveasymmetries are extreme. While the trade-off appearsto operate within this group of annuals, and almostcertainly acts as a stabilizing mechanism, it is notsufficient alone to explain their coexistence. In someyears, it appears that there is no direct advantage tolarger seeds but that, for each species, intraspecificinteractions are strongest. Spatial patterns were notdirectly related to competitive interactions, but pat-terns of association and segregation were linked toplant traits. It is possible that species specialize ondifferent microhabitat types but the nature of these‘spatial niches’ and their relationship to seed sizerequires further investigation.

Acknowledgements

This work was supported by the Natural EnvironmentResearch Council, UK, and the work was undertakenwith the permission and help of The National Trustand The Countryside Council for Wales. We thankBernhard Schmid for critical comments on the manu-script, and Ben Bolker for his advice on the expected

relationship between competition coefficients andsegregation indices. In addition, Jonathon Levine,Rob Freckleton and an anonymous referee providedextremely helpful comments on earlier versions of thismanuscript.

Supplementary material

The following material is available from http://www.blackwellpublishing.com/products/journals/suppmat/JEC/JEC856/JEC856sm.htm

Appendix S1a Classification of species recorded aspoint data into 6 categories for the neighbourhoodmodelling based on life-form considerations.

Appendix S1b Classification schemes for the back-ground matrix.

Appendix S2a The matrix of decay coefficients in 1994.

Appendix S2b The matrix of decay coefficients in 1996.

References

Aarssen, L.W. (1983) Ecological combining ability in plants:toward a general evolutionary theory of coexistence insystems of competition. American Naturalist, 122, 707–731.

Adler, F.R. & Mosquera, J. (2000) Is space necessary?Interference competition and limits to biodiversity. Ecology,81, 3226–3232.

Bolker, B.M. & Pacala, S.W. (1999) Spatial moment equationsfor plant competition: understanding spatial strategies andthe advantages of short dispersal. American Naturalist, 153,575–602.

Bolker, B.M., Pacala, S.W. & Neuhauser, C. (2003) Spatialdynamics in model plant communities: what do we reallyknow? American Naturalist, 162, 135–148.

Chesson, P.L. (1991) A need for niches? Trends in Ecology andEvolution, 6, 26–28.

Coomes, D.A. & Grubb, P.J. (2003) Colonization, tolerance,competition and seed-size variation within functionalgroups. Trends in Ecology and Evolution, 18, 283–291.

Coomes, D.A., Rees, M., Grubb, P.J. & Turnbull, L. (2002a)Are differences in seed mass among species important instructuring plant communities? Evidence from analyses ofspatial and temporal variation in dune-annual populations.Oikos, 96, 421–432.

Coomes, D.A., Rees, M., Turnbull, L. & Ratcliffe, S. (2002b)On the mechanisms of coexistence among annual-plantspecies, using neighbourhood techniques and simulationmodels. Plant Ecology, 163, 23–38.

Crawley, M.J. & May, R.M. (1987) Population-dynamics andplant community structure – competition between annuals andperennials. Journal of Theoretical Biology, 125, 475–489.

Dalling, J.W. & Hubbell, S.P. (2002) Seed size, growth rate andgap microsite conditions as determinants of recruitmentsuccess for pioneer species. Journal of Ecology, 90, 557–568.

Dixon, P. (1994) Testing spatial segregation using a nearest-neighbor contingency table. Ecology, 75, 1940–1948.

Emlen, J.M., Freeman, D.C. & Wagstaff, F. (1989) Interactionassessment – rationale and a test using desert plants. Evo-lutionary Ecology, 3, 115–149.

Eriksson, A. & Eriksson, O. (1997) Seedling recruitment in semi-natural pastures: the effects of disturbance, seed size, phenologyand seed bank. Nordic Journal of Botany, 17, 469–482.

109Competition and spatial patterns in annuals

© 2004 British Ecological Society, Journal of Ecology, 92, 97–109

Freckleton, R.P. & Watkinson, A.R. (2000) On detecting andmeasuring competition in spatially structured plant com-munities. Ecology Letters, 3, 423–432.

Freckleton, R.P. & Watkinson, A.R. (2001a) Nonmanipula-tive determination of plant community dynamics. Trends inEcology and Evolution, 16, 301–307.

Freckleton, R.P. & Watkinson, A.R. (2001b) Predictingcompetition coefficients for plant mixtures: reciprocity,transitivity and correlations with life-history traits. EcologyLetters, 4, 348–357.

Freckleton, R.P., Watkinson, A.R., Dowling, P.M. & Leys,A.R. (2000) Determinants of the abundance of invasiveannual weeds: community structure and non-equilibriumdynamics. Proceedings of the Royal Society of LondonSeries B-Biology Sciences, 267, 1153–1161.

Geritz, S.A.H., van der Meijden, E. & Metz, J.A.J. (1999) Evo-lutionary dynamics of seed size and seedling competitiveability. Theoretical Population Biology, 55, 324–343.

Goldberg, D.E. & Landa, K. (1991) Competitive effect andresponse: hierarchies and correlated traits in the earlystages of competition. Journal of Ecology, 79, 1013–1030.

Goldberg, D.E. & Werner, P.A. (1983) Equivalence of competitorsin plant-communities – a null hypothesis and a field experi-mental approach. American Journal of Botany, 70, 1098–1104.

Gross, K.L. (1984) Effects of seed size and growth form onseedling establishment of six monocarpic perennial plants.Journal of Ecology, 72, 369–387.

Gross, K.L. & Werner, P.A. (1982) Colonizing abilities of biennialplant-species in relation to ground cover – implications fortheir distributions in a successional sere. Ecology, 63, 921–931.

Grubb, P.J. (1986) Problems posed by sparse and patchily dis-tributed species in species-rich plant communities. Commu-nity Ecology (eds J. Diamond & T.J. Case), pp. 207–225.Harper & Row, New York.

Hubbell, S.P. (2001) The Unified Neutral Theory of Biodiversityand Biogeography. Princeton University Press, Princeton.

Law, R., Herben, T. & Dieckmann, U. (1997) Non-manipulativeestimates of competition coefficients in a montane grasslandcommunity. Journal of Ecology, 85, 505–517.

Leishman, M.R. (2001) Does the seed size/number trade-offmodel determine plant community structure? An assess-ment of the model mechanisms and their generality. Oikos,93, 294–302.

Levine, J.M. & Rees, M. (2002) Coexistence and relative abund-ance in annual plant assemblages: the roles of competitionand colonization. American Naturalist, 160, 452–467.

Levins, R. & Culver, D. (1971) Regional coexistence of speciesand competition between rare species. Proceedings of theNational Academy of Sciences of the USA, 68, 1246–1248.

Mack, R.N. & Harper, J.L. (1977) Interference in dune annu-als: spatial pattern and neighbourhood effects. Journal ofEcology, 65, 345–363.

Mahdi, A. & Law, R. (1987) On the spatial-organization ofplant species in a limestone grassland community. Journalof Ecology, 75, 459–476.

McConnaughay, K.D.M. & Bazzaz, F.A. (1987) The relationshipbetween gap size and performance of several colonizingannuals. Ecology, 68, 411–416.

McCullagh, P. & Nelder, J.A. (1989) Generalized LinearModels. Chapman & Hall, London.

McKane, R.B., Johnson, L.C., Shaver, G.R., Nadelhoffer,K.J., Rastetter, E.B., Fry, B., Giblin, A.E., Kielland, K.,Kwiatkowski, B.L., Laundre, J.A. & Murray, G. (2002)Resource-based niches provide a basis for plant species diver-sity and dominance in arctic tundra. Nature, 415, 68–71.

Neuhauser, C. (2001) Mathematical challenges in spatialecology. Notices of the American Mathematical Society, 48,1304–1314.

Neuhauser, C. & Pacala, S.W. (1999) An explicitly spatialversion of the Lotka–Volterra model with interspecificcompetition. Annals of Applied Probability, 9, 1226–1259.

Pacala, S.W. (1997) Dynamics of plant communities. Plant Ecology(ed. M.J. Crawley), pp. 532–555. Blackwell Science, Oxford.

Pacala, S.W. & Levin, S.A. (1997) Biologically generatedspatial pattern and the coexistence of competing species.Spatial Ecology. The Role of Space in Population Dynamicsand Interspecific Interactions (eds D. Tilman & P. Kareiva),pp. 204–232. Princeton University Press, Princeton.

Pacala, S.W. & Silander, J.A. (1985) Neighborhood models ofplant-population dynamics. 1. Single-species models ofannuals. American Naturalist, 125, 385–411.

Pacala, S.W. & Silander, J.A. (1987) Neighborhood interfer-ence among velvet leaf, Abutilon theophrasti, and pigweed,Amaranthus retroflexus. Oikos, 48, 217–224.

Pacala, S.W. & Silander, J.A. (1990) Field-tests of neighbor-hood population-dynamic models of 2 annual weed species.Ecological Monographs, 60, 113–134.

Purves, D.W. & Law, R. (2002a) Experimental derivation offunctions relating growth of Arabidopsis thaliana to neigh-bour size and distance. Journal of Ecology, 90, 882–894.

Purves, D.W. & Law, R. (2002b) Fine-scale spatial structure ina grassland community: quantifying the plant’s-eye view.Journal of Ecology, 90, 121–129.

Rees, M. (1995) Community structure in sand dune annuals. Isseed weight a key quantity? Journal of Ecology, 83, 857–863.

Rees, M., Grubb, P.J. & Kelly, D. (1996) Quantifying theimpact of competition and spatial heterogeneity on thestructure and dynamics of a four-species guild of winterannuals. American Naturalist, 147, 1–32.

Rees, M. & Westoby, M. (1997) Game-theoretical evolutionof seed mass in multi-species ecological models. Oikos, 78,116–126.

Ripley, B.D. (1981) Spatial Statistics. John Wiley, New York.Rodriguez-Girones, M.A., Sandsten, H. & Santamaria, L. (2003)

Asymmetric competition and the evolution of propagulesize. Journal of Ecology, 91, 554–562.

Silvertown, J., Dodd, M.E., Gowing, D.J.G. & Mountford, J.O.(1999) Hydrologically defined niches reveal a basis forspecies richness in plant communities. Nature, 400, 61.

Skellam (1951) Random dispersal in theoretical populations.Biometrika, 38, 196–218.

Stace, C. (1997) New Flora of the British Isles. CambridgeUniversity Press, Cambridge.

Stoll, P. & Prati, D. (2001) Intraspecific aggregation alterscompetitive interactions in experimental plant commun-ities. Ecology, 82, 319–327.

Takenaka, Y., Matsuda, H. & Iwasa, Y. (1997) Competitionand evolutionary stability of plaits in a spatially structuredhabitat. Researches on Population Ecology, 39, 67–75.

Tilman, D. (1994) Competition and biodiversity in spatiallystructured habitats. Ecology, 75, 2–16.

Turnbull, L.A. (1998) The role of spatial processes in a lime-stone grassland. PhD Thesis. Imperial College, London.

Turnbull, L.A., Rees, M. & Crawley, M.J. (1999) Seed massand the competition/colonization trade-off: a sowingexperiment. Journal of Ecology, 87, 899–912.

Van der Heijden, M.G.A., Klironomos, J.N., Ursic, M.,Moutoglis, P., StreitwolfEngel, R., Boller, T., Wiemken, A.& Sanders, I.R. (1998) Mycorrhizal fungal diversitydetermines plant biodiversity, ecosystem variability andproductivity. Nature, 396, 69–72.

Watkinson, A.R., Freckleton, R.P. & Forrester, L. (2000)Population dynamics of Vulpia ciliata: regional, patch andlocal dynamics. Journal of Ecology, 88, 1012–1029.

Weiner, J. & Conte, P.T. (1981) Dispersal and neighborhoodeffects in an annual plant competition model. EcologicalModelling, 13, 131–147.

Williams, D.A. (1982) Extra-binomial variation in logistic lin-ear models. Applied Statistics, 31, 144–148.

Received 24 July 2003 revision accepted 11 November 2003