Variation in moisture duration as a driver of coexistence by the storage effect in desert annual...

Theoretical Population Biology 92 (2014) 36–50

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Theoretical Population Biology

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Variation in moisture duration as a driver of coexistence by thestorage effect in desert annual plantsGalen Holt ∗, Peter ChessonDepartment of Ecology and Evolutionary Biology, University of Arizona, 1041 E Lowell St. Tucson, AZ 85721, United States

a r t i c l e i n f o

Article history:Received 7 May 2013Available online 14 November 2013

Keywords:Species coexistenceStorage effectEnvironmental variationGerminationDesert annual plantsSoil moisture

a b s t r a c t

Temporal environmental variation is a leading hypothesis for the coexistence of desert annual plants.Environmental variation is hypothesized to cause species-specific patterns of variation in germination,which then generates the storage effect coexistence mechanism. However, it has never been shown howsufficient species differences in germination patterns formultispecies coexistence can arise from a sharedfluctuating environment. Here we show that nonlinear germination responses to a single fluctuatingphysical environmental factor can lead to sufficient differences between species in germination patternfor the storage effect to yield coexistence of multiple species. We derive these nonlinear germinationresponses from experimental data on the effects of varying soil moisture duration. Although thesenonlinearities lead to strong species asymmetries in germination patterns, the relative nonlinearitycoexistence mechanism is minor compared with the storage effect. However, these asymmetries meanthat the storage effect can be negative for some species, which then only persist in the face ofinterspecific competition through average fitness advantages. This work shows how a low dimensionalphysical environment can nevertheless stabilizemultispecies coexistencewhen the species have differentnonlinear responses to common conditions, as supported by our experimental data.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Annual plant communities have figured prominently in boththeoretical and empirical studies of the contribution of envi-ronmental variation to the maintenance of species diversity.Theoretical work illustrates how environmentally sensitive ger-mination generates the storage effect coexistence mechanism ina temporally variable environment (Chesson, 1994, 2003; Ches-son et al., 2004). This theory has primarily focused on the coexis-tence of species competing for resources (e.g. Chesson, 1994, 2003),although more recent work has extended these results to thestorage effect acting through apparent competition (Kuang andChesson, 2008; Chesson and Kuang, 2010). Several field studieshave provided empirical support for the storage effect in annualplants arising from variable germination, variable growth, andcompetition between species (Facelli et al., 2005; Sears and Ches-son, 2007; Angert et al., 2009a).

Theoretical models of coexistence in a temporally variableenvironment normally do not model the physical environmentdirectly, but instead model fluctuations in population parameters,such as germination fraction, which are assumed to be driven

∗ Corresponding author.E-mail addresses: [email protected] (G. Holt), [email protected]

(P. Chesson).

0040-5809/$ – see front matter© 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.tpb.2013.10.007

by fluctuations in the physical environment (Chesson et al.,2004; Kuang and Chesson, 2009). The probability distributions offluctuations in these parameters (environmental responses) arethe inputs to the models. These studies have provided generalunderstanding of the ability of environmental fluctuations topromote coexistence in terms of the statistical properties ofthe environmentally dependent parameters (Chesson, 1994). Inparticular, the strength of the storage effect is determined bythe variances and correlations of these fluctuating environmentalresponses (Chesson, 1994, 2003; Angert et al., 2009b). However, akey problem is determining what these variances and correlationsare for input to the models.

Environmentally dependent germination rates provide theclearest example of fluctuating environmental responses inannual plants, although reproductive output also depends onenvironmental conditions during growth inways that can promotethe storage effect (Pake and Venable, 1995; Angert et al., 2009a).Although variances and correlations of germination fractions havebeen measured from field data (Angert et al., 2009a; Chessonet al., 2014) it is difficult to obtain long enough sequences ofobservations for much precision. In the absence of good estimatesof species differences in these respects, the phenomenologicalgermination fractions used in most theoretical studies assumea great deal of symmetry between species in the variances andcorrelations of germination fractions (Chesson et al., 2004; Kuangand Chesson, 2009). However, Angert et al. (2009a) provide good

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G. Holt, P. Chesson / Theoretical Population Biology 92 (2014) 36–50 37

evidence for substantial asymmetries in these statistical propertiesfor an annual plant community, in general agreementwith findingsfor tropical forests (Wright et al., 2005). The strong symmetriesimposed in theoretical models therefore may not provide accuratepredictions for how the storage effect operates in nature.

Because fluctuations in germination fractions are driven byenvironmental fluctuations (Grubb, 1977; Adondakis and Ven-able, 2004; Facelli et al., 2005), one possible approach to a betterunderstanding of the nature of germination fluctuations is to de-velop models of how fluctuations in physical environmental vari-ables drive fluctuations in germination. This approach, which weexplore here, potentially has great value for understanding coexis-tence in annual plants by better understanding the statistical struc-ture (variances and covariances) of the responses of the differentspecies in a community to the fluctuating environment. We arespecifically concerned with arid systems where communities ofannual plants form a major component of the flora (Venable et al.,1993; Facelli et al., 2005). In these systems, the soil surface, wheremost seeds can be found, is dry the majority of the time. Pulses ofrain dampen the soil for different periods following rainfall (Kemp,1983; Loik et al., 2004; Reynolds et al., 2004; Schwinning and Sala,2004), and depending on associated environmental factors, canlead to germination pulses in the annual plant community (Beat-ley, 1974; Chesson et al., 2004).

Field and laboratory experiments have explored links betweenthe environment and germination fraction, but in relatively limitedways. One principal finding is that variation in temperature atthe time of rainfall drives variation in the germination fractiondue to species-specific responses to temperature (Adondakis andVenable, 2004; Facelli et al., 2005; Chesson et al., 2014). It isalso known that soil moisture level, dormancy cycles, and thehistory of environmental conditions that a seed has experiencedcan affect the germination response (Grubb, 1977; Baskin andBaskin, 1986; Baskin et al., 1993; Adondakis and Venable, 2004;Facelli and Chesson, 2008). However, these studies often restrictattention to total germination at a fixed time after wetting ofthe soil (Baskin et al., 1993; Adondakis and Venable, 2004).Natural rainfall patterns lead to a wide range of durations ofsoil moisture with potential major effects on the amount ofgermination overall and the relative germination fractions ofdifferent species. Therefore, these studies potentially miss muchenvironmental variation of significance for the coexistence ofannual plant species. Here we study this important but neglectedaspect of environmental variation, the duration of sufficientmoisture for germination in the soil surface. We use our findingsto understand how species’ responses to a given environmentaldriver of germination can structure the variances and correlationsof germination fluctuations over time, and how the resultingasymmetric statistical properties affect species coexistence.

Seed germination is a biological process, and takes time asthe seeds imbibe water and begin meristematic activity (Fenner,1985; Bradbeer, 1988). As long as there is sufficient moisture,germination of a nondormant seed proceeds, but when themoisture is reduced to a sufficiently low level, germination stops(Fenner, 1985; Bradbeer, 1988). Seeds that have begun cell divisionare killed by the loss of moisture, while those that have not beguncell division dehydrate and rejoin the seed bank (Bradbeer, 1988;Baskin and Baskin, 1998). The germination fraction we considerhere is the fraction of seeds that have begun cell division, astheir germination is irreversible. The duration of soil moisturedetermines the duration of the germination process, and thereforethe number of seeds that complete the process and germinate.The germination process is a clear biological link between the soilmoisture duration and the resulting germination fractions.

To study how this link between soil moisture duration and ger-mination affects coexistence, we first experimentally determine

how desert annual plant species differ from one another in theirgermination patterns as a function of time while sufficient soilmoisture for germination is available. The patterns of germinationvariation are constrained by the one-dimensional nature of varia-tion in soil moisture duration and the germination biology of thespecies. We develop a model of germination fraction as a functionof soil moisture duration from the experimental results, and showhowgermination patterns generated by variation in the duration ofsoil moisture drive the storage effect. This analysis provides a pow-erful illustration of how a single, specific environmental variablehas the potential to cause major variation in relative germination,stabilizing species coexistence. Even though the physical environ-mental variable is one-dimensional, different nonlinear responsesto this variable allow multispecies coexistence.

2. Experiments

For temporal germination variation to affect coexistence, itmust lead to variation in relative germination fractions eitherbetween years (Chesson et al., 2004) or within years (Mathias andChesson, 2013). For variable soil moisture to generate this type ofvariation, it must lead to different communities of growing plantsat different times after rainfall. We conducted growth chambergermination studies on plants from a well-studied annual plantcommunity in the ChihuahuanDesert (Chesson et al., 2014). Detailsof the growth chamber study are provided in Appendix A, wherethe statistical tests demonstrating species-specific germinationcharacteristics are given. Here we summarize the key results.

The cumulative germination fractions of seeds of differentspecies as a function of time since the application of moistureare shown in Fig. 1. These curves show species-specific delaysbefore appreciable germination occurs. Each curve then rises withspecies-specific steepness to a plateau, which is the species-specific final germination fraction. These shapes differ significantlybetween specieswithin each temperature andmoisture treatment,and these shapes change significantly between temperature andmoisture treatments (Appendix A).

In nature, any rainfall event will be associated with a specifictemperature, moisture level, and moisture duration. Although wedid not experimentally vary moisture duration, the curves givecumulative germination as a function of time. Germination ceasesfollowing drying of the soil surface in the field, and so we interpretthe cumulative germination up to a particular time as the totalgermination that would be observed had we dried the soil atthat point. In nature the time of surface soil drying depends onmany factors such as sunshine, wind, and repeat rains, as wellas the total rain received and temperature. Our focus in thetheoretical developments here is the effect of moisture durationas revealed in the cumulative germination curves as functions oftime. The extent to which these curves lead to species-specificgermination variation as a consequence of variation in moistureduration depends on how their shapes differ between species.

Fig. 2 illustrates species-specific temporal patterns in germi-nation simulated from the experimental germination results. Al-though each species experiences the same soil moisture durationin each year, the species-specific nonlinearities in the germina-tion curves lead to fluctuating relative germination rates. Thesegermination patterns are not independent between species, asthey are often assumed to be theoretically, because the germina-tion for each species is monotonically dependent on the same un-derlying fluctuating environmental variable. This means that thedegree of statistical dependence between species is quite high.Indeed, beyond the delay before appreciable germination occurs,each species’ germination fraction can be predicted precisely fromany other species’ germination fraction from the knowledge oftheir germination curves.

38 G. Holt, P. Chesson / Theoretical Population Biology 92 (2014) 36–50

Fig. 1. Cumulative germination of desert annual plant seeds. Seeds of Erodium cicutarium (dark blue), Astragalus nuttalianus (green), Eriastrum diffusum (red), Lepidiumlasiocarpum (cyan), and Plantago patagonica (magenta) were placed in petri dishes of soil and maintained over time at the specified moisture levels and temperatures ingrowth chambers. The percent moisture is percentage of field capacity, i.e. the maximum ability of the soil to hold moisture. New germinants were counted every 48 h. Forfull details see Appendix A. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Simulated yearly germination. Yearly moisture duration, modeled as agamma random variable, determines yearly germination rate of each species fromcumulative germination curves from 16 °C, 80% moisture treatment. Species colorsas in Fig. 1. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

Fig. 1 shows that germination as a function of soil moistureduration is a sigmoid curve, which we model below. Our modelcaptures the key sorts of species differences seen in thesecurves. Different species show different delays before appreciablegermination occurs, different slopes of germination as a function ofmoisture duration, and different maximum germination fractions.However, we do not attempt to model the full variation ingermination conditions that occur in nature because far moreenvironmental data from the field would be required. Ourfocus is on how a single fluctuating environmental factor, heresoil moisture duration, can contribute substantially to speciescoexistence. This is not to deny the contributions of other factors,but to understand how multispecies coexistence can occur eventhough the environmental drivers of germination fluctuations arelow dimensional.

3. Model description

We use a standard model for the dynamics of a community ofcompeting annual plants in a variable environment (e.g. Chesson,1994), but model the germination fraction, Gj(t), of the species jin year t explicitly as a function of the soil moisture pattern in theyear t . In this model, the equation for the dynamics of species j(with population density Nj(t)) in discrete time is

Nj(t + 1) = Nj(t)λj(t), (1)

where

λj(t) = s(1 − Gj(t)) +YjGj(t)

lGl(t)Nl(t)

. (2)

Here, s is the fraction of nongerminating seeds that survive to thenext year in the soil seed bank. For simplicity, we assume thisfraction to be the same for all species. The quantity Yj is the averagenumber of seeds produced per germinated seed of species j whenthe total number of germinants per unit area (

l Gl(t)Nl(t)) is

equal to 1. We measure the magnitude of competition, C(t), as thenatural log of the number of germinants per unit area

C(t) = ln

l

Gl(t)Nl(t)

. (3)

This formula defines lottery competition, which has been used ina variety of studies of coexistence of annual plants in a variableenvironment (e.g. Chesson, 1994; Angert et al., 2009a). Other formsof competition are known to yield qualitatively similar results aslong as competition is an increasing function of the germinationfraction (Chesson, 1994, 2003; Kuang and Chesson, 2010).

To understand the consequences of moisture duration as adriver of variation in germination fraction, Gj(t), we model thegermination fraction for the simplest case, in which one rainfall


Fig. 3. Germination functions. Bold curves: cumulative germination fractions ofthree species as functions of soil moisture duration. Germination in year t is foundby evaluating these curves at u(t). Parameters: Solid line: φ1 = 0.1, ν1 = 0.25,m1 = 0.5, Dotted line: φ2 = 0.25, ν2 = 0.5,m2 = 0.4, Dashed line: φ3 = 1, ν3 = 1,m3 = 0.6. a = 0.99 for all species. Thin black line: gamma probability densityfunction defining probability of moisture durations, shape parameter α = 1, scaleparameter β = 4.

event accounts for the bulk of germination in a given year, witha small amount of germination occurring outside of this event.We define the germination fraction in year t as a function of themoisture duration in year t , whichwedesignate u(t). The followingspecific functional form captures the important qualitative aspectsof the empirical germination timeseries (Fig. 1).

Gj(t) = mj

1 − ae−νj[u(t)−φj]

+

. (4)

Here, (x)+ means x, if x is positive, and zero otherwise. This func-tion generates species-specific germination curves defining ger-mination fraction as a function of moisture duration (Fig. 3). Thegermination fraction depends on a species-specific delay, φj, un-til germination begins, a species-specific speed, νj, of adding newgerminants after germination begins, and a species-specific maxi-mumgermination fraction,mj. The constant a specifies aminimumgermination fractionmj(1− a), which is assumed to be the germi-nation arising from all sources other than the main rainfall event.Thisminimumgermination simplifies calculations because infinityis not encountered on taking logs of Gj(t). We model the moistureduration for the key rainfall event in year t , u(t), as a gamma dis-tributed random variable whose probability density function is de-picted in Fig. 3. This variation in u(t) over time generates temporalvariation in germination fraction as illustrated in Fig. 2.

4. Analysis of coexistence

We study coexistence using the technique of invasion analysis,which provides well-established methods and theory (Chessonand Ellner, 1989; Schreiber, 2012). As explained by Kuang andChesson (2008, 2010) and in Appendix B in more detail, we askwhether each species can increase following perturbation to lowdensity in the presence of its competitors. In this analysis, onespecies (the invader) is perturbed to effectively zero density andthe densities of the other species (the residents) are allowed toconverge on the stationary distribution they achieve in the absenceof the invader. The success of invasion ismeasured by the invader’slong-term low-density growth rate, ri, where rj(t) = ln λj(t). If aspecies has a positive ri, it will increase to rejoin the community.If each species has a positive invader growth rate, coexistence inthe community is stable. The signs of the invasion rates, ri, indicatewhether the species coexist and the magnitudes are an indicationof the strength of coexistence. In a model like the one here where

Table 1Notation used in the text.

Term Definition

j Index for any speciesi Denotes a species in the invader stater Denotes a species in the resident stateNj(t) Population size of species j in year tC(t) Competition in year tGj(t) Germination fraction for species j in year tu(t) Soil moisture duration in year ts Seedbank survival rateYj Per capita seed yield of species jwhen C(t) = 0mj Maximum germination fraction of species jνj Germination speed of species jφj Delay for germination of species ja Fraction of total germination outside the main rain pulse in a given yearλj(t) Finite rate of increase of species j in year t: Nj(t + 1)/Nj(t)rj(t) Per capita growth rate of species j in year t: ln(λj(t))ri Expected value of the invader growth rate: E[ri(t)]∆Ei Contribution to ri of fixed differences between species∆Ni Magnitude of relative nonlinearity∆Ii Magnitude of the storage effectEj(t) Environmental response of species j in year t : ln(Gj(t))C−i Competition when species i is invaderC∗

j Equilibrial competition for species jwhen Ej = Ejβj Rate at which seeds are lost from seedbank: 1 − s(1 − eE

∗j )

Table 2Mechanisms contributing to the invader growth rate.

Mechanism Formula

∆Ei C∗

i − C∗r

r+

12 s(1 − s)

var(Ei)

βi−

var(Er )

βr

r∆Ni

12 (βi − βr

r)var(C−i)

∆Ii scov(Er , C−i)

r− cov(Ei, C−i)

∆Ei: Average fitness difference between species i and its competitors.∆Ni: Relativenonlinearity.∆Ii: Storage effect. Subscript i: a species in the invader state; Subscriptr: resident state; Superscript −i: in the absence of the invader; Superscript r:average over the residents. These terms are measured on the natural scale of seedgenerations, 1/βj . Multiplication by βj returns these expressions to the yearly scale.The environmental response, Ej , is the natural log of the germination fraction:Ej = lnGj . C∗

j is equilibrial competition at Ej = Ej . For derivations see Appendix B.Note that relative nonlinearity is conventionally given with a negative sign, andtherefore promotes recovery of species i from low density if negative.

coexistence cannot occur in the absence of fluctuations, the ri canbe partitioned into contributions from three sources:

ri = ∆Ei − ∆Ni + ∆Ii (5)

(Chesson, 1994). The formulae for the components of Eq. (5) inthis particular application are derived in Appendix B and are givenin Table 2 using the natural timescale of seed generations, 1/βj,where βj is the probability applying independently for each yearthat a given seed of species j is lost from the seedbank (Table 1 andAppendix B).

In a constant environment, only the first term,∆Ei, is nonzero inEq. (5). The other two terms are nonzero in a variable environment,and are the magnitudes of the coexistence mechanisms relativenonlinearity (∆Ni) and the storage effect (∆Ii). These three termsare general, but their values depend on the characteristics ofparticular models. We therefore consider the effect of variation insoil moisture duration on coexistence by examining its potential toaffect each of these terms.

The first term, ∆Ei, is the average fitness difference of Chesson(2000b). Similar to a spatial version of thismodel (Chesson, 2000a),∆Ei can be related to Tilman’s R∗ rule in the case of a constantenvironment (Tilman, 1982). The quantity C∗

j corresponds to theinverse of R∗, even though the resource is not explicitly identifiedhere. The quantity C∗

j can be interpreted as the maximum amount


Fig. 4. Simulation of five species coexisting due to variable soil moisture duration.Moisture duration determines the germination fraction of each species according to(4). Soil moisture duration in each year is chosen from a gamma distribution withshape parameterα = 1 and scale parameterβ = 4. Commonparameters: s = 0.75,a = 0.99. Species-specific parameters: φ1 = 0.25, ν1 = 0.5, m1 = 0.44, Y1 = 26,φ2 = 0.5, ν2 = 0.58, m2 = 0.54, Y2 = 29, φ3 = 0.75, ν3 = 0.9, m3 = 0.61,Y3 = 30, φ4 = 1.0, ν4 = 1.5,m4 = 0.65, Y4 = 31, φ5 = 1.25, ν5 = 1.53,m5 = 0.7,Y5 = 33.

of competition that species j tolerates at equilibrium (Chesson,2000a). Only the species with the highest tolerance of competitioncan persist in an equilibrium setting, corresponding to Tilman’s R∗

rule (Tilman, 1982). When the environment varies,∆Ei is modifiedby variance in the environmental responses Ej (= lnGj), but moresignificantly, the ∆Ni, and ∆Ii terms appear.

The term ∆Ni is never very large in the situations studied here,but can be important in some circumstances (Kuang and Chesson,2008). Briefly, ∆Ni depends on nonlinearities in the populationgrowth rates, rj(t), as functions of C(t). When these nonlinearitiesare species-specific and C(t) fluctuates over time, ∆Ni is nonzero.In the present model, nonlinearity differences stem from species-specific longevities, 1/βj, of seeds in the seed bank (Kuang andChesson, 2008). These longevities are never very different in ourinvestigation, which focuses on the storage effect, ∆Ii, and so ∆Niis never very large, but it is present.

The storage effect, ∆Ii, measures the effect of interactionsbetween the response to the environment, Ej(t) and the responseto competition, C(t) (Chesson, 1994). For the storage effect topromote coexistence, species must differ in their responses tothe environment. Species-specific environmental responses arenot sufficient for coexistence, however. To promote coexistence,species-specific environmental responsesmust affect themediatorof the interaction between species, which is competition. Thisinteraction consists of two parts. First is the per unit change, γj,in the per capita growth rate, rj, as Ej and C are jointly varied:

γj =∂rj

∂Ej∂C, (6)

which here works out to be simply −s, i.e. the negative ofsurvival in the seed bank measured on the natural scale of seedgenerations. Second is a measure of the average joint variationin Ej and C , which is the covariance between them, cov(Ej, C).Multiplying this covariance by γj and comparing between species(Appendix B) gives the magnitude of the storage effect (Table 2).The contributions of γj and covariance between environment andcompetition (covEC) can be understood as follows.

CovEC measures the extent to which good germination yearscoincide with high competition. High covEC means that thebenefits of high germination are opposed by high competition,while a low covEC means that some years can be doubly good,because they have both high germination and low competition.

These doubly good years are especially favorable to populationgrowth, but it would seem that their effects might be canceledby doubly bad years that necessarily also arise with low covEC.This is where γj, determined here by survival in the seed bank,comes in. The seed bank moderates the effects of doubly bad yearsbecause when germination is low (bad E year), many seeds remainin the seed bank and are not exposed to competition. This meansthat a doubly bad year is only doubly bad for seed production.Instead, it is good for carryover of seed from previous successfulyears (Baskin and Baskin, 1978). The outcome is that low covECprovides a net benefit from environmental fluctuations relativeto high covEC where these doubly good and bad years do notarise. Therefore, invader time-average growth rates are boosted(coexistence is promoted) when invaders experience lower covECthan residents.

In many previous theoretical analyses, different species wereidentically affected by the storage effect, but this outcomedepended on the absence of species differences in the variancesof environmental responses. Such symmetry is unlikely in natureand is not a property of our present model. In such circumstances,to understand how the storage effect stabilizes the community asa whole, we study the community average storage effect (Chesson,2003, 2008), which measures the size of the coexistence region(Chesson, unpublished manuscript). The community averagestorage effect is approximated in Angert et al. (2009b) in terms ofthe variances and correlations of environmental responses as

∆I ≈

s

σ 2(1 − ρ) +

ρ

n−1

j(σj − σ )2

n − 1

. (7)

Here, σj measures the standard deviation of Ej as it fluctuatesin time. Therefore, σ 2 and σ are the averages over speciesrespectively of the variance and the standard deviation of their E’s(ln germination fraction) as they fluctuate in time. Similarly, ρ isa weighted average over pairs of species in the correlations overtime between their E’s, with the weights being the products of thestandard deviations of their E’s (Angert et al., 2009b).

Formula (7) shows that a strong community average storageeffect can be produced in two different ways: (1) low averagecorrelation with high average variance (the left term inside thebrackets), and (2) high average correlation with large differencesbetween species in standard deviation of the environmentresponse (the right term inside the brackets). In this model, σ 2, σand ρ are determined by the temporal fluctuations in soil moistureduration and the function (4) translating moisture duration intogermination fraction for each of the species. Thus, to examine howgermination variation due to variation in soil moisture durationgenerates the storage effect, we consider how these correlations,variances, and standard deviations arise. While the measurementof these statistical quantities is accomplished numerically, theirrelationship to the stabilization of coexistence by the storage effectshown in formula (7) is analytical.

5. Model results

The model shows that soil moisture duration alone is capableof supporting stable coexistence of multiple species, as illustratedin Fig. 4. Formula (7) provides the principal guidance for how thiscomes about. High correlation between the E’s of different speciesoften occurs in our model because there is a single underlyingdriver (soil moisture duration) of the environmental responses (lngermination fraction) of all species. However, different nonlinearresponses to this driver lower the average correlation. Standarddeviations of environmental responses naturally differ between


Fig. 5. Comparison of germination curves. Solid bold line, species 1, parameters fixedat m1 = 0.4, φ1 = 0.25, ν1 = 0.5. Dashed bold curve, reference curve for species2, proportional to the species 1 curve. Other curves: various possibilities for species2 representing differing departures from proportionality to the species 1 curve.Species 2 curves are created by incrementing the delay φ2 by 0.25 between curves,and adjusting the speed, ν2 , to yield the samemean (G2 = 0.3) at a fixedmaximumgermination fraction (m2 = 0.5) for each alternative species 2 curve. In all cases,a = 0.99.

species due to differences between species in sensitivity to theircommon driver. To analyze these effects, it is sufficient to considerjust two species.

Three species-specific parameters (delay, φj, speed, νj, andmaximum,mj) determine the germination fraction for each speciesby Eq. (4). Their effects on the variance can be studied by examininga single species, but correlation and standard deviation differencesrequire comparisons between species. We illustrate these effectsby fixing the germination parameters of one species while varyingthe germination parameters of the other species (Fig. 5). Becausethe variances, standard deviations and correlations are calculatedon the log scale of germination fraction (Ej = lnGj), they arenot affected by mean and maximum germination. Instead, theshapes of the curves, which are determined by φ and ν, arecritical. How these curves interact with varying moisture durationto determine the components of expression (7) is calculated bynumerical integration over the gamma distribution used for Fig. 4,and plotted against delay and speed in Figs. 6 and 7.

5.1. Community average storage effect

Formula (7) shows that increased average variance (σ 2) of theenvironmental responses increases the magnitude of the commu-nity average storage effect (∆I) as long as the average correla-tion is not 1, and increased differences in the standard deviations,(σj − σ )2, increase∆I when correlations are positive. The values ofthese statistical quantities, and the resulting strength of the com-munity average storage effect, are shown in Fig. 6 as functions ofthe delay of species 2, with fixed properties for species 1.

Species-specific delay and speed generate species-specificnonlinearities in the germination curves, which contribute tothe storage effect through both terms inside the brackets offormula (7). These species-specific nonlinearities reduce theaverage correlation between environmental responses, ρ, despitetheir dependence on a shared environmental driver (Fig. 6).Reduced correlation increases the contribution of the speciesaverage variance (σ 2) to the community average storage effect.Increasing delay and speed together increases the variance of Eover time (Fig. 6), as there are more years with no germination,and once germination starts, maximum germination is achievedsooner (Fig. 5). Thus, the differences between species in thestandard deviations of their environmental responses, asmeasuredby 1

n−1

j(σj − σ )2 in formula (7), increases as delay and

speed diverge between species (Fig. 6). The development ofthe storage effect in this model requires these species-specificnonlinearities in the germination curves as functions of theunderlying environmental driver. When species have the samedelay and speed, their environmental responses have identicalstandard deviations and a correlation of 1. Hence, formula (7) iszero, and there is no storage effect (Figs. 6–8 with φ2 = 0.25).

In the analysis above, illustrated by Fig. 5, the parameterdifferences between species can be considered to represent atradeoff between earliness of germination (shortness of the delay)and speed of germination once germination begins, affectingcorrelations between the environmental responses of differentspecies and differences in their variances. This tradeoff wasimposed by holding m2 and G2 constant as speed and delaywere varied. In our calculations, species 1 is fixed at one pointon the tradeoff curve as species 2 is varied creating varyingdifferences between the environmental responses of the two

Fig. 6. Components of the community average storage effect, ∆I , as functions of delay, φ. Components as described in Eq. (7). A: Species average variance of the Ej; B: Averagedifference between species in the standard deviation of the Ej; C: Average correlation between pairs of species in their environmental responses; D: The resulting communityaverage storage effect, ∆I . Parameters: Species 1 held at φ1 = 0.25, ν1 = 0.5. Delay and speed of species 2 vary together to maintain the same mean (G2 = 0.3) at fixedmaximum germination fraction (m2 = 0.5). Soil moisture duration modeled as gamma distributed with shape parameter α = 1 and scale parameter β = 4.


Fig. 7. Effect of speed and delay on the community average storage effect. Solid lines: Both speed and delay vary as in Fig. 6. A: Dashed line, speed held constant. B: Dashedline, delay held constant. Parameters: Species 1: φ1 = 0.25, ν1 = 0.5, m1 = 0.4. Species 2: all cases G2 = 0.3. Solid lines: φ2 and ν2 vary together to maintain m2 = 0.5. A:Dashed line, φ2 and m2 vary to maintain ν2 = 0.5. B: Dashed line, ν2 and m2 vary to maintain φ2 = 0.25. Soil moisture duration modeled as gamma distributed with shapeparameter α = 1 and scale parameter β = 4.

Fig. 8. Species-specific and community average storage effects. Dotted line: storageeffect for species 1; Dashed line: storage effect for species 2; Solid line: communityaverage storage effect. Species 1 held at φ1 = 0.25. All parameters as in Fig. 6.

species. However, there are other ways inwhich speciesmay differfrom one another affecting correlations and variance differences.We now hold delay and speed constant in turn, while allowingm2 to vary to maintain the same value of G2. Since m does notaffect the variance or correlation on the log scale, this allowsus to examine the separate effects of speed and delay on thecommunity average storage effect. We compare these results withthe community average storage effect that arises when delay andspeed can interact via a tradeoff (Fig. 7). We find that varying delayhas almost the same effect as varying both speed and delay in atradeoff. Speed alone has very little effect, and it is clear that whenthe two trade off, the change in the delay is much more importantthan the change in speed.

5.2. Species-specific storage effects

Variation in soil moisture duration does not affect the low-density growth rate of each species equally when their ger-mination rates are different nonlinear functions of this sharedenvironmental variable. We study these differences by consider-ing the species-specific storage effects (Table 2), which measurethe boost that the storage effect provides to the invader growthrate of each species individually. Appendix D shows how uniqueinvader–resident covariances arise, causing different species-specific storage effects. The species with the shorter delay has apositive storage effect. However, the species with the longer delay

Fig. 9. Coexistence regions shift as germination functions diverge. Coexistencepossible for Y1 : Y2 combinations between lines of the same type. Dashed lines:Similar germination functions. Species 1: φ1 = 0.25, ν1 = 0.5, m1 = 0.4; Species2:φ2 = 0.5, ν2 = 0.6,m2 = 0.5. Solid lines:More divergent germination functions.Species 1: φ1 = 0.25, ν1 = 0.5,m1 = 0.4; Species 2: φ2 = 1.0, ν2 = 1.0,m2 = 0.5.All cases : G2 = 0.3. Soil moisture duration modeled as gamma distributed withshape parameter α = 1 and scale parameter β = 4.

has a negative storage effect. Thus, the storage effect reduces theinvader growth rate of this species rather than boosting it. Despitethe negative storage effect of the longer delayed species, the com-munity average storage effect is always nonnegative (Fig. 8) due tothe positive correlation between environmental responses (7).

At first sight, a negative species-specific storage effect appearsto be at odds with the stabilizing effect of the nonnegativecommunity average storage effect. However, these two outcomescan be reconciled. In short, the increased stabilization measuredby the community average storage effect describes an increasein the size of the coexistence region, while the species-specificstorage effects describe a shift in the location of the coexistenceregion. As shown in Fig. 9, persistence of a longer-delayed speciesrequires an average fitness advantage, measured here as seed yieldadvantage, tomake up for the negative storage effect caused by lategermination. As the delay increases, this seed yield advantagemustincrease as well, shifting the coexistence region towards smallerY1 : Y2 ratios with increased delay. This is the effect of the differingspecies-specific storage effects. However, the range of seed yieldratios for which coexistence is possible increaseswith longer delay(Fig. 9). This increased size of the coexistence region is due to theincreased strength of the community average storage effect. Whilespecies-specific seed yields allow another mechanism to affect


Fig. 10. Coexistence regions for different sources of germination variation. Coexistencepossible for Y1 : Y2 combinations between lines of the same type. Solid lines: soilmoisture duration determines germination variation, (Gj(t) = GjDjd(t)); Dashedlines: phenomenological germination variation, (Gj(t) = GjDjr (t)); Dotted lines:germination variation from both soil moisture duration and phenomenologicalsources, (Gj(t) = GjDjd(t)Djr (t)). Parameters: Species 1: φ1 = 0.25, ν1 = 0.5,m1 = 0.4; Species 2:φ2 = 1.0, ν2 = 1.0,m2 = 0.5. Soilmoisture durationmodeledas gamma distributed with shape parameter α = 1 and scale parameter β = 4.

coexistence, viz relative nonlinearity (Appendix B), this effect isvery small compared to the storage effect (Appendix C).

5.3. Additional sources of germination variation

Soil moisture duration is only one environmental factor amongmany that generate germination variation. Normally, multiple si-multaneous environmental drivers of the environmental responsewould be expected, likely increasing the storage effect. Multipledrivers would ideally be studied with a detailed model of howall these drivers contribute jointly to the germination variation.Rarely, however, would that information be available. Instead, onemight account for additional sources of variation phenomenolog-ically to provide a more realistic picture of community dynamics.To do this, we define two multiplicative deviations, Dd and Dr , ofthe actual germination from the average germination fraction:

Gj(t) = GjDjd(t)Djr(t) (8)

where Djd(t) is the deviation due the specific soil moisture dura-tion in year t , and Djr(t) is the deviation due to other environmen-tal factors in year t . Thismultiplicativemodel for germination givesan additive model for the environmental response, Ej(t), the log ofgermination fraction, allowing the required variances and covari-ances to be readily calculated, especially if the germination devi-ations are assumed independent of one another. Unsurprisingly,variation in both Djd(t) and Djr(t) generates a larger coexistenceregion than either source of variation acting alone (Fig. 10). Theadvantage of this extended model is that it allows a more realisticexamination of the contribution to coexistence of an environmen-tal driver of germination variation in the usual case when it is oneamong many drivers, with the others not well understood.

6. Discussion

We have shown that environmental variation in just onedimension can powerfully promote species coexistence by thestorage effect. In nature, species share the temporally varyingenvironmental conditions they experience. Our experimentalresults demonstrate that while all species experience the samemoisture duration, different germination biology leads to species-specific nonlinearities in the germination response. Our modelcaptures this essential pattern, with each species’ environmental

response (ln germination fraction) being a fixed monotonicfunction of the same one-dimensional physical environmentalvariable (soil moisture duration) that varies stochastically overtime. Discounting very low germination, the environmentalresponse of any one species predicts the environmental responsesof all species. Previous models demonstrating the storage effecthave mostly assumed that the number of dimensions of thestochastic variation is equal to or larger than the number of species,and that this variation is symmetric (e.g. Chesson, 1994). Clearlythis is not necessary.

Classical equilibrium theory (Levin, 1970) emphasized thatthe number of density dependence factors limiting members ofa guild of coexisting species should be at least as large as thenumber of coexisting species. These density dependence factorsmight, for instance, be distinct resources. In variable environmentcoexistence theory the equilibrium requirement on the number ofdensity dependent limiting factors might have been replaced by arequirement on the dimensionality of the physical environment.Indeed, in most previous models of coexistence in a variableenvironment, the vector of environmental responses for the ncoexisting species is n dimensional (e.g. Chesson and Kuang, 2010).Here, however, this vector of environmental responses is a functionof a one-dimensional driver and therefore is one-dimensional.Despite this low dimensionality, coexistence occurs.

The critical nonlinearities in germination response can bedescribed by two parameters: delay, which is the maximummoisture duration for which there is no germination, and speed,which is the rate at which germination increases with moistureduration once delay has been exceeded. Of these two parameters,delay has by far the stronger effects in our results. Differencesbetween species in these parameters, which correspond todifferences in the nonlinearities of their germination responses,affectmultispecies coexistence in twodistinctways. First, althoughthe environmental responses (ln germination fractions) arefunctions of the same underlying varying environmental factor,differences in the nonlinearities of these functions mean thatan increase in moisture duration will not lead to proportionalincreases in germination across species. Hence, correlationsbetween species in environmental response are necessarily lessthan 1. Second, the environmental response fluctuations ofdifferent species have different standard deviations as a directconsequence of the fact that these responses are different functionsof the underlying environmental driver, moisture duration. Thecommunity average storage effect, given here by Eq. (7), showshow these two effects combine to promote species coexistence.

Long term observations of a desert annual plant communityshow the kinds of correlated but differentially variable germina-tion responses produced by these nonlinear responses to a com-mon environmental factor (Venable, 2007; Angert et al., 2009a).Even though the environmental responses of different species areclosely related to one another, they nevertheless lead to specieshaving large fluctuations over time in relative performance. Dif-ferences in relative performance in response to specific limitingfactors is a key feature of equilibrium coexistence (Tilman, 1982;Chesson and Kuang, 2008) and leads to the constraint on the di-mensionality of the limiting factors discussed above. That con-straint has been replaced here in this nonequilibrium setting byspecies-specific nonlinear relationships with a one-dimensionalfluctuating environmental factor.

Although different nonlinear responses to shared moistureduration variation can strongly stabilize coexistence, the species-specific delays driving these nonlinear responses mean that onespecies always germinates later than the others. We find thatthis longest-delayed species has a negative storage effect. Theenvironmental response of the longest-delayed species is mostsensitive to changes in the moisture duration, as measured by


its variance. Having the highest variance in the environmentalresponse causes the longest-delayed species’ covEC as invader tobe larger than the covEC’s of the residents (Appendix D). To persistin the community, the longest-delayed species must have anaverage fitness advantage. The storage effect stabilizes coexistenceat the community level because the positive storage effects ofshorter delayed species are of greater magnitude.

Nonlinear dependence of the environmental response on aone-dimensional environmental driver can be compared to thenonlinear dependence of population growth rates on competition,which leads to the relative nonlinearity coexistence mechanism,quantified by ∆N (Table 2). In this model, the relative nonlinearityof the growth rates as a function of competition requires differentrates of seed loss, β , from the seedbank, a feature that wasminimized in our analysis. In addition, species with larger seedloss need to generate larger fluctuations in competition than otherspecies when they are abundant (Kuang and Chesson, 2008). Thisrelationship between β and variance could arise in the modelconsidered here if the species with the higher seed loss rate alsohad a longer delay in germination in response to rain. Althoughsuch a scenario is a possibility in a model, and could promotecoexistence by relative nonlinearity, it seems maladaptive innature (Yuan and Chesson, unpublished manuscript).

Our results lend credence to previous claims from an empiricalstudy that variation in a specific one dimensional physicalenvironmental factor, growing seasonprecipitation, can contributepowerfully to diversity maintenance by the storage effect inannual plant communities (Angert et al., 2009a). However, theunderlying driver of the storage effect in the model presentedhere is precipitation available for germination rather than growth.Annual plants in the arid southwestern USA often have the bulkof their seed in the first few millimeters of soil, which dries outquickly after rain (Chesson et al., 2014). Moisture in this soil layeris essential for germination, but moisture in deeper soil layers,where plant roots are, fuels growth. Moreover, germination andgrowth are at least partially separated in time in these systems.Germinationmostly occurs in the fall and early winter, and growthlargely takes place in later winter and spring, with these processesdriven by distinct rainfall events occurring in different seasons(Ignace and Chesson, unpublished manuscript).

Levine and Rees (2004) also discuss a model and empirical sys-tem with one-dimensional environmental variation, where yearsare either good or bad for all species. The storage effect arises intheir model because one species responds to the environment onlythrough fecundity, while the other has environmentally sensitivefecundity and germination. However, in our model the storage ef-fect is generated by both one-dimensional environmental variationand nonlinearities in a single life history parameter, germinationfraction.

As increasing soil moisture duration increases the germinationfraction for all species in our model, all species rank the suitabilityof environmental conditions the same way, discounting theirinteractions with each other. Such concordant ranking of theenvironment is commonly referred to as inclusive niches, andis claimed to be common in plant communities, although mostauthors have spatial variation in mind (Westoby and Wright,2006). Our model of responses to moisture duration provides atemporal example of how this phenomenon can promote speciescoexistence. However, in the spatial case, there appear to beno comparable models of how coexistence occurs although itis frequently assumed (Keddy, 2001). Muller-Landau’s (2010)model of nested habitat suitability and Kelly and Bowler’s (2005)differential sensitivity model embody the essential idea. Likely, aspatial version of our model would provide a robust example ofcoexistence, as coexistence usually occurs more readily in spatialanalogues of models of temporal variation (Chesson, 1985, 2000a).

By explicitly considering the biology linking germination witha specific environmental variable, we can provide a basis for thevariances and covariances of the environmental responses, whichwere previously unavailable theoretically. We demonstrate thatthese biological links have strong control over the patterns of ger-mination variation, and therefore the development of the storageeffect. The resulting germination patterns are quite different thanthe symmetrical, independent germination variation commonlyassumed in theoretical work. Much asymmetry between speciesis found empirically in the variances and covariances of the envi-ronmental responses of different species of annual plants (Angertet al., 2009a). However, this empirical data is based on a limitednumber of years, and cannot determine the processes underlyingthe measured germination patterns. An explicit linkage betweenthe environmental driver and the plant responses provides a basisfor understanding such asymmetries, as we have seen here. Whilewe have focused here on the links between soil moisture dura-tion and germination fractions, the pattern of variation in any en-vironmental response depends on biological links to the physicalenvironment. We therefore expect studies of such links in othercommunities and for other environmental responses to yield simi-lar insight into the development of the storage effect in those com-munities.

It should be kept inmind that study of just onephysical environ-mental driver is an inadequate basis for precise prediction of thevariances and covariances of the environmental response. Germi-nation fractions are also critically dependent on other environmen-tal variables, such as temperature at the time of the rain pulse, asshown in our experimental data (Fig. 1 and Appendix A), as well asprevious studies (Adondakis and Venable, 2004; Facelli et al., 2005;Chesson et al., 2014). We have shown here how germination vari-ation resulting from different environmental drivers can be com-bined to determine overall diversity maintenance effects (Fig. 10),paving the way for development of more detailed and more real-istic models based on a fuller representation of key environmentaldrivers. Our experiments show that the interactions species× tem-perature and species × moisture are each significant predictors ofboth moisture duration parameters delay and speed (Appendix A).Consideration of these sorts of interactions, combined with goodestimates of real environmental variation, are likely to lead to im-proved estimation of the patterns of germination variation and thestabilization of coexistence in nature.

The particular environmental driver that we focused on herehas been much neglected in empirical studies of coexistencein annual plant communities, where total germination at afixed time with constant moisture supply has been the focusof analyses. However, descriptive studies of germination biologyoften find differences between species similar to those seen inour experimental results (e.g. Morgan, 1998; Wang et al., 1998;Li et al., 2006; El-Kassaby et al., 2008), suggesting that variationin soil moisture duration is potentially important for coexistencein many communities. This is particularly likely in arid regions,where moisture comes as distinct, highly variable pulses (Loiket al., 2004; Reynolds et al., 2004; Schwinning and Sala, 2004).Withlittle additional effort, such data could be collected in studies of therole of germination variation in species coexistence.

It is critical to note that species-specific responses to the en-vironment alone are not sufficient for coexistence to occur. Al-though species-specific responses to environmental properties cangenerally be regarded as distinguishing the response componentsof the niches of the species (Chesson, 2000b; Shea and Chesson,2002), such niche differences are not necessarily stabilizing nichedifferences, i.e. those that boost recovery of species from low den-sity (Chesson, 2000b). To do that these differential responses tothe environment must be coupled with the mediators of the in-teractions between species such that they intensify intraspecific


density-dependent interactions relative to interspecific density-dependent interactions (Kuang and Chesson, 2009, 2010).With thetemporal storage effect, that intensification arises as a long-termoutcome of integrating the effects of environmental fluctuationsover time.

In the model considered here, the mediator of the interactionbetween the species is competition. While species have both aresponse to competition and an effect on competition (Goldbergand Werner, 1983), stabilization of coexistence requires theenvironmental response to be coupled to the competitive effect.The storage effect measures this coupling between environmentalresponses and competitive effects as the difference between covECfor invaders and residents. CovEC is not the covariance between theenvironmental response and competitive effect, but the covariancebetween environmental response and competitive response.However, the necessary coupling between the environmentalresponse and competitive effect of a species leads to differencesin covEC between resident and invader states of a species. Forexample, in the extreme case of a single-species resident, thecompetitive effects and response are highly correlated, leadingto high covEC. However, an invader in this same situationwith an environmental response uncorrelated with the resident’senvironmental response will have zero covEC. Thus, species-specific responses to the environment, when combined withcoupling between the environmental responses and competitiveeffects, lead to multispecies density dependence of covEC.

One more factor is required for density dependence of covECto lead to a stabilizing effect, namely buffered population growth.This requirement is provided for here by the persistent seedbank. Buffering is expected to be a common feature of mostsystems, as it can arise from many different life histories, suchas an egg bank (Cáceres, 1997) or long lived adults (Kelly andBowler, 2002). Lower average covEC in invader versus residentstates becomes an advantage for an invader in the presence ofbuffered population growth. The invader’s growth rate is mademore variable by that lower covEC (Chesson and Huntly, 1989)but the buffered population growth means that strongly favorableconditions contribute more to population growth than stronglyunfavorable conditions, leading to a net advantage of lower covEC.

Numerous studies focus on differences between species inresponse to the environment alone without much attention tohow or whether they are coupled with the mediators of theinteractions between the species in a manner that would stabilizespecies coexistence (Siepielski and McPeek, 2010). However, suchcoupling is critical, and without it the simple finding that speciesare distinguishing by their responses to the environment cannotbe regarded as evidence of stabilizing niche differences (Chesson,2008; Siepielski and McPeek, 2010). With respect to the storageeffect, it is often straightforward to establish buffered populationgrowth (Chesson, 2008). The key then is to demonstrate thatcovEC is higher on average in resident than invader states. Thisidea suggests a program of empirical research (Chesson, 2008),but it could also receive much more attention in models toimprove understanding of coexistence mechanisms. Here we haveshown how asymmetries between species in their responses to aone-dimensional environmental driver can lead to these crucialcovariance differences and thereby stabilize coexistence by thestorage effect.

Acknowledgments

This work would not have been possible without generoussupport from NSF grant DEB-0816231. We thank Tara Woodcockfor helpwith experiments, andDanielle Ignace andAndreaMathiasfor advice and discussion. We thank three anonymous reviewersfor helpful comments on the manuscript.

Appendix A. Experimental methods

We experimentally tested the germination responses of fivespecies of desert annual plants subject to a range of environmentaltreatments. The species studied were Astragalus nuttalianus,Erodium cicutarium, Lepidium lasiocarpum, Plantago patagonica,and Eriastrum diffusum, derived from a well-studied communityof annual plants in the Chihuahuan Desert near Portal, Arizona(Chesson et al., 2014). Seeds from the study species were collectedin the spring of 2007 from plants that hadmatured their seeds. Theseeds were stored outdoors in shade in Tucson, Arizona to exposethem to a summer temperature regime.

The design of the germination study was a 3 × 4 factorial withfour replicates. It involved three moisture levels and four dailytemperature profiles. The response variables were cumulativegermination of each species as a function of time for the durationof the experiment, which we refer to as germination curves. Theinclusion of temperature and moisture treatments allows testsof whether the shapes of these curves change with temperatureand moisture, as well as comparison with previous work showingspecies-specific responses of the final germination fraction to thesefactors (Clauss and Venable, 2000, e.g. Adondakis and Venable,2004; Facelli et al., 2005). The details of the experimental designare as follows.

In September 2007, fifteen seeds of each species were placedon 75 g of paving sand (approx 1 cm depth) in each of forty-eight 9 cm glass petri dishes, and covered with 25 g of sand.Equal numbers of these plates were watered to 80%, 20%, and5% of field capacity, i.e. 16 plates for each moisture level. Plateswere wrapped in parafilm (Pechiney Plastic Packaging Company,Chicago, IL) and equal numbers of plates at each moisture levelwere placed in each of four growth chambers. Each growthchamber had a different temperature regime, defined by day/nighttemperatures of 16°/6 °C, 21°/11 °C, 26°/16 °C, or 31°/21 °C. Allchambers followed a daily cycle of 11 h of light and 13 h of dark.Germinationwas assessed for all plates every two days for 40 days,with germinants removed as observed. Ungerminated seeds wereassigned a time to germinate of 41 days.

To analyze the germination curves from this experiment,we performed separate analyses of variance for three responsevariables: delay, speed, and final germination. Each of these modelstested the explanatory variables species, temperature, moisturelevel, and the interactions species× temperature, species×moisture,and temperature × moisture. As there were multiple seeds of allspecies in each plate, plate is used as a random effect in all analysesto remove the common correlation induced by this aspect of thedesign. Analyses performed in JMP 10 (SAS Institute Inc. Cary, NC2012).

We define the delay of each seed as the time of germination. Thereciprocal of time to germinate is sensitive to small values and is anindicator of the extent to which a species tends to germinate early.We define this as speed. The final germination of each seed is 1 if theseed germinated at 40 days, and 0 if it did not, and thus measuresdifferences between treatments in the fraction of germinated seedsby the end of the experiment.

Specifying plate as a random effect means that the analysisdepends on averages at the plate level over the designatedresponse variables. Delay describes the mean time to germinateof each species in a plate, while speed describes the meanof the reciprocal. The nonlinearity of the relationship betweendelay and speed means that they describe different momentsof the distribution of germination timing of each species ina plate. The fact that the analysis depends on plate averagesmeans these variables, which at individual seed level are highlynonnormal, are partially normalized at the plate scale of theanalysis, according to the central limit theorem. Moreover, thebalanced experimental designmeans that the analysis isminimallyaffected by nonnormality.


Table A.1Significance tests of the fixed effects species, temperature, and moisture ongermination delay.

Fixed effects df den. df F ratio P

Species 4 2356 88.19 <0.0001Temperature 1 28 57.79 <0.0001Moisture level 1 28 3.75 0.0627Species × temp 4 2356 37.57 <0.0001Species × moisture 4 2356 4.83 0.0007Temp × moisture 1 28 0.65 0.4283

Table A.2Significance tests of the fixed effects species, temperature, and moisture ongermination speed.


Species 4 2356 57.35 <0.0001Temperature 1 28 19.94 <0.0001Moisture level 1 28 2.21 0.1482Species × temp 4 2356 22.32 <0.0001Species × moisture 4 2356 3.06 0.0158Temp × moisture 1 28 0.12 0.7335

Table A.3Significance tests of the fixed effects species, temperature, and moisture on finalgermination fraction.


Species 4 2356 93.98 <0.0001Temperature 1 28 71.72 <0.0001Moisture level 1 28 5.45 0.0270Species × temp 4 2356 29.88 <0.0001Species × moisture 4 2356 6.21 <0.0001Temp × moisture 1 28 1.41 0.2444

A.1. Results

No germination was observed in any petri dish with moistureat 5% field capacity, and these data are omitted from subsequentanalyses. The key result is a significant effect of species on all threeresponse variables: delay, speed, and final germination. In additionto species, temperature and the species × temperature and species× moisture interactions are also significant predictors of delay,speed and final germination (Tables A.1–A.3), while moisture is asignificant predictor of final germination (Table A.3). Fig. 1 in thetext illustrates each of these effects.

Appendix B. Invasion analysis, quadratic approximation andcoexistence mechanisms

Stable coexistence can be defined as a positive long-terminvader growth rate for each species, which can be related to otherdefinitions of stable coexistence, such as stochastic boundednessand the convergence of the probability distribution of populationsizes on a stationary distribution with all species having positivepopulation sizes (Chesson and Ellner, 1989; Ellner, 1989; Schreiberet al., 2011).

We denote this long-term invader growth rate as ri. Intuitively,when ri is positive, a species will recover if it is perturbed to lowdensity, allowing it to persist in the community. To measure theinvader growth rate, we hold the invader at a density of 0, whilethe residents are at the stationary distribution they achieve in theabsence of the invader.

As discussed in the text, rj(t) = ln λj(t) and Ej(t) = lnGj(t),therefore

rj(t) = ln[s(1 − eEj(t)) + YeEj(t)−Cj(t)]. (B.1)

This growth rate, rj, is a function of the environmental responseand competition, rj = gj(Ej, Cj). The mean invader growth

rate, ri, does not simply equal the growth rate evaluated at themean environmental condition due to properties of nonlinearaveraging (Chesson, 1994). We use methods from Chesson (1994)to approximate the invader growth rate, revealing coexistencemechanisms as quantitative contributions to long-term invadergrowth. The derivation below is a standard development similarto Kuang and Chesson (2010) and Chesson and Kuang (2010) butlike the corresponding spatial case (Chesson, 2000a) exhibits theinvasion criterion as a modification to Tilman’s R∗.

These approximations involve Taylor expansions, so we firstfind equilibrium values of C and E, which we refer to as C∗ andE∗, such that

gj(E∗, C∗) = 0, (B.2)

about which we take these expansions.We find a common C∗ for all species, and then find the E∗

j foreach species at C∗. Unlike most previous treatments, the speciesconsidered here do not have equal expected values of E, E[Ej].Therefore, we first find C∗

j , the equilibrium Cj for species j at E[Ej],

gj(E[Ej], C∗

j ) = 0 (B.3)

C∗

j = ln

YjeE[Ej]

1 − s(1 − eE[Ej])

. (B.4)

We then set the common C∗ equal to the mean of these C∗

j over allspecies,

C∗=

1n

l

C∗

l . (B.5)

To find E∗

j we solve (B.1) for E∗

j at C∗, giving

E∗

j = ln

1 − sYje−C∗

− s

. (B.6)

We now transform the right hand side of (B.1) into a function ofthe variables Ej and Cj, which are in units of growth rate. FollowingChesson (1994), we define

Ej = gj(Ej, C∗

j ) (B.7)

and

Cj = −gj(E∗

j , Cj). (B.8)

For the seedbank model used here, these give

Ej = ln[s(1 − eEj) + YeEj−C∗j ] (B.9)

and

Cj = − ln[s(1 − eE∗j ) + YeE

∗j −Cj ]. (B.10)

The growth rate can now be written in terms of Ej, Cj, and theirinteraction:

rj ≈ Ej − Cj + γjEjCj, (B.11)

where γj measures the strength of the interaction, and is given by

γj =∂2rj

∂Ej∂Cj. (B.12)

We find γj using parametric differentiation, which gives a γ that isnot species-specific since s is common for all species

γ = 1 −1

1 − s. (B.13)

The mean growth rate over time for a species can be found bytaking expected values of (B.11), which is what we will need forfinding the long term invader growth rate ri. The form of (B.11)


allows expected values to be taken of each term separately, whichrepresent the contribution of each term to the average growth rate,

rj = E[Ej] − E[Cj] + γ E[EjCj]. (B.14)

While this separates the contributions of the environment,competition, and their interaction to the average growth rate,the expected values of Ej, Cj and EjCj are not readily biologicallyinterpretable. However, their Taylor expansions lead to useful,biologically interpretable coexistence mechanisms. These Taylorexpansions follow:

Ej ≈ α(Ej − E∗

j ) −12α(1 − α)(Ej − E∗

j )2, (B.15)

Cj ≈ βj(Cj − C∗) −12βj(1 − βj)(Cj − C∗)2, (B.16)

and the interaction term, EjC−ij

EjC−ij ≈ α(Ej − E∗

j )βj(Cj − C∗). (B.17)

Here, αj and βj aredEjdEj

and dCjdCj

, respectively, giving

βj = 1 − s(1 − eE∗j ) (B.18)

and

αj = (1 − s). (B.19)Since s is common, α is equal between species. However, βj isspecies-specific when E∗

j is species-specific. Since E∗

j depends onYj, (B.6), βj is species-specific when the Yj are species-specific.Here, βj is approximately the probability of a seed being lost fromthe seedbank, βj ≈ 1 − s(1 − Gj), and therefore defines a seedgeneration.

Using small variation assumptions as in Chesson (1994), theexpected values of Ej, Cj, and EjCj simplify to functions of thevariances and covariances of E and C as follows

E[Ej] ≈ (1 − s)(E[Ej] − E∗

j ) +12s(1 − s)var(Ej), (B.20)

E[Cj] ≈ βj(E[Cj] − C∗) −12βj(1 − βj)var(Cj), (B.21)

and

E[EjCj] ≈ αβj cov(Ej, Cj). (B.22)While the average invader growth rate can be found by

evaluating these expected values for i = j, it is most informativeto write the expected invader growth rate as the sum of threecoexistence mechanisms, ∆Ei, ∆Ni, and ∆Ii, which representcomparisons between invaders and residents, as described inChesson (1994). Expressing each term as an invader–residentcomparison is possible because by definition, the average residentgrowth rate is zero, rr = 0, although each term on the righthand side of (B.14) normally will not be 0. Since ∆Ei, ∆Ni, and ∆Iiare comparisons for each species as invader while the others areresidents, each term is species-specific.

We define these mechanisms on the natural timescale of seedgenerations, i.e. 1/βj (Chesson, 2003). The yearly scale can berecovered bymultiplying these formulae byβj. The invader growthrate is now written asriβ i

= ∆Ei − ∆Ni + ∆Ii, (B.23)

where

∆Ei =E[Ei]

βi−

1n − 1

r

E[Er ]

βr, (B.24)

∆Ni =E[Ci]

βi−

1n − 1

r

E[Cr ]

βr, (B.25)

and

∆Ii = γE[EiC

−ii ]

βi−

γ

n − 1

r

E[ErC−ir ]

βr. (B.26)

Recalling that the invader has density = 0, it does not contributeto competition, and thus we use the notation −i in superscript toindicate competition in the absence of species i.

The equations for relative nonlinearity, ∆Ni, and the storageeffect, ∆Ii, can be found by straightforward subtraction of therespective expected values given in Eqs. (B.21) and (B.22).

∆Ni =12(βi − βr

r)var(C−i) (B.27)

∆Ii = scov(Er , C−i)

r− cov(Ei, C−i)

. (B.28)

The superscript r on the average means the average taken overthe residents. In this model, the variation independent parts of(B.21) cancel on subtraction, leaving only the effect of variationin competition, measured by ∆Ni (Chesson, 1994). An expressionfor ∆Ei can also be found by simply performing the subtractionin (B.24), but the result is not as useful. Instead, we recognizethat gj(E[Ej], C∗

j ) = 0 by the definition of C∗

j . To a first orderapproximation, α(E[Ej] − E∗

j ) + βj(C∗

j − C∗) ≈ 0 and thereforewe can substitute βj(C∗

j − C∗) for α(E[Ej] − E∗

j ) in (B.15) to obtain

E[Ej] ≈ βj(C∗

j − C∗) −12α(1 − α)var(Ej). (B.29)

This formula for the expected value provides a more biologicallyinterpretable form of ∆Ei,

∆Ei = C∗

i − C∗r

r+

12s(1 − s)

var(Ei)

βi−

var(Er)

βr

r, (B.30)

which compares the equilibrial competition for each species attheir respective mean environmental responses, corresponding toTilman’s R∗ rule as discussed in the text, and shown previously fora spatial version of this model (Chesson, 2000a). This comparisonbetween C∗

j ’s is modified by the difference in variation of Ebetween invaders and residents. Since the variance of Ej changeswith species identity but does not depend on whether a speciesis in the invader or resident state, these differences affect theaverage fitness differences between species, but do not affect thestabilization of coexistence.

B.1. Community average storage effect

Each of the terms ∆Ei, ∆Ni, and ∆Ii, are species-specific. Inour model, the strong asymmetry in the patterns of germinationvariation between species leads to quite different values of ∆Iifor each species, as described in the text. The community averagestorage effect, ∆I , on the other hand, measures how much thestorage effect increases the invader growth rates on average, andthus the stabilizing effect of the storage effect on the communityas a whole (Chesson, 2003, 2008).

In the following section we discuss the approximation of thecommunity average storage effect used in the text (Eq. (7)). Wefollow the methods outlined in the appendix of Angert et al.(2009a), which is for a more complex model. We present theimportant steps and quantities here for clarity.

The community average storage effect is defined in Chesson(2008) as

∆I =1n

nj=1

γj

βj

χ

{−j}j − χ

{−i}j

{i=j}

. (B.31)


Here, χj = cov(Ej, C), the superscript {−i} indicates the case withspecies i as invader, and the superscript {i = j} on the averagemeans the average over all species as invader except j. Inwords, thequantity in parentheses is the comparison between cov(Ej, C) fora species as invader with the cov(Ej, C)’s for that same species asresident averaged over all other species in turn as invader. The restof the equation averages thesewithin-species comparisons over allspecies.

This expression can be approximated by

∆I ≈−γj

βj

σ 2txs

n − 1

, (B.32)

as described inAngert et al. (2009b). Here,σ 2txs is the timeby species

variance in E. This development does not transform the equationfor rj into a function of the variables Ej and Cj, as is done above.These two methods lead to equivalent results, since E and C aretransformations of E and C onto the scale of growth rate (Chesson,1989). While the results are equivalent, their expression is notidentical, as the transformation must be accounted for. We use theform for rj given in (B.1), as this is in terms of E and C . In this case,βjand γj are defined by differentiation with respect to E and C ratherthan E and C :

βj =−∂rj∂C

= 1 − s(1 − eEj) (B.33)

and

γj =∂2rj

∂Ej∂C= −sβj. (B.34)

On this scale, βj is the same as the βj found earlier, but γj is not thesame as the γj in the previous section. With these expressions forβj and γj, (B.32) becomes

∆I ≈sσ 2

txs

n − 1. (B.35)

Nowweneed the time by species variance,which is given in Angertet al. (2009b) as

σ 2(1 − ρ) +ρ

n − 1

j

(σj − σ )2, (B.36)

where σ 2 is the species average variance of E. Here, ρij is thetemporal correlation of Ei(t) and Ej(t), making ρ the weightedaverage correlation

ρ =

i=j

ρijσiσji=j

σiσj. (B.37)

We now have the expression for the community average storageeffect used in the text, Eq. (7):

∆I ≈

s

σ 2(1 − ρ) +

ρ

n−1

j(σj − σ )2

n − 1

. (B.38)

Appendix C. Approximations and the relative strength of coex-istence mechanisms

The functional descriptions of the coexistence mechanismsgiven in Appendix B are approximations. It is therefore usefulto check whether they are close enough. We do that here bypresenting coexistence regions for simulated measurements ofinvader growth rates, which do not involve approximations,compared with the invader growth rates calculated based on the

Fig. C.1. Comparison of approximations, simulations, and strength of relativenonlinearity. Coexistence possible for Y1 : Y2 combinations between lines of thesame type. Solid lines: full approximation of invader growth rate according toC.1; Dashed lines: simulated invader growth rate; Dotted lines: approximationof invader growth rate without relative nonlinearity, according to C.2. The upperdotted line coincides with the upper solid line, obscuring it. Parameters: φ1 = 0.25,ν1 = 0.5, m1 = 0.4, φ2 = 1.0, ν2 = 1.0, m2 = 0.5.

approximations developed in Appendix B (Fig. C.1):

ri ≈ C∗

i − C∗r

r+

12s(1 − s)

var(Ei)

βi−

var(Er)

βr

r

−12(βi − βr

r)var(C−i) + s

cov(Er , C−i)

r− cov(Ei, C−i)

.

(C.39)

We present the comparison of approximate and simulatedcoexistence regions in seed yield space, as in Figs. 9 and 10 inthe text. When the Y ’s differ between species, the βj will differ.This leads to relative nonlinearity, as described in Appendix B. Wetherefore use this opportunity to compare the effect of relativenonlinearity to that of the storage effect.Wemake this comparisonusing a coexistence region calculatedwithout relative nonlinearity,where the invader growth rate is

ri ≈ C∗

i − C∗r

r+

12s(1 − s)

var(Ei)

βi−

var(Er)

βr

r

+ scov(Er , C−i)

r− cov(Ei, C−i)

. (C.40)

We see in Fig. C.1 that the region based on the full approximation(between solid lines) matches closely with the region definedby simulation (between dashed lines). While there are smallquantitative differences, these regions are qualitatively verysimilar. The region defined without relative nonlinearity is shownbetween dotted lines. Since there is little difference betweenthe regions with and without relative nonlinearity, relativenonlinearity has small effect, with the storage effect being theprimary driver of coexistence.

Appendix D. Delay length and relationships between E and Cfor invaders

Our results show species-specific storage effects, and ofparticular interest, negative storage effects for the longer-delayedspecies. The positive storage effect for the shorter-delayed speciesis always of greater magnitude than the negative storage effectof the longer-delayed species, however, leading to a nonnegativecommunity average storage effect in all cases, as seen in Fig. 8 anddiscussed in the text.


Fig. D.1. Relationship between environment and competition for a short-delayedinvader. Points are simulated E and C values for two species. Species 1 (green)invader, species 2 (blue) resident. Parameters: Species 1: φ1 = 0.25, ν1 = 0.5,m1 = 0.4. Species 2, φ2 = 1.0, ν2 = 1.0, m2 = 0.5. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. D.2. Relationship between environment and competition for a long-delayedinvader. Points are simulated E and C values for two species. Species 1 (green)resident, species 2 (blue) invader. Parameters as in D.1. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

The way the asymmetries between species’ germinationresponses generate these species-specific storage effects is notimmediately obvious, as both invaders and residents havepositive relationships between E and C . Therefore, the covariancesbetween E and C , which underlie the storage effect, bear furtherexamination. We visualize the development of these covariancesby plotting the pairs of E and C for both residents and invaders ateach timestep in a simulation.

When the species with shorter delay is the invader, itexperiences low competition until the longer-delayed speciesbegins germination. In Fig. D.1, the higher-delayed resident doesnot start germinating until the lower-delayed invader has alreadyreached about E = −2. Therefore, for moisture durations shorterthan those leading to E = −2, the shorter delayed invaderescapes competition from the resident. However, the competitionexperienced by the invader rapidly increases once the residentbegins germination. The slope of the relationship between E andC is actually higher for the invader than the resident (Fig. D.1).However, the higher slope does not lead to a greater covariance.As we saw in the text, the variance of E rapidly increases withincreasing delay because there are many more years with nogermination. In Fig. D.1, there are many more blue points at aboutE = −5.25 than green points at about E = −5.5. This increased

variance for the resident causes covEC for a long-delayed residentto be greater than covEC for a short-delayed invader, leading to apositive storage effect for the shorter-delayed species.

Since the longer-delayed invader always germinates after theresident, it always experiences competition when it germinates,seen in the high C values for all blue points above an E of about−5.25 in Fig. D.2. However, a longer delayed invader has a highervariance of E, caused by more years with no germination, seen inthe vertical set of points at an E of approximately −5.25. Thesepoints span a range of C values, because in some years, the shorter-delayed resident will have germinated and generated competitionwithout germination of the longer-delayed invader. The buildupof competition before the invader starts germinating leads it tohave a lower slope in its relationship between E and C than theless-delayed resident. However, the greater germination variancecaused by long delay leads the longer-delayed invader to havea higher covEC than the early germinating resident, giving it anegative storage effect (Fig. D.2).

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