The role of fluctuating resource supply in a habitat maintained by the competition-colonization trade-off

Ann. N.Y. Acad. Sci. ISSN 0077-8923

ANNALS OF THE NEW YORK ACADEMY OF SCIENCESIssue: Ecological Complexity and Sustainability

The role of fluctuating resource supply in a habitatmaintained by the competition-colonization trade-off

Amit Chakraborty1,2 and Bai-Lian Li1,2,3

1Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California,Riverside, California, USA. 2Center for Conservation Biology, University of California, Riverside, California, USA. 3CAU-UCRInternational Center for Ecology and Sustainability, University of California, Riverside, California, USA

Address for correspondence: Amit Chakraborty, Ecological Complexity and Modeling Laboratory, Department of Botany andPlant Sciences, University of California, Riverside, California, CA 92521-0124. Voice. +951-827-4776; fax: [email protected]

An exotic species can be a superior colonizer but inferior resource competitor relative to native species. Such speciescan spatially coexist for an extended period of time in a community maintained via competition-colonization (CC)trade-offs. Whether fluctuations in resource supply allow such exotic species to successfully invade and displace thenative species or hold the coexistence is not previously explored, and it is the focus of this study. In this article, wemodel propagule-limited spatial competition explicitly linked with resource-competition within the framework ofthe classic CC-model, while time-dependent fluctuations in resource supply are considered in a sinusoidal function.The model predicts that if the amplitude of the fluctuations is greater than the average resource supply rate, thereexist a range of values of fluctuation frequency that can allow the exotic species to successfully invade the habitatand to reduce the extent of native species. On the other hand, if the fluctuation amplitude is less than the averageresource supply rate, such exotic species can coexist with the native species, independent of fluctuation frequencies.In addition, we found that at a constant resource supply rate, the exotic species can stably coexist with the nativespecies at competitive equilibrium.

Keywords: invasibility; coexistence; competition-colonization trade-offs; fluctuating resource supply

Introduction

The fluctuating resource supply in a habitat canplay dual roles: it can promote species coexistence(Chesson 2000; Roxburgh et al. 2004; Snyder andChesson 2004) or it can increase susceptibility toinvasion by an invader species (Davis et al. 2000;Schoolmaster and Snyder 2007). While maintain-ing a constant rate of resource supply, interspe-cific resource competition would reach an equilib-rium in which a competitively superior species ei-ther excludes competitively inferior species or boththe species coexist under certain conditions, as re-vealed from modeling studies (e.g., Houston andDeAngelis 1994). But, while resource supply fluc-tuates over time, variable outcomes are possible—multispecies coexistence (e.g., Hsu 1980), domi-nance by competitively inferior species (e.g., Grover1990), increased invasibility, and successful invasion

by an exotic species (this issue). Fluctuations in re-source supply can reduce interspecific competitionintensity (Davis et al. 2000), which is then resultedin decreased rate of exclusion of inferior competitorspecies by superior competitor species from occu-pied patches and thereby can promote coexistence ofsuperior and inferior competitor species (this issue).Alternatively, that reduction in competition inten-sity creates an opportunity for an exotic species tosuccessfully establish in a new habitat, increasingsusceptibility to invasion (Davis et al. 2000). In thefluctuating resource environment, whether this suc-cessfully established exotic species (naturalized ex-otic species) would invade and displace native com-petitor species or coexist depends on the nature offluctuations, and on the ability of native and exoticspecies to respond to fluctuations. We demonstratethese phenomena using a model that includes fluc-tuations in resource supply as a sinusoidal function

doi: 10.1111/j.1749-6632.2009.05399.xAnn. N.Y. Acad. Sci. 1195 (2010) E27–E39 c© 2010 New York Academy of Sciences. E27

The role of fluctuating resource supply Chakraborty & Li

that could be associated with the actual nature ofseasonal fluctuations in nutrient supply or rainfall.

An invading exotic species often displays traitsthat favor colonization (Radford and Cousens2000; Kolar and Lodge 2001; Lloret et al. 2005;Lockwood et al. 2005). These traits allow invadersto reach at empty microsites before the native com-petitor species do, however, these are usually asso-ciated with low competitive ability (Seabloom et al.2003a; Arii and Parrott 2006). Thus, there mayexist trade-offs between species competitive abil-ity and colonization ability (CC-trade-offs). Thispossibility can allow such exotic species to coexistwith the native species. For instance, a recent ex-periment in California grasslands (Seabloom et al.2003a) found that exotic annual species are infe-rior resource competitors but superior colonizersrelative to native perennial species, and they cancoexist for an extended period of time. Therefore,it seems that the competition-colonization (CC)tradeoff-based theories could be used to explaincoexistence of such exotic and native species. Par-ticularly, the classic CC-model, formulated first byHastings (1980) and studied later by Tilman (1994),can describe spatial competition between that exoticand native species as the model relies on the emi-nent hypothesis that species that are very good atcolonization are poor competitors. With these givendifferences among the competing species, the modelpredicts stable equilibrium coexistence of superiorand inferior competitor species while there is a limitto similarity between the species which is referredas “niche shadow” by Kinzig et al. (1999). How-ever, whether fluctuating resource supply within thismodel system still holds spatial competitive coexis-tence of multiple species has not yet been explored.Here, we use a modified version of this model todetermine invasibility and coexistence criteria foran exotic species with superior colonization abil-ity but inferior competitive ability relative to nativespecies.

Model and methods

The modified CC-model has explicitly linkedpropagule-limited spatial competition and resourcecompetition without altering the basic assumptionsand structure of the original CC-model. It considersa habitat that is composed of number of micrositeslinked to each other by colonization of interacting

species, and each microsite is either empty or occu-pied and is the size of the area occupied by one adult.Species are ranked according to their strict com-petitive hierarchy, with competitive ability tradingoff against colonization ability. Species’ competi-tive ranking is assumed to be spatially invariant,that is, it does not change in space. The basic as-sumption of the model is that a superior competitorspecies can displace an inferior competitor speciesfrom occupied microsites but an inferior competi-tor species cannot displace a superior competitorspecies. Within this framework, N species spatialcompetition as described in the original CC-modelis written below

dpi

dt= cipi

⎛⎝1 −

i∑j=1

pj

⎞⎠ −

⎛⎝ i−1∑

j=1

cjpjpi

⎞⎠

−mi pi , (i = 1, 2, . . . , N), (1)

where pi is the fraction of microsites occupied byith species, which will be called abundance of thespecies, species i; ci is the colonization rate, which isgreater than cj for i > j; mi is the per capita mortalityrate, summarizing all forms of density-independentmortality of adults; and cipi refers to the rate ofpropagule production from occupied microsites. Inthe right-hand side of Eq. (1), the first term repre-sents the rate of production of newly occupied mi-crosites, and the second term represents the rate ofdisplacement by superior competitor species. Onesimplifying feature of this model is that fecundity,recruitment, dispersal tendency, and dispersal rateare subsumed into a single parameter termed as thecolonization rate.

For incorporating resource competition explicitlywithin this model framework, we employ two addi-tional assumptions: first, species are competing fora single limiting resource, which is uniformly dis-tributed within the area being considered and sec-ond, we view colonization as resource-dependent,propagule-limited recruitment of microsites. Thefirst one is a general assumption of the classic re-source competition model, which is often beingused to explain species coexistence in a homoge-nous environment in which species are different intheir abilities to use the available resource (Hous-ton and DeAngelis 1994; Grover 1997). The secondassumption is related to a local spatial competitionin which competing species are propagule limiteddue to the fact that propagule production is limited

E28 Ann. N.Y. Acad. Sci. 1195 (2010) E27–E39 c© 2010 New York Academy of Sciences.

Chakraborty & Li The role of fluctuating resource supply

by the available resource shared by the competingspecies. Since there is no effect of propagule dispersalon microsite recruitment because the CC-model as-sumed that each propagule has an equal probabilityof falling in neighboring microsites or anywhere elsein the habitat, we can approximate the recruitmentrate as the rate of propagule production limited bythe available resource level in the habitat. An adultof species i produces propagules at a rate that canbe reasonably described by Holling type II func-tion, riQ/(ai + Q), where ri is the maximum relativepropagule production rate, Q is the resource con-centration in the habitat, and ai is the half-saturationcoefficient, that is, the resource concentration levelat which propagule production rate is half of themaximum. Since species competitive ranking doesnot change in space, and the resource availabilityis spatially homogenous, the rate of propagule pro-duction from occupied microsites by the ith speciesis piriQ/(ai + Q). Thus, the colonization rate ci inthe Eq. (1) is being expressed as

ci = ri Q/(ai + Q), (2)

which will be called recruitment rate of the species,species i. Substituting Eq. (2) into Eq. (1), we obtain

dpi

dt= ri

Q(t)

ai + Q(t)pi

⎛⎝1 −

i∑j=1

pj

⎞⎠

−⎛⎝ i−1∑

j=1

rjQ(t)

a j + Q(t)pjpi

⎞⎠ − mipi. (3)

Equation (3) describes propagule-limited spatialcompetition among N number of species, wherein the right hand side of the equation, the first termrepresents the rate of production of newly recruitedmicrosites by the species, species i and the secondterm represents resource-dependent displacementrate of the species i from occupied microsites by itssuperior competitor species. Therefore, accordingto the Eq. (3), a species can be absent from a sub-set of many microsites because of limited propaguleavailability, or it has already been displaced by itssuperior competitor species or has failed to displaceoccupants, or the microsites are just released by oc-cupant’s death, or a combination of all.

The associated changes in resource concentra-tion are due to changes in resource supply rate,resource uptake rate by the species, and the rate

of resource loss from the habitat through diffu-sion or some other physical processes. As we donot know the actual mechanisms for resource sup-ply and loss from the habitat, we simply assume thatthe rate of resource loss is directly proportional toresource concentration and the resource supply ratechanges sinusoidally. Therefore, the underlying re-source concentration dynamics can be described bythe following differential equation:

dQ

dt= I (t) − �Q −

N∑j=1

1

yj

(rjQ

aj + Qpj

), (4)

where yj is the conversion factor that relates resourceconcentration to propagule production; � is the rel-ative rate of resource loss, which is constant; andI(t) is the time-dependent resource supply rate thatis described as

I (t) = I + b sin(wt), (5)

where I is the average resource supply rate in thehabitat, and b is the amplitude and w is the fre-quency of the fluctuations. For a given constancyon yj , the third term in the Eq. (4) describes thateach species uses a fixed proportion of uptaken re-source for propagule production. Solutions of theEqs. (3) and (4) are positive for all t ≥ 0 (AppendixA). Therefore, the system formed by the Eqs. (2),(3), (4), and (5) is as “wellbehaved” as one intu-its from the biological problem. A species cannotpersist in the system for a given upper thresholdon resource concentration, which is determinedby the species mortality rate, maximum relativepropagule production rate, and half-saturation co-efficient. Mathematically, that can be expressed as(Appendix B)

pi(t) → 0 as t → ∞ if Q(t)

<ai mi

ri − miand ri > mi .

While reversing this mathematical statement, itprovides a necessary condition for recruitment;in order to persist in the system with positiveabundance, each species must have a “minimumrecruitment ability”—a resource threshold deter-mined by the species to support its requisite pro-duction of propagules. This threshold is given byB∗

i = ai mi/(ri − mi ). A higher value of B∗ indicateslower recruitment ability as it requires a higher re-source level to bear up the production of propagules,

Ann. N.Y. Acad. Sci. 1195 (2010) E27–E39 c© 2010 New York Academy of Sciences. E29


whereas a lower B∗ value indicates higher recruit-ment ability as it requires relatively a lower resourcelevel. Although it seems that B∗ behaves similar likeTilman’s R∗ (Tilman 1982) but they are differentas they correspond to the life history traits CC thatare inversely correlated. Tilman’s R∗ is a quantita-tive measure of species competitive ability, whereR∗ is the equilibrium resource level determined bythe species in absence of competition; a species withlower R∗ value is regarded as competitively supe-rior to a species with higher R∗ value. According tothe R∗ rule, while multiple species intensely com-pete for a single limiting resource, the best com-petitor excludes all others at equilibrium by deplet-ing resource to the lowest level of its R∗ value. Ina resource-poor environment, such competitivelysuperior species often displays some kind of traitsthat reduce reproductive outputs, for example, pro-duction of relatively large-sized, weighty propagules(e.g., Landau 2003 and references therein) that re-quires relatively a high resource level to maximizeits propagule production. This species characteris-tic has been interpreted here by B∗ value. In otherwords, in a resource-poor habitat, a superior com-petitor species (lower R∗ value) is more propagule-limited than its inferior competitor species becauseof its higher resource requirement to produce morepropagules, which is reflected here with a higher B∗

value. Here, we characterize each species by its R∗

and B∗ value such that the R∗ value increases with thespecies ranking whereas B∗ value decreases. Giventhis inverse relationship between R∗ and B∗, a speciesthat has highest competitive ability has lowest re-cruitment ability. Therefore, in order to persist inthe system with positive abundance, a species shouldadopt some kind of recruitment strategy such thatthe habitat’s resource concentration is not reducedto that level for which it’s R∗ and B∗ requirementsare not being fulfilled. Using Eqs. (2) and (3), inthe next section, we calculate equilibrium resourceconcentration (Q) and corresponding recruitmentstrategy (c) in terms of species abundances and otherspecies-specific parameters.

Spatial competitive coexistence withoutfluctuations in resource supply

While there is no fluctuation in resource supply, thefluctuation amplitude b is equal to zero, which im-plies that the resource supply rate is constant, that

is, it does not change with time. In this situation,by exploring the system formed by the Eqs. (3) and(4), in the first part of this section, we show that at aconstant rate of resource supply competing speciesapproach toward an equilibrium point independentof their initial abundances; in the second part, weshow that there exists a limit to similarity betweenspecies that allows an inferior competitor species tostably coexist with its superior competitor speciesprovided that their R∗ and B∗ values are fulfilledby the habitat’s resource concentration; in the thirdpart, we show that how a constant resource sup-ply rate gives a limit to the number of coexistingspecies.

The qualitative behavior of two species competi-tion (Fig. 1A) has shown that for a given resourcesupply rate, superior competitor (species 1) ap-proaches equilibrium and is unaffected by inferiorcompetitor (species 2), and the inferior competitorrapidly occupied empty microsites and is broughtto equilibrium as superior competitor approachesequilibrium regardless of initial abundance. We didnumerical simulations for hypothetical species un-der different given resource supply rates, and theparametric values were chosen according to themodel conditions (Figs. 1B–D). In all the cases com-peting species attained their predicted equilibria, in-dependent of their initial abundances. This suggeststhat at a constant resource supply rate, multispeciescompetitive equilibrium is globally stable.

Competing species reach at a stable equilibriumone by one; once the best competitor attains its equi-libria, the next equation for the next best competi-tor takes on the form of the equation for the firstspecies, and the second species goes to equilibrium.This propagates down through N number of speciesindependent of their initial abundances (Fig. 1).Following this trends, we can sequentially calcu-late habitat’s resource concentration and the associ-ated recruitment rate, starting with the species 1, interms of equilibrium species abundances and otherspecies-specific parameters. When dp1/dt = 0,

Q∗ = a1�1

r1 − �1, �1 = m1

1 − p1,

where p1 is the abundance of the species 1 and Q∗ isthe habitat’s resource concentration at equilibrium.The corresponding recruitment rate, c1, is given bythe Eq. (2). Inserting c1 in the equation of dp2/dt =0, we obtain



Figure 1. Dynamics of spatial competition at a constant resource supply rate. (A) Species-1 had the parametric valuesof (r1 = 0.04 yr−1, a1 = 0.002 kg ha−1, m1 = 0.004 yr−1) and species-2 had the parametric values of (r2 = 0.8 yr−1,a2 = 0.01 kg ha−1, m2 = 0.004 yr−1); the resource supply rate ( I) was, 5 kg ha−1 yr−1, and the resource loss rate (�)was, 0.2 kg ha−1 yr−1. For both the species, yield coefficients were the same, y = 2.1 kg−1; and initial proportionalabundances were varied within the range of 0.1 to 0.3 (B) Species 1,2,3,4 had same parametric values of mortalityrate (m = 0.0001 yr−1), and yield coefficient (y = 2.1 kg−1); their relative propagule production rates (r yr−1) had thevalues in geometric progression, r i = (0.4)5−i ; half-saturation coefficients were (a1 = 0.2, a2 = 0.04, a3 = 0.06, a4 =0.08 kg ha−1), the resource supply rate ( I) was 15 kg ha−1 yr−1, and the rate of resource loss (�) was, 4 kg ha−1 yr−1;initial proportional abundances for all the species were fixed to 0.1. (C) Species 1, 2, 3, 4, 5, 6 had the same parametricvalues of mortality rate (m = 0.0001 yr−1), yield coefficient (y = 2.1 kg−1), and half-saturation coefficient (a = 0.2kg ha−1); their relative propagule production rates had the values of (r1 = 0.0256, r2 = 0.0640, r3 = 0.16, r4 = 0.4,r5 = 0.6, r6 = 0.8 yr−1); the resource supply rate ( I) and the resource loss rate (�) were fixed to 15 kg ha−1 yr−1 and4 kg ha−1 yr−1, respectively, initial abundances of all the species were fixed to 0.1. (D) Species 1, 2, 3, 4, 5, 6 had thesame parametric values of relative propagule production rate (r = 0.02 yr−1), half-saturation coefficients, a = 0.2 kgha−1, and yield coefficient, y = 2.1 kg−1; their mortality rates had the values in geometric progression, mi = (0.1)i ;the resource supply rate ( I) and resource loss rate (�) were fixed to 5 kg ha−1 yr−1 and 2 kg ha−1 yr−1, respectively;initial abundances for all the species were fixed to 0.1.

Q∗ = a2�2

r2 − �2, �2 = [ p1m1 + (1 − p1)m2]

(1 − p1)(1 − p1 − p2).

The corresponding recruitment rate of species 2is given by the Eq. (2). The process can be continued,giving a value of ith species

Q∗ = ai �i

ri − �i, �i =

i−1∑j=1

pjmj +⎛⎝1 −

i−1∑j=1

pj

⎞⎠mi

⎛⎝1 −

i−1∑j=1

pj

⎞⎠

⎛⎝1 −

i∑j=1

pj

⎞⎠

.



It could be noticed in the above equation that thecalculation of Q∗ involves the effect of superior com-petitor species, and the Q∗ is defined for ri > �i .Therefore, a species, species i, can persist in the sys-tem if R∗

i , B∗i ≤ Q∗ and ri > �i .

Considering equal mortality rate for all the com-peting species, we have

�i = m

�i−1�i

>m

�2i−1

,

where �i = 1 − ∑ij=1 pj . Alternatively, �i can be de-

scribed as

�i >�i−2

�i−1

�i−1.

This relationship defines a limit to similarity be-tween species, this limit is the same as it was calcu-lated by Tilman (1994) using the original CC-model.

In the modified model, the limit to that numberof species that coexist at equilibrium is coming fromthe Eq. (4); it gives

N∑j=1

1

yj

(rjQ∗

aj + Q∗ pj

)< I ∗

where I∗ is the resource supply rate at equilibrium.Since I∗ and � both are fixed real numbers, by

induction, there must exist a natural number sayN∗ that satisfies the following inequations:

N∗∑j=1

1

y j

(r j Q∗

a j + Q∗ p j

)< I ∗, and

N∗+1∑j=1

1

y j

(r j Q∗

a j + Q∗ p j

)≥ I ∗.

The above inequations imply that at a constant rateof resource supply, there exists a maximum N∗ num-ber of species to coexist at equilibrium point.

The analysis revealed that a constant resourcesupply rate combined with a finite lower limit onresource availability that allows survival of compet-ing species defines a limit to the number of coexist-ing species. However, the basic underlying coexis-tence mechanism remains the same as in the originalCC model, which states that a superior competi-tor species cannot occupy all microsites because ofits poor colonization ability; an inferior competitorspecies with higher colonization ability thus can in-vade and survive in the microsites left unoccupied.

An invasibility criterion for an exoticspecies

Consider an exotic species that is an inferior re-source competitor but superior colonizer relativeto native species, and has initially successfully oc-cupied some microsites within the habitat. Basedto the model formed by the Eqs. (3), (4), and (5),spatial competition between such exotic and nativespecies can be described by the following differentialequations:

dp

dt= rp

Q

ap + Qp(1 − p) − mpp, (6)

dq

dt= rq

Q

aq + Qq(1 − p − q)

− mqq − rpQ

ap + Qpq , (7)

where p is the abundance of the native species andq is the abundance of the exotic species. The as-sociated resource concentration dynamics and thefluctuations in resource supply are described by theEqs. (4) and (5), respectively, while the index j inthe Eq. (4) runs for the native and exotic species.

Qualitative behavior of solutions of the systemformed by the Eqs. (4), (5), (6), and (7) emergesthree generalities: first, at a constant resource supplyrate, the exotic species can stably coexist with the na-tive species at competitive equilibrium, independentof their initial abundances; second, if the amplitudeof the fluctuations in resource supply is less than theaverage resource supply rate, the exotic species cancoexist with the native species at unstable equilib-rium, independent of fluctuation frequencies; andthird, if the amplitude of the fluctuations is greaterthan the average resource supply rate, there existsa range of values of fluctuation frequency that canallow the exotic species to successfully invade thehabitat, and to reduce the extent of native species.On the basis of these three generalities, we can definea potential invasibility criterion of the communitymaintained via CC trade-offs specific to an exoticspecies, which is inferior resource competitor butsuperior colonizer relative to native species, that is,the amplitude of the sinusoidal fluctuations in theresource supply is greater than the average resourcesupply rate.

We have obtained above noted results by do-ing numerical simulations using the Runge–Kutta



Figure 2. Dynamics of spatial competition between the native and exotic species at a constant resource supply rate.The native species (p) is represented by black line and the exotic species (q) is represented by red line. (A) The nativespecies had the parametric values of (rp = 0.04, ap = 0.02), and the exotic species had the parametric values of(rq = 0.8, aq = 0.01); and both the species had the same initial proportional abundances of 0.1, yield coefficient (y) of2.1, and mortality rate (m) of 0.004. The average resource supply rate ( I) was 5 kg ha−1 yr−1, and the rate of resourceloss (�) was 0.2 kg ha−1 yr−1. (B) The native species had the parametric values of (rp = 0.2, ap = 0.02, mp = 0.05,yp = 2) and the exotic species had the parametric values of (rq = 0.8, aq = 0.01, mq = 0.004, yq = 2.2); the resourcesupply rate ( I) was 2 kg ha−1 yr−1, and the rate of resource loss (�) was 0.2 kg ha−1 yr−1; the initial proportionalabundances of p varied from 0.1 to 0.4 and q varied from 0.1 to 0.6.

method of the fourth order. Recruitment rates ofthe exotic and native species are typified accordingto the conditions, r p < rq and a p ≥ aq , which im-plies higher recruitment rate of the exotic speciesq relative to the native species p. In the simulationexperiment, a hypothetical species is represented bythe parametric values of maximum relative propag-ule production rate, r (yr−1), half-saturation coef-ficient, a (kg/ha), yield coefficient, y (kg−1), andmortality rate, m (yr−1). If a test species’ abundancedecreases and its competitor species’ abundance in-creases with time, it is assumed that the competitorspecies will displace the test species from the habi-tat. If species abundances oscillate around a fixedpoint or do not change as time increases, competi-tive coexistence is assumed. We have continued thesimulation until such trends are evident, for at least250 time steps.

At a constant resource supply rate (i.e., b = 0),the exotic species approaches equilibrium while thenative species already attained its equilibrial abun-dance. The simulation experiments show that theexotic species can stably coexist with the nativespecies at that equilibrium point (Fig. 2). For deter-mining conditions that allow either the coexistenceor exotic invasion under sinusoidal fluctuations inresource supply, we have fixed the parametric val-

ues of the competing species: the native species (p)is represented by the parametric values of, rp = 0.1,ap = 0.7, mp = 0.04, yp = 1.1, and the exotic species(q) is represented by the parametric values of, rq =0.7, aq = 0.7, mq = 0.04, yq = 1.2. Two sets of simu-lation experiments have been carried out. In the firstset, we have kept the amplitude of the fluctuationsbelow the average resource supply rate and consid-ered different values of fluctuation frequency. In allthe cases, both the competing species have showncoexistence at unstable equilibrium, independent offluctuation frequencies (Fig. 3). In the second set,the amplitude of the fluctuations has kept above theaverage resource supply rate; and corresponding toeach numerical value of the amplitude, we find arange of values of the fluctuation frequency, whichshow that the exotic species successfully invade anddisplace the native species (Fig. 4).

Figure 3 depicts coexistence of exotic and nativespecies. While the fluctuation amplitude, b, has keptbelow the average resource supply rate, the avail-able resource concentration is maintained at a levelthat both the species’ R∗ and B∗ requirements arefulfilled. Given this condition, the superior nativecompetitor species fails to response significantlywhich can outweigh the response of the competi-tively inferior exotic species. This occurred because



Figure 3. Spatial coexistence of the native species (p), drawn in black, and the exotic species (q) drawn in red, whilethe resource supply rate (I) fluctuates sinusoidally, where b is the amplitude and w is the frequency of the fluctuations.For all the simulations, the average resource supply rate ( I) and the rate of resource loss (�) were fixed to 5 kg ha−1

yr−1 and 1.2 kg ha−1 yr−1, respectively, and the initial proportional abundances of the species were P = q = 0.3.

of the trade-off between species’ competitiveability and colonization ability, and thereby both thespecies were capable to adopt a specific colonizationstrategy (c), which is limited by the resource con-centration, that allowed partitioning the space withvariable abundance over time.

Figure 4 portraits a potential invasibility crite-rion at which an exotic species successfully reducesthe extent of native species. While the fluctuationamplitude has kept above the average resource sup-ply rate, the exotic species shows increasing abun-dance whereas the native species has displayed adecreasing trend. This occurs possibly because ofoverwhelming response of the exotic species to in-creased resource availability as it has higher recruit-ment ability and higher recruitment rate (c) relativeto native competitor species. Increasing exotic abun-dance over time reduces the resource level betweenthe B∗ values of exotic and native species, that is,B∗

q ≤ Q(t) < B∗p , which drives native abundance

toward zero, that is, p(t) → 0 as t → ∞.

Discussions and conclusions

The concept of invasibility has emerged in invasionecology to describe susceptibility of a habitat (orcommunity) to invasion by a putative exotic species.It is essentially a habitat- and species-specific con-cept (Lonsdale 1999; Davis 2005), that is, the habitatinvisibility can be interpreted as an emergent prop-erty determined by the interaction of traits of thehabitat’s environment (e.g., Alpert et al. 2000), thenative community (e.g., Crawley 1987), and the pu-tative exotic species (e.g., Rejmanek and Richard-son 1996). Indeed, most of the suggested invasibil-ity conditions (e.g., fluctuating resource availability[Davis et al. 2000], propagule pressure [Rejm�aneket al. 2005; Colautti et al. 2006]), unstated the as-sociated traits of native and exotic species, mainlyexplain initial establishment success in a new habi-tat rather than the successful invasion by an exoticspecies. As a result, several other studies, mainlyexperimental, solely focused on determining the



Figure 4. The exotic species invasion under sinusoidal fluctuations in resource supply, where b and w are theamplitude and the frequency of the fluctuations, respectively. The exotic species (q) is drawn in red and the nativespecies (p) is drawn in black. For all the simulations, the resource loss rate (�) was 1.2 kg ha−1 yr−1 and the averageresource supply rate ( I) was 5 kg ha−1 yr−1, and the initial proportional abundances were P = q = 0.3.

traits of exotic species allowing invading and dis-placing native species (species invasiveness) (e.g.,Goodwin et al. 1999). In this study, habitat inva-sibility has been determined by the interaction oftraits of the habitat’s environment, and the nativeand exotic species, that is, the considered habitat(local community) is maintained via CC trade-offs,with the fluctuating availability of resource due tofluctuations in resource supply, and the putative ex-otic species is superior colonizer but inferior re-source competitor relative to native species. Themodeled species interaction via the CC trade-offwell fits with the grassland community. For ex-ample, in California grasslands, invaded success-fully by exotic annual grasses, field experiment hasshown that the native perennial species are supe-rior resource competitors but poor colonizers rela-tive to exotic grass species (Seabloom et al. 2003a).However, the employed integrating approach canalso be applied to other real communities providedthe modeling assumptions well matched with fieldconditions.

It is almost impossible today to avoid the intro-duction of species that do not naturally appear ina habitat (referred here as exotic species) becauseof increasing human activities that are resulting inthe transport of variety of species across regions.And, once an introduced species is naturalized intoa new environment, which is essential for its sur-vival, the species can coexist or invade and displacenative species depending on different sets of con-ditions. Thus, for management and conservationof native species diversity, thorough understandingof invasibility and coexistence criteria specific tothe habitat and species characteristics are needed;here invasibility criteria refer to the conditions thatallow an exotic species to thrive, and the coexis-tence criteria refer to the conditions that promotecoexistence of exotic and native species. In this con-nection, we demonstrated here that the fluctuatingresource supply is one important factor of diver-sity maintenance, and it can play dual roles: pro-mote coexistence or increase invasibility. Invasibilityand coexistence criteria for a putative exotic species



depend on the nature of fluctuations, and on thetraits of native community and putative exoticspecies. For sinusoidal-type fluctuations in resourcesupply, which could be associated with actual sea-sonal fluctuations in nutrient supply or rainfall, thisstudy has shown that the invasibility and coexis-tence criteria of a local community maintained viaCC trade-offs can be determined by amplitude andfrequency of the fluctuations. Specifically, while am-plitude of the fluctuations is greater than the averageresource supply rate, there exists a range of values offrequency that can allow a naturalized exotic speciesto invade and displace native species. On the otherhand, if amplitude of the fluctuations is less thanthe average resource supply rate, the exotic speciescan coexist with native species independent of fluc-tuation frequencies.

Superior competitive ability of an invader speciesrelative to native species might allow invasions to oc-cur because if all available resources in a habitat areused in full capacity by native species then any gainfor an invader is a loss for natives. Such competitiveability of an invader enhances exclusion of nativespecies with increased populations of own (Hobbsand Humphries 1995; Seabloom et al. 2003b). How-ever, a successful invader species can be competi-tively inferior relative to native species, noticed inmultiple field studies (e.g., Seabloom et al. 2003a,b); but underlying mechanisms of the invasion suc-cess are relatively poorly understood. In this respect,this study suggests that invasion success of a com-petitively inferior species can be explained by the na-ture of fluctuations in resource supply—amplitude,frequency, and type (e.g., sinusoidal, random) ofthe fluctuations, and its interaction with traits ofthe native community and the exotic species. Oneanother possibility is that the coexistence of in-vader species with native species, which can be ex-plained by the same factors, with different set ofconditions.

Recently, community ecology theories are promi-nently being operationalized to understand theconditions that promote invasion by an exoticspecies (e.g., Huston 1994; D’Antonio and Levin1999; Davis et al. 2000; Shea and Chesson 2002;Huston 2004). In that respect, the classic CC-modelhas often been used to explain multispecies coexis-tence at local or regional spatial scale (Tilman 1994;Yu and Wilson 2001); our study here is the firstthat explored the theory linked with interspecificresource competition for determining invasibility

criteria by the amplitude and frequency of the fluc-tuations in resource supply.

The model, we formulated here, is a modi-fied version of the classic CC model that wasformulated first by Hastings (1980) and stud-ied later by Tilman (1994). It explicitly inte-grates propagule-limited spatial competition withresource-competition. While there is no resourcelimitation, species recruitment rate is determinedby the maximum relative rate of propaguleproduction (i.e., ci → ri as Q → ∞). In this sit-uation, the modified model has retained the sameproperties as the original CC-model. There existsa limit to similarity between species—a minimumvalue for this recruitment rate termed as “nicheshadow” by Kinzig et al. (1999), which allows sta-ble coexistence of superior and inferior competitorspecies. If species satisfy this condition, the modelpredicts coexistence of unlimited number of species,that is, there is no limit to the number of coexist-ing species (Tilman 1994). While available resourceis limited, the modified model predicts a limit tothis number, where the limit to similarity betweenspecies has remained unchanged. It describes thatat a constant rate of resource supply, the competingspecies reach at an equilibrium point irrespectiveof their initial abundances (Fig. 1) in which theypartition the space according to their equilibrial re-cruitment rate, ci, such that both R∗ and B∗ valuesare fulfilled by the habitat’s resource concentration(i.e., R∗

i , B∗i ≤ Q∗). Thus, in this case, the resource

quantity defines a limit to the number of coexistingspecies.

Understanding the role of disturbances, such asfire or clear cutting, is essential for proper manage-ment and maintenance of native species diversitybecause disturbances often release resources or alterthe rate of resource supply. The model results, men-tioned above, may be useful to understand someof their specific roles by associating the proposedinvasibility and coexistence criterion to the situ-ations that occur following a disturbance. How-ever, the applicability of these results needs prede-termination of the requisite CC tradeoff and thespatial scale at which the model may apply. Ap-parently, the model and model results are moreappropriate at local spatial scale rather than theregional scale because it does not include any re-gional processes, such as emigration or immigra-tion of propagules across localities, and the resourceheterogeneity.



Acknowledgments

We highly appreciate Edith Allen and her group forsome useful discussions on this subject. We thankKurt E. Anderson for his valuable comments onthe earlier version of this manuscript. This researchwas partially supported by the U.S. National Sci-ence Foundation’s Biocomplexity Program (DEB-0421530) and the University of California Agricul-tural Experiment Station.

Conflicts of interest

The authors declare no conflicts of interest.

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Appendix A

Statement: pi (t) ≥ 0, Q(t) ≥ 0 for all t ≥ 0, andQ(t) is bounded by I/� when b = 0.

Rearranging Eq. (3), we have

dpi

dt≥ −mi pi −

i−1∑k=1

(rk pk) pi , i.e.,

pi (t) ≥ pi0 exp

[∫ t

0

(−mi −

i−1∑k=1

rk pk

)dt

]≥ 0,

where pi (0) = pi0 ≥ 0.Similarly rearranging Eq. (4) we have

dQ

dt≥ −�Q −

N∑k=1

rkpk

ykakQ , i.e.,

Q(t) ≥ Q(0) exp

[∫ t

0

(−� −

N∑k=1

rkpk

ykak

)dt

]≥ 0,

where Q(0) ≥ 0.At b = 0, Eq. (4) can be written as

dQ

dt+ �Q ≤ I , that is,

0 ≤ Q(t) ≤ I

�+ Q(0) exp(−�t),

which implies that,

0 ≤ Q(t) ≤ I

�, as t → ∞.

Appendix B

Statement: pi (t) → 0 as t → ∞ if either mi > ri ,or Q(t) <

mi airi −mi

and mi < ri for all t ≥ 0.Rearrangement of Eq. (3) yields

dpi

dt≤ (ri − mi ) pi , i.e.,

pi (t) ≤ pi0 exp[(ri − mi )t], therefore,

pi (t) → 0 as t → ∞ if mi > ri .

Let us assume that mi < ri for all t ≥ 0, the anotherrearrangement of Eq. (3) yields

dpi

dt≤

(ri

Q

ai + Q− mi

)pi = (ri − mi )

ai + Q(Q − mi ai

ri − mi

)pi . (B1)

Therefore, dpidt ≤ 0 if Q(t) ≤ mi ai

ri −mi.

We now used a well-established lemma which isgiven below

Lemma: Let x(t) ∈ C 2[t0,∞), x(t) ≥ 0, K > 0.If x ′(t) ≤ 0 and x ′′(t) ≥ −K > −∞ for all t ≥ t0,then x ′(t) → 0 as t → ∞.

Existence of solutions of Eq. (3) gives the guar-antee of existence of p′′

i (t), and the boundednessof p′′

i (t) follows from the fact that pi(t) is alwaysbounded, which is the physical assumption of themodel.

Let us consider, Q(t) <mi ai

ri −miand mi < ri for

all, t ≥ 0. Under this condition our claim is that,pi (t) → 0 as t → ∞.

We established this result by contradiction. If pos-sible let, pi (t)lim t→∞ = p∗

i > 0. With the help ofabove lemma, we have from the inequation (B1)



0 ≤ (ri − mi ) limt→∞

(1

ai + Q

)lim

t→∞

×(

Q − mi ai

ri − mi

)p∗

i .(B2)

This holds if,lim

t→∞(Q − mi airi −mi

) ≥ 0, i.e., Q(t) ≥ mi airi −mi

for all

t ≥ t0 for some t0, which contradict our considera-tion that Q(t) <

mi airi −mi

for all t ≥ 0. Therefore, theexpression only holds, if p∗

i = 0. Hence the resultfollows.


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The role of fluctuating resource supply in a habitat maintained by the competition-colonization trade-off