An R0 theory for sourcesink dynamics with applicationto Dreissena competition

Martin Krkoek & Mark A. Lewis

Received: 23 May 2008 /Accepted: 29 March 2009 /Published online: 28 May 2009# Springer Science + Business Media B.V. 2009

Abstract Sourcesink dynamics may be ubiquitous inecology. We developed a theory for sourcesink dynamicsusing spatial extensions of the net reproductive value, R0,which has been used elsewhere to define fitness, diseaseeradication, population growth, and invasion risk. R0decomposes into biologically meaningful componentslifetime reproductive output, survival, and dispersalthatare widely adaptable and easily interpreted. The theoryprovides a general quantitative means for relating funda-mental niche, biotic interactions, dispersal, and speciesdistributions. We applied the methods toDreissena and founda resolution to a paradox in invasion biologycompetitivecoexistence between quagga (Dreissena bugensis) and zebra(D. polymorpha) mussels among lakes despite extensiveniche overlap within lakes. Sourcesink dynamics withinlakes between deepwater and shallow habitats, which favorquagga and zebra mussels, respectively, yield a metacom-munity distribution where quagga mussels dominate largelakes and zebra mussels dominate small lakes. The sourcesink framework may also be useful in spatial competitiontheory, habitat conservation, marine protected areas, andecological responses to climate change.

Keywords Competition . Dispersal . Survival .

Sourcesink dynamics . Niche theory .Dreissena

Introduction

Sourcesink dynamics are central to ecology for theirinfluence on population dynamics (Brown and Kodric-Brown 1977; Pulliam 1988), species ranges (MacArthur1972; Holt 2003), and competitive coexistence (Amarasekareand Nisbet 2001; Snyder and Chesson 2004). Dispersal maybe as important as competition in explaining differencesbetween fundamental and realized niche (Hutchinson 1957;Pulliam 2000) because sourcesink dynamics can maintainpopulations in poor habitat (Pulliam 1988) and extinguishthem in suitable habitat (Amarasekare and Nisbet 2001).Understanding sourcesink dynamics is also of appliedimportance for conserving species on fragmented landscapes(Hanski and Gyllenberg 1993; Hanski 1998) and designingmarine protected areas (Lubchenco et al. 2003; Neubert2003). Despite the fundamental and applied importance ofsourcesink dynamics, there is some confusion surroundingwhat actually constitutes a source or a sink and how toquantitatively distinguish between the two. The centralconfusion for population dynamics is whether sources shouldreflect the fundamental niche (Pulliam 2000) or thecontribution a local population makes to the metapopulation(Figueira and Crowder 2006; Runge et al. 2006).

In this paper, we introduce a new theory for sourcesinkdynamics and apply it to a long-standing problem ininvasion biologycompetition between Dreissena mussels.The theory is based on the net reproductive value, R0,which measures the number of progeny produced in thelifetime of a single individual (Heesterbeek 2002). Webegin by briefly reviewing the theory of sourcesink

Theor Ecol (2010) 3:2543DOI 10.1007/s12080-009-0051-7

M. Krkoek (*) :M. A. LewisCentre for Mathematical Biology,Departments of Mathematical and Statistical Sciencesand Biological Sciences, University of Alberta,Edmonton, AB T6E 2E7, Canadae-mail: mkrkosek@u.washington.edu

M. A. Lewise-mail: mlewis@math.ualberta.ca

Present Address:M. KrkoekSchool of Aquatic and Fishery Sciences,University of Washington,Seattle, WA, USA

dynamics, with a focus on their relation to niche, dispersal,and distribution. We then develop the new theory for bothsingle species and competing species cases, using a mutualinvasibility analysis to assess scenarios of species persistence,exclusion, and coexistence. Our review and theoreticaldevelopment brings together and quantifies concepts fromcompetition theory, spatial ecology, and mathematical biologywhose interrelatedness had not been fully appreciated. Wefollow by examining the Dreissenid paradox, which is thecompetitive coexistence of quagga (Dreissena bugensis) andzebra (D. polymorpha) mussels among lakes despite exten-sive niche overlap. Application of the theory to a spatialmodel of zebra and quagga mussel competition revealssourcesink dynamics between shallow and deepwaterhabitats that leads to within-lake exclusion and among-lakecoexistence.

Theory of sourcesink dynamics

Pulliam (1988) considered a population distributed over afragmented landscape and suggested that population flowfrom source habitats can maintain populations in sinkhabitatsechoing the rescue effect proposed a decadeearlier (Brown and Kodric-Brown 1977). Pulliam (1988)proposed two definitions for sources and sinks based onequilibrium and invasion (low density) conditions. The firstdefinition considers a population at equilibrium (allsubpopulation densities are constant through time) anddefines a source as a net exporter and a sink as a netimporter of individuals. The second definition considers apopulation at low density and defines a source as habitatpatches in which a population can grow in the absence ofimmigration and emigration. The equilibrium definition hasbeen used in the evolutionary theory of sourcesink dynamics(Kawecki 1995) and the invasion definition has been used inpopulation dynamics (Pulliam 1988), where it has recentlybeen modified and redefined (Figueira and Crowder 2006;Runge et al. 2006). Our focus is on the invasion definitionused in population dynamics, which allows us to beconsistent with previous works and also allows us to uselinear approximations that simplify the analysis.

In his original article, Pulliam (1988) used the localpopulation growth rate (at low population size and nodensity dependence) of an annually breeding species,

l PA PJb 1

to classify source and sink habitat patches. The parameterPA is adult survival, PJ is juvenile survival, and is thenumber of juveniles produced per adult. All of PA, PJ, and are habitat specific and so each habitat can have adifferent value of . Source patches were those with >1and sinks were those with

outcomes of competition such as species extirpation fromwhat is otherwise suitable habitat (Amarasekare and Nisbet2001; Schreiber and Kelton 2005). Many of these papers(e.g., Snyder and Chesson 2004; Schreiber and Kelton2005) use the traditional definition of source or sink basedon local growth rates similar to Pulliam (1988), and sosuffer from the same limitations identified by Runge et al(2006) in that they may not reflect the contribution a localpopulation makes to the metapopulation. There is largevariation in the structure of competition models in whichsourcesink dynamics arise, ranging from continuous timemodels on patches (Amarasekare and Nisbet 2001) todiscrete time models on continuous landscapes (Snyder andChesson 2004). As such, there is no general mathematicalframework for sourcesink dynamics to study how distri-butions of sources and sinks are affected by competitionand dispersal as well as how sourcesink dynamics mediatethe relations among niche, competition, and distribution.

A classification of source and sink habitats shouldaccommodate dispersal to reflect the flow of individualsfrom a focal subpopulation to the larger population(Figueira and Crowder 2006; Runge et al. 2006). Such atheory should also be general in its application byextending beyond classical metapopulation formulationsfor single species with annual lifecycles. It should accom-modate age- or stage-structured lifecycles as well asfragmented and continuous landscapes within a frameworkthat is biologically intuitive. Further, it should also addressclassical problems such as relating niche and distribution aswell as explaining competitive coexistence. We proposethat such a theory exists when sourcesink dynamics areviewed through the lens of the net reproductive value, R0.R0 theory offers the advantage of integrating demographicprocesses over an entire lifecycle, and because of its longhistory in demography and epidemiology (Heesterbeek 2002),it offers a well-developed quantitative toolkit for treatingstructured populations. We begin by briefly reviewing R0and then describe its application to sourcesink dynamics insingle species models, competitive coexistence, and therelationship between niche and distribution.

An R0 theory for sourcesink dynamics

A new perspective on sourcesink dynamics comes viathe net reproductive value, R0, defined as the number ofadult progeny produced in the lifetime of a single adult(Heesterbeek 2002). R0 is prevalent in epidemiology,where it differentiates disease persistence and eradication(Diekmann et al. 1990; Anderson and May 1991; Diekmannand Heesterbeek 2000). It also appears in life history theory(Metz et al. 1992; Mylius and Diekmann 1995), age- orstage-structured population dynamics (Caswell 2001), andinvasion biology (de Camino-Beck and Lewis 2007). In all

cases, if R0>1 a small population will establish/persist but ifR0 1. Locations where thepopulation cannot establish

bR0 x < 1 correspond tounsuitable habitat and are outside the fundamental niche.

To see the effects of dispersal on sourcesink dynamics,we introduce a dispersal kernel, k(x, z), which defines the

Theor Ecol (2010) 3:2543 27

probability density that a propagule released from locationx will settle at location z. The net reproductive value of anaverage individual at location x must account for thedispersal from x to z and subsequent survival of offspringto adulthood at z throughout a spatially continuous landscape

R0(x) = (x)reproductionat location x

(z)k(x,z)dzsurvival

after dispersal from x

.{ 4For the case where habitat is fragmented among discrete

patches, we have

R0(x) = (x)reproductionat location x

(z)k (x,z)survival

after dispersal from x

{ 5where k(x, z) is the probability of moving from patch x topatch z. For organisms with no mortality during dispersal,the dispersal kernel gives unity when summed over allpossible settling locations z. If, in addition, the speciesbecomes immediately reproductive, the survival term (z)equals unity and so the sum in Eq. (5) also equals unity andthe source or sink nature of patch x is determined solely bylocal dynamics. Locations where R0(x)>1 function assources because individuals at location x on averageproduce more than one adult offspring somewhere in thespatial domain. Locations where R0(x)

A single species with discrete non-overlapping gener-ations and annual reproduction and dispersal events can bemodeled with an integrodifference equation

n x; t 1 Z

f n z; t ; z n z; t k x; z dz 9

where n(x, t) is the abundance of individuals at location xand time t, and f is a density-dependent reproductionfunction such as Ricker or BevertonHolt populationgrowth with nonnegative geometric growth rate f(0, x). Atlow population densities, dynamics from one generation tothe next are approximated by the linear equation

n x; t 1 Z

f 0; z n z; t k z; x dz n x; t 10

For this model, the reproduction and survival terms forthe R0 theory are (x)= f(0, x) and (x)=1, and, with these,the next generation operator takes the form given in Eq. (6).If there is some density-dependent mortality of individualsfollowing dispersal, with fraction s(n, x) surviving, then Eq.(10) can be extended to

n x; t 1 s n x; t ; x Z

f n z; t ; z n z; t k z; x dz;11

or

nx; t 1 Z

f nz; t; z snz; t; znz; tkz; xdz12

depending whether mortality is before or after census. Bothyield (x)= f(0, x) and (x)=s(0, x).

How can this model be extended further to includeoverlapping generations? The simplest case, applicable to someplant and insect populations, involves dispersal of juvenilesprior to a sedentary adult stage. This extends Eq. (9) to

n x; t 1 s n; x; t ; z n x; t

Z

f n z; t ; z s n z; t ; z n z; t k z; x dz13

where the survival term s is applied prior to reproduction. Atlow densities, dynamics from one year to the next areapproximated by the linear equation

n x; t 1 s 0; z n x; t

Z

f 0; z s 0; z n z; t k z; x dz: 14

Here, the lifespan of a reproducing adult is geometricallydistributed with expectation 1 s 0; x 1 (see Appendix 3).

Thus, is the expected lifespan of an individual at location xin the absence of density-dependent mortality times its annualreproductive output x 1 s 0; x 1f 0; x and isthe probability of newly produced juveniles surviving toadulthood (x)=s(0, x). In this case, the next generationoperator (6) takes the form

n x; t s 0; x Z

1 s 0; z 1f 0; z n z; t k z; x dz15

An extension of this model will be considered for theDriessenid competition dynamics in later sections of thepaper.

A metapopulation example where R0 theory may bepreferable to Cr is when the population is structured bystages and the number of patches is relatively large.Consider a lifecycle consisting of dispersing young, sessilejuveniles, and sessile adults. The corresponding model is

ny x; t 1 Pzf zsazna z; t k' z; x

nj x; t 1 syxny x; t ;na x; t 1 sjxnj x; t saxna x; t

16

where ny, nj, and na are the three population stagesyoung,juveniles, and adults. The annual survival of each stage issy, sj, and sa, respectively. Only adults reproduce, producingf(x) young each year. To apply Cr theory, one mustconstruct a transition matrix that connects the network ofnodes representing every combination of population stageand patch location (Runge et al. 2006). The large number ofpossible combinations (in this case, three stages times thenumber of patches, squared) means the transition matrixcan quickly become very large, encumbering calculationsfor the contribution metric or the metapopulation growthrate. Also, the homogenization of space and populationstage into one transition matrix loses some biologicalappeal because the intuitive structuring of space andpopulation stage is lost. However, application of R0 theoryis straightforward. The quantities of interest are lifetimereproductive output of young per adult x f x1 sax 1 and survival, the probability a young willsurvive to reach the adult stage, (x)=sy(x)sj(x)sa(x). If thenetwork of patches was replaced with a continuouslandscape of varying habitat quality, Cr theory would notapply whereas R0 theory remains straightforward.

R0 for spatial competition models

Sourcesink dynamics are one mechanism that can facili-tate competitive coexistence (Amarasekare 2003). The R0framework can be extended to analyze spatial competitionmodels and provide insights into the relations among niche,

Theor Ecol (2010) 3:2543 29

competition, dispersal, and distribution. To do so, we turnto a mutual invasibility analysis and consider multiplecompeting species each of which has its own non-zeroequilibrium distribution in the absence of interspecificcompetition. The mutual invasibility analysis considers thenet reproductive value of a focal species introduced at lowabundance into a community of resident species at theirequilibrium densities. In this situation, the focal speciesexperiences competition with resident species, which mayaffect reproduction and/or survival. The invader is consid-ered sufficiently rare to escape intra-specific competition.Below, we summarize how the principles of R0 theory forsourcesink dynamics can be applied to spatial competitionmodels. We follow with a detailed application of the theoryto Dreissena competition, which involves stage-structuredintegrodifference models.

The principles for the competing species case followsimilarly from the single species case. We modify thenotation replacing and with i and i to denote theeffects of competition with resident species r on reproduc-tion and survival of the focal invading species i. Interspe-cific competition acts to reduce the values of i and i

relative to the single species case, but the exact effect isdependent on the nature of competition. Because theresident is at steady state, the values of i and i aretemporally constant but may be spatially variable. Theeffects of competition on distributions relative to funda-mental niche can be seen by mathematically or graphicallycomparing bR0x and bRi0x. The effect of competition onsourcesink dynamics can be viewed by comparing R0(x)and Ri0x. The global net reproductive value defines if thefocal invading species will grow if Ri0 > 1 or declineif R

i0 < 1. Coexistence occurs when R

i0 > 1 for both

species.The behavior when one or both R

i0 < 1 depends upon

the nature of the non-linear dynamics and requires moredetailed analysis. General conditions for existence of aglobally stable equilibrium value are that the dynamics aremonotone (typical of two-species competitive systems), thatthe basin of attraction for the steady state for extinction ofboth species is trivial, that, in the absence of the otherspecies, each species has a globally stable resident steadystate (thus excluding possible Allee effects), and that atmost one coexistence equilibrium exists (Smith and Theime2001). Under these conditions, we expect competitiveexclusion to occur if one species has R

i0 > 1 and the other

has Ri0 < 1, and if both species have R

i0 < 1 then the

dominant species is one that first becomes established(bistability). The more exotic case of semi-stability of thecoexistence equilibrium is theoretically possible, but degen-erate from a mathematical perspective, and thus unlikely tooccur for biologically realistic models (see Discussion ofSmith and Theime 2001). As described above, this

relationship between R0 and competitive outcomes implic-itly relies upon the absence of Allee effects. If an Alleeeffect is present, there is the possibility of establishmentwhen R

i0 < 1 but introduction must occur at sufficiently

high levels to overcome depensation (Boldin 2006).

Summary of R0 theory for sourcesink dynamics

To summarize, we have shown how to map the boundariesof a fundamental niche by solving Eq. (3)when there isno dispersal or competitionfor bR0x 1. Areas wherebR0 x > 1 are within the fundamental niche and correspondto positive equilibrium densities. Areas where bR0x < 1are outside the fundamental niche and cannot support apopulation. When dispersal is included, a fundamentalniche may be exceeded through the rescue effect (immi-gration prevents local population extinction) (Brown andKodric-Brown 1977). This occurs when the global netreproductive value is greater than unity, R0 > 1, becausepopulation growth occurs everywhere inside or outside thefundamental niche. The population growth can be sustainedat sink locations (R0(x)1). Alternatively, thefundamental niche may not be filled because of strongdispersal coupling to extensive sink habitat. This occurswhen the global net reproductive value is less than unity,R0 < 1; global population growth is negative everywhere,inside and outside the fundamental niche, and the popula-tion will eventually reach extinction. Regardless of whetherthe outcome is population persistence or extinction, one canmap the spatial distributions of source and sink habitats asthey function in response to the environment, competition,and dispersal. This is done by solving Eqs. (4)(5) for R0(x)=1. Regions where R0(x)>1 function as sources and regionswhere R0(x)

ecology, they differ markedly in their patterns of spread.After being introduced to the Great Lakes in the late 1980s(Hebert et al. 1989; Griffiths et al. 1991; May and Marsden1992; Mills et al. 1993), zebra mussels quickly spreadthrough temperate eastern North America whereas quaggamussels displaced zebra mussels from a few large lakes(Erie, Ontario, Michigan, and Simcoe) and there theyremain (H. J. MacIsaac, Biological Sciences, University ofWindsor, personal communication; Mills et al. 1999;Stoeckmann 2003; Wilson et al. 2006). Widespreadtransport of Dreissena can occur by boater traffic (Johnsonand Carlton 1996) and has probably occurred for severalyears (Wilson et al. 1999). Quagga mussels may be superiorcompetitors based on energetics (Stoeckmann 2003), butthere is no satisfying explanation for their absence fromsurrounding smaller lakes. One clue is the apparentdifference in depth adaptation: reproduction and bodygrowth at cold temperatures favor quagga mussels (Roeand MacIsaac 1997; Claxton and Mackie 1998; Thorp et al.1998); survival at high temperatures favor zebra mussels(Spidle et al. 1995; Thorp et al. 1998); and quagga musselsfirst colonized deepwater habitats before moving to shore(Mills et al. 1993; H.J. MacIsaac, personal communication).We show that sourcesink dynamics between deepwaterand shallow habitats, which favor quagga and zebramussels, respectively, resolve the differences in historicalspread and contemporary distributions.

A Dreissenid model

Quagga and zebra mussels, like many other aquaticinvertebrates, algae, and terrestrial plants, have a sessileadult (A) stage that reproduces annually and a juvenile (Y)stage that disperses before settling. The lifecycle, for anyspecies j with this lifecycle can be expressed by the graph

where s1,j and s2,j are the basal survival rates for juvenilesand adults, respectively, fj is the number of juvenilesproduced per adult, and is a function accounting fordensity-dependent mortality of juveniles and adults. Formussels, we assume that individuals are distributed alongan environmental gradient, x, that, in this case, correspondsto a lake cross-section stretching along benthic habitatsfrom the shoreline at x=0 to the center of the lake at x=L.

Dispersal links locations x and z according to a dispersalkernel k(x, z), which results in a stage-structured integro-difference equation model

Aj x; t 1 x; t s1;jxYj x; t s2;jxAj x; t

Yj x; t 1 R L0 z; t s2;jzfjzAj z; t k z; x dz

17where k(x, z) defines the probability of moving fromlocation z to location x. Note that survival and fecundityare dependent on the local environment and that integrationoccurs over the entire domain of x.

Equation (17) indicates that competition is dependent ontime and space, and this is based on the effects ofenvironment and local population densities. We assumeindividuals compete for a limiting resource that is localrelative to the scale of dispersal and so do not include acompetition kernel that weights the strength of competitionamong individuals according to their separation distances.This is relevant, for example, to competition for nutrients orlight in plants, algae, or aquatic filter feeders. We assumethat competitive ability is proportional to some trait such asbody size and that competition does not induce overcom-pensation and so choose a modified BevertonHolt density-dependent survival term

x; t 11 bP

jj;yxYj x; t j;axAj x; t 18

(Caswell 2001; Kot 2001). Here, is the competitioncoefficient that relates competitive ability to a phenotypictrait, j,y(x) and j,a(x), which we take to be shell lengths ofjuveniles and adults, respectively. We assume that is thesame for each species and life-history stage and thatvariation in competitive ability among species and stagesis accounted for in j,y(x) and j,a(x).

We compiled statistical models of mussel growth,survival, and reproduction from the literature and linkedthese to position along benthic habitat via a mean summertemperature profile, T=30e0.05x. Details of the parameter-ization are given in Appendix 2 and the spatial dependencyof model parameters are shown in Fig. 1. Results arequalitatively the same for most temperature functions weconsidered, including those with thermal stratification.Dispersal is determined by the diffusiondecay equation

du

dt D d

2u

dx2 au 19

where u is the density of larvae, D is the diffusioncoefficient governing dispersion of larvae and is themortality rate of larvae. We assume that at the shorelinelarvae are reflected back into the lake and at x=L, larvaeleaving the domain are replaced by larvae entering the

Theor Ecol (2010) 3:2543 31

domain from the opposite side of the lake. This leads toreflecting boundary conditions and dispersal kernel with aFourier series representation (van Kirk and Lewis 1999).

Single species dynamics

We begin by deriving expressions for R0 and its reproduc-tion and survival components for the single species case byignoring competition (by setting =1). We follow byexamining the consequences of dispersal for sourcesinkdynamics using Eq. (4) and species persistence using Eqs.(6) and (8). The single species model without dispersal orcompetition is a linear matrix population model

Nj x; t 1 BjxNjt 20where

Bjx 0 s2;jxfjxs1;jx s2;jx

21

and Nj is a vector containing the abundances of adults andjuveniles. To calculate the net reproductive value, we divideBj into transition and fecundity components, Bj=T+F,where

Tjx 0 0s1;jx s2;jx

22

and

Fjx 0 s2;jxfix0 0

23

Any founding population will produce F offspring in thefollowing year. T of the founding population will survive tothe next year and produce F offspring. After 2 years, T2 ofthe founding population will produce F offspring, and soon. The lifetime reproductive output of any foundingindividual will be (F+TF+T2F+T3F+)=[kT

k]F=F(IT)1 (Appendix 3). This expression is another example of anext generation operator and the net reproductive value, R0,is its dominant eigenvalue (Cushing and Zhou 1994; Caswell2001). For our model, the net reproductive value of a smallintroduced population when dispersal is ignored is

bR0 r F I T 1h i s1;js2;j fj 1 s2;j 1h i

shoreline get recruited into favorable habitat at shallow tointermediate depth. The opposite effect occurs near thedeepwater boundary between source and sink regions.There, as dispersal increases and connects source and sinkhabitats the deepwater edge of source habitats decline intheir function as sources and the sink regions movesslightly into shallower depths. At a lake-wide scale, theglobal net reproductive value declines as dispersal increases(but remains greater than unity) because of increased lossesto extensive sink regions in deepwater habitats. Examina-tion of the stable population distribution, (x), and theequilibrium distributions reveal that, at low dispersal, theinvading population remains concentrated in the mostfavorable habitats but then spreads out as dispersalincreases (Fig. 2).

Competing species dynamics

Consider two competing species described by Eq. (17)where the species differ in their environmental response. Ofinterest here are the interactions among competitiveinteractions, differential environmental response, and dis-persal in determining sourcesink dynamics and speciescoexistence. We use a mutual invasibility analysis toevaluate the conditions under which coexistence and/orexclusion results from competition. The logic is thatcompeting species can coexist if each species, when rare,can invade the resident community. The analysis entailsfirst calculating the equilibrium densities of a residentspecies with no interspecific competition and, second,calculating the net reproductive value of the invader as it

Fig. 2 Effects of dispersal onsourcesink dynamics, spatialdistributions, and persistencecriteria of single speciesdynamics of quagga and zebramussels. Top panels showspatial distributions of R0componentsreproductiveoutput (x) and survival (x).These components are thenconnected by dispersal accord-ing to the dispersal kernel,k(x, z), whose probability densi-ty is plotted at z=0.5 for threedifferent kernels (thick graylines with parameters L=10, 0.4,0.05 where L is the non-dimensionalized parameter inequation 10 in van Kirk andLewis 1999, which measures thecharacteristic dispersal distancerelative to lake size, rather thanthe habitat length of our model).The resulting stable spatial dis-tribution of population density,(x) (dashed lines), is plottedfor each corresponding dispersalkernel. Sourcesink dynamicsare described by the thin lines inthe bottom three rows of panels,where the division betweensource and sink, R0(x)=1, isshown by the horizontal dottedlines. Sources are regions whereR0(x)>1 and sinks are regionswhere R0(x)

experiences competition with residents. The invader isconsidered to be sufficiently rare to escape intra-specificcompetition but it may be excluded by interactions withresident competitors. Details of the competition dynamics,given by Eq. (18), make the system monotone andsublinear. Hence, one-way invasibility (one species caninvade the second, but the second cannot invade the first)will result in a globally stable equilibrium with only onespecies present, and hence competitive exclusion.

We begin by simplifying the model by rescaling the statevariables. Then we investigate the effects of competitionand dispersal on sourcesink dynamics, species distribu-tions, and global coexistence. First, we simplify the modelby non-dimensionalizing the state variables such thatyi bi;yYj;aj bj;aAj where i;y and i;a are the pheno-typic traits (e.g., body size) determining competitive abilityof species j juveniles and adults, respectively. The survi-vorship and fecundity terms are already dimensionless andthey remain unchanged and so Bj also remains unchanged.The competition term is simplified to

x; t 11P

jyj x; t aj x; t 25

where j is summed over species. To complete the rescaling,we define nj as the vector containing yj and aj.

The invasibility analysis begins with finding the equi-librium densities of a resident species when it exists inisolation, n*r y*r ; a*r , where the subscript r signifies theirresident status in the lake. The invader, i, competes with the

resident, r, for a limiting resource, and this interaction maybe sufficiently strong to prevent establishment. We areinterested in the net reproductive value of the invader, but itmust account for competitive interactions with residents.Because the resident is at equilibrium and the invader is atlow population density, competition will reduce survivor-ship of the invader by

i 1

1 y*r a*r26

and the population dynamics of the invader will bedescribed by Eq. (17) with =i. From here, we canquickly see the effect of competition on realized nichespace. Recall that the fundamental niche of a species isdefined by areas where bR0x > 1. Because the terms in Bare now weighted by i1 the realized niche will be lessthan or equal to the fundamental niche (Figs. 3, 4). This canbe seen by calculating the net reproductive value of theinvader in the absence of dispersal

bRi0 fi1 s2;i 1 y*r a*r

10B@

1CA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}lifetime reproductive output

s1;is2;i

1 y*r a*r 2

0B@

1CA:

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}survival

27

As resident equilibrium densities of the invader increasefrom zero, bRi0 declines. The convenient separation of thenet reproductive value into spatially independent reproduc-

Fig. 3 Effects of competitionon quagga (solid lines) andzebra (dashed lines) equilibriaand local net reproductivevalues in the absence of dis-persal. Equilibria and net repro-ductive values are plotted forspecies in isolation (ab) andfor species in competition (cd).The horizontal dotted line atR0(x)=1 distinguishes locationsthat can support a population( bR0x > 1) from those thatcannot ( bR0x > 1) when theeffects of dispersal are ignored.The shore is at x=0 and thelakes center is at x=1

34 Theor Ecol (2010) 3:2543

tion and survival components remains. By defining the netreproductive value through the spatially separated processesof reproduction

iz fiz1 s2;iz 1 y*r z a*r z

h i1 28and survival

< ix s1;ixs2;ix1 y*r x a*r xh i2 29

we can make the same calculations as in the single speciescase, including source and sink regions (Ri0x for Eq. (4)),and global persistence criteria (R

i0 for Eq. (8)). The mutual

invasibility analysis determines species coexistence whenRi0 > 1 for both species, exclusion occurs when R

i0 > 1 for

one species but Ri0 < 1 for the other, and when R

i0 < 1 for

both species then dominance depends on initial conditionsthe first species to reach equilibrium excludes the other.

Here, we see how species can be extirpated from locallyfavorable habitat bR0x > 1, even if it remains a sourcelocation Ri0x > 1, if the global net reproductive value isless than one R

i0x < 1. One can directly map the source

sink dynamics that underlie all these scenarios of exclusionand coexistence (Figs. 5, 6).

It is evident from Fig. 2 that zebra mussels and quaggamussels overlap substantially in fundamental niche space.This is detailed by overlapping distributions of singlespecies equilibria and regions of suitable habitat (wherebR0x > 1). However, there were also interspecific differ-ences. The peak single species equilibria of quagga musselsoccurred at a greater depth and distance from shore than itdid for zebra mussels. Further, the equilibrium abundanceof quagga mussels was greater than zebra mussels for thedeeper part of the lake but in the shallows zebra musselsattained higher densities than quagga mussels. Competitioninduced a shift in equilibria and regions of suitable habitatsuch that zebra mussels occupied nearshore depths andquagga mussels occupied deepwater habitats. Thiscompetition-induced shift from fundamental to realizedniche is summarized in Fig. 4 which shows the deepwaterboundary for zebra mussels shifting to shallower habitatsand the shallow water boundary for quagga mussels shiftingto greater depths. This effect corresponds nicely with theexpected zonation of species distributions given the relativeadaptations of quagga mussels to deepwater habitats andzebra mussels to shallow habitats. Finally, there remained anarrow band of coexistence where the ranges of thecompeting species met.

There were complex interactions among dispersal,competition, and habitat size that affected sourcesinkdynamics which departed strongly from the mapping ofeach species fundamental niche (Fig. 4). For example, theentire fundamental niche of a species can become a sink.This is exemplified by zebra mussels in large lakes (Fig. 5),where dispersal coupling to large regions of unsuitabledeepwater habitat was sufficiently strong to turn the zebramussel fundamental niche into a sink. The effect was alsoevident for quagga mussels in small lakes where thefundamental niche declined sharply in its function as asource (Fig. 6). These effects can be understood firstthrough competition inhibiting reproduction and survival,second by dispersal coupling suitable and unsuitablehabitat, and finally by habitat size regulating the productionvalue of source habitat and the absorptive value of sinkhabitat (Figs. 5, 6). Dispersal also acted to produce sourceregions outside a species fundamental niche, in particularnearshore habitats, for the same reasons as described in thesingle species case; there, competition was not sufficientlystrong to counteract the effects of dispersal because singlespecies equilibria of residents were low. When dispersalwas relatively local, there was global competitive coexis-

Fig. 4 Effects of competition on niche space of quagga (a, light gray)and zebra (b, dark gray) mussels in the absence of dispersal. Shadedareas in (a)(b) represent fundamental niche space (defined by areaswhere bR0x > 1) and the arrows indicate a competition-induced shiftin niche boundaries to the dashed lines. Competition results inzonation of species distributions (c) with quagga mussels (light gray)occupying shallow habitats, zebra mussels (dark gray) occupyingdeep habitats, and a small area of coexistence (black) at intermediatedepth. The shore is at x=0 and the lakes center is at x=1

Theor Ecol (2010) 3:2543 35

tence through niche partitioning such that zebra musselsdominated shallow habitats and quagga mussels dominateddeepwater habitats. As dispersal increased, species zonationwas lost and the species with the greatest amount offavorable habitat dominated the entire spatial domain(Figs. 5, 6).

The combined effects of dispersal, competition, andhabitat size on global exclusion and coexistence aresummarized in Fig. 7. When dispersal was local, therewas a large region of global coexistence which wascharacterized by zonationzebra mussels occupied shallowhabitats and quagga mussels occupied deep habitats.However, as dispersal coupling between shallow anddeepwater habitats increased, coexistence declined, andthe dynamics moved towards lake-wide domination by one

species. The superior species was determined by therelative amount of favorable habitat: quagga musselsoccupied large lakes with extensive deep/cold habitat andzebra mussels occupied small lakes where there was alarger proportion of shallow/warm habitat (Fig. 7). Thiseffect can be understood by the underlying sourcesinkdynamics (Figs. 5, 6), where under extensive dispersal, therelative amounts of favorable habitat determine the sourceor sink functioning of fundamental niche space of eachspecies. These effects scale up to a metapopulationdistribution where quagga mussels occupy large lakes andzebra mussels occupy small lakes. This distribution isconsistent with the observed spread and present distribu-tions of quagga and zebra mussels across temperate easternNorth America.

Fig. 5 Effects of competitionand dispersal on sourcesinkdynamics and Ri0x componentsin a large lake. As dispersalincreases among columns fromleft to right, the responses in thespatial distributions are plottedfor single species equilibria,survival (x), lifetime repro-ductive output (x), spatial netreproductive values for theinvader Ri0x, and equilibria ofcompeting species. Solid linesare quagga mussels and dashedlines are zebra mussels. Regionswhere Ri0x > 1 function assources and Ri0x < 1 functionas sinks for the invader. Thehorizontal dotted line in thebottom row of panels corre-sponds to R0(x)=1. Shadedareas correspond to fundamentalniche space (panel rows 56)and zones of dominance atequilibria (bottom row ofpanels) for quagga (light gray)and zebra (dark gray) mussels.The shore is at x=0 and thelakes center is at x=1

36 Theor Ecol (2010) 3:2543

Discussion

The R0 theory for sourcesink dynamics can accommodatea diversity of spatial population models and provides aquantitative means for relating niche, sourcesink dynamics,and species distributions. Insights come from an ability to trackthe influence of environmental heterogeneity, biotic inter-actions, and dispersal on sourcesink dynamics. Traditionalapproaches to sourcesink dynamics classified habitat patchesaccording to the local population growth rate in the absence ofdispersal (Pulliam 1988) whereas recent advances classifysources and sinks according to the contribution a local patchmakes to the metapopulation (Figueira and Crowder 2006;Runge et al. 2006). The contribution metric, Cr, provides ageneral means for classifying patches as sources and sinks(Runge et al. 2006), but limitations include a loss of

biological interpretation from the calculations, rapidly accel-erating complexity of the calculations, and no application forcontinuous landscapes or competitive coexistence. Forresource competition models, R* theory provides a meansfor predicting the outcome on species persistence (Tilman1982), but the theory breaks down when dispersal isintroduced (Abrams and Wilson 2004). In spatial competitionmodels, there are often model-specific criteria for coexistenceand exclusion, which are sometimes difficult to interpretbiologically (Snyder and Chesson 2004). Among spatialcompetition models, there has been no general framework forclassifying sources and sinks according to ideas of contribu-tion metrics (Runge et al. 2006) or for quantifying therelations among niche, dispersal, competition, and distribu-tion. The R0 sourcesink theory overcomes many of theselimitations. It is broadly applicable to single and competing

Fig. 6 Effects of competitionand dispersal on sourcesinkdynamics and Ri0x compo-nents in a small lake (20% ofthe lake in Fig. 5). The shore isat x=0 and the lakes center is atx=0.2. See Fig. 5 caption fordetails

Theor Ecol (2010) 3:2543 37

species models. The calculations are biologically intuitive. Itis applicable to patch metapopulations or metacommunities aswell as on continuous landscapes. Finally, it provides ageneral framework for sourcesink dynamics in the relationbetween niche and distribution.

However, the R0 theory for sourcesink dynamicsbecomes more complex when individuals can both repro-duce and disperse for many years, for example, giving apopulation dynamics model of the form

n x; t 1 Z

s n z; t ; z n z; t kA z; x dz

Z

f n z;t ; z s n z ; t ; z n z ; t k J z ; x d z30

(compare with Eq. (13)). In this situation, the formulae forR0 become complicated by the need to integrate annualdispersal and spatially dependent survival and reproductionover the lifetime of an individual (Appendix 4). This isrelatively easy if the lifecycle has a discrete number ofyears, but if there is simply an annual survival probabilityfollowed by dispersal, such as in Eq. (30), the calculationsbecome complex. We leave this situation for futuremathematical development.

To see the utility and limitations of R0 theory, it is usefulto apply it to previous theory in sourcesink dynamics. ThePulliam (1988) sourcesink model, in its linearized form, is

n x; t 1 Xz

PAz PJzbz n z; t k' z; x 31

where x denotes patch identity, PA(z) is adult survival inpatch z, PJ(z) is juvenile survival in patch z, B(z) is thenumber of juveniles produced per adult in patch z, andk(z, x) is the probability an individual disperses frompatch z to patch x in 1 year. Pulliam (1988) used the localreproductive rate (x)=PA(x)+PJ(x)(x), Eq. (1), to iden-tify source patches (x)>1 and sink patches (x) 1. However, notethat Pulliams model (Eq. (31)) similar to Eq. (30) in thatthere is annual survival, dispersal, and reproduction ofadults. Hence, it suffers the same difficulty in calculatingbR0x and R0 easily, because one must track the spatiallifecycle trajectory of each individual starting at somelocation z when calculating lifetime reproductive output(Appendix 4).

The R0 theory developed for the Dreissena competitionmodel assumes an absence of depensation or Allee effects,which, if included, may change predictions on exclusionand coexistence based on a mutual invasibility analysis(Boldin 2006). Extension of our framework to includeAllee effects is an area for future work. We also did notconsider matrix population models where different popula-tion stages have different dispersal kernels. This is another

Fig. 7 Declining coexistence (shaded regions) of quagga and zebramussels with increasing dispersal. Dispersal increases among panelcolumns from left to right and is represented by the dispersal kernelk(x, z) plotted for z=0.5. In the second row of panels, speciespersistence (determined by the global net reproductive value R

i0) is

plotted against habitat size (not position along habitat) for quagga(solid lines) and zebra (dashed lines) mussels for the corresponding

dispersal kernel. The invader can establish in lakes where Ri0 > 1 and

coexistence occurs in those lakes where Ri0 > 1 for both species. The

horizontal dotted line in the bottom row of panels marks Ri0 1. Note

that the region of lake sizes permitting coexistence (shaded regions)declines with increasing dispersal leading to small lakes occupied byzebra mussels and large lakes occupied by quagga mussels. The shoreis at x=0 and the lakes center is at x=1

38 Theor Ecol (2010) 3:2543

mathematical challenge for future developments in sourcesink theory. The model we considered in detail had sessileadults and a single dispersal event associated withproduction of offspring. For this kind of model, R0 theoryprovides biologically pleasing calculations. The R0 theoryfor sourcesink dynamics may not be the simplestframework for studying spatial population dynamics forall types of models. However, our results indicate that thetheory is broadly applicable and may provide a generalframework for studying sourcesink dynamics under manymodel types on fragmented and continuous landscapes.

In the analysis of competition between zebra and quaggamussels, the R0 sourcesink theory showed how environ-ment, dispersal, and habitat size can influence the distribu-tions of source and sink habitat and the resulting speciesdistributions. In this case, an environmental gradientcreated variation in habitat quality for the competingspecies. Dispersal across habitats of varying size andquality then affects the local and global outcome ofcompetition. Environmental gradients have been shown, inother studies using cellular automata, to affect the distribu-tions of single species as well as competitive andcooperative communities (Wilson et al. 1996; Wilson andNisbet 1997). However, only recently have environmentalgradients been considered to produce sourcesink dynamicsthat affect species distributions in nature (Rex et al. 2005).While (Rex et al. 2005) hypothesized that sourcesinkdynamics along depth gradients may regulate biodiversityin the deep sea benthos, it is immediately apparent thatenvironmental gradients are ubiquitous, from the deep seato the marine intertidal to elevation gradients in mountainranges to latitudinal gradients across continental land-scapes. The R0 sourcesink theory provides a frameworkfor quantitatively characterizing sourcesink dynamics andspatial competition on such continuous landscapes.

Consistent with other studies, our analysis showed thatcompetition can reduce a species range (Case et al. 2005),and that dispersal can cause species distributions to differfrom fundamental niche space due to immigration fromsource populations (the rescue effect) (Brown and Kodric-Brown 1977; Pulliam 1988) or emigration to sink popula-tions (Amarasekare and Nisbet 2001). Underlying theseeffects were changes in source and sink regions thatdeviated from fundamental niche space due to dispersalbetween and reflecting boundaries near favorable orunfavorable habitat. Competition enhanced these effectson source and sink regions, to the extent that an entirefundamental niche can become a sink through strongdispersal coupling to unsuitable habitat. At a global(whole-lake) scale, coexistence was enhanced by localdispersal that maintained source and sink regions, andresulted in niche partitioning through species zonation.

Similar to other sourcesink models (Amarasekare andNisbet 2001, Snyder and Chesson 2004), large-scaledispersal had a homogenizing effect. This either decreasedor increased source regions depending on the relativeamounts of favorable and unfavorable habitat and resultedin a single speciesthe one with the greatest amount offavorable habitatdominating the entire domain.

The sourcesink dynamics underlying zebra and quaggamussel competition resolve a long-standing paradox ininvasion biology. The invasion of North America byDreissena began with an initial rapid spread of zebramussels across most of the temperate East followed by theappearance of quagga mussels and their subsequentdisplacement of zebra mussels in a few large lakes but notin surrounding smaller lakes (Mills et al. 1996, 1999;Wilson et al. 2006; HJM, personal communication). Theestimated fundamental niches of quagga and zebra musselsoverlapped substantially indicating that competitive exclu-sion should apply, as others like Stoeckmann (2003) havesuggested. Quagga mussels may have an energetic advan-tage (Stoeckmann 2003) but this fails to explain theirabsence from surrounding smaller lakes. Dispersal limitationis not an adequate explanation either because Dreissena arereadily transported by the same vectors (boat traffic) or thishas probably occurred for many years (Johnson and Carlton1996; Wilson et al. 1999). Closer inspection of quagga andzebra mussel fundamental niches reveals subtle inter-specific differencesquagga mussels are better adapted todeeper habitats and zebra mussels are better adapted toshallower habitat. This subtle difference in adaptation,measured directly as fitness through R0, leads to sourcesink dynamics between deepwater and shallow habitats thatresolves the paradox of Dreissena competition. Quaggamussels are predicted to exclude zebra mussels from largedeep lakes and zebra mussels are predicted to excludequagga mussels from small shallow lakes.

Differences in depth adaptation between quagga andzebra mussels are explained by subtle interspecific differ-ences in survival and growth responses to temperature(Thorp et al. 1998) and scaling relations between body sizeand fecundity (Walz 1978; Stoeckel et al. 2004; Strayer andMalcom 2006). Competition alone induced niche partition-ing by narrowing each species realized niche into a band ofquagga mussels in deep habitats, a narrow band ofcoexistence at intermediate depth, and a band of zebramussels in shallow habitats. Competitive interactions wereassumed to be mediated by body sizea passive form ofinterference competition where smaller mussels can onlyaccess food resources in the water column that have beenpartially pre-filtered by larger mussels. When we intro-duced dispersal at local scales, these patterns of specieszonation remained over a broad range of lake sizes.

Theor Ecol (2010) 3:2543 39

However, Dreissena have vast dispersal potential owing tolong-lived planktonic larvae (Mackie and Schloesser 1996;Mills et al. 1996). As dispersal increased the range of lakesizes capable of supporting both species declined; patternsof species zonation were lost and a single species typicallydominated the entire lake. The partitioning of lake-widedominance to quagga or zebra mussels depends on anassumption that larger lakes have a larger proportion ofcolder benthic habitat and that smaller lakes have a largerproportion of warmer benthic habitat during summer. Sincelakes typically have a thermal depth gradient duringsummer, either continuous or discontinuous, this causesshallow benthic habitats to be warmer than deepwaterbenthic habitats. On average then, larger lakes would have alarger proportion of deepwater and colder benthic habitatthat favors Quagga mussels. These effects scale to predict ametacommunity distribution where quagga mussels domi-nate large lakes and zebra mussels dominate small lakes,which is consistent with their patterns of spread and presentdistributions.

The classical result for sourcesink dynamics and spatialcompetition is that sourcesink dynamics among habitatpatches support the coexistence of competing species(Levin 1974; Amarasekare and Nisbet 2001). The land-scape coexistence of zebra and quagga mussels has adifferent mechanismsourcesink dynamics within lakesundermine within-lake coexistence. Subtle differences infitness between zebra and quagga mussels in response totemperature creates a sourcesink dynamics along thecontinuous habitat of a lake bottom. Competitive inter-actions with quagga mussels yield source habitat for zebramussels in shallow habitats and sink habitat in deephabitats. In contrast, competition with zebra mussels leadsto sink regions in shallow habitat and source regions indeep habitats for quagga mussels. Lake size then mediatesthe outcome of sourcesink dynamics between deepwaterand shallow habitats. Because dispersal is large, couplingbetween the two habitat types is strong, leading todominance of the species with the greatest amount of habitatand exclusion of the species with the least. Dominance ofone species and exclusion of the other depends on lake size,where strong connectivity to competition-determined sinkhabitat extinguishes species from what is otherwise suitablehabitat, a result theorized previously by Amarasekare andNisbet (2001) and Schreiber and Kelton (2005). Theamong-lake pattern of coexistence arises then because thelandscape includes small lakes where zebra mussels prevailand large lakes where quagga mussels prevail, not becauseof sourcesink dynamics among lakes.

While the relative adaptations to deep and shallowhabitats for quagga and zebra mussels are well supported,there are some factors we did not consider. For example,temperature control of gametogenesis and spawning allows

reproduction by quagga mussels at greater depths thanzebra mussels (Roe and MacIsaac 1997; Claxton andMackie 1998). There are also interspecific differences inadaptations to wave swept environments that may favorzebra mussels in shallow habitats (Mills et al. 1993). Thesefactors would act to increase the interspecific partitioning oflakes into source and sink regions and further substantiateour results. Other factors may act in neutral or unknownways. For example, mussels can deplete food supply atwhole-lake scales (Bridgeman et al. 1995; Fahnenstiel et al.1995; Idrisi et al. 2001), which implies that competitionmay occur among individuals other than immediateneighbors. Further, if food limitation induces interspecificdifferences in survival, growth, or fecundity, then this couldaffect the model results. These types of differences, if theyexist, could be directly incorporated into the model if theyare sufficiently documented over a range of temperatures.Temperature was incorporated in the model as a meansummer spatial profile and we did not consider within-season temporal variation. This assumes that competitiveinteractions, growth, and reproduction occur primarilyduring summer months. Further, we used data frommesocosm experiments to parameterize temperature depen-dency of survival and growth (Thorp et al. 1998) and theseof course depart from field conditions. Nevertheless, whilethe model made many simplifying assumptions, we believeit captured the essential features of Dreissena biologyinparticular interspecific differences in habitat adaptationand allowed us to examine how these differences interactwith dispersal, competition, and habitat size to producesourcesink dynamics and explain species distributions.

Sourcesink dynamics may be ubiquitous in nature.They may occur along depth gradients and regulatebiodiversity in the deep sea benthos (Rex et al. 2005).They may occur along depth gradients in lakes andstructure benthic communities, as we have suggested here.Sourcesink dynamics may occur on other environmentalgradients down intertidal shorelines, up mountain ranges, oracross continental scales. For fragmented habitats, sourcesink dynamics among habitat patches may facilitatecoexistence of competing species (Levin 1974; Amarase-kare and Nisbet 2001). The general sourcesink theory wehave presented here can accommodate these various spatialstructures yet yield general and coherent criteria that arebiologically intuitive for understanding relationships amongniche, dispersal, biotic interactions, sourcesink dynamics,and species distributions. It is important to note that ourmethods are not limited to theory but can be directly portedto Geographic Information Systems to accommodateenvironmental information from real habitats. The applica-tion of sourcesink theory to competition between quaggaand zebra mussels resolves their paradoxical distributionsand shows that sourcesink dynamics rather than dispersal

40 Theor Ecol (2010) 3:2543

limitation explain the absence of species from suitablehabitat. There are numerous other possible applications ofour theory, perhaps most notably spatial coexistence,habitat conservation, marine protected areas, and ecologicalresponses to climate change.

Acknowledgements We thank Franz Weissing, William Wilson,Brad Anholt, Marjorie Wonham, Hugh MacIsaac, and David Lodgefor helpful discussions and comments. We are grateful to HughMacIsaac and Ed Mills for updating us on the distributions ofDreissena and to James Thorp for sharing parameter values ofstatistical models. This work was supported by the US NationalScience Foundation grant DEB 02-13698, by the Natural Science andEngineering Research Council of Canada Collaborative ResearchOpportunities program, by the Canadian Aquatic Invasive SpeciesNetwork, and by a Canada Research Chair to MAL.

Appendix 1

Here we show how Eq. 8, which applies to the general case,can be used to find intuitively pleasing solutions to twospecial cases: when the initial individual is introduced atone location and when dispersal is uniform across thedomain.

For the point release, we must introduce the Dirac deltafunction, (zx), which is a distribution (a) equals zero forall xz, (b) is unbounded when x=z, and (c) integrates tounity. For example, if an individual disperses according tothe Dirac delta function, it will remain where it isprobability oneit does not disperse. Similarly, if theinitial individual is introduced at some location , then ithas an initial distribution 0(x)=(x). So if we consideran adult individual that is introduced at location , Eq. (6)simplifies to

d x x < xZL0

zd z x k x; z dz

< x x k x; x 32We define R0(x) to be the total number of offspring

produced at any spatial location by the single individual byintegrating Eq. (32) over all spatial locations x to yield

R0 x ZL0

< x x k x; x dx

x ZL0

< xk x; x dx

33

as given in Eq. (4). In the case where there is no dispersalk(, x)=(x) and Eq. (33) simplifies to Eq. (3).

When dispersal operates on the largest scale possible, thedispersal kernel becomes a uniform distribution. The lake isessentially homogenized by dispersal so that larvae cansettle anywhere with equal probability regardless of wherethey were released. In this case, the next generationoperator, Eq. (4), becomes

f z < z 1L

Z L0

xf0xdx: 34

If we choose (z)=(z), then we see that (z) is theeigenfunction and that the dominant eigenvalue is

R0 1LR L0 x< xdx

1LR L0 R0xdx

35

which is pleasingly intuitive. When dispersal operates on alarge enough scale, the lake-wide net reproductive rate, R0,becomes the spatial average of the local R0(x).

Appendix 2

In this section, we describe the Dreissena model parame-terization. Equation (9) is our model with two speciesquagga and zebra mussels. We compiled statistical modelsof mussel growth, survival, and reproduction from theliterature and linked these to position along benthic habitatfrom the shore at x=0 to the center of a lake at x=L via amean summer temperature profile, T=30e0.05x. Results arequalitatively the same for most temperature functions weconsidered, including those with thermal stratification. Wetake , the phenotypic trait determining competitive ability,to be shell length, assuming that mussels must compete withimmediate neighbors for access to the water column. Shellgrowth is related to temperature through a logistic regression: T ; t exp a b1T b2T 2 b3 t 1 1, where isshell length, T is temperature in degrees centigrade, and t istimehere taken to be years but in the source study it is a 3-month mesocosm experiment (Table 1) (Thorp et al. 1998).Together with our model, this assumes that body growthoccurs primarily during a 3-month summer season. Shell sizeis related to egg production by 0.4( )4.4 (Walz 1978;Stoeckel et al. 2004), which we assume is the same for bothspecies. We assume that 0.1% of larvae survive to settle onthe lake bottom, similar to other models (Strayer andMalcom 2006) and consistent with other observations(Sprung 1989). Fecundity is then f x 0:001 0:4 x 4:4,which assumes an excess of sperm. Basal survival rates arealso related to temperature by sj,1(T)=sj,2(T)=[exp(a+b1T+b2T

2)][exp(a+b1T+b2T2)]1 based on the same mesocosm

experiments (Table 1) (Thorp et al. 1998) plus oursimplifying assumption that sj,1(T)=sj,2(T).

Theor Ecol (2010) 3:2543 41

Appendix 3

Here, we calculate the expected lifespan of an adult musselinvading a resident community. Let the annual survivorshipof an adult be p. Let be a random variable for the durationof an adult lifespan. The probability of dying after 1 year isP[=1]=(1p). The probability of dying after 2 years isP[=2]=p(1p) and after 3 years is P[=3]=p2(1p). Theprobability of dying after n years is P[=n]=pn1(1p). Themean duration of an adult is then given by the expectation

E t Pt tP t t 1 p 1 2p 3p2 4p3 ::: 1 p ddp 1 p 1

h i 1 p 1

36

Appendix 4

Here we give the next generation operators for Eqs. (30)and (31). Equation (30) has next generation operator

n x; t R f 0; z s 0; z n0 z; t n1 z; t K kJ z; x dzn0 x; t n x; t ni x; t

Rs 0; z ni1 z; t kA z; x dz; i 1; 2;K

37and Eq. (31) has next generation operator

n x; t PzbzPJz n0 z; t n1 z; t K k' z; x

n0 x; t n x; t ni x; t

PzPAzni1 z; t k 0 z; x ; i 1; 2;K

38The global net reproductive value R0 is the spectral

radius of . The fundamental niche and sourcesink R0scan be calculated using the methods of Appendix 1.

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Table 1 Parameter values of Dreissena survival and shell growthmodels (Thorp et al. 1998)

Model Species a b1 b2 b3

Shell growth Quagga 1.39 0.49 0.0095 0.058Zebra 1.39 0.34 0.015 0.058

Survival Quagga 10.81 1.046 0.023 Zebra 20.46 1.88 0.039

42 Theor Ecol (2010) 3:2543

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An R0 theory for sourcesink dynamics with application to Dreissena competitionAbstractIntroductionTheory of sourcesink dynamicsAn R0 theory for sourcesink dynamicsR0 for single species modelsExamples of single species modelsR0 for spatial competition modelsSummary of R0 theory for sourcesink dynamicsDreissenid competitionA Dreissenid modelSingle species dynamicsCompeting species dynamics

DiscussionAppendix 1Appendix 2Appendix 3Appendix 4References

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