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Interspecific Variation in Critical Patch Size and Gap‐Crossing Ability as Determinants ofGeographic Range Size Distributions.Author(s): William F. Fagan, Robert Stephen Cantrell, Chris Cosner, andSubramanian RamakrishnanSource: The American Naturalist, Vol. 173, No. 3 (March 2009), pp. 363-375Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/10.1086/596537 .

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vol. 173, no. 3 the american naturalist march 2009 �

Interspecific Variation in Critical Patch Size and Gap-CrossingAbility as Determinants of Geographic Range

Size Distributions

William F. Fagan,1,* Robert Stephen Cantrell,2 Chris Cosner,2 and Subramanian Ramakrishnan2

1. Department of Biology, University of Maryland, College Park, Maryland 20742; 2. Department of Mathematics, University of Miami,Coral Gables, Florida 33124

Submitted May 16, 2008; Accepted September 24, 2008; Electronically published January 20, 2009

Online enhancements: appendixes.

abstract: How biological processes such as reproduction and dis-persal relate to the size of species’ geographic ranges constitutes amajor challenge in spatial ecology and biogeography. Here we developa spatially explicit theoretical framework that links fundamentalpopulation-level ecological traits (e.g., rates of dispersal and popu-lation growth or decay) with landscape heterogeneity to derive es-timates of species’ geographic range sizes and, further, distributionsof geographic range sizes across species. Although local (patch-scale)population dynamics in this model are completely deterministic, weconsider a fragmented landscape of patches and gaps in which thespatial heterogeneity is itself stochastic. This stochastic spatial struc-ture, which juxtaposes landscape-level patch and gap characteristicsagainst population-level critical patch sizes and maximum gap-crossing abilities, determines how far a novel species can spread fromits evolutionary origin. Given reasonable assumptions about land-scape structure and about the distribution of critical patch sizes andcritical gap lengths among species, we obtain distributions of geo-graphic range sizes that are qualitatively similar to those routinelyfound in nature (e.g., many species with small geographic ranges).Collectively, our results suggest that both interspecific differences inpopulation-level traits and the landscapes through which speciesspread help determine patterns of occupancy and geographic extent.

Keywords: critical patch size, gap-crossing ability, geographic rangesize, range expansion, spatial heterogeneity.

Introduction

Interactions among habitat availability, biological pro-cesses, and physical factors determine the size of species’geographic ranges and the location of species’ geographicrange boundaries. Empirical data on geographic range sizesmake clear that environmental thresholds (e.g., Cabrera1996), gradients (e.g., Virgos and Casanovas 1999), and

* Corresponding author; e-mail: [email protected].

Am. Nat. 2009. Vol. 173, pp. 363–375. � 2009 by The University of Chicago.0003-0147/2009/17303-50463$15.00. All rights reserved.DOI: 10.1086/596537

spatially varying physical processes (e.g., Gaylord andGaines 2000) can all act as determinants of species’ bound-aries. Of these issues, environmental gradients have re-ceived the most attention from theoreticians, often in con-junction with studies of the spatial dynamics ofevolutionary clines (Barton and Bengtsson 1986; Pialekand Barton 1997).

In contrast, the effects of discontinuous spatial variation(i.e., patchy landscapes) on species’ boundaries have re-ceived relatively less attention. Nonetheless, some perti-nent results about range limits in discontinuous landscapesare available. For example, metapopulation dynamics act-ing across an environmental gradient may enforce a spe-cies’ range limit (Carter and Prince 1981, 1987; Holt etal. 2005). Likewise, a partial dispersal barrier may preventcompetitive exclusion and facilitate stable parapatricboundaries (Goldberg and Lande 2007). In a discretizedlandscape, species’ range limits can emerge even in theabsence of clear environmental gradients via the mecha-nism of “invasion pinning” (Keitt et al. 2001), in whichAllee dynamics, coupled with fine-scale patchiness, limitthe spatial advance of a spreading species. In an importantstep toward unification, Holt et al. (2005) proposed thata small set of demographic mechanisms related to spatialvariation in population dynamic parameters (e.g., birth,death, and dispersal rates) may collectively determine thespatial position of species’ range boundaries under manyconditions.

Overall, the issue of species’ boundaries can be viewedfrom two complementary perspectives: (1) what processesinitially limit the spatial expansion of a novel species(Webb and Gaston 2000; Keitt et al. 2001) and (2) whatprocesses govern the long-term size and occupancy of aspecies’ geographic range (Gaston 2003). Building on ideasof Keitt et al. (2001), we adopt the “initial expansion”perspective on geographic range dynamics to explore howlandscape heterogeneity influences species range bound-



364 The American Naturalist

aries. However, unlike Keitt et al. (2001), who assumedthe environment to be homogeneous or to have a direc-tional gradient in quality, we assume here that the envi-ronment is heterogeneous, with the heterogeneity itselfbeing stochastic. As a result, our efforts converge on a“landscape ecology of species spread,” as outlined by With(2002).

Because the initial expansion perspective on species’boundaries shares its theoretical foundations with modelsof invasion dynamics (Shigesada and Kawasaki 1997), re-sults from invasion theory are highly relevant here. A com-mon theme in theoretical investigations of invasion dy-namics is to quantify the speed of the invasion, that is,the rate at which spatial spread occurs. For example, usingreaction-diffusion (RD) models, Shigesada et al. (1986)demonstrated how habitat heterogeneity (characterized asan alternating sequence of good and bad habitat patchesof fixed sizes) could slow the progress of a traveling waveof invasion. Dewhirst and Lutscher (2009) have analyzeda similar spatial setup for cases of discrete-time integro-difference equations, obtaining an estimate of the frag-mentation threshold necessary for an invasion to propa-gate (i.e., the minimum percentage of good habitat thatallows population spread; see also With 2002). Severalother studies have examined aspects of invasion dynamicsin situations with spatial variation in population growthbut not dispersal ability (Van Kirk and Lewis 1997; Bots-ford et al. 2001; Kawasaki and Shigesada 2007).

However, unlike most previous investigations of spatialspread, we focus here not on quantifying the rate of spreadbut instead on the maximum distance covered by a speciesthat is expanding its range in a heterogeneous landscape.As we discuss below, we eschew the traveling-wave for-malism that is standard in theoretical studies of invasiondynamics in favor of the critical patch size perspective onecological dynamics in patchy systems (Skellam 1951;Kierstead and Slobodkin 1953; Cantrell et al. 2001). Inwhat follows, we introduce a modeling approach that com-bines basic life-history traits with information on thepatchiness of a landscape to estimate the size of a particularspecies’ geographic range. Our efforts speak directly to theproblem of nonequilibrial geographic range dynamics, in-cluding effects of landscape heterogeneity and Allee effects,a set of topics that has been identified as a major challengein spatial ecology and biogeography (Holt et al. 2005)

We then extend the analysis to illustrate how one canobtain a multispecies distribution of geographic range sizesfor a particular landscape. As such, our modeling effortsprovide a novel route linking population biological traitsto geographic range size distributions via the initial ex-pansion perspective on geographic ranges. Consequently,our analysis complements but contrasts with models suchas those of Gaston and He (2002) that focus on stochastic

variations in range size occurring after the initial phase ofspreading.

Empirical syntheses suggest that within major taxa (i.e.,groups of species [such as waterfowl or pine trees] that,by virtue of shared evolutionary heritage, exhibit a diversebut not extreme range of life-history traits), geographicrange sizes are strongly right (positively) skewed. In his-tograms of geographic range size for major taxa, the left-most category is typically the modal category (Brown etal. 1996; Gaston 2003). Consequently, as a final step weidentify some conditions under which particular distri-butions of life-history traits will yield geographic rangesize distributions like those commonly observed in nature.

Model Setup

We seek to understand how ecological attributes of a spe-cies, specifically the minimum patch size needed for pop-ulation persistence and the maximum gap length acrosswhich dispersal is possible, interact with landscape het-erogeneity to determine how far that species can be ex-pected to spread from its area of evolutionary origin andthus how big a geographic range that species is likely toattain. In this scenario, species expected to spread onlyshort distances would likely be “endemics,” whereas spe-cies expected to spread great distances would likely attainlarge geographic ranges.

Our modeling approach envisions a landscape as a one-dimensional environment consisting of alternating patchesand gaps whose sizes are drawn from some specified dis-tribution. This is a more generalized perspective than thatused in previous studies of spatial spread in heterogeneouslandscapes; those studies treated habitat heterogeneity asalternating pairs of good- and bad-quality patches of fixedsize (Shigesada et al. 1986; Van Kirk and Lewis 1997; Bots-ford et al. 2001; Berestycki et al. 2005; Kawasaki and Shi-gesada 2007; Dewhirst and Lutscher 2009).

Given a set of biological parameters such as dispersalrates and birth or death rates, we derive a distribution forthe distance that a species with those parameters couldspread. How far, specifically, such a species could spreadwould depend on the particular landscape (and startingpoint). Finally, by considering the population parametersfor different species as being drawn from some distribu-tion, we derive a distribution for the frequency of species’geographic ranges along the axis from endemic to wide-spread. Our approach allows us to link expected range sizeof a species to parameters describing that species’ dispersaland demography. We make this link by using simple RDmodels to obtain characteristics of minimum patch sizeand maximum gap length that we can then feed into thelandscape model and thus predict the distance that thespecies can spread in any given landscape realization. The



From Life Histories to Geographic Ranges 365

combined RD-landscape model can be parameterized us-ing distributions of species’ life-history traits together withinformation about the spatial heterogeneity of a landscape.

Our model development hinges on several foundationalconcepts in spatial ecology. First, we require that thereexists a “minimum patch size” needed for population per-sistence. This concept is a standard one in theoretical ecol-ogy, going back to Skellam (1951) and Kierstead and Slo-bodkin (1953), and a wide variety of empirical studiesdocument the importance of patch size for populationpersistence (Groom 1998; Hanski 1998). Second, we re-quire that the connectivity between two habitat patchesdepends on the size of the gap between those patches.Urban and Keitt (2001) clearly describe how dispersal be-tween patches can depend on gap length. Third, we requirethat the founding population of a species in a habitat patchreach some minimum density to successfully colonize thatpatch. This idea, which corresponds to a kind of Alleeeffect, has been observed in a variety of empirical systems(Veit and Lewis 1996; Johnson et al. 2006; Tobin et al.2007) and has been built into spatially explicit models forisland colonization (Cantrell et al. 1996) and for the es-tablishment of geographic range boundaries via invasionpinning (Keitt et al. 2001).

Analysis

To characterize minimum patch size we will use a simpleRD model (Skellam 1951; Kierstead and Slobodkin 1953).If a population inhabits a patch of length L (distance),disperses through it with diffusion rate D (distance2

time�1), increases with local intrinsic growth rate r(time�1), and has local carrying capacity K, then the pop-ulation density u within the patch satisfies the relation

2�u �u up D � ru 1 � , (1)

2 ( )�t �x K

for . If individuals that reach the boundary of the0 ! x ! Lpatch always exit the patch, then equation (1) is aug-mented with the boundary condition

u(0, t) p u(L, t) p 0. (2)

Solutions of equations (1) and (2) decay toward 0 if r �and grow toward a unique positive density if2 2Dp /L ! 0

(Cantrell and Cosner 2003). Thus, for pop-2 2r � Dp /L 1 0ulation persistence we must have

D∗ �L 1 L p p , (3)r

where is the minimum patch size needed for successful∗Lpersistence of the population.

Although we have focused this particular derivation ofa species’ critical patch size (i.e., diffusion-mediated lossacross the patch boundary), we note that there are severalalternative routes to a critical patch size that might besubstituted for species that possess dispersal behaviorsmore sophisticated than simple diffusion. These mecha-nisms include (1) scenarios where there is within-patchhabitat heterogeneity (Cantrell and Cosner 2001) or wherea species tends to move toward or away from edges or upor down spatial gradients (Cantrell and Cosner 1991), (2)situations where the regrowth capacity of a species scaleswith patch size (Cantrell et al. 2005), and (3) cases wherea species as a whole suffers from an Allee effect but in-dividuals defend home ranges, placing limits on the num-ber of individuals that can be packed into a given patch.

We use a model similar to equation (1) to describe theexpected density of a population in a gap of matrix habitatbetween patches, but we also incorporate the idea thatthere is a minimum rate at which individuals must arriveat the next patch for colonization to succeed. The notionthat propagule pressure influences colonization success iswidely supported from both theoretical and empiricalviewpoints (Lockwood et al. 2005; Drake and Lodge 2006).Suppose that a population inhabits an isolated patch atdensity K, disperses through the matrix outside the patchby diffusion at rate d, and experiences mortality in thematrix at rate s. The propagule density in the matrix canvthen be described by

2�v �vp d � sv, (4)

2�t �x

with boundary condition

v(0, t) p K. (5)

The unique bounded equilibrium of equations (4) and (5)is . We use m to indicate the min-1/2v(x) p K exp [�x(s/d) ]imum average propagule density needed for successful col-onization of a new patch. As such, m can thus be thoughtof as a type of Allee threshold necessary for invasion suc-cess (Lewis and Kareiva 1993; Veit and Lewis 1996). If thenew patch is at distance l from the original patch, suc-cessful colonization requires , which1/2K exp [�l(s/d) ] 1 mcan be rewritten as

K d∗ �l ! l p ln , (6)( )m s

where is the maximum gap length that the species can∗l




Figure 1: Schematic of the model setup for our analysis of geographic range size. Patch sizes and gap widths vary stochastically across the landscape.As it expands outward from its evolutionary origin, species A cannot persist on patch L3 (minimum patch size ). In contrast, species B has∗L 1 LA 3

a different suite of life-history traits and can persist on patch L3 but cannot cross gap l4 (maximum gap width ). Consequently, species A has∗l ! lB 4

a geographic range smaller than that of species B (lines).

be expected to cross successfully. A model similar to equa-tions (4) and (5) was used by Cantrell et al. (1996) as partof a spatially explicit model for a species colonizing anisland. The idea that the rate of successful colonizationdecreases exponentially with distance from a source, whichwe have adopted here, is a common assumption in eco-logical theory and is a mainstay of both island biogeog-raphy theory (MacArthur and Wilson 1967) and meta-population theory (Hanski 1998).

To describe the landscape that a spreading species mightencounter, we assume that the overall environment con-sists of an alternating series of patches and gaps whoselengths are drawn from distributions that reflect the large-scale structure of the landscape. Specifically, we assumethat each realization of the landscape starts with a patchof length L1, which is followed by a gap of length l1, whichis followed by a patch of length L2, then a gap of lengthl2, and so on, with patches and gaps alternating (fig. 1).We assume that the species we are considering will spreaduntil it reaches a patch too small to support a population,that is, , or a gap too large to cross, that is,∗L ! L l 1n n

.∗lNote that as we develop it here, the full model does not

allow a species to “jump over” (i.e., bypass) a nearby smallpatch and still colonize a sufficiently large patch down-stream. However, we address this issue in the examplecases, where we work with what amount to upper andlower bounds on a species’ geographic range size. In cases1 and 2, we consider the situation where there are nocritical patch size effects and any patch can be effectivelybypassed by a spreading species. In these upper-boundscenarios, we develop geographic range size distributionsusing an overestimate of each species’ geographic rangesize. Later, in case 3, we consider a corresponding lower-bound scenario, in which a spreading species cannot jump

over a nearby small patch. We return to this importanttopic in “Discussion.”

Assume that and are sequences of inde-(L ) (l )n n≥1 n n≥1

pendent identically distributed random variables. We de-fine

∗t p min {n : L ! L },1 n

∗t p min {n : l 1 l }, (7)2 n

t p min {t , t }.1 2

The distance that a species will spread in the positive di-rection from is then . The initial

tx p 0 V p � (L � l )1 j jjp1

geographic range of species i after it has spread across thelandscape would then be , where V2 is aV p V � Vi i1 i2

second random variable, independent of V1 but with thesame distribution, representing spread in the negative di-rection. For any given parameter set describing the min-imum patch size and the maximum gap length for∗ ∗L li i

species i, the geographic range size Vi will be a randomvariable with a distribution depending on the life-historyparameters that determine the critical patch size and max-imum gap length for species i.

In table 1 we recapitulate all of the key life-history pa-rameters that together determine a species’ critical patchsize and maximum gap length, along with a qualitativesummary of how an increase in the magnitude of a givenlife-history trait would affect a species’ critical patch size,maximum gap length, and geographic range size. Note thatall of the life-history parameters enter the calculations ofcritical patch size and maximum gap length (eqq. [3], [6])in a sublinear fashion, such that a doubling (or halving)of the magnitude of a particular trait will translate into aless than linear increase or decrease in the patch size orgap length thresholds.




Table 1: Key parameters linking life history to geographic range size

Parameter Interpretation

Consequences of an increase inparameter value on CPS,

MGL, and GRS

r Population growth rate in habitat patches Decreases CPS and increases GRSD Diffusion rate in habitat patches Increases CPS and decreases GRSs Death rate in matrix gaps Decreases MGL and decreases GRSd Diffusion rate in matrix gaps Increases MGL and increases GRSK Carrying capacity (as a density) in patches Increases MGL and increases GRSm Minimum propagule density at patch edge necessary for

successful colonizationDecreases MGL and decreases GRS

∗ �L p p D/r Minimum patch size to support a population Decreases GRS∗ �l p ln (K/m) d/s Maximum gap length across which colonization is possible Increases GRS

E0 Mean patch size in the landscape Increases GRSe0 Mean gap length in the landscape Decreases GRSVi Geographic range size of species i

Note: CPS p critical patch size; MGL p maximum gap length; GRS p geographic range size.

Next, we assume that the parameters describing mini-mum patch size and maximum gap length are distributedamong a pool of species; thus, quantities such as the meanand variance of the range size ( and , respec-E(V ) V(V )i i

tively) will themselves be treated as random variables thatdiffer among species. Given a distribution of dispersal anddemographic parameters across a species pool, we canthen, in principle, compute the probability Pr (a ≤

, which can be interpreted as giving a distri-E(V ) ≤ b)i

bution of the probability that a species i drawn from thepool of candidates will be endemic, widespread, or in be-tween. Table 1 summarizes all parameters involved in thedevelopment of this model.

A Brief Sketch of Model Analysis Necessary toDerive a Species’ Geographic Range Size

The analysis and justification of the model presented above(eqq. [1]–[7]) depend on results from probability theory.We will sketch the results of the analysis in this section,leaving detailed derivations and justifications for appendixA in the online edition of the American Naturalist. We willdenote the probability of an event e as . The first keyPr (e)observation is that , , and in equation (7) are stoppingt t t1 2

times, so they can be treated as random variables. Let Eand V denote the mean and variance, respectively. Forindependent random variables X and Y, E(X � Y ) p

and . Thus,E(X) � E(Y ) V(X � Y ) p V(X) � V(Y )and . If LE(V) p E(V � V ) p 2E(V ) V(V) p 2V(V )1 2 1 1

denotes a random variable with the same distribution asthe patch sizes in our model and l is a random variablewith the same distribution as the gap lengths, then

E(V) p 2[E(L) � E(l)]E(t), (8)

with

1E(t) p , (9)i ∗ ∗1 � Pr (L ≥ L ) Pr (l ≤ l )i i

where and are the minimum patch size (eq. [3]) and∗ ∗L li i

maximum gap length (eq. [6]) of species i, respectively.(This assumes that ; otherwise∗ ∗Pr (L ≥ L ) Pr (l ≤ l ) ! 1i i

is infinite.) Thus,E(t)

2[E(L) � E(l)]E(V ) p . (10)i ∗ ∗1 � Pr (L ≥ L ) Pr (l ≤ l )i i

Similarly, we have

2V(V ) p 2{[E(L) � E(l)] V(t) � [V(L) � V(l)]E(t)}, (11)i i i

with

∗ ∗Pr (L ≥ L ) Pr (l ≤ l )i iV(t) p . (12)i ∗ ∗ 2[1 � Pr (L ≥ L ) Pr (l ≤ l )]i i

Using equations (8)–(12), we can compute the mean andvariance of the geographic range size for species i, Vi, ifwe know , , and the landscape distributions of L and∗ ∗L li i

l.

Extending the Model to Multiple Species toCharacterize Geographic Range Size Distributions

Equations (8)–(12) are general results that hold for a widerange of landscapes (defined by distributions of L and l)and for a wide range of species (whose critical patch sizeand gap-crossing ability are determined for species i by ri

and Di in habitat patches, si and di in the matrix, and the




colonization constraints ). To explore the effects thatm /Ki i

particular assumptions about landscapes and species poolshave on the distribution of geographic range size, we pre-sent three cases here. In each of these cases we make as-sumptions about the distributions of patch and gap lengthsin the landscapes and the distributions of critical patchsizes and maximum gap lengths among species. Most com-monly we assume that these landscape components andinterspecific thresholds are distributed exponentially, andbefore proceeding, we provide some justification for theseassumptions.

Situations in which patch sizes and/or gap lengths areapproximately exponentially distributed appear repeatedlyin the landscape ecology literature, both from remote sens-ing analyses of real landscapes and as a consequence ofstandard assumptions in landscapes generated for perco-lation theory and other spatial analyses. Examples of ex-ponential distributions occurring in binary (patch, non-patch) landscapes include the size of snowfields (Bahr andMeier 2000), the size of patches created by forest fires(Cumming 2001), analyses of forest patch (gap) structurein temperate (Keitt et al. 1997) and tropical (Lawton andPutz 1988) forests, and patch sizes and interpatch distancesin one-dimensional planktonic landscapes (Curriel et al.1998). Patch size distributions that are approximately ex-ponential in form are to be expected in random landscapemodels with low to moderate percent habitat cover and ahigh degree of habitat fragmentation (R. Gardner, personalcommunication). In a detailed analysis of a southern Cal-ifornia landscape classified into many patch types (SanBernardino Mountains), frequency is a monotonic andstrongly declining function of patch size on an arithmeticaxis (M. Neel, unpublished data). Distance to nearestneighboring patch (of the same patch classification) ex-hibits a similar shape.

Moving from landscapes to the distributions of inter-specific traits, recent analyses yielding estimates of criticalpatch size for 474 resident bird species and 163 mammalspecies that live in native habitats in Costa Rica (Pereiraet al. 2004; Pereira and Daily 2006) both indicate mono-tonic, steeply declining distributions of critical patch sizeacross species, so our assumption here that critical patchsizes are distributed exponentially appears reasonable. Ourassumptions about how maximum gap-crossing ability isdistributed among species are arguably the most tenuous.In a compilation of data on gap-crossing ability for selectedtropical species, Dale et al. (1994) reported many morespecies with small or intermediate maximum gap lengthsthan with large maximum gap lengths. Creegan and Os-borne (2005) reported similar data for a suite of temperatebirds. Numerous other studies have examined gap-crossing ability for individual taxa (e.g., Bakker and VanVuren 2004; van der Ree et al. 2004; Castellon and Sieving

2006). However, we are not aware of a systematic assess-ment of the distribution of maximum gap length acrossmany species, as has been done for critical patch size.

Case 1: Exponentially Distributed Gap Lengths

If gap lengths follow an exponential distribution withmean e0, the probability density function for gap length is

andl ∼ (1/e ) exp (�x/e )0 0

∗l i

1 �x∗Pr (l ≤ l ) p exp dxi � ( ) ( )e e0 00

∗�lip 1 � exp , (13)( )e 0

where is the maximum gap length for species i. Let E0∗li

denote the mean patch size in the landscape. We next makethe further assumption that all habitat patches are ade-quately large with probability 1. (Note that this assumptionimplies that all species’ in-patch dispersal rates Di are smallenough and/or all species’ reproductive rates ri are largeenough to avoid any critical patch size constraints on spe-cies’ geographic range sizes; eq. [3]). With this assumption,equation (10) reduces to

∗liE(V ) p 2(E � e ) exp .i 0 0 ( )e 0

In formulating our models we assume that any speciesstarts out by spreading into the initial patch and gap ineither direction. Thus, the expected range size is always atleast as large as the expected size of two patches plus twogaps, that is, . Constraining a landscape thresh-2(E � e )0 0

old B such that , thenB ≥ 2(E � e )0 0

B∗Pr (E(V ) 1 B) p Pr l 1 e ln . (14)i i 0{ [ ]}2(E � e )0 0

We now move from considering the landscape to fo-cusing on the species pool. Equations (8)–(12) and henceequations (13) and (14) should now be interpreted asgiving the conditional expectation and variance of Vi given

and . With that interpretation we will retain our pre-∗ ∗L li i

vious notation. If we assume that the critical gap lengthsare exponentially distributed among species with mean∗li

, then equation (14) can be rewritten as∗e




Figure 2: Geographic range size distribution derived from case 1 (eqq.[15], [16]). Three parameter combinations are plotted that examine howrange size distribution depends on the ratio between the mean gap lengthof the landscape (e0) and the mean critical gap length among species( ). These combinations are (1) (diamonds), (2)∗ ∗ ∗e e /e p 1 e /e p 0.50 0

(triangles), and (3) (circles). In combination 2, where ,∗ ∗e /e p 2 e ! e0 0

fully 20% of species in the pool are geographically widespread (1200range size units), whereas in combination 3, where , almost all∗e 1 e0

species are very narrowly distributed (!50 range size units). In all cases,. Combinations 1 and 2 equal 0 below ; combination 3E p 2 B p 60

equals 0 below .B p 8

�e B0Pr (E(V ) 1 B) p exp lni ∗{ [ ]}e 2(E � e )0 0

∗�e /e0

Bp . (15)[ ]2(E � e )0 0

Provided , the full multispecies distributionB ≥ 2(E � e )0 0

of geographic range sizes can be obtained from equation(15) and has the probability density function

d[Pr (E(V ) ! B)] e ∗ ∗i 0 e /e �(1�e /e )0 0p [2(E � e )] B , (16)0 0∗( )dB e

which is a Pareto distribution (Johnson et al. 1994). Thefunction in equation (16) declines monotonically suchthat, in a histogram, the leftmost category would be themodal category, a pattern that agrees with empirical ob-servations on geographic range size distributions (Brownet al. 1996; Gaston 2003). Initially equation (16) declinesfaster than exponential, but the function has a heavy tailcompared with an exponential distribution that (eventu-ally) becomes apparent for large range sizes. The ratio

, which appears three times in the probability density∗e /e0

function of range size given in equation (16), relates e0,the mean gap length in the landscape, to , the mean∗ecritical gap length among species, and is a crucial factorthat affects both the shape of the geographic range sizedistribution and its spatial scale. In figure 2, we show howa mere fourfold change in this key ratio may shift theshape of the geographic range size distribution from ascenario where almost all species are very narrowly dis-tributed (!50 range size units) to one in which fully 20%of species in the pool are geographically widespread (1200range size units). Note that because this case does notallow for any constraints on geographic range size due tocritical patch size, from the perspective of the underlyinglife-history traits themselves, the key determinants of theserange size distributions are the ratio of in-matrix diffusionrate to in-matrix mortality rate ( ) and the ratio of car-d/srying capacity to colonization threshold ( ; table 1; eq.K/m[6]).

Case 2: An Alternative Distribution for Gap Lengths

Suppose instead that gap length l has the distribution

a

a � 1 a � 1a �1�aa e x on e , � (17)0 0( ) [ )a a

for . Expression (17) is a Pareto distribution witha 1 1mean e0 and shape parameter (Johnson et al. 1994).a

Pareto distributions have been used as models for the sizedistributions of lakes (Downing et al. 2006) and othernatural features (Vidondo et al. 1997), so this is at leasta plausible choice for a distribution of gap or patch sizes.Then,

∗la

a � 1∗ a �1�aPr (l ≤ l ) p a e x dx� 0( )ae [(a�1)/a]0

aaa � 1 e 0p 1 � . (18)∗ a( )a (l )




Figure 3: Geographic range size distribution derived from case 2 (eqq.[20], [21]). Three parameter combinations are plotted that examine howrange size distribution depends on the ratio between the mean gap lengthof the landscape (e0) and the mean critical gap length among species( ). These combinations are (1) (diamonds), (2)∗ ∗ ∗e e /e p 1 e /e p 0.50 0

(triangles), and (3) (circles). In all cases, and .∗e /e p 2 E p 2 a p 20 0

Thus, using the same approach as in case 1,

∗ a2(E � e )(l )0 0Pr (E(V ) 1 B) p Pr 1 Bi a( ){[(a � 1)/a]e }0

1/a

1 �1 a � 1 Bp exp e . (19)0∗ ∗{( )[( ) ][ ] }e e a 2(E � e )0 0

So

Pr (E(V ) ! B) pi

1/a

1 �e a � 1 10 1/a1 � exp B , (20)∗ ∗{( )( )[ ] }e e a 2(E � e )0 0

and by taking the derivative , we obtain the multi-d/dBspecies geographic range size distribution as

1/a

d[Pr (E(V ) ! B)] �1 �e a � 1 1i 0p ∗ ∗( )( )[ ]dB e e a 2(E � e )0 0

1/a

�e a � 1 1 10 1/a �1�(1/a)# exp B B .∗{( )( )[ ] }( )e a 2(E � e ) a0 0

(21)

Because , equation (21) is not an exponential dis-a 1 1tribution, but if is close to 1, then the resulting distri-a

bution of geographic range sizes is approximately expo-nential in form. As was the situation in case 1, the keyratio , which relates the mean gap length in the land-∗e /e0

scape to the mean critical gap length among species, helpsdetermine both the shape of the geographic range sizedistribution and its spatial scale. In figure 3, we againdemonstrate how a fourfold change in this key ratio mayshift the shape of geographic range size distribution. Inthis case, however, the same three parameter combinationsall yield range size distributions with fairly heavy tails (i.e.,a large proportion of widespread species), and moreover,the curves for the cumulative distribution functions ac-tually cross. Note again that because this case does notallow for any constraints on geographic range size due tocritical patch size, from the perspective of the underlyinglife-history traits themselves, the key determinants of theserange size distributions are the ratios and (tabled/s K/m1; eq. [6]).

Case 3: Exponential Distributions for Patch Size and GapLength in the Landscape and for Critical Patch Size

and Critical Gap Length among Species

In the cases we have considered so far, we have assumedfor simplicity that all patches are larger than the minimalsize needed to support a population of the focal species,but it is natural to allow the possibility that some patchesare too small to support populations. Suppose that thesizes of the patches as well as the lengths of the gaps areexponentially distributed. If patch sizes follow an expo-nential distribution with mean E0, then ∗Pr (L ≥ L ) pi

, where is the minimum patch size for spe-∗ ∗exp (�L /E ) Li 0 i

cies i. Combining this with equation (13) yields




2(E � e )0 0E(V ) p . (22)i ∗ ∗1 � exp (�L /E )[1 � exp (�l /e )]i 0 i 0

Note that . Continuing as in the earlierE(V ) ≥ 2(E � e )i 0 0

cases, the probability density function for the distributionof geographic range sizes is

d[Pr (E(V ) ! B)] 2(E � e ) e Ei 0 0 0 0p2 ∗ ∗( )( )dB B e E

∗(E /E )�10

2(E � e )0 0# 1 � (23)[ ]B

[2(E �e )]/B0 0

∗ ∗�(E /E ) (e /e )�10 0# (1 � w) w dw�0

for and 0 otherwise (app. B in the onlineB ≥ 2(E � e )0 0

edition of the American Naturalist). The integral in equa-tion (23) cannot generally be evaluated in terms of ele-mentary functions; however, for a range of parameter val-ues, it can be expressed in terms of incomplete betaintegrals (see app. B; Abramowitz and Stegun 1965), andfor certain parameter values, it can be computed explicitly.For example, if , then the density∗ ∗(E /E ) p (e /e ) p 10 0

function is given by

2(E � e ) 2(E � e )0 0 0 0� ln 1 �2 [ ]B B

for and is 0 otherwise. SinceB ≥ 2(E � e ) ln (1 � x) ≈0 0

when x is small, this density is approximately�xwhen B is large. Thus, in this situation, the2 34(E � e ) /B0 0

distribution with density given by equation (23) is qual-itatively similar to Pareto distributions for large B (i.e., thedistribution would decline monotonically but have a heavytail compared with an exponential distribution). The ex-pression in equation (23) can be computed explicitly inother cases where and are integers, again lead-∗ ∗E /E e /e0 0

ing to distributions with asymptotic behavior similar toPareto distributions.

Note that because this case allows for constraints ongeographic range size due to both critical patch size andmaximum gap length, all six of the key life-history pa-rameters (table 1) will influence the shape of the geo-graphic range size distribution. In this case, the key life-history determinants of these range size distributions are(1) the ratio of in-patch diffusion to reproductive rate( , which determines ), (2) the ratio of in-matrix∗D/r Ediffusion rate to in-matrix mortality rate ( ), and (3) thed/s

ratio of carrying capacity to colonization threshold ( ;K/mtable 1; eq. [6]).

Discussion

Using equations like those typically used to study invasiondynamics (e.g., Skellam 1951), we have developed an an-alytical framework that links fundamental population-levelecological traits with the distribution of geographic rangesizes among species. This is admittedly a large conceptualgulf to bridge with just one mechanistic model. Never-theless, because of the potentially strong link between de-mographic processes and species boundaries (Holt et al.2005), it seems appropriate to construct a model that linksspecies’ life-history traits with the distribution of geo-graphic range sizes for a species pool. Although the local(patch-scale) population dynamics in this model are com-pletely deterministic, we have considered a fragmentedlandscape of patches and gaps in which the spatial het-erogeneity is itself stochastic. This stochastic spatial struc-ture, which juxtaposes landscape-level patch and gap char-acteristics against population-level critical patch sizes andmaximum gap-crossing abilities, determines how far anovel species can spread from its evolutionary origin.Given reasonable assumptions about the distribution ofpatch and gap lengths in the landscape and about thedistribution of critical patch sizes and critical gap lengthsamong species, we can obtain distributions of geographicrange sizes that are qualitatively similar to those found innature (Brown et al. 1996; Gaston 2003).

Geographic Ranges and Geographic RangeSize Distributions

The analyses we have outlined afford insights on both thearea occupancy and extent of occurrence measures of geo-graphic range size (see Gaston 1991). A species’ expectedgeographic range size using the extent of occurrence metricis given by equation (8), whereas the corresponding ex-pectation for area occupancy would be obtained subtract-ing the intervening gap lengths, that is,

2E(L)E(t)E(V) p . (24)∗ ∗1 � Pr (L ≥ L ) Pr (l ≤ l )

In this framework, endemic species would be those withsmall geographic ranges as determined by their encounterswith too-small patches or too-large gaps as the speciesexpanded from their origin. In these scenarios, endemismwould be expected to be more common for species witha high rate of leakage across patch boundaries relative totheir population growth rates (high ; eq. [3]) or forD/rhighly habitat-specific species that could not tolerate being




outside a habitat patch and thus had a low rate of move-ment through the matrix relative to their in-matrix deathrate (low ; eq. [6]). In addition, species with low averaged/sdensities (low K) would be expected to be endemics, aswould species that have a high threshold density for col-onization (high m). Of course, endemism in nature mayarise in many ways, including the evolution of habitat orresource specificity and range contraction by ancient lin-eages (Nekola 1999). However, it also seems likely that abroad suite of biotic and abiotic (or physical and evolu-tionary) mechanisms may act through only a few demo-graphic parameters to enforce range boundaries (Holt etal. 2005). Consequently, it is interesting to note that manyof the factors associated with endemism in this model arelikewise associated with different types of spatial rarity innature (e.g., Gaston and Blackburn 1996; Cofre et al.2007).

We emphasize, however, that this modeling frameworkwould allow species with identical life-history traits to at-tain very different geographic range sizes, depending onthe specific landscape through which the species werespreading (which would dictate the distributions of patchsizes and gap widths that the species would have to contendwith) and depending on the species’ evolutionary origins(which would dictate the particular sequence of patchesand gaps that species encountered). From this “context-dependent” perspective, our analytical results concerningthe interplay between landscape structure and dispersalbehavior confirm findings from simulation studies on sto-chastic neutral landscapes (e.g., Fahrig 1997; King andWith 2002).

Links to Invasion Biology

In this article, we have focused our model developmenton the joint problems of geographic range boundaries andgeographic range size. However, given the conceptual andbiological links between the dynamics of biological inva-sions and the expansion of geographic ranges (e.g., Keittet al. 2001), key aspects of our results are equally pertinentto understanding the spread of invasive species in patchylandscapes. Our efforts demonstrate that the interplay be-tween population-level and landscape-level traits can de-termine not just how fast a species may spread (e.g., Shi-gesada et al. 1986; Van Kirk and Lewis 1997; Dewhirst andLutscher 2009) but also the spatial limits of that spread.

Our results also pertain to the urgent call for improvedunderstanding of the “landscape ecology of invasivespread” (With 2002). Nearly 2 decades ago, Mooney andDrake (1989) argued that the spread of a species througha patchy environment is likely to depend on the degreeof habitat heterogeneity, size and distribution of patches,distance between suitable patches, and population char-

acteristics such as growth rate and dispersal ability. Theframework we have presented captures all of these featuresand combines them to quantify how a species’ spatialspread depends on measurable population- and landscape-level characteristics. Future work should explore howclosely results predicted by this “population dynamics”perspective of invasive spread in stochastically fragmentedlandscapes correspond to predictions from other modelingframeworks.

Opportunities for Extending the Model

Last, we would emphasize that our model may be devel-oped further in several ways. For example, as mentionedin “Model Setup,” our full model does not allow for asituation in which a species may jump over a nearby too-small patch and still successfully colonize a sufficientlylarge patch remote from the species’ origin. This is animportant limitation of our model because it means thatthe geographic range size we calculate for a species thathas both patch size and gap length constraints will be anunderestimate of the true range size that would occur ifjumping over were permissible. As a consequence, a mul-tispecies distribution of range sizes calculated from ourfull model (e.g., case 3) will be necessarily biased down-ward compared with the true range size distribution (i.e.,the size distribution will be left shifted). A more detailedconsideration of this issue must wait for another article,but we note that we have already addressed this issue inthis article in a limited fashion. We did this by constructingcases 1 and 2 as upper bounds on the general model (i.e.,even a tiny patch may be successfully bypassed by an ex-panding species, and the full extent of a species’ range sizewill be constrained only by the presence of a too-big gapin the landscape). Consequently, for the case of a landscapewith exponentially distributed patch sizes and gap lengths,the true geographic range size distribution will lie, for aparticular parameter combination, somewhere betweenthe range size distribution predicted by case 1 and thatpredicted by case 3.

This “jumping over” limitation of our model arises asa direct mathematical consequence of our reliance on thestopping time approach to the stochastic nature of frag-mented landscapes and our choice to present results fora general model. At a technical level, the difficulty emergesbecause addressing the jumping over issue would requireknowing how the probability density function for the sumof the combined gaps compares with the species’ maxi-mum gap-crossing ability. Such a calculation requires tak-ing the convolution of the probability densities of the gaplengths, and explicit calculations are possible only in cer-tain special cases. A mathematically similar issue arises ifwe try to determine the full probability density function




for the range size, as opposed to just computing its meanand variance.

Another key difference between our modeling approachand more typical (empirical or theoretical) studies ofspecies boundaries is that we have completely ignoredthe potentially important roles played by biogeographicgradients of various types (e.g., temperature extremes, eva-potranspiration potential, densities of resources, compet-itors, or natural enemies). For example, species interac-tions may certainly influence species geographic range sizesin the real world (Galen 1990), and from a modeling per-spective, there may be some ways forward on this frontbecause species interactions, such as consumer impacts,may help set a species’ critical patch size (Cantrell et al.2001). Alternatively, using the same modeling frameworkthat we established here, one could impose spatial gra-dients in any of several life-history traits. For example, inmany species, both local reproductive rate and local den-sity may vary systematically over large spatial scales (e.g.,Brown et al. 1995, 1996). Because these measures corre-spond closely to the parameters r and K in our model andthese in turn help determine both a species’ critical patchsize and its maximum gap-crossing ability (table 1), itseems that a reasonable next step may be to consider theconsequences of gradients in life-history traits for modeledgeographic range sizes. Likewise, gradients in patch sizeor landscape “gappiness” also occur in nature, and theimposition of such specific spatial structure in a modelwould certainly affect the expectation and variance of geo-graphic range sizes for particular species (and for distri-butions of range sizes across species). Many of these spatialgradient scenarios could not be treated analytically butcould be systematically explored using numerical simu-lation techniques.

Moreover, similar modeling efforts should also be pos-sible by exchanging the partial differential equation frame-work that we adopted for one of several alternative ap-proaches. For example, our results should be extensible todiscrete-time problems (e.g., seasonal systems) because in-tegrodifference equations (Kot et al. 1996) can yield criticalpatch sizes just as RD models can (Van Kirk and Lewis1997; Fagan and Lutscher 2006; Fagan et al. 2007). Like-wise, there may be opportunities for developing a similarmodel for the case of geographic ranges in reaction-ad-vection-diffusion models, which also generate criticalpatch sizes (e.g., Lutscher et al. 2005; Pachepsky et al.2005). Such a model could shed light on the factors gov-erning the distribution of geographic range sizes in reeffish (Lester and Ruttenberg 2005; Mora and Robertson2005; Kiflawi et al. 2006) and other species that inhabitpatchy landscapes in current-dominated systems.

Other extensions of the model, such as considering spa-tial spread and geographic range size distributions in two

dimensions, would present substantial analytical chal-lenges because of the greater difficulty of determining stop-ping times in two dimensions for generic probability dis-tributions. Nevertheless, such generalizations are, at leastin principle, possible. Given the potential conceptual andquantitative integration that is possible with models thatlink life-history traits to multispecies geographic range sizedistributions, future work on such models seemswarranted.

Acknowledgments

We thank C. Che-Castaldo, E. Goldberg, H. Lynch, andL. Ries for comments that improved the manuscript. W.F.F.was supported by National Science Foundation (NSF)grant DEB-0624163. R.S.C. and C.C. were partially sup-ported by NSF grant DMS-0514839.

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