Predation and the Evolutionary Dynamics of Species Ranges.

vol. 178, no. 4 the american naturalist october 2011

Predation and the Evolutionary Dynamics of Species Ranges

Robert D. Holt,1,* Michael Barfield,1 Ido Filin,2 and Samantha Forde3

1. Department of Biology, University of Florida, Gainesville, Florida 32611; 2. Biomathematics Group, Department of Mathematics andStatistics, University of Helsinki, PL 68 FIN-00014 Helsinki, Finland; 3. Ecology and Evolutionary Biology Department, University ofCalifornia, Santa Cruz, California 95064

Submitted November 24, 2010; Accepted June 3, 2011; Electronically published August 19, 2011

Online enhancement: appendix.

abstract: Gene flow that hampers local adaptation can constrainspecies distributions and slow invasions. Predation as an ecologicalfactor mainly limits prey species ranges, but a richer array of pos-sibilities arises once one accounts for how predation alters the in-terplay of gene flow and selection. We extend previous single-speciestheory on the interplay of demography, gene flow, and selection byinvestigating how predation modifies the coupled demographic-evolutionary dynamics of the range and habitat use of prey. Weconsider a model for two discrete patches and a complementarymodel for species along continuous environmental gradients. Weshow that predation can strongly influence the evolutionary stabilityof prey habitat specialization and range limits. Predators can permitprey to expand in habitat or geographical range or, conversely, causerange collapses. Transient increases in predation can induce shifts inprey ranges that persist even if the predator itself later becomesextinct. Whether a predator tightens or loosens evolutionary con-straints on the invasion speed and ultimate size of a prey rangedepends on the predator effectiveness, its mobility relative to its prey,and the prey’s intraspecific density dependence, as well as the mag-nitude of environmental heterogeneity. Our results potentially pro-vide a novel explanation for lags and reversals in invasions.

Keywords: range limits, gene flow, gradient, habitat specialization,predation.

Introduction

The study of species range limits is pivotal to ecology,drawing on many different subfields and providing a uni-fying theme for a wide range of ecological investigations(Antonovics et al. 2006; Gaston 2009). Understanding thedrivers of range limits is increasingly urgent, given theneed to project how species and communities will respondto global change and to predict the ultimate distributionof invasive species. There is increasing interest in inte-grating evolutionary and ecological perspectives into ex-planations of range dynamics (Case et al. 2005; Bridle and

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

Am. Nat. 2011. Vol. 178, pp. 488–500. � 2011 by The University of Chicago.

0003-0147/2011/17804-52642$15.00. All rights reserved.

DOI: 10.1086/661909

Vines 2007; Sexton et al. 2009). The interaction betweendemography and evolution—mediated in part by the in-terplay of gene flow and selection—has emerged as a po-tentially key driver of range dynamics (e.g., Kirkpatrickand Barton 1997; Gomulkiewicz et al. 1999; Kawecki2008), local adaptation (Garant et al. 2007), and nicheconservatism (Wiens et al. 2010).

Traced back to Haldane (1956) and Mayr (1963), theessential idea is that a flow of individuals from abundant(e.g., central) populations to sparsely populated rangemargins can restrain local adaptation at the margins, con-straining geographical distributions. Asymmetric gene flow(potentially overwhelming selection) could be particularlystrong when genetic variation is limited and low- and high-density populations are spatially close, for instance, alongsharp environmental gradients (Kirkpatrick and Barton1997; Barton 2001), or where distinct habitats are closelyjuxtaposed. Population density should reflect how welladapted a species is to local environments, so spatial var-iation in density could itself reflect the interplay of geneflow and selection. The implications of this positive feed-back between demography and selection has been the focusof considerable attention for single species evolving in het-erogeneous landscapes (e.g., Kirkpatrick and Barton 1997;Gomulkiewicz et al. 1999; Antonovics et al. 2001; Holt2003; Filin et al. 2008; Polechova et al. 2009).

Yet few species live in isolation. How do interspecificinteractions influence evolutionary range limitation bygene flow? Interspecific competition can constrain species’ranges (MacArthur 1972; MacLean and Holt 1979; Rough-garden 1979; Case et al. 2005) and, at times, magnify theimpact of gene flow. Case and Taper (2000) adapted themodel of Kirkpatrick and Barton (1997) to include a com-petitor and demonstrated that competition could sharpenspatial gradients in density and magnify the inhibitoryimpact of gene flow on selection, particularly given sharpor abrupt changes in selective optima along gradients. Spe-cies borders and replacements of related species often oc-cur across transitions in vegetation structure (Boone andKrohn 2000), consistent with this prediction.

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Predation and Prey Range Evolution 489

Another major class of interspecific interactions influ-encing range limits consists of those between natural en-emies and their victims, such as predator-prey and host-pathogen interactions. Prey and hosts are resources fortheir respective natural enemies, so the ranges of theseresources obviously could constrain predator and patho-gen distributions (Holt 1997a; Alexander et al. 2007). Con-versely, generalist predators and other natural enemies canlimit their victim’s distributions (Case et al. 2005; Holtand Barfield 2009). This is not a new observation; in hisclassic article introducing the niche concept, Grinnell(1917) suggested that the California thrasher (Toxostomaredividum) is restricted to chaparral because of high pre-dation from hawks in woodland. Hawks are generalistpredators presumably sustained in their abundance by al-ternative woodland prey, permitting them to exclude thisvulnerable bird species from a portion of its potential geo-graphical range.

By contrast, specialist predators at first glance seem un-likely to govern prey range limits. The reason is simple:beyond the prey range limit, such a predator has nothingto eat and so tends to become extinct. However, this first-order expectation of the unimportance of specialist ene-mies as a cause of range limits of their victims may notalways hold (Antonovics 2009; Holt and Barfield 2009).For instance, dispersal by the predator from sites withproductive, abundant prey can sustain predators in mar-ginal populations of prey, imposing a range limit on theprey (Hochberg and Ives 1999; Owen and Lewis 2001; Holtand Barfield 2009).

Beyond such purely ecological effects, and pointing inthe converse direction, Holt and Barfield (2009) conjec-tured that predators could alter the constraining influenceof gene flow on prey range limits via effects on prey abun-dance, at times facilitating habitat generalization and rangeexpansion in prey. Specialist predators can grow and de-press prey numbers to a level that just keeps the predator’sown growth in check. If attack rates and other predatorparameters are spatially invariant, at equilibrium the pred-ator should reduce its prey to uniform abundances ev-erywhere the predator persists, flattening spatial variationin prey abundance that would otherwise exist. This couldweaken asymmetries in net rates of prey movement, reducethe effect of gene flow relative to selection, and therebyindirectly permit a prey species to expand its habitat rangeor distribution along a gradient. The indirect evolutionaryeffect of predation on prey habitat use and ranges couldthus qualitatively differ from its direct ecological effect.

Here we examine this conjecture in detail using modelsof coupled ecological and evolutionary dynamics. We ex-tend models explored earlier (Kirkpatrick and Barton1997; Ronce and Kirkpatrick 2001; Filin et al. 2008; Holtand Barfield 2009) by splicing a model of prey evolution

with models of coupled predator-prey population dynam-ics, which alter the interplay of demography, gene flow,and local adaptation in a prey species. We first considerhow a predator can influence the evolutionarily stable hab-itat range of its prey in landscapes comprised of two dis-crete habitats. We then turn to predator-prey interactionsalong smooth gradients and explore how predation altersevolutionary stability of range limits, as well as transientrange dynamics. The actual landscapes in which rangedynamics unfold are complex. We seek to identify commonfeatures of coupled ecological and evolutionary dynamicsin two idealized landscapes—a pair of linked discrete hab-itats, and smooth continuous gradients—in the hope ofreaching robust conclusions about predator impacts onprey ranges.

The Evolution of Species Ranges in Two-HabitatLandscapes with Predation

Consider a landscape containing two distinct habitats ofequal area coupled by dispersal. To examine how predationinfluences the evolution of prey habitat specialization andgeneralization, we use a model presented by Ronce andKirkpatrick (2001) and extended by Filin et al. (2008) toencompass a broader range of nonlinear density depen-dence in prey growth. In this model, a species experiencesstabilizing selection on a quantitative trait with differentoptima in each habitat. Specialists have trait values nearone habitat’s optimum, while generalists are nearer theaverage of both. A specialist has high density in the habitatto which it is adapted and low density in the other, whilea generalist population has more nearly equal densities inboth. In a specialist, the mating of immigrants and resi-dents in the suboptimal habitat can overwhelm selection,constraining adaptation there and maintaining habitat spe-cialization. This model predicts alternative equilibria, thelikelihood of which varies with the degree of habitat het-erogeneity, movement rates, and the nonlinear form ofdensity dependence. Ronce and Kirkpatrick (2001) notedthat habitat generalists may lose adaptation to one habitatas a result of the swamping effects of gene flow after num-bers are perturbed to low levels, a phenomenon they called“migrational meltdown.” Predation, by altering habitat-specific prey abundance, can shift the relative strengths ofgene flow and selection and thus influence evolutionarystability of different patterns of prey habitat use.

We generalize the model developed by Ronce and Kirk-patrick (2001) in two respects (for more details, see ap-pendix in the online edition of the American Naturalist):the prey can experience nonlinear density dependence andpredation in one or both habitats i and j. Prey abundancein habitat i is Ni, and its maladaptation (the difference

490 The American Naturalist

between its phenotype and the habitat optimum) there isYi. The model for the prey habitat i is as follows:

dY Ni jp �GY � M (H � Y � Y ), (1)i i jdT Ni

dN Gi v 2p (1 � N )N � Y N � M(N � N ) � D N . (2)i i i i j i i idT 2

(The equations for habitat j are the same, with a switchof subscripts.) Equation (1) describes phenotypic evolu-tion and matches equation (6a) of Ronce and Kirkpatrick(2001). The first term on the right is selection, driving Yi

toward 0 (and therefore the prey phenotype toward thehabitat optimum). The quantity G is the rate of evolutionand includes (fixed) genetic variation and the strength ofselection. Without movement, the population in each hab-itat equilibrates at the local optimum. The two habitatsdiffer in their optima by H (a measure of habitat hetero-geneity). Per capita movement at rate M (the same in bothdirections) leads to maladaptive gene flow (the secondterm). This flux is biased toward the habitat with lowernumbers, producing greater maladaptation there. The ex-pectation of greater maladaptation in habitats with lowabundance is a general prediction of models for evolutionin heterogeneous landscapes (Kawecki 2008). Note thatpredation does not directly drive prey evolution. We as-sume the predator does not evolve.

Equation (2) describes prey population dynamics. Timeis scaled by the prey intrinsic growth rate and density byits carrying capacity. The first term is inherent prey growth(including nonlinear density dependence), the second ex-presses demographic load from maladaptation, the thirdgives the direct effect of dispersal on population size, andthe last is mortality from predation at a habitat-specificper capita rate of Di. The form of density dependence istheta logistic (Sæther and Engen 2002; Filin et al. 2008).The larger v is, the weaker density dependence is at lowdensity. Equation (2) generalizes equation (6b) of Ronceand Kirkpatrick (2001) to include nonlinear density de-pendence and mortality due to predation.

We first consider the impact of predation acting simplyas a density-independent source of mortality (fixed Di)and then allow predator numbers to vary dynamically withprey availability, driving variation in Di. The former as-sumption could describe a generalist predator, sustainedby alternative prey. The latter is appropriate if the predatoris a specialist, dependent on our focal prey species.

Predation as a Density-Independent Mortality Factor

Predation can alter the likelihood of habitat generalizationor specialization in a prey species. With fixed predation,the above model can have multiple equilibria, including

generalists and specialists. For some parameters, especiallyat low M, both kinds of equilibria can be evolutionarilystable alternative states, while for others, only one type ofequilibrium is stable (generalist for low H, specialist forhigh H). With symmetric predation, one can solve ex-plicitly for the symmetric equilibria (with equal abun-dances and degrees of maladaptation in both habitats; i.e.,generalists). Asymmetric equilibria are populations spe-cialized to some extent on one habitat. At high M, someequilibria have modest differences in numbers and mal-adaptation between habitats. In these cases, it may not beclear whether the prey should be considered a generalistor a slight specialist.

Both predation and prey movement strongly affectwhether the prey evolves to be a habitat specialist or ageneralist. As an example, figure 1a shows evolutionarilystable equilibria of prey density as a function of movementrate without predation at (a moderately large dif-H p 7ference in the phenotypic optima; this difference is 7 ge-netic standard deviations, 3.5 phenotypic standard devi-ations for a heritability of 0.25). The system always goesto one of the stable equilibria (discontinuities result wherea stable equilibrium becomes unstable). The degree of mal-adaptation Yi varies inversely with local density (notshown). Low movement ( ) permits three stableM ! 0.22equilibria: a generalist (dashed line is density in each hab-itat) and two specialist equilibria (solid black line is densityin the habitat where the species is better adapted, gray lineis its density in the other). At intermediate movement, theonly stable equilibria are specialists (solid lines), but athigh M, the species is a generalist (dashed lines).

Now add a generalist predator to habitat 1, where itimposes density-independent mortality at a per capita rateD1. The symmetry is broken, so equilibria of equations (1)and (2) are found numerically. Figure 1b shows how pre-dation in habitat 1 (at a per capita rate of 0.1) alters preyhabitat use. Equilibrium densities differ between habitats.As makes intuitive sense, predation restricted to habitat 1decreases the range of parameters favoring generalization(dashed lines) in favor of specialization to the predator-free habitat (habitat 2; gray lines), including at high move-ment (although for increasing M, the degree of speciali-zation decreases). In this example, specialization to habitat2 is evolutionarily stable at much higher movement ratesthan without predation; specialization to habitat 1 (solidblack lines) can still occur with the predator present butonly for M up to 0.46, while it occurs for M up to 0.55without predation.

Localized predation can cause the generalist strategy todisappear for many parameter values. Predation often con-verts a generalist to a specialist on whichever habitat lackspredation (e.g., cf. fig. 1a, 1b for ). Figure 2aM 1 0.55shows an example of the effect of gradually increasing


Figure 1: Population sizes for habitat generalists and specialists forthe generalized Ronce-Kirkpatrick model when a generalist predatoris absent (a) or present only in habitat 1 (b). Stable equilibria ofequations (1) and (2) are shown as a function of M (per capita rateof prey movement). “Stable” here denotes evolutionary as well asecological stability. The difference in phenotypic optima between thetwo habitats is ; intensity of selection is ; density de-H p 7 G p 0.1pendence has . Predator-induced mortality in b is .v p 2 D p 0.11

a, For low movement (and a tiny range of moderate movement),the prey has both stable generalist (dashed line) and specialist equi-libria (solid black line for habitat to which prey is adapted, solidgray line for the other). At other movement rates, there are stableequilibria of one type or the other (specialization can be to eitherhabitat). b, Predation eliminates the generalist equilibrium at highM, replacing it with a specialist to habitat 2 (solid gray lines). Ageneralist occurs only for very low M and is asymmetric (in termsof abundance and maladaptation). A specialist to habitat 1 occursfor a smaller range of movement rates than does a specialist to habitat2. Figure 2: a, Gradual increases in predation can cause abrupt shifts

in prey habitat use. Equilibrial densities in habitats 1 (solid line) and2 (dashed line) as functions of predation in habitat 1 (D1). D p 01

has a symmetric generalist equilibrium. At sufficiently high predation,the generalist shifts to specialization to the predator-free habitat (hab-itat 2). , , , . b, Example of transientH p 7 G p 0.1 v p 2 M p 0.15increases in predation leading to permanent switches in prey habitatuse. The system is initially at the asymmetric equilibrium of spe-cialization to habitat 1; at , predation ( ) is added toT p 0 D p 0.11

habitat 1, leading to a permanent shift in prey use to habitat 2. At, this predation is removed, and the system remains spe-T p 800

cialized to habitat 2. Parameters as in a, except .M p 0.5

predation (restricted to habitat 1) for a low movementrate. The prey exhibits habitat generalization in the absenceof predation ( ). As predation increases, the gen-D p 01

eralist equilibrium remains but with shrunken numbersin habitat 1 and increased abundance in habitat 2 (as aresult of lower gene flow from habitat 1). Beyond a thresh-old predation level, the generalist equilibrium disappears,

and the system abruptly shifts to specialization on habitat2.

Transient increases in predation in one habitat can in-duce permanent shifts in the habitat to which a prey spe-cies is specialized. Figure 2b shows an example. The systemstarts without predation, with the prey specialized to hab-itat 1 (x in fig. 1a). At , a predator (imposing aT p 0mortality rate of 0.1) is added to habitat 1, shifting thebalance of gene flow and selection in the prey. There is asmall initial drop in abundance in habitat 1, reducing geneflow to habitat 2 and allowing the individuals there toslowly adapt. Eventually increasing movement from hab-


itat 2 (where numbers increase after an initial small drop,as fitness rises) degrades adaptation in habitat 1, depressingprey abundance and strengthening gene flow relative toselection. The prey shifts in its specialization to habitat 2(x in fig. 1b). Predation is removed at , but theT p 800prey remains specialized to habitat 2. We show below thatpermanent shifts in prey habitat use can also occur witha dynamic predator population coupled to the prey. (Inthe appendix, we explore how prey habitat specializationvs. generalization is influenced by intraspecific densitydependence.)

Generalized Ronce-Kirkpatrick Modelwith a Specialist Predator

The mortality imposed by specialist predators should notbe fixed but instead varies with changes in predator num-bers. For simplicity, we assume linear functional and nu-merical responses by the predator to its prey. If Pi is pred-ator density in habitat i, the mortality imposed on the preyis , where r is the attack rate. The predator movesD p rPi i

between habitats at a constant per capita rate of Mp anddies at a constant rate of m. With an appropriate scalingfor predator numbers (see appendix), these assumptionslead to the following model for the predator:

dPi ( )p rN � m P � M (P � P). (3)i i p j idT

The full model consists of equations (1) and (2) withequation (3).

One equilibrium is for the prey to be a symmetricalhabitat generalist:

HM∗ ∗ ∗Y p Y { Y p , (4)1 2 2M � G

m∗ ∗ ∗N p N { N p , (5)1 2r

∗21 GY∗ ∗ ∗ ∗vP p P { P p 1 � N � . (6)1 2 ( )r 2

The quantity is unchanged by predation (cf. Ronce and∗YKirkpatrick 2001; Filin et al. 2008), so predation in thiscase alters prey densities without changing the evolution-ary equilibrium. Solving for the maximum value of het-erogeneity H allowing gives a condition for pred-∗P 1 0ator persistence:

v�2[1 � (m/r) ](2M � G)H ! . (7)�M G

(Predator persistence also requires .) High H causesm ! r

predator extinction. The prey alone persists if

�2(2M � G)H ! H p (8)v �M G

(Filin et al. 2008). Condition (7) implies condition (8).The difference between these two conditions is larger atsmaller v. Thus, the magnitude of habitat heterogeneitytolerable by a specialist predator is less than for its prey,particularly when prey density dependence is stronger atlow densities. This is because heterogeneity depresses max-imal prey abundance and productivity (via maladaptivegene flow), making it harder to sustain the predator.

The appendix contains results of a stability analysis ofthe generalist equilibrium for stationary and moving pred-ators and different strengths of density dependence (fig. A2).With weak density dependence (fig. A2a), predation oftendestabilizes habitat generalization, leading to specialization.With somewhat stronger density dependence (fig. A2b), theprey alone is less likely to be a generalist, and the predatorhas little impact on this. For even stronger density depen-dence (fig. A2c, A2d), adding a predator tends to stabilizeprey habitat generalization, given that the predator can per-sist. The effect of a predator on the evolutionary stabilityof prey habitat use is thus rather complex, but in general,prey habitat generalization is more likely to be stable witha dynamic predator than without, at low prey movement,and less likely to be stable at higher prey movement or withweak density dependence.

Figure 3 demonstrates two distinct ways the introduc-tion of a specialist, dynamic predator can alter the evo-lutionary habitat distribution of an initially specializedprey. In figure 3a, after predator colonization, prey spe-cialization is lost and replaced by a generalist distribution.Predation can thus indirectly induce habitat generalizationin its prey. By contrast, in figure 3b, the prey is initiallyspecialized to habitat 1; at , a stationary predatorT p 0( ) invades habitat 1, leading to a reversal of preyM p 0p

habitat specialization. This reduction in the predator’sfood supply leads to its own extinction, leaving behind apermanent shift in prey habitat use.

The Evolution of a Prey Species Rangealong Continuous Gradients

These results demonstrate that predation can substantiallyalter asymmetric gene flow among habitats in a prey spe-cies and thus the evolution of habitat use. Comparableeffects should occur for prey along smooth gradients. Toexamine this, we generalize a model of coupled demo-graphic and evolutionary dynamics first explored by Kirk-patrick and Barton (1997) and later expanded to encom-pass a wider range of functional forms for densitydependence (Filin et al. 2008). The strength of densitydependence influences the steepness of spatial gradients


Figure 3: Examples of predators leading to both habitat generali-zation and specialization. a, Initially, the prey is a habitat 1 specialist.Adding a dispersing predator leads to evolution of generalization.Prey (solid lines) and predator (dashed lines) population densitiesin habitat 1 (black lines) and habitat 2 (gray lines) are shown.

, , , , , , . b, AH p 7 G p 0.1 M p 0.4 v p 2 r p 10 m p 1 M p 0.01p

predator restricted to habitat 1 colonizes at time 0. The prey, initiallya habitat 1 specialist, switches its equilibrial habitat use, remainingspecialized to habitat 2 even after the predator becomes extinct.

, , , , , .H p 7 G p 0.1 M p 0.5 v p 2 r p 1 m p 0.5

in density and thus the strength of asymmetries in geneflow.

The model tracks changes in density and the mean ofa quantitative trait for a prey species along a continuousone-dimensional gradient in the trait’s phenotypic opti-mum, assuming fixed genetic variance and spatial unifor-mity in maximal intrinsic growth rate, carrying capacity,and strength of stabilizing selection. We assume that thepredator has linear functional and numerical responses toits prey, disperses at a constant per capita rate along thegradient, and does not evolve. These assumptions lead tothe following model:

2�N � N 12p � U(N) � (Z � Z ) � P N, (9)opt2 [ ]�T �X 2

2�Z � Z �Z� ln Np � 2 � A(Z � Z ), (10)opt2�T �X �X �X

2�P � Pp d � (aN � m)P, (11)

2�T �X

where T, X, N, , and P are dimensionless variables ofZtime, space, prey population density, prey mean pheno-type, and predator population density, respectively; a con-verts consumed prey to predator reproduction, and m ispredator mortality (for initial equations and normaliza-tion, see appendix). Both species disperse diffusively. Theprey diffusion constant is absorbed into X, while d is thepredator’s diffusion rate, relative to that of its prey. Equa-tions (9) and (10) match equations (4a) and (4b) of Filinet al. (2008), with predation added.

The first term of equation (9) describes changes in preydensity due to dispersal. The second term (in square brack-ets) is density-dependent local growth (through the func-tion U(N)), reduced by maladaptation (dependent on howmuch the local mean phenotype differs from the localoptimum Zopt(X)) as well as predation. We consider onlynegative density dependence, that is, U(N) monotonicallydecreases with N. Prey evolution (eq. [10]) combines re-mixing of phenotypes along the gradient by dispersal (thefirst two terms on the right-hand side) and stabilizingselection toward a local optimum (scaled by A, the geneticpotential for adaptation, a parameter combining geneticvariance and intensity of selection; Kirkpatrick and Barton1997; Case and Taper 2000). We assume genetic varianceis constant and that the selective optimum changes linearlyalong the gradient; that is, .Z (X) p BXopt

A useful limiting case for understanding the full systemis for the prey to experience no direct density dependence(e.g., early in invasion or at range margins where densityis low). We briefly review what happens with the prey alone(see Filin et al. 2008) and then examine how predationalters prey invasion dynamics. Without the predator, ona shallow gradient ( , where DINP1/2B ! B p A/(2 )L(DINP)

denotes the density-independent, no-predator case), rangeevolution in the prey permits an unlimited, constantlygrowing uniform distribution at steady state, with expo-nential growth rate everywhere of (for details, seeR p 1Kirkpatrick and Barton 1997; Filin et al. 2008). Expansionoccurs at a constant velocity (similar to Skellam’s [1951]classic result for homogenous environments), spatial den-sity gradients flatten through time, and levels of malad-aptation continually decrease. If the gradient is very steep( ), the species becomes extinct.1/2B 1 B p (A � 2)/(2 )U

For intermediate steepness ( ), the steadyB ! B ! BL(DINP) U


state prey density has a Gaussian shape, the height of whichgrows exponentially. Maladaptation increases toward theexpanding range margin. After a phase of initial transients,the range edge (defined as density below some fixed lowlevel) is found at position , where R is de-1/2X(t) ∝ (RT)pressed because of maladaptation. With a steeper gradient,invasion is slower as a result of maladaptive gene flow.Before this pattern emerges, one can see quasi-stable rangelimits (i.e., invasion lags) in the initial density-independentgrowth phase, even if the steady state is unlimited expan-sion (Filin et al. 2008).

How does a specialist predator alter prey invasion? Thedirect effect of predation is reduced prey growth rates,which should always reduce invasion speeds. Without preyevolution, for a reaction-diffusion predator-prey system ina homogeneous environment, Dunbar (1983; see alsoOwen and Lewis 2001; Murray 2003) showed that theLotka-Volterra predator-prey system has traveling wave so-lutions. At a fixed location, initially a prey front is chasedby a predator front, after which local densities settle downto periodic oscillations (Okubo and Levin 2001, pp. 326–328). Predation reduces the velocity of this prey wavefront.

Specialist predators also indirectly alter prey growthrates by shifting the relative strengths of gene flow andlocal selection (and hence the degree of local maladap-tation) via impacts on prey abundance (e.g., see figs. 4,5). Predator-prey oscillations emerge when a mobile pred-ator is introduced at the range center of an exponentiallygrowing invasive species moving along a gradient (fig. 4).The predator reduces prey density as it invades (fig. 4a)and in so doing creates reverse gradients in prey densitybehind the wavefront. The result is a traveling wave wherea predator front chases a prey front (fig. 4b). Gene flowoccurs toward the prey’s range center, not its edge, overthe zone where prey numbers decline because of increasingpredation pressure. This boosts prey growth rates by fa-cilitating local adaptive evolution. These adaptive increasesin growth rates near the invasion front in turn shouldincrease the speed of the prey invasion.

Without predation, density-independent prey growthoccurs only during transient low-density phases (e.g., earlyin invasion). Eventually, density dependence checksgrowth. A prey species growing alone along a mild gradientstill has a traveling wave solution (see Filin et al. 2008),where an advancing wavefront connects a region of highpopulation density (near carrying capacity) and near com-plete local adaptation ( ) with a region of low pop-Z ≈ Zopt

ulation density and higher maladaptation. On steeper gra-dients, the range is limited. Density dependence increasesthe magnitude of gradient steepness needed to limit therange (i.e., [where DDNP denotes density-depen-B L(DDNP)

dent prey growth, no predator] is higher than

, particularly at high v; see Filin et al.1/2B p A/(2 )L(DINP)

2008).Imagine the prey has settled into a traveling wave so-

lution ( ), that is, expansion and adaptation atB ! BL(DDNP)

a constant speed, where well within the range andN p 1the phenotype is at its optimum, everywhere. Now add apredator in the range interior. The predator sees a ho-mogeneous environment, since prey density is (locally)uniformly at . Given (required for predatorN p 1 a 1 m

persistence), a traveling wave of predators forms. After thewavefront passes, prey density is depressed from N p 1to . Prey density is still at locations∗N p m/a ! 1 N p 1invaded by prey but not yet by the predator. Over a tran-sition zone, prey density increases from m/a to 1, andpredator density drops from (p ; eq. [9]) to 0.∗P U(m/a)Within this zone, prey maladaptation emerges because theprey density gradient generates asymmetric gene flow backtoward the range center. Prey individuals in the transitionzone should be better adapted to locations farther fromthe range center, as a result of the density gradient set upby the predator. Overall, we obtain an expanding range ofthe prey as it invades and adapts, with a lagged expansionof the predator range and a transient, moving zone of preymaladaptation.

For a more detailed analysis, it is useful to consider thepredator invasion speed (i.e., how fast the predator ex-pands within the occupied prey range) given by v p

and the length scale of the predator spillover1/22[d(a � m)]effect (i.e., how far a predator penetrates areas where preyare absent or scarce), which is scaled by . An-1/2y p (d/m)other key compound variable is equilibrial prey density,

, which measures the efficacy of the predator in∗N p m/alimiting prey. Figure 5 shows examples (with logisticgrowth) of a predator tracking a prey adapting to a gra-dient. If steady state invasion speed of the prey exceeds, the predator front never reaches the prey front, and thev

distance between the two increases indefinitely. This canoccur along a shallow gradient where prey adaptation isnot markedly impeded by gene flow (e.g., fig. 5a). On asteeper gradient (see fig. 5b), prey invasion slows and thepredator front now catches up to the prey, with a constantspatial lag. The range is still unlimited, but the rate of preyexpansion is greatly slowed. Even steeper gradients lead toevolutionarily stable range limitation of the prey (and thepredator) caused by maladaptive gene flow in the prey (seefig. 5c). Hence, the ecological effect of predator “spillover”causing stable prey range limits shown by Hochberg andIves (1999) is magnified by the indirect effect of predationon prey adaptive evolution.

By depressing prey numbers, predation can alter thestrength of prey density dependence (eq. [9]). The effectsof predation on invasion dynamics and conditions for preyrange limitation depend on the form of density depen-


Figure 4: Interplay of predation and local adaptation in prey can create reversals in prey abundance along a gradient, facilitating invasion.a, Prey density across space for density-independent predator-prey dynamics at four time points in an invasion by both species: T p 0(solid black line), 14 (dashed gray line), 28 (dotted line), and 43 (solid gray line). Parameters: , , , ,a p 0.7 d p 0.7 m p 0.35 A p 0.4

. Initial conditions at for both prey and predator are narrow Gaussian distributions centered at , with maximumB p 0.34 T p 0 X p 0prey density of 1 and maximum predator density of 0.01. As the prey expands, its abundance is maximal near the range margin, where itescapes predation; predation can reverse density gradients, altering gene flow away from the invasion front. b, Heterogeneity in dynamicsalong a gradient. Prey (solid lines) and predator (dashed lines) densities across space at (black lines) and (gray lines).T p 62 T p 89Parameters and initial conditions as in a, except and . Note that densities are scaled separately for each species (ford p 0.35 m p 0.36details, see appendix in the online edition of the American Naturalist). The predator lags the prey; the magnitude of population oscillationsvaries greatly across the range, with instability pronounced near the range edge but with stabilized dynamics toward the interior.

dence. Whether predation slows the asymptotic rate ofprey invasion depends on gradient steepness, the evolu-tionary potential A, and predator dispersal. Using thetheta-logistic model ( ) in equation (9), wevU(N) p 1 � Ncarried out numerical studies to assess the effect of dif-ferent forms of density dependence. We start with thelimiting case of no prey density dependence. Figure 6ashows for a slice of parameter space that predation canenhance prey invasion (in this example, predator spilloveris rather weak). Between the solid and lower dashed lines,predation permits the growth rate to reach its maximalvalue ( ), when without predation growth rate wouldR p 1be constrained below this by gene flow. Thus, predatorssometimes enhance the rate of invasion by prey.

Density dependence permits the prey to settle into aspatially bounded range at evolutionary and ecologicalequilibrium. Figure 6b and 6c shows examples of howpredation alters the transition between range-limited evo-lution, and an unlimited range, with prey density depen-dence. The solid gray line in figure 6b describes rangelimitation for the prey alone. When predation has only aminor effect on prey abundance (dotted line), there is littleeffect on the transition line. When predation has a moresubstantial impact on prey numbers, increases in spilloverlead to prey range limitation in a swath of parameter space

where otherwise the range would be unlimited (dashedand dotted-dashed lines). In such cases, if a prey speciesis expanding so that on its own it would occupy the en-tirety of a gradient, predator invasion can first halt andthen reverse this invasion, so that the range collapses toa much narrower slice of the gradient (fig. A4 provides anumerical example).

Conversely, when predation has a strong impact on preynumbers but weak spillover (solid black line), prey rangesmay expand that otherwise would be constrained by geneflow. Finally, figure 6c assumes strong density dependenceat low prey numbers and moderate reduction of prey num-bers. In this case, predation has a relatively minor effectof increasing the range of parameter space where prey canexpand along all of a gradient. However, even in thesecases, a predator may strongly influence the realized equi-librial range of its prey. If a prey species has settled intoa limited-range equilibrium, introducing a specialist pred-ator in just a portion of that range may induce transientdynamics, after which the prey settles into a new limited-range equilibrium, with a different limit along the gradient.If the predator is constrained in the part of the gradientit can occupy, for reasons other than prey availability, itmay be vulnerable to driving itself to extinction as its preyshifts largely out of reach (a gradient analogue of the sce-


Figure 5: Predator-prey invasion along gradients of increasing steepness. Prey densities are solid lines, predator densities are dashed lines.The gradient becomes steeper (i.e., higher B) from a to c. Initial conditions are as in figure 4a. Prey density dependence is logistic (v p

). a, Both prey and predator expand along a gentle gradient, with the prey expanding faster than the predator (densities shown for two1times, [black lines] and [gray lines]). , , (giving , , ), , .∗T p 58 T p 110 a p 0.5 d p 0.1 m p 0.1 v p 0.4 y p 1 N p 0.2 A p 0.5 B p 0.35b, For a steeper gradient ( ), prey and predator fronts move at the same rate (shown for [black lines] and [grayB p 0.7 T p 67 T p 134lines]). c, Along an even steeper gradient ( ), both species settle into a limited range (shown for [black lines] andB p 1.2 T p 15 T p 68[gray lines]), with evident predator spillover into the prey range margin.

nario of fig. 3b). Such examples suggest that even if pre-dation does not alter the fact that gene flow constrains therange of the prey, predation can strongly affect where,exactly, range limits occur.

Discussion

Interspecific interactions can influence species’ ranges andhabitat specialization both directly and indirectly via evo-lutionary mechanisms such as gene flow and selection. Theecological impact of predation is usually to constrain thehabitat distribution and geographical ranges of prey. Ourmodels show that evolutionary responses of prey to spatialvariation in selection, coupled with gene flow, can implya rich array of alternative outcomes, including dramaticshifts in habitat use and expansion of prey species alonggradients where otherwise ranges would be limited (asconjectured by Holt and Barfield [2009]). Facilitative ef-fects of predation on prey ranges are particularly likelyearly in invasions, at least if predator dispersal is rathermodest and prey growth is largely density independent.Alternatively, sometimes predators that are effective dis-persers sharpen the impact of gene flow on prey rangelimitation. Such predators can even induce prey range col-lapse to a much greater extent than expected on purelyecological grounds.

We have deliberately considered two idealized extremesof landscape structure: a pair of discrete habitats andsmoothly varying, continuous gradients. There are manycommonalities in the evolutionary patterns emerging inboth landscapes. For instance, in general, an increase inhabitat heterogeneity in both landscapes makes the re-striction of range and habitat breadth by gene flow more

likely, with and without predation. Introducing predatorsin both cases can lead to dramatic shifts in habitat use orthe position of a range edge. But there are differences inthese scenarios as well. The continuous gradient case can,for example, exhibit dynamics with no obvious parallel forthe discrete habitat case, such as predator and prey wave-fronts either moving at the same rate or with prey out-racing the predator across a landscape. Another differenceis that qualitative changes in the range seem to be morelikely in the two-patch system because of the frequentpresence of multiple stable equilibria. In this case, addinga predator can dramatically shift the system from one sta-ble state to another. However, it is difficult to quantify thisqualitative impression. The discrete habitat case has onlya small number of possible qualitative equilibria (a preymay be specialized to one habitat or the other, or a com-plete generalist, or in between). Along a continuous gra-dient, there are an infinite number of alternative stablerange limits, depending on initial conditions.

The impact of the rate of movement on the evolutionof habitat specialization and range dynamics is complex(Holt et al. 2005; Garant et al. 2007; Kawecki 2008) andis made more so by predation. For instance, for the discretelandscape with a generalist predator in one habitat, for aprey species with a low v to be either a specialist to thehabitat with predation or a generalist, its movement ratemust be quite low (fig. 1b, left side). This result sharplycontrasts with what is expected if the prey is alone, whereat high movement only habitat generalization is stable(Filin et al. 2008). This effect of habitat-specific predationis altered at high v. Now generalization can again be evo-lutionarily stable at high but not intermediate movementrates (see fig. A1).


Figure 6: Predation and density dependence influence the potential for prey range limitation by gene flow along a gradient. Equilibrialstates are depicted in terms of BL curves, the gradient steepness B needed for range constraint as a function of the evolutionary potentialfor adaptation, A. In a–c, the bottom gray dashed line is the BL curve for density-independent, single-species dynamics. The top gray dashedline is an extinction threshold; above it, the gradient is too steep for prey persistence. Between those two lines, either range expansion(below a line) or a limited range (above) may occur. a–c all have predation but differ in the presence and functional form of prey densitydependence. a, Density independence ( , , ; , , ). The solid line shows how predation expands∗a p 1.25 d p 0.25 m p 0.25 v p 0.5 y p 5 N p 1the potential for prey range expansion at its maximal rate, , unconstrained by gene flow. Predation increases the A-B combinationsR p 1where invasion occurs at maximal speed. b, Density dependence with (weak at low N) for several cases (lines in b and c indicatev p 8range limitation): prey alone (solid gray line); predator-prey dynamics (dashed line; , , ); weak predation:∗ ∗v p 1 y p 1 N p 0.2 N p 0.9(dotted line; , , hardly distinguishable from the single-species case); the length scale of predator spillover y is increased (dotted-v p 1 y p 1dashed line; , , ); highly effective predation: , , while keeping (solid line). c, Density1/2 ∗ ∗v p 1 y p (10) N p 0.2 N p 0.01 y p 0.1 v p 1dependence with (strong at low N): prey alone (solid gray line); with predator (dotted-dashed line; , , ).∗v p 0.25 v p 1 y p 1 N p 0.2

We have made simplifying ecological assumptions thatshould be relaxed in future studies, such as linear functionalresponses. Predators with saturating functional responsescan generate unstable dynamics (Case and Taper 2000). Theconsequences of unstable dynamics for the interplay of geneflow and selection in defining range limitation have notbeen addressed in the literature. Movement in our modelsis symmetric and random. Yet physical transport processescan be asymmetric, and many organisms are not at themercy of air currents and water flows but directionally dis-perse. Asymmetric movement, including habitat selection,can strongly alter the direction of evolution in heteroge-neous landscapes (Kawecki and Holt 2002; Holt 2003) andwould doubtless influence our conclusions.

Interspecific interactions can create historical contin-gencies in determining species ranges and do so in a waydifficult or impossible to discern by examining currentdistributions. For instance, in the discrete habitat model,predation can induce permanent switches in habitat spe-cialization of a prey species, even if the predator thendisappears (fig. 3b), and similar effects arise in the gradientmodel. In a spatially closed community, evolutionary re-sponses by a victim to a specialist natural enemy seemrather unlikely a priori to lead to the enemy’s extinction.As a prey species evolves to withstand predation, the pred-ator declines, and selection on the prey to avoid predationweakens. But in a heterogeneous landscape, prey evolutioncan lead to predator extinction if the prey utilizes habitats


inaccessible to a predator. Nuismer and Kirkpatrick (2003)explored a model in which a host and a specialist parasitecoevolve in a landscape of two discrete habitats. They as-sumed that the parasite depends demographically on thehost but that the host, conversely, is not regulated by par-asitism (and has a fixed distribution). They showed thathost gene flow significantly affects the parasite range and,in some circumstances, even leads to parasite extinction.In our model, by contrast, only one species (the prey)evolves, and its density is strongly impacted by predation.As in the model explored by Nuismer and Kirkpatrick(2003), prey habitat evolution in response to specialistpredation can lead to predator extinction. Such evolu-tionary mechanisms for the “ghost of predation past”(Brown and Vincent 1992) could provide a substantialsource of historical contingency in community assemblyand the realized ranges of prey or host species.

We made a number of simplifying assumptions aboutevolution, such as the assumption that genetic variationis fixed. If genetic variation freely evolves, negative effectsof gene flow on local adaptation can be counteracted bydispersers providing genetic variation (Gomulkiewicz etal. 1999; Barton 2001). Range limits in this case are morelikely to reflect purely organismal and ecological factors,rather than gene flow. Conversely, our models ignore ef-fects of drift and demographic stochasticity, which depletegenetic variation. When prey are productive and predatorshave saturating functional and numerical responses, large-amplitude limit cycles arise. This could force recurrentbottlenecks, amplifying the impact of genetic drift even ifaverage abundance is high. In general, the implications ofunstable population dynamics for range limits are poorlyunderstood. Moreover, we have ignored evolution in theprey to reduce predation itself (via evolution of the attackparameter) and coevolutionary responses by the predator.Coevolutionary dynamics can lead to complex mosaics(Thompson 2005), and the processes we have exploredcould contribute to this complexity. Prey with a long evo-lutionary history of exposure to particular predators mayevolve successful adaptations to escape high rates of pre-dation. Effects of predation on limiting a prey’s rangeshould be unimportant if the prey has evolved effectivecounter defenses so that prey densities are not significantlydepressed by predation (until the predator then evolves inresponse). A more detailed account of generalist predatorimpacts on the evolution of prey ranges will also need toconsider evolution and coevolution across multiple preyspecies, which can experience apparent competition viashared predation with the focal prey species (Case et al.2005).

Possibly the most important evolutionary assumptionin our models is that gene flow hampers local adaptation.Though long assumed (e.g., Mayr 1963), this may not

always hold (Gomulkiewicz et al. 1999; Barton 2001). In-terspecific interactions in isolation can lead to range limitswith a strong evolutionary component. Price and Kirk-patrick (2009) have recently shown that at times rangelimits generated by competition can be evolutionarily sta-ble, even if gene flow does not constrain local adaptation,because of how interspecific competition in effect main-tains stabilizing selection on resource use. A competitorcan constrain the shape of the adaptive landscape, so theoptimal phenotype is spatially uniform—so gene flow can-not perturb local adaptation. Holt (1997b) sketched aresource-consumer model that implied that spatial vari-ation in resource supply rates (including effects of resourcecompetition) can lead to range limits without altering theoptimal trade-off between consumption rates and mor-tality rates. In this model, stabilizing selection is spatiallyuniform, so again gene flow should not be a significantdeterminant of range limits. It would be illuminating tocraft comparable models including predation.

Despite these cautionary remarks, this study stronglysuggests that predators—even specialist predators—canprofoundly affect habitat use and geographical distribu-tions in their prey via a modulation of the effect of geneflow on local adaptation. Several empirical arenas we sug-gest could provide testing grounds and applications forour theories.

First, our extension of the Ronce-Kirkpatrick model(Ronce and Kirkpatrick 2001) to include predation couldbe used to analyze patterns of host plant use in herbivorousinsects, particularly when host plants are spatially segre-gated in different habitats. Many insect ecologists haveargued that natural enemy impacts have a strong influenceon insect community structure (Jeffries and Lawton 1984).Our models suggest that predation could have a partic-ularly strong impact on host plant specialization if theirprey have weak density dependence at low densities.

Second, an intriguing finding in island biogeography isa pattern of evolutionary habitat shifts called the taxoncycle. Wilson (1959) suggested that southwest Pacific antscolonized coastal habitats and then shifted to interior hab-itats such as montane forests. Such evolutionary shifts canbe driven by asymmetric gene flow and local adaptationin a single species (Filin et al. 2008) or competitive dis-placement by later colonists (the explanation favored byWilson [1965]; see also MacLean and Holt [1979]), butpredation and parasitism could also play a role. The dy-namics of figure 3b can occur during island communityassembly and evolution. The solid lines are prey densityin two island habitats, one near the sea (habitat 1) andthe other interior (habitat 2). The prey, initially betteradapted to habitat 1, occurs in habitat 2 but remains mal-adapted there as a result of gene flow from habitat 1. Aspecialist predator invades but is constrained to habitat 1


(e.g., by thermal adaptation). Reducing prey numbers inhabitat 1 shrinks gene flow into habitat 2, facilitating ad-aptation and increased abundance there. A backflow ofdispersers to habitat 1 grows; gene flow degrades adap-tation there, reducing local growth. The predator in habitat1 experiences an inexorable decay in its environment, withdeclining prey abundance and productivity eventuallyleading to predator extinction. The net effect is that thetaxon cycle has turned a notch, with the island colonizerhaving shifted to the interior habitat. Ricklefs and Ber-mingham (2007) suggest that pathogens introduced withsister taxa may drive island extinctions, preventing sec-ondary sympatry. Our results suggest that habitat shiftsmay provide an alternative outcome to pathogen and pred-ator introductions, at least if natural enemies are con-strained to the initial habitat range of the victim.

Third, one potential source of application of our resultsis invasion biology. Sometimes species show puzzling lagsafter establishment, remaining in a small area for a whilebefore range expansion. There are a number of potentialexplanations for this (Crooks 2005), and our theoreticalresults suggest yet another.

Finally, if gene flow constrains a prey species to remainspecialized to a particular habitat, the introduction of abiological control agent may have at times the perverseconsequence of unleashing the target species to expandover a much larger landscape. We do not argue that thisis the norm but, rather, that it is one possibility that shouldbe kept in mind in biological control. Our models suggestthat this outcome is most likely if the natural enemy is aneffective local control agent but a rather poor disperser,and if the target species has weak density dependence atlow numbers. Failed biological control attempts could pos-sibly occur for this reason (though of course there aremany other reasons for such failure).

Our results show that the interplay of predation andevolutionary dynamics in prey species in theory can leadto a rich range of potential effects, such as shifts in habitatuse and expansions in prey range size, that go well beyondconsidering the purely ecological effects of predation. Be-yond further theory development, we suggest that empir-icists interested either in the basic biology of range limitsor in applied questions based on such understanding—such as controlling invasions, crafting biological controlprograms, or projecting impacts of climate change—should consider the intertwined roles of trophic interac-tions and evolution in determining range limits.

Acknowledgments

We gratefully thank T. Price and an anonymous reviewerfor their helpful comments. I.F. acknowledges support by

Academy of Finland funding to the Finnish Centre of Ex-cellence in Analysis and Dynamics Research. R.D.H., M.B.,and S.F. thank the National Science Foundation (DEB-0515598 and -0525751), and R.D.H. and M.B. thank theUniversity of Florida Foundation for support. R.D.H. alsovery much thanks the University of Helsinki (in particular,M. Gyllenberg, I. Hanski, and V. Kaitala) for the warmhospitality shown to him and his wife during a visit toHelsinki needed to complete this manuscript.

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