spatially coupled larval supply of marine predators and their...

vol. 171, no. 5 the american naturalist may 2008

E-Article

Spatially Coupled Larval Supply of Marine Predators and Their

Prey Alters the Predictions of Metapopulation Models

J. Wilson White*

Department of Ecology, Evolution, and Marine Biology, Universityof California, Santa Barbara, California 93106

Submitted August 2, 2007; Accepted December 27, 2007;Electronically published March 13, 2008

abstract: Oceanographic forces can strongly affect the movementof planktonic marine larvae, often producing predictable spatial pat-terns of larval delivery. In particular, recent empirical evidence sug-gests that in some coastal systems, certain locations consistently re-ceive higher (or lower) larval supplies of both predators and theirprey. As a consequence, rates of settlement and predation may becoupled spatially, a phenomenon I term the “coupled settlementeffect.” To investigate the metapopulation consequences of this phe-nomenon, I created discrete-time, patch-based analytical and sim-ulation models with a common larval pool and uneven larval supplyamong patches. Using two complementary measures of subpopu-lation value as a basis of comparison, I found that models with andwithout the coupled settlement effect yielded strikingly different pre-dictions. When prey and predator larval supplies were not coupled,patches supplied with a larger proportion of the larval pool made agreater contribution to the metapopulation. When settlement of preyand predator was strongly coupled, however, the opposite was true:subpopulations with lower rates of larval supply (above some min-imum) were more essential to metapopulation persistence. Theseconsiderations could facilitate more effective selection of sites forprotection in marine reserves.

Keywords: metapopulation, predator-prey interactions, source-sinkdynamics, spatially coupled dispersal.

The dynamics of many animal populations are stronglyinfluenced by the spatial distribution of predators (Hasselland May 1974; Holt 2002). This is especially true of speciesin fragmented or spatially discontinuous habitats in which

* Present address: Department of Wildlife, Fish, and Conservation Biology,

University of California, One Shields Avenue, Davis, California 95616; e-mail:

[email protected].

Am. Nat. 2008. Vol. 171, pp. E179–E194. � 2008 by The University ofChicago. 0003-0147/2008/17105-42785$15.00. All rights reserved.DOI: 10.1086/587079

both predators and prey are found in discrete patcheslinked by dispersal into metapopulations. For example, theabsence of predators from certain patches can providerefuge habitat that stabilizes the prey metapopulation(Murdoch et al. 1992), while high rates of predator im-migration into a particular patch may greatly depress and/or destabilize the local prey population (Holt 2002).

Most theoretical investigations of predator-prey meta-populations center on scenarios in which predator dis-persal is either random and diffusive (e.g., Hassell et al.1994; Hosseini 2005; Amarasekare 2006) or behaviorallydirected, so that predators actively search for prey (e.g.,Holt 1985; Murdoch et al. 1992; Bernstein et al. 1999;Jackson et al. 2004). In either case, prey population dy-namics are shaped by the resulting spatial heterogeneityin predation rates (Briggs and Hoopes 2004). However,neither dispersal assumption is likely to hold for systemsin which movement is strongly affected by the advectivephysical forces, such as wind (Bell et al. 2005) or flowingwater (Fagan 2002), a dispersal scenario that is rarely ex-amined in the terrestrial metapopulation literature (butsee Lett et al. 2003, 2005). This scenario is particularlyrelevant for benthic marine metapopulations (Kritzer andSale 2004) in which dispersal between patches of adulthabitat is achieved by planktonic larvae whose movementis driven predominately by oceanic currents (Norcross andShaw 1984; Shanks 1995). Consequently, theoretical in-vestigations of marine metapopulation dynamics must in-voke specific assumptions regarding the specific type ofdispersal being modeled, for example, well-mixed larvalpools (Roughgarden et al. 1985) versus alongshore advec-tion (Alexander and Roughgarden 1996). In some cases,oceanographic data have been used to create detailed sim-ulations of larval trajectories (James et al. 2002; Gilg andHilbish 2003; Cowen et al. 2006). Regardless of the levelof oceanographic detail used, the majority of these effortshave taken a single-species (i.e., prey) perspective and as-sumed that within-patch predation rates are spatially in-variant, so prey population dynamics are shaped only bypatterns of larval delivery (James et al. 2002; Cowen et al.2006; Karlson 2006; Mumby and Dytham 2006; for notable

E180 The American Naturalist

exceptions, see Gaines and Lafferty 1995; Connolly andRoughgarden 1999). However, the importance of predatormovement to population dynamics of species in both ter-restrial and aquatic systems suggests that marine meta-population models could profit from a consideration oftrophic interactions.

Although specific modeling approaches may vary, thegoals of most metapopulation studies are generally thesame: to identify those patches that are most importantto the stability, persistence, or abundance of the meta-population (Carr et al. 2003). In many cases, the objectiveis to inform decisions regarding the placement of reserveareas. For marine reserves, placement recommendationsvary depending on the specific goals of the reserve (Hal-pern and Warner 2002; Hastings and Botsford 2003) butcommonly emphasize the importance of sites where ocean-ographic conditions favor consistently high larval supplyof target species (Roberts et al. 2003). When the primaryreserve goal is biodiversity conservation, protected sitesshould receive enough locally produced larvae for the re-serve population to be self-sustaining (Botsford et al. 2003;Roberts et al. 2003). With this need in mind, it is usuallysuggested that reserves be placed near oceanographic re-tention zones or spawning grounds in order to maximizelarval delivery (Shepherd and Brown 1993; Warner et al.2000; Sala et al. 2002). A second common goal is theenhancement of local fisheries by the export of larvae fromreserves to fished zones (Halpern and Warner 2002). Inthis case, retention of all larvae in the reserve is undesirable(Roberts et al. 2003), and some high-settlement sites mayremain unprotected in order to maximize the biomassavailable for harvest (Crowder et al. 2000; Parnell et al.2006). These recommendations follow from the predic-tions of single-species metapopulation models (Gerber etal. 2003) and generally rely on the observation that evenwhen mortality in reserves is compensatory and densitydependent (Hixon and Webster 2002), adult populationdensity will be positively correlated with the level of re-cruitment, at least over a wide range of recruit densities(Hixon et al. 2002).

The importance ascribed to the sites with high rates oflarval replenishment in existing marine metapopulationmodels may be an artifact of the single-species approachand the accompanying assumption of spatially uniformmortality rates. New recruits of many marine organismssuffer heavy mortality soon after settling to the benthos(Glynn 1988; Hixon 1991), and in most cases, the pred-ators responsible for this mortality also have planktonicallydispersed larvae. Oceanographic forces often affect larvaeof different species in similar ways, so taxonomically di-verse assemblages of fish and invertebrate larvae are com-monly concentrated in coastal eddies and retention zones(Boehlert and Mundy 1993; Wing et al. 1998; Nishimoto

and Washburn 2002). Coastal areas near such oceano-graphic features generally experience consistently high set-tlement of many different species (Cowen and Castro 1994;Swearer 2000; Connolly et al. 2001; Valles et al. 2001; Maceand Morgan 2006). Consequently, certain habitat patchesare likely to be settlement hotspots for both predators andtheir prey, and predation rates may be higher in thesepatches as a result. For example, at the Caribbean islandof St. Croix, leeward reefs near an oceanographic conver-gence experience consistently higher settlement of bothbluehead wrasse (Thalassoma bifasciatum) and its majorpredator, the coney (Cephalopholis fulva), relative to moreexposed, windward reefs (Hamilton et al. 2006; White2007). As a result, bluehead wrasse recruits experiencestrong, density-dependent mortality immediately after set-tlement on the leeward, high-settlement reefs but not onthe windward, low-settlement reefs (fig. 1; White 2007).

Metacommunity models predict that a high rate of pred-ator immigration will depress the prey density in a patch(Holt 2002), but single-species models of marine meta-population dynamics do not incorporate the potential forhigher predator densities in patches with high rates oflarval supply, a phenomenon I term the “coupled settle-ment effect.” Here I construct a series of simple meta-population models to illustrate the importance of ac-counting for spatial correlations in the dispersal ofpredators and their prey and the resulting spatial variationin predation rates. While not as sophisticated as manyrecent models grounded in oceanographic flow simula-tions, these models permit a straightforward examinationof the consequences of coupled settlement. A commongoal of metapopulation modeling is to identify key“source” subpopulations, so I compare models with andwithout the coupled settlement effect on the basis of theirpredictions regarding the importance of a focal deme tothe larger metapopulation. Although the models describea nonexploited system, I used measures of subpopulationimportance that would be useful in assessing the costs andbenefits of including a particular site in a reserve network.

Quantifying the Value of a Subpopulation

The role played by individual demes in a metapopulationcontext is commonly described in terms of sources andsinks (reviewed inmean larvalDias 1996; Runge et al.2006). The definitions of these terms vary by author butare often framed in terms of self-sustainability: sourceswould persist in isolation, but sinks would not (Pulliam1988; Armsworth 2002). This definition is favored becausethe presence of a self-sustaining source is sufficient for thepersistence of a metapopulation; if at least one deme is aself-sustaining source, that deme (and, by extension, themetapopulation) will persist even if the remaining sinks

Coupled Supply of Marine Predators and Prey E181

Figure 1: Empirical evidence for the coupled settlement effect for a coralreef fish (bluehead wrasse Thalassoma bifasciatum) and its predator (co-ney grouper Cephalopholis fulva) from White (2007). A, Relationshipbetween mean prey recruitment and mean recruitment and adult densityof the predator on seven reefs at St. Croix, U.S. Virgin Islands, in June–September 2003. Reefs were separated by distances greater than typicaldistances moved by adult coneys, so the pattern of adult predator abun-dance reflects long-term trends in recruitment. Error bars are 1 SE. B,Per capita strength of density-dependent prey mortality (measured as theslope of the density-mortality relationship and equivalent to 1/g in eq.[6]; Schmitt and Holbrook 2007) as a function of predator density. In-dependent estimates of transect-wide mortality and settler density wereobtained on three transects at each of five St. Croix reefs in two monthsin 2005; points indicate a running average of the slope of the (approx-imately linear) density-mortality relationship as a function of predatordensity. Each point represents the slope calculated using the transect-wide density and mortality estimates from a subset of seven transectswith similar predator densities; adjacent points have overlapping subsets.

become extinct (Armsworth 2002). However, under thiscriterion, a self-sustaining source need not export indi-viduals to any other patch and so may not be sufficientfor the persistence of a multiple-patch metapopulation.Futhermore, a metapopulation may consist entirely ofnon-self-sustaining demes if interdeme connectivity is suf-ficient (Hastings and Botsford 2006). The focus of thisarticle is on the contribution of a focal subpopulation tothe larger metapopulation, so I eschew potentially unin-formative self-replenishment-based definitions of source-sink status in favor of two alternative metrics of patchvalue (sensu Ovaskainen and Hanski 2003) drawn fromnetwork graph theory (Urban and Keitt 2001).

The first metric is a point estimate of the net repro-ductive contribution of subpopulation i to the metapop-ulation, which I term the “export ratio” Ei . Consider anetwork of equal-sized subpopulations thati p 1 … mare separated by barriers to adult movement but linkedby the exchange of dispersing larvae. Each subpopulationhas a population density ni and per capita fecundity bi

(for convenience, all variables and parameters used in thisarticle are summarized in table 1). Larvae spawned insubpopulation i disperse to subpopulation j with proba-bility fij ; in transit they have survivorship hS. The netreproductive contribution of patch i in a given time in-terval is the difference between the total number of larvaeproduced in i recruiting anywhere in the metapopulationand the number of larvae produced anywhere in the meta-population recruiting to i, which is most conveniently rep-resented as a log ratio:

m� n bh fi i S ijjp1

E p log . (1)i m� n b h fj j S jijp1

By this criterion, “source” subpopulations that producemore larvae than they receive will have , whileE 1 0i

“sinks” will have . In a reserve context, Ei addressesE ! 0i

the goal of fishery enhancement, capturing the degree towhich reserves export larvae to exploited populations. Thismeasure is analogous to network flux (Urban and Keitt2001) and is similar to patch contribution metrics usedby other authors (Tuck and Possingham 2000; Figueiraand Crowder 2006; Runge et al. 2006; Samhouri 2007).

The second metric considers the consequences of re-moving subpopulation i from the metapopulation, thuscapturing both the importance of the total reproductiveoutput of i and its potential function as a stepping-stonelinking other sources and sinks. A less formal version ofthis approach was taken by James et al. (2002). Using thesame notation as above, let the mean population densityof the metapopulation at equilibrium be


Table 1: Symbols and parameter values used in population models

Symbol Value Description (units)

f Proportional larval supplyp Larval supply correlationn Local prey density (fish m�2)N Metapopulation-wide prey density (fish m�2)S Prey settler density (fish m�2)R Prey recruit density (fish m�2)g Maximum recruit density (recruits m�2)a Subpopulation area (m2)hS 1.7 # 10�10–2.6 # 10�1 Prey larval survivorshipa

hP1 1.7 # 10�12–2.6 # 10�3 Predator larval survivorshipb

hP2w .015 Mean predator density–open population model (predators m�2)c

P .015 Mean predator density–metapopulation model (predators m�2)c

b 1.1 # 106–3.0 # 106 Prey reproductive rate (larvae adult�1)d

dS Settler survivorshipdA .39–.96 Adult survivorshipe

dg .14–.51 Density-independent settler survivorshipf

Kd 51 Density-independent survivorship scaling factor (predator�1)g

Kg 17–109 Density-dependent survivorship scaling factor (predator�1)h

Note: Unless otherwise noted, parameter values were randomly drawn from uniform distribution defined by range of values.

Values not given for state variables, independent variables, and variables not used in numerical simulations.a Larval mortality rates used by Cowen et al. (2000) were converted to monthly survivorships assuming a larval duration of

approximately 30 d (the model time step). Values were sampled from uniform distribution of and then back transformed.log hS

b Fixed at 0.01hS to match empirical relationship between Thalassoma bifasciatum settler density and Cephalopholis fulva density

at St. Croix, U.S. Virgin Islands (White 2007).c Mean density of C. fulva at St. Croix (White 2007).d Calculated from range of values given by Caselle et al. (2003), assuming that per capita fecundity is constant.e Monthly survivorships calculated from range of weekly mortality rates for T. bifasciatum at San Blas, Panama (Warner and

Chesson 1985).f 95% confidence interval on density-independent component of per capita T. bifasciatum settler mortality (White 2007).g This value is scaled so that dS p 0.23 in the case with p p 0; this corresponds to the mean density-independent survivorship

observed for T. bifasciatum settlers at St. Croix (White 2007).h 95% confidence interval on slope of relationship between density-dependent component of T. bifasciatum recruit mortality

and C. fulva density (White 2007).

m1∗ ∗N p n ,�● im ip1

where the bullet subscript indicates that all subpopulationsare present. If subpopulation i is deleted from the meta-population so that all individuals dispersing to i die im-mediately on arrival (this could be imagined to be sometype of “scorched earth” harvesting), the equilibrium den-sity of the metapopulation will be (where the mean∗N●�i

is still taken over all m patches). Now I define the deletionindex Di as a function of the ratio between these twovalues:

∗N●�iD p 1 � .i ∗N●

This index produces a continuum rather than a binaryclassification, and subpopulations with higher values of Di

make a larger contribution to the metapopulation. At theextremes, indicates a patch that is essential forD p 1i

metapopulation persistence; this could be termed a “globalsource.” Under most conditions, the minimum value ofDi is 0, although strongly overcompensatory density de-pendence could theoretically produce . In a reserveD ! 0i

context, this metric addresses both the biodiversity goal(will the species persist if the focal patch is overexploited?)and the fishery enhancement goal (will population den-sities in other patches be higher if a focal patch is leftunexploited?).

Metapopulation Models

Marine metapopulation models are frequently framedaround the well-studied life history of small-bodied reeffish (James et al. 2002; Cowen et al. 2006). My effortfollows suit and reflects to a large degree the empiricalrelationship between predator recruitment and prey mor-tality observed in Thalassoma bifasciatum and Cephalo-pholis fulva (White 2007). Nonetheless, the general resultsshould be broadly applicable to any system in which small,


short-lived prey are vulnerable to a longer-lived, generalistpredator. Here I consider a series of metapopulation mod-els of increasing complexity; for simplicity and analyticaltractability, I begin with a single prey subpopulation nestedwithin a larger metapopulation (cf. Roughgarden et al.1985) before moving on to simulation models of entiremetapopulations.

Many reef fish spawn and settle in monthly pulses (Rob-ertson 1991, 1992), so the models operate in discrete time.The length of each time step corresponds to the intervalbetween spawning and recruitment, spanning the pelagiclarval period and the initial bout of heavy mortality thatoccurs at the time of settlement.

Analytical Model of an Open Local Population

In this initial scenario, a subpopulation receives St settlinglarvae per unit area at each time t, and a negligible fractionof these larvae were spawned locally. Larval delivery is afunction of some physical factor f, so that , whereS p h ft S

hS is a constant. A fraction dS of the settlers survive theinitial bout of mortality to recruit into the adult popu-lation, after which they have per capita survivorship dA.The adult population density ni follows the equation

n p d S � d n .t�1 S t A i, t

Assuming constant larval supply, so , this systemS { St

reaches a globally stable steady state at

d SS∗n p . (2)1 � dA

Reef predators are generally omnivorous (Randall 1967),and predatory fishes are usually longer lived and havelower annual reproductive investment than their prey(Shuter et al. 2005), so predator dynamics should be slowerthan and independent of the dynamics of any one preyspecies. Indeed, Murdoch et al. (2002) showed that a widevariety of generalist predatory fishes exhibited populationfluctuations inconsistent with strong coupling to the dy-namics of their prey. As such, I preserve the simplicity ofthe models presented here by avoiding an explicit treat-ment of predator dynamics. Instead, a subpopulation isassumed to have constant predator density P, which couldrepresent the combined density of multiple predatory spe-cies. This density will be a function of larval supply, whichmay be driven by the same physical factor f that deliversprey larvae. Alternatively, predator settlement may be un-related to f if, for example, predator larvae differ markedlyfrom prey in swimming ability or behavioral response tooceanographic cues. Therefore, it is convenient to express

P in terms of the relative proportions of individuals de-livered by different physical processes:

( )P p ph f � 1 � p h w, (3)P1 P2

where wi is some alternative process that does not affectprey larvae, p is the proportion of predators delivered byf, and the hP are constants of proportionality. Hereafter, p

will be used as a proxy for the strength of the coupledsettlement effect. Equation (3) can be recast in terms of S:

phP1 ( )P p S � 1 � p h w, (4)P2hS

where phP1/hS describes the slope of the relationship be-tween predator density P and prey settlement. Predatordensity could affect prey mortality in any number of ways,but mortality is often most intense immediately after set-tlement, so I focus on cases in which dS is a function ofP. There is no evidence that predators affect the settlementof reef fish larvae (Almany 2003), so I consider postset-tlement predation only.

In this open local population scenario, it is not possibleto calculate D. Calculating E is also problematic; the de-nominator of equation (1) is equal to S, but without ex-plicit consideration of the rest of the metapopulation, thereis no sensible way to calculate “export.” However, for anopen population, it is informative to consider the size ofthe population relative to the larval import, on the as-sumption that the number of exported larvae (whatevertheir destination) is roughly proportional to populationsize. Consequently, for this scenario I define a revised met-ric, , which is equivalent to equation (1) with′E p log (n/S)the unspecified terms related to export removed. Whilethe actual values of E and D are unknown, it is possibleto investigate the effect of changes in p on the relativevalue of E ′.

I explore the dynamics of the open local population fortwo different scenarios: first, the highly illustrative butrelatively unrealistic case of density-independent postset-tlement mortality; second, the case of strong postsettle-ment density dependence, which is more representative ofmost reef fish populations studied to date (Hixon andWebster 2002; Osenberg et al. 2002).

Case 1: Density-Independent Mortality. In the simplestscenario, dS is independent of prey density but linearlyrelated to predator density:

d p 1 � k P, (5)S d

where kd is a constant describing the (linear) functional


Figure 2: Effect of larval supply correlation f on the dynamics of anopen population with density-independent (A, C, E) or density-depen-dent (B, D, F) survivorship. A, B, Per capita survivorship dS; C, D, equi-librium population density n∗; E, F, export ratio E ′ for a population withthe indicated density of settlers S (higher values of E indicate greaterexport). Population densities are dimensionless and scaled by the factorh2hSw/h1. Mean values from table 1 were used for all parameters.

response of the predator. Because P and S are both functionsof f when , this effectively makes prey mortalityp ( 0strongly dependent on prey density, despite the nominallydensity-independent local dynamics (fig. 2A). Combiningequations (2), (4), and (5) and simplifying yields

2 [ ][(�k ph )/h ]S � 1 � k wh (1 � p) Sd P1 S d P2∗n p .1 � dA

If (i.e., predator and prey settlement are uncoup-p p 0led), the S2 in the numerator of the right-hand side dis-

appears and n∗ is an increasing function of S. However,if , n∗ is a quadratic function of S. This function isp 1 0negatively concave for all values of S, with a maximum at

. As S increases from 0,S p h [1–k h (1–p)w]/(2k h p)S d P2 d P1

population size increases from 0 to the maximum; if larvalsupply increases further, the subpopulation declines to ex-tinction (fig. 2C). This counterintuitive relationship be-tween larval supply and local population size is the essenceof the coupled settlement effect.

The magnitude of p has a strong effect on the exportratio. When , E ′ is constant for all values of S. Whenp p 0

, E ′ is substantially higher than in the case atp 1 0 p p 0low values of S but then declines at an accelerating rateas S increases (fig. 2E).

Case 2: Density-Dependent Mortality. The details of therelationship between S, p, and E ′ change only slightly asthe model becomes more realistic. Most reef fish experi-ence strong density-dependent mortality at the time ofsettlement (Hixon and Webster 2002), and in most casesthis mortality can be approximated well by a Beverton-Holt relationship (Schmitt et al. 1999; Osenberg et al.2002). In discrete time, this can be expressed as

dgd p ,S 1 � (d /g)Sg

where dg represents density-independent survivorship andg represents the maximum possible density of recruits. Inthe continuous-time version of this function, the maxi-mum recruit density is inversely proportional to the percapita density-dependent mortality rate (Osenberg et al.2002), which can be strongly affected by local predatordensity. In order to approximate the empirical relationshipbetween predator density and the density-dependent mor-tality rate documented by Schmitt and Holbrook (2007)and White (2007; also see fig. 1B), I made the maximumrecruit density inversely proportional to predator density:

1g p . (6)

k Pg i

If there is correlated settlement ( ), the positive re-p 1 0lationship between Pi and S causes g to be inversely pro-portional to S as well. As with parameter kd in equation(5), the parameter kg describes the predators’ functionalresponse. Model results are quite sensitive to the strengthof this response; for all results shown here, I used valuesestimated from empirical data (e.g., fig. 1B plots 1/g vs.P for the T. bifasciatum–C. fulva system; the slope of thatrelationship corresponds to kg. Also see table 1).


With density-dependent settler mortality and ,p p 0n∗ increases with S to an asymptotic maximum of

. When , the expression for n∗ is quiteg/(1 � d ) p 1 0A

complex but has a relationship with S similar to that ofthe density-independent case. As S increases from 0, n∗

rises to a maximum at before de-1/2S p (h /d k h p)S g g P1

clining toward extinction. Unlike the density-indepen-dent case, there is an inflection point in n∗ to the rightof the maximum, and n∗ approaches 0 asymptoticallywith increasing S (fig. 2D).

The behavior of the export ratio E ′ differs from thedensity-independent scenario. When mortality is densitydependent, E ′ declines with S, even when . This isp p 0a straightforward consequence of the maximum densityimposed by the Beverton-Holt function: as S increases, asmaller proportion of settlers survive to enter the adultpopulation, reproduce, and contribute larvae to the meta-population. When , populations with higher S havep 1 0lower maximum densities than populations with lower S,so E ′ declines more steeply with S (fig. 2F). In both thedensity-independent and density-dependent cases, P (andthus E ′) is the same for all values of p at .S p h h w/hS P1 P2

Models not accounting for the coupled settlement effectwill underestimate E ′ for populations with values of S lessthan this value and overestimate E ′ for populations withS greater than this value.

Numerical Simulations of Metapopulation Dynamics

In order to fully capture the effects of coupled settlementon the value of a patch within a metapopulation, it be-comes necessary to model the dynamics a full metapop-ulation. Such models no longer afford easily interpretableanalytical solutions, so I used numerical simulations tocharacterize the relationship between Si, p, and subpop-ulation value (defined by the export ratio Ei and deletionindex Di).

Consider an entire metapopulation consisting of m sub-populations n1 … nm, each with area a. The metapopu-lation is closed to external inputs of larvae, so it is nec-essary to introduce a parameter for per capita fecundityb. At each time t, the ani, t adult fish in population i pro-duce abni, t larvae, which then enter a larval pool with theoffspring of all other subpopulations. The size of the larvalpool is

m

L p ab n�t i, tip1

As before, each subpopulation is associated with a param-eter fi that describes the intensity of larval supply to thatlocation. In the context of an entire metapopulation, fi is

most easily defined as the fraction of the larval pool de-livered to subpopulation i. Thus, the number of settlersper unit area arriving at each population is S pi, t

, where hS now represents larval survivorship andfh L /ai S i, t

the sum of fi across all populations is unity. This for-mulation is equivalent to having an connectivitym # mmatrix F, with elements fij defining the proportion oflarvae produced at i that disperse to j; but in this case,each row of F is identical, so that each subpopulationreceives the same fraction fi of larvae from each subpop-ulation. This type of connectivity scheme is appropriatefor some but not all marine metapopulations; I outlinethe justification for this approach in the “Discussion.”

Now that fi has been recast as a proportion, the ex-pression for the local density of predators (eq. [3]) mustbe revised. Defining to be the mean density of predatorsPacross all subpopulations, the predator density in sub-population i is now

( )P p pfmP � 1 � p P.i i

Under this definition, Pi exhibits the same behavior as inthe case of density-dependent mortality: when p p 0,

for all subpopulations i; when p 1 0, Pi approachesP p Pi

as fi increases toward unity and approaches 0 as fimPshrinks toward 0. Because variation in fi (and, by exten-sion, S) is of most interest here, this assumes that the onlyspatial variation in Pi is due to the oceanographic processrepresented by fi. The strength of the coupled settlementeffect is still indicated by p, but the slope of the relation-ship between Pi and Si changes slightly and is now

. Postsettlement mortality remains density depen-pmP/hP1

dent, which ensures that the metapopulation has a stablenonzero equilibrium (Armsworth 2002), and the intensityof density dependence is a function of predator density,as in the open population model (eq. [3]).

In this model, gi effectively sets the spatial scale of thesimulation by defining the maximum number of recruitsin a unit of area, so the absolute value is less meaningfulthan the relative difference in gi among demes within thesame metapopulation. For the purpose of parameterizingthe model for simulations, I used units of fish/m2 for gi.Biologically reasonable values were assigned to the otherdemographic parameters by using published data for T.bifasciatum (table 1). I created 100 unique parameter setsby taking uniform random draws from the range of plau-sible values for each parameter and then used each pa-rameter set to simulate model dynamics across the rangeof p and fi. For each p-fi combination, I report the meanvalue of Di and Ei taken across all 100 parameter sets.Simulations were run for 500 time steps, sufficient timefor the model to reach equilibrium (to calculate Di, an


Figure 3: Effect of larval supply correlation, p, on relative patch im-portance in two-deme metapopulations. Each point is the equilibriumvalue of (A) log equilibrium population density n∗, (B) export ratio E,or (C) deletion index D for a subpopulation with the indicated value ofproportional larval supply fi. Vertical dashed lines indicate the meanvalue of f. The horizontal dashed line indicates E p 0, the boundarybetween net larval export and net larval import.

additional set of 500 step simulations were run with thefocal population deleted).

I focused on the dynamics of the simplest possible meta-population, which consists of two subpopulations, n1 andn2; the results can be extended to larger, multipatch meta-populations without a loss of generality (see appendix).Taking subpopulation 1 as the focal deme, mean D1 andE1 were calculated for values of f1 ranging from 0.01 to0.99.

Simulation Results

The dynamics of the simulated metapopulations werelargely similar to those of the open population model withdensity-dependent mortality. When predator and prey set-tlement were uncoupled ( ), equilibrium populationp p 0density was constant for most values of larval supply∗ni

f1 (fig. 3A), but when , was highest at low values∗p 1 0 ni

of f1 and decreased with increasing f1 (as in the density-dependent open population, when and∗n p 0 f p 0i 1

then quickly rises to a maximum; this behavior is onlybarely visible for and in fig. 3A). Changesp p 0 p p 0.5in the export ratio E1 were nearly identical to the openpopulation case: E1 declined with increasing larval supplyfor all values of p, but when , E1 was higher for low-p 1 0settlement populations (indicating greater relative larvalexport and more sourcelike behavior) and lower for high-settlement populations (indicating lower relative larval ex-port and more sinklike behavior; fig. 3B). The effect ofcoupled settlement on the metapopulation contribution ofsubpopulation 1 was even more evident in the values ofthe deletion index D1 (fig. 3C). When settlement was un-coupled ( ), D1 increased modestly and monotoni-p p 0cally with f1, indicating that the population receivinggreater larval supply was more important to the meta-population. For or 1, by contrast, D1 was highestp p 0.5for low values of f1 and declined with increasing f1, soimportance of a patch declined as larval supply increased.This decline was not monotonic, however. At extreme val-ues of f, the low-settlement subpopulation received solittle larval supply that it was no longer completely self-sustaining, causing a slight upturn in the deletion indexof the high-settlement subpopulation and a correspondingdownturn in the deletion index of the low-settlement sub-population. Nonetheless, the value of D1 when settlementwas coupled ( ) remained far below that of the un-p 1 0coupled case ( ) for all high values of f1. The valuesp p 0of , E1, and D1 were the same for all values of p only∗ni

at , when larval supply to both demes wasf p 1/m p 0.51

exactly equal.For all metapopulations, the export ratio Ei was less than

0 (indicating net larval import) for large values of larvalsupply (fi), regardless of the degree of larval coupling (fig.


3B). This was a consequence of density-dependent mor-tality in these subpopulations (the same phenomenon wasevident in the open local population with density-depen-dent mortality; fig. 2F). When , these high-settle-p p 0ment subpopulations were the most important demes inthe metapopulation (as indicated by Di) despite their lowvalues of Ei. This seeming disparity between Ei and Di

reflects a fundamental difference between the two mea-sures: Ei describes only the relative efficiency of larval pro-duction (regardless of its overall magnitude), while Di isa measure of the absolute contribution to population den-sity in the metapopulation. It was striking, then, that thesecomplementary measures were in accord when ;p 1 0high-settlement populations had low values of both Ei andDi relative to those of lower-settlement subpopulations.

Discussion

It is well known that the dynamics of predator-prey in-teractions can be strongly affected by the existence of spa-tial refuges that predators cannot access (Sih 1987). Re-leased from predation, the prey populations in refuges haveincreased reproductive success and can replenish high-predation patches elsewhere in the metapopulation (Mur-doch et al. 1992). Metapopulation models often incor-porate refuges to represent some sort of habitat unsuitablefor predators, but a similar effect is produced when phys-ical forces distribute predators unevenly through the meta-population. This phenomenon was first noted by Lett etal. (2003, 2005), who studied a Nicholson-Bailey modelof host-parasitoid dynamics in a system with fixed mi-gration rates between patches. They found that the stabilityand equilibrium density of the host population were oftenmaximized when the hosts and parasitoids dispersed inopposite directions, so that most hosts tended to arrive atpatches with few parasitoids. However, they acknowledgedthat such divergent dispersal patterns were unlikely to befound in nature (Lett et al. 2003, 2005). Here I investigatedthe opposite situation, which is far more likely to occurin environments with physical forcing: predators and preytend to disperse in the same direction. Furthermore, Ifocused on the effects of coupled dispersal on measuresof patch value that are particularly relevant for conser-vation planning and the placement of marine protectedareas (Roberts et al. 2003), and one of the primary resultsof this article was the development of two novel metricsfor patch value. When prey and predator dispersal werestrongly coupled ( ) in the reef fish metapopulationsp p 1modeled here, the few settlers arriving at low-settlementpatches were effectively entering a predation refuge wherethey could maintain a much greater adult population den-sity and have increased value to the metapopulation. Thiseffect is unique to a system with physically forced dispersal

and would not occur if prey were able to emigrate fromhigh-density patches (Amarasekare 2004) or if predatorscould redistribute themselves after depleting the prey in apatch (Bernstein et al. 1999; Jackson et al. 2004). Scenarioswithout dispersal coupling ( ) recapture the type ofp p 0system more commonly examined in metapopulationmodels (reviewed in Briggs and Hoopes 2004), in whichhigh-settlement subpopulations were more valuable to themetapopulation. The opposite was true when : bothp 1 0the export ratio and the deletion index decreased withproportional larval supply. This contrast is the second pri-mary result of this article: metapopulation models notaccounting for the coupled settlement effect in systemswhere it exists will incorrectly predict the relative meta-population value of subpopulations.

The key feature of the coupled settlement effect is mosteasily visualized in the unimodal settler-adult relationshipin the open population model (fig. 2C, 2D): subpopula-tions receiving a very small proportion of the larval poolexperience approximately density-independent mortality,while subpopulations with high proportional larval supplyexperience intense density-dependent mortality. The un-imodal shape and resemblance to a Ricker stock-recruit-ment function (Hilborn and Walters 1992) makes it tempt-ing to think of this phenomenon in terms of over-compensating density dependence. It is important to note,however, that this relationship does not describe the tem-poral relationship between settlement and mortality withina single population (as a Ricker function does) but rathervariation in mortality across a spatial gradient of settle-ment. With the coupled settlement effect, mortality is afunction of the average larval supply at a site, not thedensity of any particular prey cohort. Recruit mortalitywithin each cohort is always a saturating Beverton-Holtfunction that produces dynamic stability (Armsworth2002). The importance of this distinction becomes evidentif one considers the consequences of allowing prey larvalsupply to vary stochastically through time while keepinglocal predator densities a function of the mean larvalsupply. In this scenario, sporadic large pulses of larvae tosites with high mean supply will experience strong density-dependent mortality, while large pulses at sites with lowmean supply will experience low density-independentmortality. These low-settlement sites will thus be able tostore large settlement pulses (Warner and Chesson 1985),buffering them against future settlement failures. Conse-quently, sites with low mean larval supply could be im-portant reservoirs of reproductive adults.

Larval pools are a common feature of marine meta-population models (Roughgarden and Iwasa 1986). Thisapproach probably assumes an unrealistic degree of mixingand homogeneity (Flowers et al. 2002; Planes et al. 2002)and is clearly an inappropriate description of highly di-


rectional systems with stepping-stone dispersal (Carr andReed 1993). Nonetheless, it can capture the essential fea-tures of systems in which frequent current shifts facilitatebidirectional larval exchange among multiple subpopu-lations, as in the uniform metapopulation scenario (e.g.,Southern California Bight; Carr and Reed 1993; Selkoe etal. 2006), or in which currents favor larval accumulationand delivery to particular geographical locations, with lim-ited or infrequent larval delivery to other sites in the meta-population (e.g., headland accumulation zones: Botsford2001; eddies and accumulation zones near oceanic islands:Lipcius et al. 2001; Harlan et al. 2002). It is important torecall that because my model assumes that all larvae areidentical, larval mixing per se is not important, and thelarval pool could be equally well described by a connec-tivity matrix of a particular form (see “Numerical Simu-lations of Metapopulation Dynamics”). In any case, thelarval pool assumption provided a simple way to modelthe correlation between prey settlement and predatorabundance without explicitly modeling predator dynamics,and using a more complex connectivity matrix would re-quire a dynamic model of the predator population. Theresults of such a model would probably depend on thespecifics of the connectivity matrix, but so long as thereis some bidirectional dispersal between most patches andsome patches tend to receive higher larval supply thanothers, the general patterns described here should hold.

Most predators on coral reefs have diverse diets (Randall1967), and omnivory is a common strategy in many othersystems as well (Polis and Strong 1996), so making pred-ator densities independent of the abundance of a singleprey species should be a reasonable approximation to re-ality (Murdoch et al. 2002). However, the effects reportedhere could differ in models incorporating a full comple-ment of prey species and predator dynamics, especially ifthe predator population is size structured. In general, as-suming additional prey species follow the same generalspatial pattern of settlement, predators at high-settlementsites will have access to a greater and more consistent influxof prey, which should lead to higher predator survival,enhancing the general pattern presented here. This effectwould be dampened if territoriality produced density de-pendence in the predator population, but numerical den-sity dependence could be offset by a developmental re-sponse if per capita prey consumption increases withbiomass in a size-structured predator population (Mur-doch 1971). However, these predictions will also dependon the degree to which predators influence adult preymortality rates. In this model, predator abundance affectedonly settler mortality, but if predators can accumulate bio-mass by feeding on both new settlers and adults, low-settlement sites may no longer accumulate high densitiesof adult prey. For example, a biomass-based model of

predator-prey dynamics suggests that the accumulation ofpredator biomass due to feeding outstrips the influx ofbiomass due to settlement (Sandin and Pacala 2005). Allelse being equal, this would negate spatial differences inpredation created by larval delivery patterns. However,asymptotic limits on the size of individual fish shouldensure that there is still a numerical constraint on thepotential predation pressure at low-settlement sites, as inthe model presented here. An additional consideration isthe strength of the predator functional response, whichwas linear and given by the k parameters (kd and kg) inthis model. If the relationship between predator densityand predation rate were weak or rapidly asymptotic, therewould be no effect of coupled settlement; indeed, themodel results are just as sensitive to k as to the settlementcoupling parameter p. However, the available field datasuggest a substantial and approximately linear functionalresponse (Schmitt and Holbrook 2007; White 2007), andempirically derived values were used in the results pre-sented here.

This study was inspired by the observation of correlatedsettlement and predation rates in a coral reef fish meta-population (White 2007), but this phenomenon may bea common feature of many metacommunities, both ma-rine and terrestrial. Intertidal mussels along the Californiacoast experience significantly higher predation rates at siteswith high settlement of both mussels and their sea starpredators, and mussel densities in these sites are similarto densities in locations with far lower settlement (Mengeet al. 2004). Movement in riverine metacommunities isstrongly affected by the intensity of stream discharge,which tends to concentrate resources and organisms indownstream confluences (Schlosser 1995; Power and Die-trich 2002). In turn, these wide, deep, downstream loca-tions are thought to act as source populations that canrecolonize upstream headwaters after disturbances (Os-borne and Wiley 1992; Schlosser 1998). However, predatordensities and predation rates also increase with down-stream distance and stream size (Schlosser 1987; Powerand Dietrich 2002), which could dilute the efficacy oflarger stream regions as “sources” and may partially ex-plain some mismatches between observed recolonizationrates and single-species metapopulation model predictions(Gotelli and Taylor 1999). In some terrestrial metacom-munities, interpatch dispersal is wind driven. Zooplanktonand their invertebrate predators form aquatic metacom-munities in the leaves of pitcher plants (Sarracenia pur-purea; Miller and Kneitel 2005). Both predators and preydisperse among pitcher plants aerially, and larger plantsare likely to receive higher inputs of both trophic levels(Heard 1994; Caceres and Soluk 2002; Vanschoenwinkelet al. 2008), so these metacommunities may also be struc-tured by coupled dispersal of predators and their prey. In


general, this sort of relationship should arise whenevercertain patches are equally attractive to dispersing prop-agules of both species, either because of similar habitatpreferences, in the case of active dispersers (e.g., van derMeijden and van der Veen-van Wijk 1997), or similar sus-ceptibility to abiotic forcing, in the case of water- or wind-dispersers (e.g., Cowen and Castro 1994; Bell et al. 2005).It should be noted that many existing single-species meta-population models with passive dispersal assume thatlarger patches receive more colonists and have lower ex-tinction rates (e.g., Hanski and Ovaskainen 2003), but theextinction rates of those patches might be higher thanexpected if predators also colonize those patches at a highfrequency. Alternatively, coupled dispersal is not likely tooccur if predators and prey differ greatly in larval behavioror the seasonality of dispersal (in locations where currentsvary seasonally) or if adult movement occurs on a scalelarge enough to disrupt patterns established at settlement(Kritzer and Sale 2006).

The source-sink concept is widely applied in metapop-ulation ecology, and the criteria used to differentiate sourcesand sinks can vary widely (Runge et al. 2006). For terrestrialstudies, investigators generally calculate the difference be-tween local birth and death rates (Pulliam 1988); in a marinecontext, this requires that a sufficient number of locallyproduced larvae return to replenish the natal population(Armsworth 2002; Hastings and Botsford 2006). Evidenceis beginning to accumulate that shows populations oftenretain at least some fraction of locally produced larvae(Swearer et al. 2002), so that many populations may be self-replenishing as long as fecundity exceeds larval mortalityby a sufficient margin (Byers and Pringle 2006). Indeed,most of the subpopulations simulated in this study wereself-replenishing sources even when the values of Di and Ei

were quite low. The focus of this article, however, was onthe contribution of a given subpopulation to an equilibrialmetapopulation rather than a near-extinction metapopu-lation. This can be a crucial distinction; a self-replenishingsource that contributes few recruits to the rest of the meta-population would persist in isolation but cannot be reliedon to replenish sinks where self-replenishment is low ormortality is high. Likewise, subpopulations for which localoceanography precludes larval retention and self-replenish-ment would be sinks under Armsworth’s (2002) criterionbut might be nonetheless essential to the persistence andequilibrial size of the larger metapopulation. Such subpop-ulations could occupy favorable habitat that permits themto export large numbers of larvae to other demes (garneringa high export ratio Ei; Figueira and Crowder 2006) or actas stepping-stones, linking distant patches through subse-quent generations of dispersal (Hastings and Botsford 2006)and yielding a high value of deletion index Di. Such con-siderations underscore the value of the export- and deletion-

based statistics as measures of patch contribution. The ex-amples I chose also highlight the complementarity of thesestatistics. A net larval exporter (high Ei) might not be asessential to metapopulation connectivity as a stepping-stone(high Di), and vice versa, but taken together, the two sta-tistics readily identify such distinct roles. One caveat is thatdemes with extremely low larval supply could produce abiologically insignificant number of larvae yet have a largevalue of Ei; in such cases, the near-zero value of Di will bemore informative. Nonetheless, in a conservation context,both metrics may provide more information than traditionalsource-sink definitions regarding the consequences of de-stroying or overharvesting a particular subpopulation andso may be more useful in matching reserve functions todesired goals (Halpern and Warner 2002).

Accounting for correlated settlement effects should im-prove our ability to predict marine metapopulation dy-namics in general but could also be quite important toconservation planning and reserve design. In the case ofsingle-species management, the coupled settlement effectmay temper the value ascribed to certain high-settlementsites by models assuming spatially constant predation (re-viewed in Gerber et al. 2003). In reality, sites with lowersettlement may be more useful as reserves because of theirelevated reproductive potential (Caselle et al. 2003). Thismay be true even when the species of interest is a toppredator, since most marine organisms are vulnerable topredation when they are young and small. However, forsome marine predators, population dynamics are stronglyinfluenced by cannibalism of recruits by older conspecifics(Claessen et al. 2004). The predictions of the models pre-sented here are not likely to hold in that situation, andthe spatially explicit metapopulation dynamics of a can-nibalistic species would be a productive avenue for futureresearch (Claessen et al. 2004; Olson et al. 2005). An at-tractive alternative to single-species conservation strategiesis ecosystem-based management (EBM). Species interac-tions and spatial variability are nominally included in theEBM rubric but are rarely considered in practice (Arkemaet al. 2006). This study provides additional evidence forthe need to move beyond the single-species approach(Botsford et al. 1997). Indeed, other authors have recog-nized that because top predators are protected from har-vest, reserves may have higher predation rates than sur-rounding habitat (Micheli et al. 2004; Baskett et al. 2006;Langlois et al. 2006). This extends the lesson noted byCrowder and colleagues (Crowder et al. 2000; Crowderand Figueira 2006; Figueira and Crowder 2006) that spatialvariability in habitat quality and resource availability mustbe taken into account when evaluating subpopulation im-portance. The future of informed marine population ecol-ogy and conservation may lie in embracing the metacom-munity concept (Guichard et al. 2005) and recognizing


Figure A1: Effect of larval supply correlation, p, on relative patch im-portance in multiple-deme metapopulations. Each point is equilibriumvalue of (A) log population density n∗, (B) export ratio Ei, or (C) deletionindex Di for a subpopulation with the indicated value of fi. Verticaldashed lines indicate the mean value of f. The horizontal dashed lineindicates Ei p 0, the boundary between net larval export and net larvalimport.

the importance of oceanographically driven spatial vari-ation in species interactions to the role individual patchesplay in the larger network.

Acknowledgments

This work benefited greatly from discussions with L. Bots-ford, R. Nisbet, M. Steele, S. Teo, and especially my dis-sertation committee, S. Gaines, S. Holbrook, and R. Warner.R. Warner, C. White, and two anonymous reviewers pro-vided helpful comments on the manuscript, and discussionswith J. Samhouri about the proper way to measure source-sink status were particularly valuable. Financial support wasprovided by a National Science Foundation PredoctoralFellowship, a University of California Regents Fellowship,and a University of California, Santa Barbara, AffiliatesFellowship. This is contribution 279 from the Partnershipfor Interdisciplinary Studies of Coastal Oceans, fundedprimarily by the Gordon and Betty Moore Foundation andthe David and Lucile Packard Foundation.

APPENDIX

Simulation of Multiple-DemeMetapopulation Dynamics

The main text describes simulations of a simple, two-dememetapopulation, but those results can be extended to morecomplex metapopulations as well. To demonstrate thispoint, I present results for a 10-deme metapopulation.

To model the scenario, I created con-m p 10 m # mnectivity matrices F with elements fji, as described in themain text. Each matrix had identical rows (in keeping withthe larval pool assumption), so generating F simply in-volved randomly generating a single row vector of length10, . The elements of f sum to unityF p (f , … , f )1 10

and define the proportional larval supply to each sub-population; for brevity, I drop the subscript j and referonly to the value of fi for each subpopulation. Note thatimport and export connectivities to and from each patchare still explicitly defined in this case: a particular sub-population q will have different rates of export to eachother subpopulation ( ; ) but will havef p “f ” f p “f ”q1 1 q2 2

the same rate of import from every other subpopulation( ).f p f p “f ”1q 2q q

Metapopulations with a range of larval supply config-urations were simulated by drawing 10 values of fi uni-formly on the interval (0, 1), with the constraint that theysum to unity. I generated 500 vectors f; for each one,metapopulation dynamics were simulated using 100 pa-rameter sets generated in the same way as in the two-demecase and with the strength of settlement coupling ,p p 00.5, or 1. For each f and each value of p, the mean values


of Ei and Di (taken across all 100 parameter sets) werecalculated for each subpopulation. To illustrate the be-havior of Ei and Di across the range of potential fi values,for each of the 500 connectivity vectors (f), I present themean values of both metrics for a single, randomly selectedfocal subpopulation ni. The resulting relationship betweenfi and Ei (or Di) taken across all simulations is identicalto that present within each individual 10-dememetapopulation.

The effects of larval supply f and settlement couplingp on mean population density , export ratio Ei, and the∗n1

deletion index Di in the multiple-deme metapopulationwere similar to those observed in the two-deme meta-population. When larval supply was uncoupled ( ),p p 0Ei was highest at low values of fi and declined steadily(fig. A1B), and Di increased monotonically with fi (fig.A1C). For , however, both Ei and Di declined rapidlyp 1 0with fi, and Di was highest for subpopulations with verylow values of fi (fig. A1C).

As in the two-deme case, in the multiple-deme meta-∗n1

population was equal for all values of p at the mean valueof fi, which is 1/m, or 0.1 for a 10-deme metapopulation(fig. A1A). However, the values of Ei and Di were not equalfor all p at this point (fig. A1B, A1C). Unlike the two-deme scenario, all subpopulations in the multiple-dememetapopulation did not have equal larval supply when thefocal subpopulation had , and the presence off p 1/mi

additional subpopulations with depressed the val-f ! 1/mi

ues of Ei and Di for that focal subpopulation. This effectwas barely noticeable for but quite pronouncedp p 0.5for .p p 1

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