Identifying critical regions in small-worldmarine metapopulationsJames R. Watsona,1,2, David A. Siegela,b, Bruce E. Kendallc, Satoshi Mitaraid, Andrew Rassweillere, and Steven D. Gainesc

aEarth Research Institute, bDepartment of Geography, cBren School of Environmental Science and Management, and eMarine Science Institute, Universityof California, Santa Barbara, CA 93106; and dMarine Biophysics Unit, Okinawa Institute of Science and Technology, 1919-1 Tancha Onna-son, Okinawa904-0412, Japan

Edited by James Hemphill Brown, University of New Mexico, Albuquerque, NM, and approved September 13, 2011 (received for review July 14, 2011)

The precarious state of many nearshore marine ecosystemshas prompted the use of marine protected areas as a tool formanagement and conservation. However, there remains substan-tial debate over their design and, in particular, how to best accountfor the spatial dynamics of nearshore marine species. Many com-mercially important nearshore marine species are sedentary asadults, with limited home ranges. It is as larvae that they dispersegreater distances, traveling with ocean currents sometimes hun-dreds of kilometers. As a result, these species exist in spatiallycomplex systems of connected subpopulations. Here, we explicitlyaccount for the mutual dependence of subpopulations and ap-proach protected area design in terms of network robustness. Ourgoal is to characterize the topology of nearshore metapopulationnetworks and their response to perturbation, and to identify criti-cal subpopulations whose protection would reduce the risk forstock collapse. We define metapopulation networks using realisticestimates of larval dispersal generated from ocean circulationsimulations and spatially explicit metapopulation models, andwe then explore their robustness using node-removal simulationexperiments. Nearshore metapopulations show small-world net-work properties, and we identify a set of highly connected hubsubpopulations whose removal maximally disrupts the metapopu-lation network. Protecting these subpopulations reduces the riskfor systemic failure and stock collapse. Our focus on catastropheavoidance provides a unique perspective for spatial marine plan-ning and the design of marine protected areas.

connectivity | larval dispersal | protected area design | network theory |Lagrangian simulations

Nearshore marine ecosystems are some of the most pro-ductive and diverse environments on earth, maintaining

a wide variety of organisms and providing essential food andservices to the global population. Unfortunately, they are alsounder increasing stress from perturbations, such as oil spills,climate change, and overfishing, and are at risk for losing theirproductive output (1, 2). Mitigating the impacts of these per-turbations is difficult because most nearshore marine speciesexist in a spatially complex system of connected subpopulations;stress placed at one location may detrimentally reduce stocklevels at many others (3, 4). Quantifying patterns of connectivityis crucial to our ability to manage these systems effectively. Forexample, marine protected areas (MPAs) have emerged as animportant tool for conservation and fisheries managers taskedwith maintaining nearshore systems and accounting for the mu-tual demographic dependence of populations (3–9). Currentapproaches to their design center on either protecting essentialhabitats (5, 6, 9) or maximizing economic goals (10, 11). Weprovide an alternative and consider the design of MPAs as aproblem of metapopulation network robustness, here defined asthe ability of a system to withstand perturbation (12). Our goal isto account for patterns of connectivity and identify key regionswhose protection from perturbation safeguards against systemicfailure and stock collapse (12–14).

Many nearshore marine species are sedentary as adults, withhome ranges on the order of 0.1–10 km (15). Longer distancesare traveled as newly spawned larvae (15). During this stage,larvae advect primarily by means of ocean currents, potentiallytraveling distances orders of magnitude greater than thosetraveled as adults (i.e., ∼100 km) (15–18). As a direct conse-quence of this larval dispersal stage, nearshore marine speciesexist in systems of interconnected subpopulations (3, 4). Typi-cally, larval connectivity has been estimated using simple statis-tics of ocean circulation and Euclidean distance (e.g., 3, 4, 19).This approach has been used extensively to gauge criteria formetapopulation persistence and in the design of protected areas(3, 4), but it ignores several important details (17, 18). For in-stance, theoretical studies have shown that heterogeneity,asymmetry, and temporal variability in larval connectivity pat-terns are key properties that affect the persistence of meta-populations (20–23). To a large degree, these properties are notcaptured by standard empirical methods and remain elusive inrealistic settings.To capture the complexity of larval connectivity patterns,

we combine ocean circulation simulations with realistic meta-population modeling. Over the past decade, simulation ap-proaches to quantifying larval connectivity have grown in prom-inence and have been used to investigate the scales of larvaldispersal (16–18, 24), the population genetic structure of near-shore species (25, 26), and processes affecting the compositionof nearshore communities (27), to name a few examples. Here,our goal is to use ocean circulation simulations to explore hownearshore metapopulations respond to perturbation and toidentify specific nearshore regions that are key to metapopulationrobustness. We focus our attention on the dynamics of severalcommonly fished species in the Southern California Bight (Fig.1A): kelp bass (Paralabrax clathratus), kelp rockfish (Sebastesatrovirens), ocean whitefish (Caulolatilus princeps), red sea ur-chin (Strongylocentrotus franciscanus), opaleye (Girella nigricans),California halibut (Paralichthys californicus), and Californiasheephead (Semicossyphus pulcher), whose life history charac-teristics span a broad range. These species were used as part ofa recent MPA planning program in the Southern California Re-gion (9), and our methods use the numerical models developedfor this process.Patterns of larval connectivity for each study species are

quantified in a two-step process. First, Lagrangian probability

Author contributions: J.R.W., D.A.S., B.E.K., S.M., A.R., and S.D.G. designed research;J.R.W., S.M., and A.R. performed research; J.R.W. analyzed data; and J.R.W., D.A.S., B.E.K.,S.M., A.R., and S.D.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1Present address: Atmospheric and Oceanic Sciences Program, Princeton University,Princeton, NJ 08544.

2To whom correspondence should be addressed. E-mail: jrwatson@princeton.edu.

See Author Summary on page 17583.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1111461108/-/DCSupplemental.

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density functions (PDFs), which summarize water parcel trajec-tories derived from high-resolution ocean circulation simulations(28), were used to quantify the probability of a larva, of a givenspecies, traveling from one subpopulation to another. Thisquantity is termed “potential connectivity” (17, 18). An exampleof a Lagrangian PDF for a particular production site on SanNicolas Island in the center of the domain, consistent with thespawning season of kelp bass, is shown in Fig. 1B (larval lifehistory parameters are provided in Table S1). Determinations ofpotential connectivity were then used in realistic metapopulationmodels to estimate the number of larvae spawned at a givensource that subsequently recruit to a certain destination sub-population. Because this quantity accounts for such demographicprocesses as larval production and mortality, it is termed “re-alized connectivity” (18). Network metrics were then used toanalyze these estimates of realized connectivity, guiding node-removal simulation experiments (29, 30). These simulate a seriesof catastrophic events at particular nearshore locations and as-sess metapopulation robustness, answering the question of whichnodes (subpopulations) should be protected in order to fortifythe entire network (metapopulation) against systemic failure(stock collapse) (12, 29–32).

ResultsThe potential connectivity for all species in the SouthernCalifornia Bight (Fig. 2A for kelp bass and Fig. S1 for all otherspecies) is heterogeneous and asymmetric, with differences indispersal parameters (spawning season, spawning period, andpelagic larval duration) creating differences among species. Thisinterspecific variation is also evident in the realized connectivity(Fig. 2B for kelp bass and Fig. S2 for all other species), where themetapopulation models and parameters (fecundity, naturalmortality, and density-dependent larval mortality) transform thepotential connectivity. Among species, potential and realizedconnectivity show stark differences, emphasizing the importanceof these demographic processes in spatially structuring meta-population networks. For example, for kelp bass, there are strongpotential connections from southern mainland subpopulations tothose on the northern islands (Fig. 2A, source indexes 1–63,destination indexes 63–96). However, because these sub-populations are characterized by areas of low suitable habitat,and hence low larval production, realized connectivity is rela-tively weak (Fig. 2B). Equilibrium spatial distributions of bio-mass also reflect these differences in dispersal and demographicparameters (contrast Fig. 2C for kelp bass and Fig. S3 for allother species).Plotted on a log-log axis, the frequency distributions for realized

connectivity across all nonzero i-j pairs are reasonably linear, al-

beit with slight curvature (Fig. 3A for kelp bass and Fig. S4 for allother species), indicating that metapopulations are characterizedby many weak and few strong larval connections (33). In this way,nearshore metapopulations resemble small-world networks,which typically have long-tailed degree distributions (31, 33, 34).Further, across all species investigated, only a small fraction of thecoast (10–23%) accounted for 50% of the total number of recruitsin the metapopulation (Fig. 3B, this metric is henceforth termedF0.5). Together, these two properties of realized connectivityindicate that “hub” subpopulations exist. These are the mostconnected subpopulations in the metapopulation. Interestingly,we found that F0.5 was smaller for species with summer spawningmonths (Fig. 3B, colors). This variation reflects the seasonalityfound in patterns of ocean circulation in the Southern CaliforniaBight (17, 18, 28).For all species, sequentially removing subpopulations follow-

ing the network metric, eigenvector centrality, produced themost rapid declines in metapopulation biomass (35–37) (Fig. 4Aand Fig. S5, red line), collapsing metapopulations after 40–65%of the coast was removed. In contrast, the expectation of re-moving subpopulations randomly was shallower, even though theset of random removal curves theoretically includes the eigen-vector centrality curve itself (Fig. 4A and Fig. S5, light blue linesare 100 random realizations, and the dark blue line is the meanover 2,000 realizations). The difference between the central andmean random removal curves reveals that these nearshoremetapopulations are, in general, robust to random perturbationsyet vulnerable to directed disturbances.A greedy search algorithm was also used to identify critical

subpopulations and for comparison with the eigenvector cen-trality results. At each removal stage, the locally optimal solu-tion, the subpopulation that would create the largest decline inmetapopulation biomass when removed, was found numerically.The greedy algorithm removal curves were matched by eigen-vector centrality in the rate at which the metapopulation biomassdeclined to collapse (Fig. 4A and Fig. S5, compare the blackcircles and red lines). However, the exact order in which sub-populations were removed was not the same, revealing that thereis redundancy in the importance of subpopulations; it is notnecessarily the order in which important subpopulations are re-moved, merely that they are (note the initial mismatch in the redline and black circles for kelp rockfish; Fig. S5F). Removingsubpopulations following the area of suitable habitat createdremoval curves that were slightly shallower than both the greedyand eigenvector centrality curves (green lines) but steeper thanthe mean random removal curve.We defined a small-world metric as the difference between the

eigenvector centrality and the mean of the random removal

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Fig. 1. (A) Southern California Bight with bathymetry and 135 subpopulations (red circles). (B) Example Lagrangian PDF for kelp bass. Lagrangian particlesare released from the pink subpopulation during the kelp bass spawning period and are transported with ocean velocities, produced from a regional oceanmodel, for a given pelagic duration (Table S1). Given these time constraints, Lagrangian PDFs quantify the probability of a Lagrangian particle reaching otherlocations (color scaling; ref. 17).

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curves in the fraction of removed coastline required to reducethe metapopulation biomass by 50%. This was termed C0.5.When C0.5 is high, there is a large difference between the ran-dom and central removal curves, indicating that the studiedmetapopulation resembles a small-world network whose ro-bustness is reliant on a few key hub subpopulations. This metricshows a significant negative linear relationship with F0.5, thefraction of the metapopulation responsible for 50% of themetapopulation recruits (reference to this second metric in Fig.3B and relationship in Fig. 4B). Differences in C0.5 and F0.5

highlight the interspecific variation in metapopulation robustnessand in network topologies. For example, the urchin meta-population maintained 50% of its biomass after ∼18% of itssubpopulations had been removed and is relatively robust com-pared with the opaleye metapopulation, which maintained 50%of its biomass after only ∼10% of its subpopulations had been

removed (Fig. 4B). Like F0.5, seasonality was found in C0.5, withwinter-spawning species more robust than summer-spawningspecies (Fig. 4B, colors).The spatial distribution of the most critical subpopulations

covering ∼13% of the coastline is shown in Fig. 5 for kelp bass,kelp rockfish, and halibut and in Fig. S6 for the remainder of thespecies examined. This fraction of the coastline is consistent withthat of the State of California’s MPA planning for the SouthernCalifornia Bight (9). Colors denote the fractional drop in met-apopulation biomass when that subpopulation was removed.There are large differences in the spatial locations of key sub-populations among the species. Most notably, halibut’s keysubpopulations reside along the mainland, whereas the majorityof every other species’ key subpopulations reside along thesouthern islands. This disparity can be explained by the differ-ences in suitable habitat; halibut is the only species examinedthat is associated with soft substrate, which is distributed mostlyalong the mainland (Fig. S7A). Interestingly, there are key kelprockfish subpopulations in the south of the domain, even thoughadult abundances are highest in the northern islands (compareFig. 5B and Fig. S3F). These northern subpopulations have lowpotential connectivity; hence, in the node removal experiments,they were not found to contribute to the overall robustness of thekelp rockfish metapopulation.

DiscussionWe have found that patterns of larval connectivity in nearshoremarine metapopulations have small-world network properties(14, 33, 34). The most important characteristic is the existenceof hub subpopulations (21, 29, 35, 36). These are the most

opaleye(10%)sheephead(10%)kelpbass(13%)rockfish(16%)urchin(19%)whitefish(19%)halibut(23%)

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Fig. 3. (A) Log-log frequency plot for the kelp bass realized connectivity.Plots for the six other species can be found in Fig. S4. (B) Cumulative numberof larval recruits as a function of the fraction of the metapopulation/coast-line. Percentage values are the fraction of the metapopulation that accountsfor 50% of the total number of recruits, termed F0.5 (black line and crosses).Values of F0.5 indicate that a small fraction of the metapopulation accountsfor half of the larval recruits for all species. Colors denote spawning season:those species that spawn during the summer are shown in orange, and thosethat have winter spawning months are shown in green.

A

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Fig. 2. (A and B) Kelp bass potential and realized connectivity matrices,respectively. Potential connectivity is the probability of larval transport be-tween a source (y axis) and destination (x axis) subpopulation, and realizedconnectivity values are the number of dispersing larvae normalized by thetotal number of larvae dispersing in the system. The numbering of axesrelates to the subpopulation labels in (Fig. 1A). (C) Spatial distribution ofequilibrium biomass for kelp bass. Subpopulation biomasses are fractions ofthe total metapopulation biomass. Sites with zero biomass are those withlittle or no suitable habitat. The potential connectivity, realized connectivity,and distributions of biomass for the six other species can be found in Figs.S1–S3, respectively. Isl, island; N., north; S., south.

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connected subpopulations, and they produce the majority oflarval recruits in the metapopulation. We used realistic meta-population models to estimate patterns of realized connectivity

and eigenvector centrality to identify the locations of hub sub-populations for a range of species in the Southern CaliforniaBight. Across species, we found that the hub subpopulations arespatially aggregated (Fig. 5 and Fig. S6), suggesting that a local-ized catastrophe, such as an oil spill, has the potential to affectmultiple sites and species. If these sites are already challenged byongoing perturbations (e.g, from fishing), this additional eventcould effectively “remove” these subpopulations and increasethe risk for domain-wide metapopulation collapse. With regardto spatial management, MPAs designed to cover these key sub-populations would be an important step in avoiding this worst-case scenario. This perspective reflects the fortification of keynodes in other network systems, such as keystone species in foodwebs (37) or server machines in computer networks (32).The metapopulation models can be thought to balance the

demographic influence of the amount of habitat and potentialconnectivity. The amount of habitat determines a given sub-population’s carrying capacity, but it alone does not guaranteethat a subpopulation will be important to the robustness of themetapopulation. For that to happen, the subpopulation mustalso have high potential connectivity, with the larvae that itproduces dispersing to other subpopulations. Thus, although thehabitat removal curves tracked those of eigenvector centrality(Fig. 4A and Fig. S5), they were never precisely the same andalways shallower. Indeed in a direct comparison of the removalcurves, the areas of suitable habitat and eigenvector centrality donot show strong correspondence (Pearson’s r, range: 0.44–0.81;Table S2). On its own, the area of suitable habitat cannot ac-count for patterns of larval connectivity; hence, it identifies notonly subpopulations that are highly central but also those thatare sinks of larval connectivity. These sinks do not contribute tothe overall robustness of the metapopulations (22).A long-held tenet of spatial management is that the number

of recruits produced from a given subpopulation or “sourcestrength” identifies areas that are important to metapopulationpersistence (3, 4, 6, 8, 11, 22). In the Southern California Bight,source strength removal curves show extremely high correspon-dence with those of eigenvector centrality (Pearson’s r, range:0.92–0.99 across species; Table S2). The strong correlations re-veal that the key hub subpopulations are exactly the majorsources of recruits. In the Southern California Bight, the spatial

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Fig. 5. Spatial distribution of key subpopulations, covering ∼13% of thecoast, for kelp bass (A), kelp rockfish (B), and halibut (C). Colors are thefractional drop in metapopulation biomass when that subpopulation wasremoved. Maps for the other species are provided in Fig. S6.

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F 0.5increasing robustness

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RKelp bass removal

Fig. 4. (A) Node removal curves for kelp bass. Metapopulation biomass, expressed as a fraction of the virgin biomass (y axis), declines as subpopulations areremoved (x axis) following eigenvector centrality (red), the area of suitable habitat (green), randomly (100 realizations are shown in light blue, and the meanover 2,000 realizations is shown in dark blue), and using a greedy algorithm (black circles; lines are not drawn for clarity). Removal curves for the six otherspecies can be found in Fig. S5. For all species, removing subpopulations following eigenvector centrality matched most closely the greedy algorithm, col-lapsing metapopulations with the fewest removed subpopulations. We define a small-world metric C0.5 as the difference between the central and randomremoval curves in the fraction of removed coastline required to reduce the metapopulation biomass by 50% (black line). (B) Plotting C0.5 against F0.5 (Fig. 3B)reveals a negative linear relationship. Both metrics quantify the interspecific variability in metapopulation robustness and network topologies. Colors denotespawning season: those species that spawn during the summer are shown in orange, and those that have winter spawning months are shown in green. It isevident that winter spawners are more robust to disturbance.

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scales of the domain and of potential connectivity overlap. Theresult is that a larva spawned anywhere has the potential todisperse to all other subpopulations in one step. The overlap inspatial scales results in the first-order connections being the mostimportant; hence, the strong correspondence between eigen-vector centrality and source strength. However, being a goodsource is not necessarily the same as being a central hub. Sourcestrength only captures the influence of first order larval con-nections, whereas eigenvector centrality is designed to take intoaccount higher order “stepping-stone” connections. If a largerdomain were considered, for example, the whole western coast ofthe United States, larvae would not be able to reach every sub-population in one step. Potential connectivity would be moresparse, and the importance of higher order connections wouldincrease, resulting in greater differences between source strengthand eigenvector centrality.The area of suitable habitat deserves further consideration

because it controls the intensity of density-dependent larvalmortality and the carrying capacity of a given subpopulation (38,39) (Eq. 3 and Methods). Theoretically, at some extremely highlevel of density dependence, spatial heterogeneity in larvalconnectivity, which creates differences in the importance ofsubpopulations, will diminish (40). We tested the effect of dif-ferent levels of larval density dependence by altering de-mographic parameters (Methods) and found that the robustnessof metapopulations to subpopulation removal was indeed af-fected by these changes; removal curves were steeper at lowercompensation ratios (CRs) and less intense larval density de-pendence (Fig. S7B). However, the location of the hub sub-populations did not change greatly, suggesting that the locationof key subpopulations is robust to this demographic uncertainty(Fig. S7 C and D).A network perspective is extremely useful for understanding

metapopulation structure and dynamics (14, 21, 29, 33, 35, 41–44). However, there are several caveats to its application. Al-though both the potential and realized larval connectivity showlarge degrees of structure, with asymmetry and heterogeneity inconnections, it is the edge weights that hold all the information.Both the potential and realized connectivity matrices have nozero elements, with all nodes in the metapopulation networkconnected to each other. Thus, it is not the lack of connectionsthat defines nearshore metapopulation networks but the strengthof the connections. This is in contrast to most studied networks(e.g., 31, 32, 37, 41, 44), which do not show such extreme con-nectance. Thus, caution should be applied when using networkmetrics that were originally designed for binary (i.e., networkswhere pairs of nodes are simply connected or not) and moresparse networks. For example, betweenness centrality has beenused previously to identify critical subpopulations in meta-population networks (21). However, in the node removalexperiments, betweenness centrality did not match the greedyalgorithm as closely as eigenvector centrality and producedshallower removal curves (Pearson’s r, range: 0.65–0.95; TableS2). Thus, it did not identify the most critical nodes in thenearshore metapopulation networks.We must note that eigenvector centrality is a metric designed

for linear systems (36, 37), whereas models of metapopulationdynamics, like ours, are typically nonlinear (45). As a result, ei-genvector centrality is only a heuristic metric for identifyingcritical subpopulations. It does not identify the exact solution tothe problem of finding the smallest set of subpopulations whoseremoval maximally reduces metapopulation biomass. However,the close match-up with the greedy algorithm indicates that ei-genvector centrality performs well, allowing us to conclude that itis indeed hub subpopulations that are critical to metapopulationrobustness. Another caveat is that our analysis focuses on thedynamics of one species at a time. An important developmentwill be to understand how perturbations, occurring as a result of

overfishing or climate change, affect the trophic linkages be-tween species as well as their spatial dynamics [i.e., meta-community dynamics (46)]. Needless to say, our analysis alsodepends on the accuracy of the Lagrangian particle simulationsin capturing the spatial patterns in larval dispersal. Going for-ward, it will be important to gauge the impact of known com-putational issues, such as domain edge effects (17, 18), andbiological processes, such as larval behavior (47), mortality (48),and environmental stochasticity (23, 49).In summary, coupling high-resolution oceanographic simula-

tion products with realistic metapopulation modeling is a pow-erful tool for understanding the spatial dynamics of nearshoremarine species. We have found that nearshore marine meta-populations exhibit small-world characteristics, being robust torandom perturbations yet highly vulnerable to directed dis-turbances. Numerous complex systems exhibit this “robust yetfragile” property (12, 50), and we have shown that by knowingmetapopulation network quantities, such as realized connectivityand eigenvector centrality, critical subpopulations can be iden-tified. These subpopulations are hubs of larval connectivity andare key to the robustness of metapopulation networks. Hence,they are ideal candidates for MPAs placed with the aim of re-ducing the risk of stock collapse.

MethodsPotential Connectivity. Potential connectivity (Dij) is defined as the probabilityof a larva, spawned at a given source subpopulation (i ), traveling to adestination subpopulation (j ). Estimates of potential connectivity wereproduced using Lagrangian particle simulations (17). Over the model in-tegration period, 1996–2002, ∼400,000 Lagrangian particles were releasedfrom a number of discrete patches (5-km radius; Fig. 1A, red circles) duringeach annual spawning season of a given species (a total of over 50 millionparticles). The size of these patches reflects limitations in both the numericalsimulations of larval dispersal and the node removal experiments. Particlesthen advect passively for a specified pelagic larval duration (Table S1), beingmoved by ocean velocities produced from a high-resolution (1 km horizon-tally, 40 layers vertically) regional ocean modeling system solution for theSouthern California Bight (28).

Lagrangian PDFs were then used to calculate the site-to-site transitionprobabilities that define potential connectivity (17). Potential connectivitieswere calculated for the spawning periods of each case-study species and foreach of the 7 y of modeled Lagrangian particle data. These data were thenaveraged across years. Potential connectivity values between all pairs ofnearshore patches were then stored in a matrix (Fig. 2A and Fig. S1). Thismethod is described in detail by Mitarai et al. (17), and the reader is referredto that study for further information on Lagrangian PDFs.

Metapopulation Model and Larval Connectivity. Dynamic and spatially realisticmodels for several nearshore species were developed frommodels created aspart of the State of California’s Marine Life Protection Act (MLPA) (9). For allspecies, the model domain was the Southern California Bight and the 5-kmpatches used to calculate potential connectivity were utilized to define thespatial extent of the subpopulations (Fig. 1A, red circles). The state variableis biomass (kg), and the models are deterministic with equilibrium valuesinsensitive to initial conditions. The model is general to all species, and itis the parameterization that leads to species-specific dynamics. Speciesparameters were estimated as part of the MLPA process (9) and can befound in Table S1. Population dynamics are iterated forward in discreteyearly intervals until equilibrium biomass values are reached (examples ofspatial distributions of biomass are shown in Fig. 2C and Fig. S3):

Atþ1j;nþ1 ¼

(At

j;nð1−mÞRtj

if n≥1if n ¼ 0

; [1]

where At+1j,n+1 is the number of adults of age n+1, at subpopulation j, at

time t +1 (next year). Atj,n is the number of adults of age n, at subpopulation

j, at time t. m is the natural mortality and, with the exception of kelprockfish, is constant across patches and through time. The kelp rockfish isa northern species, and we modeled its natural mortality as a negative linearfunction of the long-term average sea-surface temperature, where thewarmest subpopulation has m = 0.4 and the coolest has m = 0.2 (9, 18, 51).The long-term mean sea surface temperature for each subpopulation was

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extracted from a 4-km resolution Pathfinder 5 Advanced Very-High Reso-lution Radiometer monthly sea surface temperature dataset (NASA), overthe period 1997–2008 (52). Temperatures were then linearly interpolated tothe centroid of each kelp rockfish subpopulation. This gradient in kelprockfish natural mortality was chosen to align the spatial distribution ofmodeled adult biomass (Fig. S3F) with its known biogeography (9, 18, 51).

Rjt is the number of recruits (zero age) at subpopulation j at time t, and it

is defined as

Rtj ¼ σtj ∑

i∑nWt

i;n fn Dij ; [2]

where Wti,n is the biomass of adults of age n (kg), at subpopulation i, at time

t. fn is the fecundity (number of larvae produced per kilogram of adultbiomass) of age group n, and Dij is the potential connectivity describedpreviously (i.e., the probability of a larva traveling from a source sub-population i to a destination subpopulation j). Fecundity is zero if n < age atmaturity (Table S1). The number of larvae arriving at a given destinationpatch (j) is a result of larvae produced by all age groups (summation over n)and from all source patches (summation over i). σtj is the postsettlement,density-dependent, recruitment probability of a larva arriving at patch j,modeled using a Beverton–Holt function:

σtj ¼1

1þ βltjHj

; [3]

where Hj is the area of suitable habitat in the spatial extent of a givensubpopulation, β is a constant, and lj

t is the number of larvae arriving atpatch j at time t. lj

t is simply the double-summation term in Eq. 2. σtjdescribes larvae-on-larvae competition and is used widely in fisheries science(e.g., 9, 38, 45); adult-on-larvae competition was assumed to be negligible.For halibut, suitable habitat (Hj) was defined as nonrocky substrate, whereassuitable habitat was defined as rocky habitat for all other species (9, 18, 51)(Fig. S7A). β is a nondimensional parameter that, along with fecundity,controls the strength of density dependence. It also accounts for larvalmortality as larvae disperse, which is assumed to be constant. We measuredβ using the settler-recruit curve, where the slope at the origin can be de-scribed as a nondimensional CR (9, 38, 39). This is the ratio of the per capitasettler survival rate at very low densities to the per capita settler survivalrates at the highest possible density of settlers. We chose β and fecundityvalues to create a range of CRs (3, 4, and 5) for all species. The figurespresented in the main text and SI Text are from simulations using a CR of 4;this is a commonly chosen strength of density dependence, reflecting therecent MLPA process (9). The affect of varying CR is presented in theDiscussion and in Fig. S7.

We further calculated realized connectivity (Rij), which is defined as thenumber of larvae that travel from a source subpopulation (i) and sub-sequently recruit to a destination location (j) (18). Hence, it takes into ac-count larval production and density-dependent larval mortality. Realizedconnectivity is found by knowing both the number of larvae dispersingbetween patches and the density-dependent larval mortality rate:

Rij ¼ σi ∑nWj;n fn Dij [4]

Like potential connectivities, realized connectivities between all pairs ofnearshore patches are stored as a matrix.

The lengths of adults of each age were found using the von Bertalanffyequation:

Ktn ¼ K∞

�1− e− kðn−n0Þ

�; [5]

where Ktn is the length of an individual at age n at time t for any given

subpopulation. K∞, k, and n0 are constants related to asymptotic length, size

at settlement, and growth rate, respectively (each specific value is presentedin Table S1). The mass of individuals of age n at each subpopulation wasthen found through the following allometric relationship:

Wti;n ¼ a

�Kti;n

�b; [6]

where Wti,n is the mass of an individual (kg) of age n, at time t, at patch i.

The terms a and b are constants (Table S1). The total biomass (Bti, kg) at

a given patch i at time twas then found by multiplying the number of fish ofeach age by their mass and summing over all ages (n):

Bti ¼ ∑

nAt

i;nWtn [7]

Node Removal Experiments. Node removal experiments were used to exploremetapopulation network robustness and to identify key subpopulations. Foreach modeled species, subpopulations were sequentially removed, allowinga new equilibrium to be obtained at each iteration. This was achieved bychanging the appropriate value of Hj , the area of suitable habitat at thetarget subpopulation (j), to zero, essentially creating 100% larval mortalityat a particular patch. Removal continues until the total metapopulationbiomass is less than an arbitrary “collapsed” state, which we define as 1% ofthe virgin, before node removal, metapopulation biomass.

Subpopulations were chosen for removal using metrics of subpopulationimportance. In age-structured models, the “reproductive value” quantifiesthe contribution of individual age classes to future generation (53). Thismetric has been generalized to metapopulations by Ovaskainen and Hanski(35) and Jacobi and Jonsson (36), and it identifies the contribution of anindividual subpopulation to the growth rate of the entire metapopulation.This metric is otherwise known as “eigenvector centrality” in network the-ory and has been used to identify keystone species in food webs (36) andimportant Web pages by the Google search algorithm (54). Here, eigen-vector centrality is calculated as the left eigenvector associated with theleading eigenvalue of the realized connectivity matrix (Rij) and identifiessubpopulations that are hubs of realized connectivity. We present eigen-vector centrality as a dimensionless metric by normalizing values to the in-terval [0, 1]. In this way, this metric is used simply to rank patches by theirimportance. More details on eigenvector centrality and its ecological appli-cations are provided elsewhere (35–37).

A greedy search algorithm was also used to identify critical subpopulationsand for comparison with the eigenvector centrality results. At each removalstage, the locallyoptimal solution (i.e., thesubpopulation thatwouldcreate thelargest decline in metapopulation biomass when removed) was found nu-merically. This is a heuristic approach and does not necessarilyfind the globallyoptimal solution (i.e., the smallest set of subpopulations that, if removed,collapses themetapopulation). This, however, is notour goal. By implementingthegreedyalgorithm,wesimplycreateanumerical solutionthatweknowtobenear optimal. This provides an objective measure with which to compare theeigenvector centrality removal experiments. Subpopulations were also re-moved in descending order of the area of suitable habitat, source strength[defined as the number of recruits produced from a given patch (18)], andbetweenness centrality (21, 55); as a null case, subpopulations were also re-moved at random, with 2,000 replicates to generate measures of variance.For source strength and betweenness centrality, the order in which sub-populations were removed was compared directly (using Pearson’s correlationcoefficient); results are presented in the Discussion and Table S2.

ACKNOWLEDGMENTS. We thank the Flow Fish and Fishing research group,R. Warner, C. Costello, R. Hilborn, C. White, J. McWilliams, C. Dong, E. Fields,S. Levin, J. Elliott, and the anonymous reviewers for their help in preparingthis material. Financial support was provided by the National ScienceFoundation and the National Aeronautics and Space Administration.

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