Metapopulation extinction risk: Dispersal’s duplicity

Theoretical Population Biology 76 (2009) 146–155

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Theoretical Population Biology

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Metapopulation extinction risk: Dispersal’s duplicityKevin HigginsDepartment of Biological Sciences, University of South Carolina, Columbia, SC 29208, United States

a r t i c l e i n f o

Article history:Received 22 October 2007Available online 6 June 2009

Keywords:DispersalHabitat fragmentationDensity- dependenceExtinction riskMetapopulation

a b s t r a c t

Metapopulation extinction risk is the probability that all local populations are simultaneously extinctduring a fixed time frame. Dispersal may reduce a metapopulation’s extinction risk by raising itsaverage per-capita growth rate. By contrast, dispersal may raise a metapopulation’s extinction risk byreducing its average population density. Which effect prevails is controlled by habitat fragmentation.Dispersal in mildly fragmented habitat reduces a metapopulation’s extinction risk by raising its averageper-capita growth rate without causing any appreciable drop in its average population density. Bycontrast, dispersal in severely fragmented habitat raises a metapopulation’s extinction risk because therise in its average per-capita growth rate is more than offset by the decline in its average populationdensity. The metapopulation model used here shows several other interesting phenomena. Dispersalin sufficiently fragmented habitat reduces a metapopulation’s extinction risk to that of a constantenvironment. Dispersal between habitat fragments reduces a metapopulation’s extinction risk insofaras local environments are asynchronous. Grouped dispersal raises the effective habitat fragmentationlevel. Dispersal search barriers raise metapopulation extinction risk. Nonuniform dispersal may reducethe effective fraction of suitable habitat fragments below the extinction threshold. Nonuniform dispersalmaymake demographic stochasticity amore potentmetapopulation extinction force than environmentalstochasticity.

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1. Introduction

Metapopulation (Hanski and Gilpin, 1997) extinction risk is theprobability that all local populations are simultaneously extinctduring a fixed time frame. Dispersal may affect a metapopulation’sextinction risk in two ways as the habitat becomes morefragmented. Dispersal may reduce a metapopulation’s extinctionrisk by raising its average per-capita growth rate. For dispersalto raise a metapopulation’s average per-capita growth rate therandom per-capita population growth rates in the patchesmust beasynchronous. I call this reduction in ametapopulation’s extinctionrisk the metapopulation rescue effect. By contrast, dispersal mayraise a metapopulation’s extinction risk by reducing its averagepopulation density. The reduction in a metapopulation’s averagepopulation density occurs when some patches spontaneouslyreceive too many propagules while other patches spontaneouslyreceive too few propagules. In a patch receiving too manypropagules the competition to obtain a portion of the limitingresource is analogous to a musical chairs game. I call this rise ina metapopulation’s extinction risk themusical chairs effect.As the habitat becomes more fragmented there is tension be-

tween the metapopulation rescue effect and the musical chairseffect for control of a metapopulation’s extinction risk. Dispersal

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inmildly fragmented habitat reduces themetapopulation’s extinc-tion risk because dispersal raises its average per-capita growth ratewithout causing an appreciable drop in its average population den-sity. By contrast, dispersal in severely fragmented habitat raises themetapopulation’s extinction risk because the rise in its per-capitagrowth rate is more than offset by the decline in its average popu-lation density.Dispersal’s ability to raise a metapopulation’s average per-

capita growth rate was investigated in earlier work on dispersalin asynchronous environments (Roff, 1974a,b; Strathmann, 1974;Palmer and Strathmann, 1981; Ives et al., 2004). In a given gener-ation, dispersal across the habitat fragments of a metapopulationcauses the metapopulation census to grow as if all of the individ-uals were in just a single environment, with just one per-capitagrowth rate. The ‘‘single environment’’ per-capita growth rate fora generation is found by spatially averaging the random per-capitapopulation growth rates from all of the habitat fragments. Im-portantly, the single environment per-capita growth rate displaysinherent random variation from one generation to the next. It isexactly this type of generation-to-generation random variation ina single population’s per-capita growth rate that was exploredby Lewontin and Cohen (1969) who showed it can be devas-tating to the census of a single-population over time. Here, thesame phenomenon appears in the dynamics of a metapopulationwhen the metapopulation is viewed as a whole. Dispersal over anincreasingly fragmented habitat raises a metapopulation’s averageper-capita growth rate by making the random single-environment

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K. Higgins / Theoretical Population Biology 76 (2009) 146–155 147

per-capita growth rate less variable over generations. Interest-ingly, a similar mechanism underlies modern portfolio theory,where the goal is allocate an investor’s balance across a collectionof assets so that the expected rate of return ismaximized for a givenlevel of risk (Tobin, 1958, Section 3.6).Dispersal’s ability to reduce a metapopulation’s average popu-

lation density was investigated in earlier work on patchy single-species systems with localized density-dependent survival (deJong, 1979; Ives and May, 1985; Chesson, 1996, 1998). In thesesystems random dispersal places too many propagules in somepatches and not enough propagules in other patches, causing areduction in average survival that reduces average populationdensity. By contrast, in systems with an Allee effect, the samenonuniform dispersal pattern causes a rise in average survivalthat raises average population density (Chesson, 1998). Further,in systems where multiple species compete for resources withinpatches, nonuniform dispersal may level their competitive abili-ties, promoting coexistence (Atkinson and Shorrocks, 1981; Mayand Hassell, 1981; Ives and May, 1985; Klopfer and Ives, 1997; Leiand Hanski, 1998). Importantly, in both single-species (no Allee ef-fect) and multispecies systems, competition in the overpopulatedpatches exacts a toll on average survival that is not fully repaid byreduced competition in the underpopulated patches. The net resultis a reduction in average population density.There are at least two mechanisms that cause dispersal to be

spatially nonuniform or lumpy. First, an organism’s life historymay cause it to place more propagules in some patches than inothers, either centered where the organism itself grows, or atother locations. The tendency to produce eggs/seeds in clutches,or for seeds/eggs to travel in clumps, or for dispersal to beshort range, are all factors that drive dispersal lumpiness. Second,habitat fragmentation (breaking the same total area into smallerfragments) may make dispersal lumpiness more pronounced. Asthe habitat is broken into smaller fragments it becomes less likelythat the number of propagules arriving in a fragment are a goodfit for the carrying capacity of that fragment. Some fragmentsreceive too many propagules while other fragments receive toofew propagules. Increasing the degree of habitat fragmentationmakes larger mismatches more common. Finally, the joint actionof these two mechanisms compounds the potential for dispersallumpiness.Themetapopulationmodel usedhere shows several other inter-

esting phenomena. First, dispersal in sufficiently fragmented habi-tat reduces a metapopulation’s extinction risk to that of a constantenvironment even though environmental stochasticity causes theper-capita growth rates in the habitat fragments to fluctuate in-tensely. Second, dispersal between habitat fragments reduces ametapopulation’s extinction risk insofar as local environments areasynchronous. Third, dispersing propagules in groups or bundlesraises the effective habitat fragmentation level. Fourth, barriersthat prevent disperser search behavior, or cause dispersers to bespatially aggregated, raise ametapopulation’s extinction risk. Fifth,sufficiently nonuniform dispersal reduces the effective fraction ofsuitable habitat fragments below the extinction threshold, whereextinction becomes certain. Sixth, sufficiently nonuniform disper-sal makes demographic stochasticity a more potent metapopula-tion extinction force than environmental stochasticity.The rest of the paper is organized so that those wishing to focus

on the biological mechanisms and their consequences can avoidsome of the mathematical details. Those readers may want to justglance at Sections 4 and 5.

2. Metapopulation model

The mathematical metapopulation model involves a number ofbiological assumptions that areworthy of discussion. Alternatively,

a succinct simulation version of the model is given in Table 1,where the key source-code parts are shown.The mathematical metapopulation model consists of p local

populations of discrete individuals living on discrete patches thatare connected by dispersal. Propagule dispersal is handled by oneof the mechanisms to be described later. The patch locations arenot specified, so spatial structure is implicit. The total area of thepatches is α and the area of a patch is α/p.

2.1. Population dynamics

Generations are non-overlapping with a life cycle that alter-nates between a sedentary adult phase and a dispersive juve-nile phase. The per-capita population growth rate in a constantenvironment is r , where r is the average number of propagulesper-female that both survive the dispersal process and the pre-recruitment period in the destination patch. In other words, r ac-counts for all of the density-independent propagulemortality priorto recruitment.Environmental stochasticity is modeled by assuming that the

local random per-capita population growth rate, R, is gamma dis-tributed (Johnson et al., 1994), with expectation r and varianceσ 2. The between patch correlation of the R’s is ρ. The gamma dis-tribution is a biologically realistic model of per-capita populationgrowth driven by environmental stochasticity. Biologically realis-tic parameterizations produce a humped probability density func-tion that approximates a Gaussian distribution on the positive realnumbers (Johnson et al., 1994, pp. 340). An important feature of thegamma distribution is the ready availability of algorithms to pro-duce spatially correlated samples that preserve themean and vari-ance if the correlation between patches is altered (Schmeiser andLal, 1982). Alternatively, the lognormal distribution should pro-duce similar results. In fact, the lognormal distribution is also usedto approximate the Gaussian distribution on the positive real num-bers (Johnson et al., 1994, pp. 239).Each generation, in a patch, R is sampled to get the expectation

of a Poisson distribution (Johnson et al., 1993). The Poissondistribution is sampled for each female to get the number ofpropagules that survive to pre-recruitment. Thus, the gammadistribution sets the expected per-capita population growth ratein the patch and the Poisson distribution generates demographicstochasticity by varying the propagule number produced by eachfemale.Most single-population models in discrete time can be written

in the form,

n(t + 1) = rn(t)︸︷︷︸ f (n(t))︸︷︷︸,propagules×survival

(1)

where r is the density-independent per-capita population growthrate prior to recruitment, f is the density-dependent survival ratethrough recruitment, and n(t) is the adult population densityat time t . Such models imply that propagule survival duringrecruitment is a function of the adult population density at themoment the propagules were generated. However, for manyspecies recruitment occurs after dispersal within a destinationpatch, as the propagules compete for resources. If propagulesdisperse between t and t + h, and recruitment occurs betweent + h and t + 1, then the adult population density in patch i afterrecruitment, ni(t + 1), is,

ni(t + 1) = ni(t + h)f(ni(t + h)

), (2)

where the propagules making up the local post-dispersal density,ni(t + h), may have arrived from any patch.Among those working on extinction the favorite choice

for density-dependent competition is a cap-on-density function

148 K. Higgins / Theoretical Population Biology 76 (2009) 146–155

Table 1Metapopulation simulationmodel C++ source code. p is the number of patches in themetapopulation. delta*alpha/p is a patch’s adult carrying capacity. R is the gamma-distributed per-capita population growth rate in patch i, with expectation, r, standard deviation, sigma, and spatial correlation, rho. gamma(r,sigma,rho) returnsR. ranint(p) returns a uniformly distributed random integer between [0, p). poisson(x) returns a Poisson sample with expectation x. binomial(n,s) returns thenumber of survivors from n individuals, where the survival probability is s. Adults and DisProp are vectors that tally adult and dispersed-propagule numbers in eachpatch.

Generate & disperse propagules

Standard dispersal Gravid-female dispersal Cohort dispersal

Density-dependent recruitment

(Henle et al., 2004). The Henle et al. survey found that density-independent survival up to a cap was most common (91 casestudies), logistic survival up to a cap was next (17), followedby Ricker (14), and Beverton–Holt (11). Other investigationsusing cap-type models include, Lande (1993), Foley (1994, 1997)and Hanski et al. (1996). The cap-on-density model, with density-independent survival up to a cap, is arguably the most biologicallyrealistic extinction model, among these alternatives. A commonsituation when a metapopulation is near extinction is to havejust a few individuals in some patches, where resources areabundant. If resources are abundant, in the absence of Alleeeffects, density should not cause mortality. The cap-on-densitymodel, with density-independent survival up to a cap, meets therequirement. By contrast, the Beverton–Holt, Ricker, and logisticmodels do not. In fact, the classic models cause self-competition,reducing the survival probability of a lone individual because ofthe density it creates.To model post-dispersal survival I use the density-dependent

survival function from the cap-on-density hockey-stick model(Barrowman and Myers, 2000). The probabilistic hockey-stick isimplemented as follows. If the propagule density in a patch is D,then the probability that a propagule survives competition to reachadulthood ismin(δ/D, 1), where the population density at carryingcapacity is δ (units: adults/area). In other words, if D ≤ δ then allpropagules in the patch are recruited to the adult population; andif D > δ then the patch fills to carrying capacity with adults, onaverage.Importantly, the hockey-stick has no Allee effect. If propagules

are dispersed uniformly, the expected number of propagulessurviving recruitment in highly fragmented habitat is the same asin unfragmented habitat. There is no survival penalty just becauseindividuals live in small patches.A simple example demonstrates how lumpy dispersal causes

a reduction in both survival and average density. Two dispersalmechanisms are considered, but first a few assumptions must bestated. Consider a two-patch metapopulation where the densityat carrying capacity is δ. Further, let density-dependent survivalbe δ/x, where x is the post-dispersal propagule density in a patch(i.e., a simple cap on density). To avoid making my example overlycomplex, only situations where x > δ are considered. The firstdispersal mechanism, which has the most spatial variance, sendsall of the propagules from both patches to just one patch. If the

post-dispersal density in the occupied patch is D then survivalis δ/D. The second dispersal mechanism, which has no spatialvariance, divides the propagules equally between the patches. Ifthe total number of propagules is the same as before, then the post-dispersal density in both patches isD/2, and survival is 2δ/D. Thus,uniform dispersal doubles survival. As for average density, underthe variable dispersal mechanism it is δ/2 and under the uniformdispersal mechanism it is δ. Thus, uniform dispersal doubles theaverage population density.

2.2. Dispersal mechanisms

Taxa show great variety in the many ways that propagules aredispersed to suitable habitat. The adults of some species literallycast their propagules to the wind, while the females of otherspecies actively search for a suitable location to place all of theirpropagules. And for yet other species, propagules are transportedby ocean currents, with a significant amount of correlation, fromone intertidal location to another. In these examples, dispersal isstructured on the individual, clutch, and patch levels—illustratingthe potential for dispersal structure on various levels of ecologicalorganization. Given the potential of dispersal to generate spatialvariance, it is important to know how dispersal generates spatialvariance and how much spatial variance to expect from commondispersal mechanisms.In important early work on the interaction between dispersal

and localized density-dependence, de Jong (1979) used a collectionof dispersal probability distributions, spanning a range of spatialvariances (i.e., dispersal lumpiness is a tunable parameter). Themean density of the dispersed propagules is identical for alldistributions and for any number of patches, only the spatialvariance of the dispersed propagule densities changes from onedistribution to another. Here, I take a similar approach.Uniform dispersal. All patches receive an equal number of dis-

persers (i.e., propagules). This spatial distribution is determinis-tically uniform, with zero spatial variance (in practice, there issome insignificant variance if the propagule number does not di-vide evenly into the patch number). In a natural system, propagulesthat search for low density patches could produce a uniform distri-bution. Alternatively, as I will show later, uniform dispersal is ob-tained from standard dispersal if the per-capita population growthrate is very high.


A B C

Fig. 1. Metapopulation extinction risk versus habitat fragmentation level (p). Metapopulation extinction risk is estimated by repeatedly simulating the model for 256generations, and counting the number of metapopulation extinctions. Metapopulation extinction occurs when all patches are simultaneously extinct. The x-, y-, and z-regions are discussed in the text. α = 16,384 ha, δ = 1 adults/ha, ρ = .4, CV(R) = .5, and, top to bottom, E[R] = 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09, 1.1.(A) Standard dispersal. (B) Gravid-female dispersal. (C) Cohort dispersal.

Standard dispersal. All patches have probability, 1/p, of receivinga disperser (i.e., a propagule), where p is the patch number.Dispersal is uniform from a probabilistic perspective, as opposedto a deterministic perspective.Gravid-female dispersal. All patches have probability, 1/p, of

receiving a disperser (i.e., a gravid-female), where p is the patchnumber. A gravid-female carries a Poisson distributed number ofpropagules.Cohort dispersal. All patches have probability, 1/p, of receiving a

disperser (i.e., a propagule cohort), where p is the patch number. Adisperser ismadeup fromall of the propagules produced in a patch.Physical transport or behavior are processes that could producecohort dispersal.Finally, a comment on an alternative dispersal mechanism.

Typically, in work on coexistence in multispecies systems, thenegative binomial distribution is used to model aggregateddispersal to ephemeral habitat patches. The negative binomial isobtained when dispersers carry a Poisson distributed number ofpropagules, and the expected values of the Poisson distributions,associated with the dispersers, are gamma distributed (Johnsonet al., 1993, p. 204). The gamma distribution is called the mixingdistribution (Johnson et al., 1993, p. 328). If the habitat patchesare permanent, as they are here, the correct mixing distributionis the binomial (or multinomial). And the correct dispersal modelis a binomial mixture of Poisson distributions (Johnson et al.,1993, p. 333). Often, in studies that use the negative binomial,dispersal lumpiness is tuned by a single parameter of the negativebinomial distribution. Here, similarly, gravid-female and cohortdispersal are the endpoints on an array of parameter values thattune dispersal lumpiness.

3. Metapopulation extinction risk

The metapopulation extinction risk curves in Fig. 1A–Bare rather interesting because they are U-shaped, rather thanmonotonic. TheU-shape indicates that mild habitat fragmentationreduces metapopulation extinction risk, but that severe habitatfragmentation raises metapopulation extinction risk. ConsiderFig. 1A; starting from a single large patch (p = 1), fragmenting thepatch into a few pieces causes the metapopulation extinction riskto decline (x-region). Further fragmentation produces a relativelyconstant metapopulation extinction risk (y-region). Finally, evenfurther fragmentation causes the metapopulation extinction riskto rise (z-region). The curve in Fig. 1B is more or less similar toFig. 1A. At first glance, Fig. 1C (note Metapopulation extinction riskscale change) appears to be quite different, however, the z-regionin Fig. 1C is similar to the z-region in Fig. 1B. In fact, Fig. 1 showsthat going rightward through the dispersal mechanisms (i.e., A→B→ C), shifts the extinction curves leftward.

The U-shape suggests there may be two mechanisms control-ling the metapopulation extinction risk — one mechanism in con-trol at low fragmentation and the second mechanism in controlat high fragmentation. As I show later, the low-fragmentationmechanism is the metapopulation rescue effect and the high-fragmentation mechanism is the musical chairs effect. To showhow these mechanisms drive metapopulation extinction risk,some essential tools are developed now. Those wishing to take aless mathematical path may want to just glance at Sections 4 and5.

4. Quantifying dispersal lumpiness

The amount of spatial variability produced by dispersal is a con-sequence of both the dispersal mechanism and the fragmentationlevel of the habitat. Independent propagule dispersal produces rel-atively low spatial variance, while bundled propagule dispersalproduces relatively high spatial variance. An alternative route fromlow to high spatial variance is produced by increasing the habitatfragmentation level, for a given dispersal mechanism.Both simulation and the moments of appropriate probability

distributions are used to quantify the spatial variance generated bydispersal in fragmented habitat. The expected propagule numberin a patch after dispersal is, rδα/p, for all of the dispersalmechanisms.

4.1. Uniform dispersal

Using simulation, Fig. 2A (©) shows that uniform dispersalproduces zero spatial variance for any patch number. The densitieswere generated by having each propagule search for the patchwiththe lowest population density, at the time it dispersed.

4.2. Standard dispersal

Using simulation, Fig. 2A (�) shows spatial variance as afunction of the patch number. By comparison, Fig. 2B (�) showsthe analytical result. The curves are identical.The analytical spatial variance assumes the disperser numbers

in the patches are multinomial. However, rather than use thecumbersome multinomial distribution directly, I take advantageof a convenient fact. From the perspective of a given patch, adisperser either landed there or it did not, and so the dispersersare binomially distributed (i.e., the marginal distribution isbinomial, Johnson et al. (1997, pp. 32–34)). The probability massfunction of the binomial distribution (Johnson et al., 1993) is,


A

B

Fig. 2. Spatial variability of dispersed propagule densities (i.e., lumpiness), asmeasured by CV(D), versus habitat fragmentation level (p). The expected numberof dispersed propagules is rδα; r = 1.02, δ = 1 adults/ha, α = 16,384 ha. (A)Simulated uniform dispersal (©), standard dispersal (�), gravid-female dispersal(�), and cohort dispersal (N). To estimate CV(D), one generation of dispersal wassimulated 100,000 times, CV(D)was calculated for each simulation, and the CV(D)values were averaged to obtain CV(D). (B) Graphs of the analytical expressions forCV(D) (see text). Symbols same as (A).

Pr[N = n] =(dn

)qn(1− q)d−n, n = 0, 1, 2, . . . , d, (3)

where N is a binomial random variable, q is the probability adisperser arrives in the patch and d is the number of dispersersin the metapopulation. The expectation, E[N] = dq, the variance,Var(N) = dq(1 − q), and the coefficient of variation, CV(N) =√(1− q)/(dq). If, d = rδα, and, q = 1/p, then, E[N] = rδα/p,and, CV(N) =

√(p− 1)/(rδα). Further, the coefficient of variation

is the same if the local population state is density, D, rather than acount, that is, CV(D) = CV(N).

4.3. Gravid-female and cohort dispersal

Simulation and theory, for both gravid-female and cohort dis-persal, produce curves that are in excellent agreement (Fig. 2A–B,�, N).Both the gravid-female and cohort dispersal mechanisms pro-

duce disperser numbers in the patches that follow a multinomial

distribution. However, the disperser is either a gravid-female ora propagule cohort, rather than a bare propagule. Either a gravid-female or a cohort carries a Poisson distributed number of propag-ules. Because the multinomial marginal distribution is binomial(see above), the probabilitymass function is that of a binomialmix-ture of Poisson distributions (Johnson et al., 1993, p. 333),

Pr[N = n] =d∑j=0

(dj

)qj(1− q)d−je−jφ(jφ)n/n!,

n = 0, 1, 2, . . . , (4)

where N is a mixture random variable, q is the probability that adisperser arrives in the patch and d is the number of dispersers inthe metapopulation. The expectations of the Poisson distributionsare sampled from the random variable, φV , where φ is a constantand V is binomially distributed with parameters, d and q. Themixture distribution has expectation, dqφ, variance, dqφ+ dq(1−q)φ2, and CV(D) = CV(N) =

√(1+ (1− q)φ)/(dqφ).

Gravid-female dispersal. For dispersed gravid-females, d =δα, and, q = 1/p. Each female contains a Poisson distributedpropagule number, with expectation, r . If the dispersed propaguleexpectation in a patch is δαr/p, then δαφ/p = δαr/p, implyingφ = r . Therefore, CV(D) = CV(N) =

√(p+ rp− r)/(rδα).

Cohort dispersal. For dispersed cohorts, d = p, and, q = 1/p.Each cohort contains a Poisson distributed propagule number,withexpectation, rδα/p. If the dispersed propagule expectation in apatch is δαr/p, then pφ/p = φ = δαr/p. Therefore, CV(D) =CV(N) =

√(p+ rδα − rδα/p)/(rδα).

4.4. Sensitivity analysis

An important question is whether a high per-capita populationgrowth rate reduces, or eliminates, the spatial variance that isgenerated by dispersal. In other words, do the patches becomemore uniformly occupied as the per-capita population growth rateis increased. The spatial coefficient of variation clearly shows thatthe answer depends on the dispersal mechanism.Standard dispersal. Simulations show that the spatial variance

vanishes as the per-capita population growth rate is exponentiallyincreased (Fig. 3A). Further, letting the per-capita populationgrowth rate go to infinity in the analytical coefficient of variationcauses spatial variance to go to zero. Thus, if r →∞ then CV(D)→0. This limit shows that standard dispersal converges to uniformdispersal when the per-capita population growth rate is high.Gravid-female dispersal. Simulations show that high spatial

variance is maintained as the per-capita population growth rateis exponentially increased (Fig. 3B). Further, letting the per-capitapopulation growth rate go to infinity in the analytical coefficient ofvariation causes spatial variance to approach a positive limit. Thus,if r →∞ then CV(D)→

√(p− 1)/(δα).

A B C

Fig. 3. Sensitivity of dispersal lumpiness, CV(D), in Fig. 2A, to increases in the per-capita population growth rate, r . Cases with symbols are from Fig. 2A, as are otherparameters. Top to bottom, r = 1.02, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. (A) Standard dispersal. (B) Gravid-female dispersal. (C) Cohort dispersal.


Cohort dispersal. Simulations show that high spatial varianceis maintained as the per-capita population growth rate isexponentially increased (Fig. 3C). Further, letting the per-capitapopulation growth rate go to infinity in the analytical coefficient ofvariation causes spatial variance to approach a positive limit. Thus,if r →∞ then CV(D)→

√1− 1/p.

5. The scale transition

Chesson (1996) calls ‘‘changes that take place in populationdynamics when the view shifts from one scale in space or timeto another’’ the scale transition. In a metapopulation, importantdynamical properties, such as the average population densityor the average per-capita growth rate, may change in valueas the measurement scale moves from the patch level to themetapopulation level, or from a single generation to long expansesof time.Metapopulation extinction is a macro-scale event produced by

the local population dynamics that ultimately drive the system.A mechanistic explanation of metapopulation extinction riskrequires some measure of the macro-scale population dynamicsin terms of the local population dynamics. The two macro-scalemetrics that I use are the average population density and theaverage per-capita growth rate.

5.1. Average population density

The interaction between spatially localized density-dependentsurvival and spatially variable local-population densities may re-duce the average population density (de Jong, 1979; Ives and May,1985; Chesson, 1996, 1998). Here, the average adult populationdensity is computed using nonlinear averaging (Chesson, 1996,1998), as follows. A single generation is completed between (t ,t + 1), with dispersal between (t , t + h) and competition between(t+h, t+1). After dispersal, in patch i,Di(t+h) is the local propag-ule density, and after competition, Di(t + 1) is the local adult den-sity. The average adult population density, D(t + 1), is,

D(t + 1) =1p

p∑i=1

Di(t + 1) =1p

p∑i=1

Di(t + h)f(Di(t + h)

), (5)

where f is some density-dependent survival function and p is thepatch number. If survival follows the hockey-stick,

D(t + 1) =1p

p∑i=1

min(Di(t + h), δ

). (6)

The next step could be skipped, however, it shows how variablepropagule densities and density-dependent survival act togetherto reduce the average adult population density. Using algebra, (6)yields,

D(t + 1) = min(D(t + h), δ

)−min(B, A), (7)

B =1p

∑Di(t+h)≤δ

δ − Di(t + h),

A =1p

∑Di(t+h)>δ

Di(t + h)− δ.

In the absence of spatial variance, min(B, A) is zero, and theaverage adult population density is just the survival functionapplied to the spatially averaged post-dispersal propagule den-sity (i.e., min(D(t + h), δ)). Importantly, this means that in theabsence of spatial variance, density-dependent survival in frag-mented habitat is identical to density-dependent survival in non-fragmented habitat. However, if there is spatial variance, min(B, A)

B

A

Fig. 4. (A) Average per-capita growth rate, R̂, from Eq. (9), versus habitatfragmentation level (p) and correlation between habitat fragments (ρ). E[R] = 1.1,CV(R) = .5, and t = 500,000. (B) Average adult population density, D(t + 1), fromEq. (8), versus habitat fragmentation level (p). α = 16,384 ha, δ = 1 adults/ha,E[D(t + h)] = 1.1 propagules/ha. Standard dispersal (�), gravid-female dispersal(�), and cohort dispersal (N).

reduces the average adult population density because of localdepartures from the carrying capacity, δ. B measures the aver-age density-departure of propagule populations below carrying-capacity, while A measures the average density-departure ofpropagule populations above carrying capacity. Although the av-erage density-departure is not a true variance, it is close in form,providing similar insight. Importantly, the metapopulation nowsuffers from a possibly devastating reduction in average adult pop-ulation density because of intense localized competition for re-sources by propagules.At this point the post-dispersal propagule densities are

required. de Jong’s (1979) approach uses a dispersal distribution’sprobability mass function to provide the probability of observingany given population count at time t + h. Using the probabilitiesas weights, and the expected dispersal density, E[D(t + h)] =pE[N(t + h)]/α, yields the average adult population density,

D(t + 1) = min(pE[N(t + h)]/α, δ

)−min(B, A), (8)

B =δα/p∑n=0

(δ − pn/α) Pr[N(t + h) = n],

A =pE[N]∑

n=1+δα/p

(pn/α − δ) Pr[N(t + h) = n],

where Pr[N(t + h) = n] is the probability that a patch containsn dispersed propagules. Using the probability mass functionsdeveloped earlier, Fig. 4B shows the average adult populationdensity for standard (�), gravid-female (�), and cohort (N)dispersal, as functions of habitat fragmentation.

5.2. Average per-capita growth rate

The average per-capita growth rate of a metapopulation ac-counts for two effects — the effect of dispersal within a genera-tion and the effect of multiplying serial random per-capita growth


rates across generations. For themetapopulation as awhole,withina generation, dispersal produces a per-capita growth rate equal tothe spatial arithmetic average of the random per-capita popula-tion growth rates in the patches. Over the course of generations themetapopulation grows as if these spatial arithmetic averages weremultiplied. Further, the average per-capita growth rate over thosegenerations is the geometric average of the spatial arithmetic av-erages (Roff, 1974a,b; Strathmann, 1974; Palmer and Strathmann,1981; Ives et al., 2004). Taking both effects into account, the aver-age per-capita growth rate, R̂, is,

R̂ =

[t∏j=1

R(j)

]1/t=

[t∏j=1

(1p

p∑i=1

Ri(j)

)]1/t, (9)

where Ri(j) is the random per-capita population growth rate inpatch i at time j; R(j) is the spatial arithmetic average of the Ri(j)’s,over p patches, at time j; and R̂ is the geometric average of the spa-tial arithmetic averages over the time span [1, t]. Fig. 4A showshow R̂ responds to habitat fragmentation, and to spatial correla-tion in the local random per-capita population growth rates.I now derive (9). In patch i, within generation t , the life cycle

implies the following sequence of expected values,

Ri(t)Ni(t) propagules, (10)

1p

p∑j=1

Rj(t)Nj(t) dispersed propagules, (11)

f

(1p

p∑j=1

Rj(t)Nj(t)

)1p

p∑j=1

Rj(t)Nj(t) adults. (12)

If dispersal is uniform, then at the beginning of a generation, theexpected adult numbers in the patches are equal, that is, N1(t) =· · · = Np(t) = N(t), where N(t) stands for the common expectednumber. Then,

1p

p∑j=1

Rj(t)Nj(t) =N(t)p

p∑j=1

Rj(t) = N(t)R(t). (13)

After competition the expected adult number in a patch, N(t + 1),is,

N(t + 1) = f(N(t)R(t)

)N(t)R(t). (14)

Finally, the recursion implies,

N(t + 1) = N(1)t∏j=1

f(N(j)R(j)

)R(j)

= N(1)̂Rtt∏j=1

f(N(j)R(j)

), (15)

where, R̂ =[∏t

j=1 R(j)]1/t. The product of the f ’s must be

≤1, because they are probabilities. Therefore, metapopulationpersistence requires R̂ ≥ 1.Remarkably, even though the derivation of (9) assumes uniform

dispersal, it can be shown using simulation that R̂ is invariantacross all of the dispersal mechanisms used here. An analyticalderivation of R̂ for the nonuniform dispersal mechanisms canbe done using the probability mass function approach employedearlier. Unfortunately, at a key step in the derivation themultinomial coefficients of a vast number of possible spatialconfigurations are required, making the analytical approach lessuseful. However, for cases with just a few patches and individuals,it is clear from the symmetry of the multinomial coefficientsthat the nonuniform dispersal mechanisms also produce spatiallyaveraged growth that is equal to the arithmetic average of the

random per-capita population growth rates in the patches. Thus,for all of the dispersal mechanisms investigated here the averageper-capita growth rate is given by (9).The reason that all of the dispersal mechanisms produce the

same average per-capita growth rate, in spite of their differingspatial variabilities, is that all of the dispersal mechanisms arespatially isotropic. That is, under all dispersal mechanisms theexpected number of propagules dispersed to any given patch is thesame. Because of this assumption the expected adult numbers inthe patches are all the same, too. Therefore, (13) holds for all of thedispersal mechanisms, and then the rest of the derivation follows.

6. The mechanisms driving metapopulation extinction risk

Dispersal in an increasingly fragmented habitat both raises theaverage per-capita growth rate (Fig. 4A) and reduces the averagepopulation density (Fig. 4B). In this section I will show how thesecounteracting extinction-forces shape metapopulation extinctionrisk.

6.1. Metapopulation rescue effect versus musical chairs effect

Metapopulation extinction risk shows a U-shaped pattern inFig. 1A–B (�,�). The x, y, and z labels denote three important placeson the U where habitat fragmentation has altered the balance ofpower between the metapopulation rescue effect and the musicalchairs effect.The x-region shows that metapopulation extinction risk is

declining as the habitat becomes mildly fragmented. The causeof the declining risk is the rising average per-capita growth rateassociated with dispersal in an increasingly fragmented habitat(Fig. 4A, ρ = .4). Further, dispersal is relatively smooth (CV(D) ≈0; Fig. 2; �, �), so that there is almost no decline in the averagepopulation density when the habitat is mildly fragmented (Fig. 4B;�, �).The y-region shows that metapopulation extinction risk is

relatively constant as the habitat passes through a range ofmoderate fragmentation levels. The risk is constant because therise in the average per-capita growth rate is nearly over (Fig. 4A,ρ = .4) and the decline in the average population density isjust beginning (Fig. 4B; �, �). The onset of the decline in averagepopulation density is driven by the onset of appreciable dispersallumpiness (Fig. 2; �, �).The z-region shows that metapopulation extinction risk rises

as the habitat becomes severely fragmented. The cause of therising risk is the declining average population density associatedwith dispersal in a severely fragmented habitat (Fig. 4B; �, �). Inseverely fragmented habitat dispersal is highly lumpy (Fig. 2;�,�),and it is high dispersal lumpiness that directly causes the declinein the average population density.

6.2. Effective fragmentation level

Metapopulation extinction risk shows anunusual pattern undercohort dispersal (Fig. 1C, N; note Metapopulation extinction riskscale change). At first glance the form of the extinction-risk curvesappears to be novel, when compared to standard or gravid-femaledispersal, with almost all levels of habitat fragmentation producingnear certain extinction. However, the cohort extinction-risk curves(Fig. 1C, N) are a far left-shifted version of the standard andgravid-female extinction-risk curves (Fig. 1A–B; �, �). Cohortdispersal left-shifts the extinction-risk curves by generating verylumpy dispersal at much lower levels of habitat fragmentationthan either standard or gravid-female dispersal do (compare: �,�, N; Fig. 2). Because cohort dispersal is extremely lumpy evenin mildly fragmented habitat, cohort dispersal causes the average


population density to decline precipitously when the habitat isbroken into just two fragments (Fig. 4B; N). By contrast, anequivalent decline in average population density under eitherstandard or gravid-female dispersal requires that the habitat bebroken into thousands of fragments (compare: �, �, N; Fig. 4B).The dashed line in Fig. 2B shows that the three dispersal mech-

anisms produce identical levels of dispersal lumpiness at threedifferent levels of habitat fragmentation. From the perspective ofdispersal lumpiness, the effective habitat fragmentation level alongthe dashed line is the same for all three dispersal mechanisms,although the actual habitat fragmentation levels are quite differ-ent. Importantly, raising the propagule-bundle size is equivalent toraising the habitat fragmentation level. Therefore, it is the joint ac-tion of propagule-bundle size and habitat fragmentation level thatsets the dispersal lumpiness level, the average population density,and the metapopulation extinction risk.

6.3. Turning on lumpy dispersal raises metapopulation extinction risk

Uniform dispersal spreads propagules across habitat fragmentsin an even layer, with zero spatial variance (Fig. 2, ©). By con-trast, standard dispersal spreads propagules across habitat frag-ments with uniform probability, generating a level of lumpinessthat grows with the level of habitat fragmentation (Fig. 2, �).The metapopulation extinction-risk curves for these two dispersalmechanisms, where all parameters are otherwise identical, showvery marked divergence (Fig. 5A–B). The curves are identical atlow habitat fragmentation levels, indicating the metapopulationextinction risk is the same for both dispersal mechanisms. By con-trast, the curves are entirely different at high habitat fragmenta-tion levels, indicating that the metapopulation extinction risk un-der smooth dispersal is far below the metapopulation extinctionrisk under lumpy dispersal. This divergence of risk for the smoothand lumpy dispersal mechanisms provides a striking example ofthe power of the musical chairs effect to raise metapopulation ex-tinction risk.

6.4. Recovering the metapopulation extinction risk of a constantenvironment

The convergence of the metapopulation extinction risk curvesat high fragmentation levels in Fig. 5 suggests somethinginteresting may be happening. In the convergence region themetapopulation extinction risk in the random environment casesis approaching the metapopulation extinction risk of the constantenvironment case. From the perspective of metapopulationextinction risk, the random metapopulation environment iseffectively constant, even though the local environments in thepatches are fluctuating intensely. Remarkably, themetapopulationrescue effect has recovered the metapopulation extinction risk ofa constant environment.The recovery of themetapopulation extinction risk of a constant

environment is driven by the rise of the average per-capitagrowth rate to the constant environment per-capita growthrate, as the habitat becomes sufficiently fragmented. Given asufficiently fragmented habitat, with independent random per-capita population growth rates in the fragments (ρ = 0), theaverage per-capita growth rate, R̂, rises to the expected value of therandom per-capita population growth rates in the patches, E[R],(Fig. 4A, ρ = 0), where E[R] = r , the constant environment per-capita growth rate.Finally, an important aside. The flatness of the w-region curve

in Fig. 5A confirms that there is no Allee effect in the model — themetapopulation extinction risk is the same at all levels of habitatfragmentation when dispersal is uniform.

A

B

Fig. 5. Turning on lumpy dispersal raises metapopulation extinction risk.Metapopulation extinction risk versus habitat fragmentation level (p). (A) Uniformdispersal (no lumps). (B) Standard dispersal (lumpy). Bottom curve in (A) or (B)shows metapopulation extinction risk in a constant environment; all higher curvesare for random environments. Bottom to top, CV(R) = 0, .025, .05, .1, .2. E[R] =1.01, ρ = 0, α = 256 ha, δ = 1 adults/ha. Extinction risk estimated as in Fig. 1. Thex-,w-, and z-regions are discussed in the text.

6.5. Environmental correlation impairs the metapopulation rescueeffect

The metapopulation rescue effect relies on the power ofdispersal to raise the average per-capita growth rate as the habitatfragmentation level rises. For the metapopulation rescue effectto properly function there must be some degree of asynchronybetween habitat fragments. Fig. 4A shows the average per-capitagrowth rate, R̂, for three levels of environmental correlation,as functions of habitat fragmentation. If the patch number issufficiently high and if the local environments are independent(ρ = 0), the average per-capita growth rate, R̂, rises by themaximum amount to E[R], the per-capita growth rate in a constantenvironment. By contrast, in a partially correlated environment(ρ = .4) the rise in R̂ falls short of themaximum. In the worst case,in an exactly correlated environment (ρ = 1) there is no boostto R̂ from dispersal in fragmented habitat — at all levels of habitatfragmentation the metapopulation grows as if it were a singlepopulation in a random environment. In an exactly correlatedenvironment there is no metapopulation rescue effect. In relatedwork, Palmqvist and Lundberg (1998) show that environmentalcorrelation raises metapopulation extinction risk in models withlocal density-dependent dynamics.

6.6. Extinction threshold

Metapopulation extinction is certain if the fraction of suitablehabitat patches drops below a critical value called the ‘‘extinctionthreshold’’ (Lande, 1987). In the metapopulation model developedhere, from the perspective of a dispersing propagule, a destinationpatch with too many propagules has a degree of unsuitability thatincreases with the level of overcrowding there. Raising dispersallumpiness is tantamount to reducing the quality of the suitablehabitat patches in the metapopulation. If dispersal is very lumpy,


the effective fraction of suitable habitat patches drops below theextinction threshold. The extinction threshold behavior predictedby Lande is self-evident in the gravid-female and cohort dispersalz-regions of Fig. 1B–C (�, N).

7. Discussion

The metapopulation model developed here brings togetherseveral important mechanisms to investigate metapopulationextinction risk. The joint action of random environments, density-dependent survival, dispersal, and habitat fragmentation; af-fect metapopulation extinction risk in ways that might not beanticipated from the action of these forces in isolation. Two impor-tant mechanisms are identified that provide a complete explana-tion for themetapopulation extinction risk produced by themodel.Metapopulation extinction risk is reduced by dispersal in an in-creasingly fragmented habitat when there is some degree of asyn-chrony between local randomper-capita growth rates. Dispersal inan increasingly fragmented habitat raises the average per-capitagrowth rate of the metapopulation by reducing the generation-to-generation variability of metapopulation growth. Lewontin andCohen (1969) showed that such generation-to-generation vari-ability may be devastating to single-population growth; and hereit is no less devastating to metapopulation growth. Metapopula-tion extinction risk is raised by lumpy dispersal in an increas-ingly fragmented habitat. Lumpy dispersal reduces the averagepopulation density in fragmented habitat with localized density-dependent competition for resources. Because survival is densitydependent, reduced survival in overcrowded patches is not offsetby improved survival in underpopulated patches. Although bothmechanisms are driven by dispersal, the counteracting mecha-nisms affect metapopulation extinction risk in opposite fashion,with the habitat fragmentation level deciding which mechanismprevails. If habitat fragmentation is mild, dispersal raises the av-erage per-capita growth rate, lowering metapopulation extinctionrisk. If habitat fragmentation is severe, dispersal raises the per-capita growth rate by the maximum amount, but this time lumpydispersal reduces the average population density below the extinc-tion threshold,makingmetapopulation extinction certain. That thejoint action of dispersal and habitat fragmentation might eitherraise or lower metapopulation extinction risk is consistent withthe results from 17 empirical investigations on the effects of habi-tat fragmentation on biodiversity — more than half showed thathabitat fragmentation had positive effects (Fahrig, 2003).In a sufficiently large single-population, demographic stochas-

ticity is a relatively weak extinction force, by comparison to en-vironmental stochasticity (Lande, 1993). In the metapopulationmodel developed here, in a sufficiently fragmented habitat, theordering is reversed, with demographic stochasticity becoming astronger extinction force than environmental stochasticity. De-mographic stochasticity comes from a surprising source — globaldispersal. Global dispersal of discrete propagules in fragmentedhabitat has great potential to generate highly variable propaguledensities, that then interact with localized density-dependenceto generate high systemic risk for the metapopulation. Thus, de-mographic stochasticity, usually associated with local events in asmall population, here generated by a global process, threatensglobal extinction.That demographic stochasticity might cause systemic effects

in large spatially subdivided populations was shown in importantearly work by Chesson (1978, 1981). Chesson found that the jointaction of patchiness, localized variability, and density-dependencehas great potential to thoroughly alter population dynamics atthe system level. More recent work, on the extinction of isolatedlocal populations, unconnected by dispersal, finds that the joint

action of patchiness, localized variability, and density-dependenceis, indeed, a potent extinction force (Burkey, 1999).In population viability analyses, it may be just as important

to model propagule dispersal structure, or its absence, as it is tomodel habitat fragmentation, given that both features contributeto dispersal lumpiness. Importantly, dispersal lumpiness may beintensified by barriers that prevent, or hinder, efficient dispersalto suitable habitat fragments. The upshot is that dispersal barriersmay cause the effective habitat fragmentation level to be fargreater than the actual habitat fragmentation level. Dispersalbarriers, especially those erected by humans, may make anotherwise highly resilientmetapopulation extremely vulnerable toextinction.In models of metapopulation genetics, the genetic effective

size is greatly reduced by intrapopulational structure (Sugg et al.,1996; Chesser et al., 1993; Chesser, 1991). Lumpy dispersal mayhave similar potential to reduce the genetic effective size, eventhough the biology is quite different. Lumpy dispersal producesa propagule-bundle rain over all habitat fragments, like an islandmodel. However, the local populations created by lumpy dispersalare composed of propagules that are more likely to have a com-mon geographic origin than a random sample from the pool of allpropagules, similar to a stepping-stonemodel. Thus, lumpy disper-sal makes mating between close relatives more likely; and eventhough dispersal is global, inbreeding occurs as if dispersal weremore local. Such novel behavior suggests that a metapopulationgenetics model with lumpy dispersal is an island/stepping-stonehybrid.The dynamically rich and well-studied wasp–butterfly–plant

system in Southwest Finland (Ehrlich and Hanski, 2004) affordsseveral opportunitieswhere both themetapopulation rescue effectand the musical chairs effect are likely to be important driversof metapopulation extinction dynamics. The pairwise interactionsbetween constituent species form a hierarchy with severallevels (Lei et al., 1997; Kuussaari, 1998; Lei and Hanski, 1998). Thebutterfly larvae feed on two host plant species that live on about4000 dry meadows on the Åland islands in the Baltic Sea. In turn,the butterfly larvae are parasitized by two specialist parasitoidwasps. Finally, two hyperparasitoidwasps parasitize the two larvalparasitoids. For a given exploiting-species (one of the fourwasps orthe butterfly), dispersal over the spatially variable patch-densitiesof its host may raise the exploiting-species’ average per-capitagrowth rate. The actual rise will be sensitive to the amountof spatial correlation in the host’s local population densities.Further, for a given exploiting-species, its average populationdensity may be reduced by lumpy dispersal. However, dispersallumpiness is highly sensitive to search efficiency, a property thatvaries markedly between exploiting-species. The great potentialof the Åland system for systemic growth and density effects –caused by patchiness, localized variability, density-dependenceand dispersal – make Åland an exemplary system for the study ofmetapopulation extinction.

Acknowledgments

I thank Oscar Gaggiotti, Ilkka Hanski, Jerry Hilbish, Beth Krizek,Don Krizek, Simon Levin, Dave Wethey, and the anonymousreviewers for their generous help. I am grateful to JinhuaWu; whodid a yeoman’s job on the simulations. This workwas supported byNational Science Foundation Grant DEB–0238354 and by start-upfunding from the University of South Carolina.

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