Theoretical Population Biology 76 (2009) 146–155
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Theoretical Population Biology
journal homepage: www.elsevier.com/locate/tpb
Metapopulation extinction risk: Dispersal’s duplicity
Kevin Higgins
Department of Biological Sciences, University of South Carolina, Columbia, SC 29208, United States
article
info
Article history:
Received 22 October 2007
Available online 6 June 2009
Keywords:
Dispersal
Habitat fragmentation
Density- dependence
Extinction risk
Metapopulation
abstract
Metapopulation extinction risk is the probability that all local populations are simultaneously extinct
during a fixed time frame. Dispersal may reduce a metapopulation’s extinction risk by raising its
average per-capita growth rate. By contrast, dispersal may raise a metapopulation’s extinction risk by
reducing 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 average
per-capita growth rate without causing any appreciable drop in its average population density. By
contrast, dispersal in severely fragmented habitat raises a metapopulation’s extinction risk because the
rise in its average per-capita growth rate is more than offset by the decline in its average population
density. The metapopulation model used here shows several other interesting phenomena. Dispersal
in sufficiently fragmented habitat reduces a metapopulation’s extinction risk to that of a constant
environment. Dispersal between habitat fragments reduces a metapopulation’s extinction risk insofar
as local environments are asynchronous. Grouped dispersal raises the effective habitat fragmentation
level. Dispersal search barriers raise metapopulation extinction risk. Nonuniform dispersal may reduce
the effective fraction of suitable habitat fragments below the extinction threshold. Nonuniform dispersal
may make demographic stochasticity a more potent metapopulation extinction force than environmental
stochasticity.
© 2009 Elsevier Inc. All rights reserved.
1. Introduction
Metapopulation (Hanski and Gilpin, 1997) extinction risk is the
probability that all local populations are simultaneously extinct
during a fixed time frame. Dispersal may affect a metapopulation’s
extinction risk in two ways as the habitat becomes more
fragmented. Dispersal may reduce a metapopulation’s extinction
risk by raising its average per-capita growth rate. For dispersal
to raise a metapopulation’s average per-capita growth rate the
random per-capita population growth rates in the patches must be
asynchronous. I call this reduction in a metapopulation’s extinction
risk the metapopulation rescue effect. By contrast, dispersal may
raise a metapopulation’s extinction risk by reducing its average
population density. The reduction in a metapopulation’s average
population density occurs when some patches spontaneously
receive too many propagules while other patches spontaneously
receive too few propagules. In a patch receiving too many
propagules the competition to obtain a portion of the limiting
resource is analogous to a musical chairs game. I call this rise in
a metapopulation’s extinction risk the musical chairs effect.
As the habitat becomes more fragmented there is tension between the metapopulation rescue effect and the musical chairs
effect for control of a metapopulation’s extinction risk. Dispersal
E-mail address: [email protected].
0040-5809/$ – see front matter © 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.tpb.2009.05.006
in mildly fragmented habitat reduces the metapopulation’s extinction risk because dispersal raises its average per-capita growth rate
without causing an appreciable drop in its average population density. By contrast, dispersal in severely fragmented habitat raises the
metapopulation’s extinction risk because the rise in its per-capita
growth rate is more than offset by the decline in its average population density.
Dispersal’s ability to raise a metapopulation’s average percapita growth rate was investigated in earlier work on dispersal
in asynchronous environments (Roff, 1974a,b; Strathmann, 1974;
Palmer and Strathmann, 1981; Ives et al., 2004). In a given generation, dispersal across the habitat fragments of a metapopulation
causes the metapopulation census to grow as if all of the individuals were in just a single environment, with just one per-capita
growth rate. The ‘‘single environment’’ per-capita growth rate for
a generation is found by spatially averaging the random per-capita
population growth rates from all of the habitat fragments. Importantly, the single environment per-capita growth rate displays
inherent random variation from one generation to the next. It is
exactly this type of generation-to-generation random variation in
a single population’s per-capita growth rate that was explored
by Lewontin and Cohen (1969) who showed it can be devastating to the census of a single-population over time. Here, the
same phenomenon appears in the dynamics of a metapopulation
when the metapopulation is viewed as a whole. Dispersal over an
increasingly fragmented habitat raises a metapopulation’s average
per-capita growth rate by making the random single-environment
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
147
per-capita growth rate less variable over generations. Interestingly, a similar mechanism underlies modern portfolio theory,
where the goal is allocate an investor’s balance across a collection
of assets so that the expected rate of return is maximized for a given
level of risk (Tobin, 1958, Section 3.6).
Dispersal’s ability to reduce a metapopulation’s average population density was investigated in earlier work on patchy singlespecies systems with localized density-dependent survival (de
Jong, 1979; Ives and May, 1985; Chesson, 1996, 1998). In these
systems random dispersal places too many propagules in some
patches and not enough propagules in other patches, causing a
reduction in average survival that reduces average population
density. By contrast, in systems with an Allee effect, the same
nonuniform dispersal pattern causes a rise in average survival
that raises average population density (Chesson, 1998). Further,
in systems where multiple species compete for resources within
patches, nonuniform dispersal may level their competitive abilities, promoting coexistence (Atkinson and Shorrocks, 1981; May
and Hassell, 1981; Ives and May, 1985; Klopfer and Ives, 1997; Lei
and Hanski, 1998). Importantly, in both single-species (no Allee effect) and multispecies systems, competition in the overpopulated
patches exacts a toll on average survival that is not fully repaid by
reduced competition in the underpopulated patches. The net result
is 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 history
may cause it to place more propagules in some patches than in
others, either centered where the organism itself grows, or at
other locations. The tendency to produce eggs/seeds in clutches,
or for seeds/eggs to travel in clumps, or for dispersal to be
short range, are all factors that drive dispersal lumpiness. Second,
habitat fragmentation (breaking the same total area into smaller
fragments) may make dispersal lumpiness more pronounced. As
the habitat is broken into smaller fragments it becomes less likely
that the number of propagules arriving in a fragment are a good
fit for the carrying capacity of that fragment. Some fragments
receive too many propagules while other fragments receive too
few propagules. Increasing the degree of habitat fragmentation
makes larger mismatches more common. Finally, the joint action
of these two mechanisms compounds the potential for dispersal
lumpiness.
The metapopulation model used here shows several other interesting phenomena. First, dispersal in sufficiently fragmented habitat reduces a metapopulation’s extinction risk to that of a constant
environment even though environmental stochasticity causes the
per-capita growth rates in the habitat fragments to fluctuate intensely. Second, dispersal between habitat fragments reduces a
metapopulation’s extinction risk insofar as local environments are
asynchronous. Third, dispersing propagules in groups or bundles
raises the effective habitat fragmentation level. Fourth, barriers
that prevent disperser search behavior, or cause dispersers to be
spatially aggregated, raise a metapopulation’s extinction risk. Fifth,
sufficiently nonuniform dispersal reduces the effective fraction of
suitable habitat fragments below the extinction threshold, where
extinction becomes certain. Sixth, sufficiently nonuniform dispersal makes demographic stochasticity a more potent metapopulation 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 avoid
some of the mathematical details. Those readers may want to just
glance at Sections 4 and 5.
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 that
are connected by dispersal. Propagule dispersal is handled by one
of the mechanisms to be described later. The patch locations are
not specified, so spatial structure is implicit. The total area of the
patches is α and the area of a patch is α/p.
2. Metapopulation model
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
The mathematical metapopulation model involves a number of
biological assumptions that are worthy of discussion. Alternatively,
2.1. Population dynamics
Generations are non-overlapping with a life cycle that alternates between a sedentary adult phase and a dispersive juvenile phase. The per-capita population growth rate in a constant
environment is r, where r is the average number of propagules
per-female that both survive the dispersal process and the prerecruitment period in the destination patch. In other words, r accounts for all of the density-independent propagule mortality prior
to recruitment.
Environmental stochasticity is modeled by assuming that the
local random per-capita population growth rate, R, is gamma distributed (Johnson et al., 1994), with expectation r and variance
σ 2 . The between patch correlation of the R’s is ρ . The gamma distribution is a biologically realistic model of per-capita population
growth driven by environmental stochasticity. Biologically realistic parameterizations produce a humped probability density function that approximates a Gaussian distribution on the positive real
numbers (Johnson et al., 1994, pp. 340). An important feature of the
gamma distribution is the ready availability of algorithms to produce spatially correlated samples that preserve the mean and variance if the correlation between patches is altered (Schmeiser and
Lal, 1982). Alternatively, the lognormal distribution should produce similar results. In fact, the lognormal distribution is also used
to approximate the Gaussian distribution on the positive real numbers (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 Poisson
distribution is sampled for each female to get the number of
propagules that survive to pre-recruitment. Thus, the gamma
distribution sets the expected per-capita population growth rate
in the patch and the Poisson distribution generates demographic
stochasticity by varying the propagule number produced by each
female.
Most single-population models in discrete time can be written
in the form,
n(t + 1) = rn(t ) f n(t ) ,
|{z} | {z }
(1)
propagules×survival
where r is the density-independent per-capita population growth
rate prior to recruitment, f is the density-dependent survival rate
through recruitment, and n(t ) is the adult population density
at time t. Such models imply that propagule survival during
recruitment is a function of the adult population density at the
moment the propagules were generated. However, for many
species recruitment occurs after dispersal within a destination
patch, as the propagules compete for resources. If propagules
disperse between t and t + h, and recruitment occurs between
t + h and t + 1, then the adult population density in patch i after
recruitment, ni (t + 1), is,
ni (t + 1) = ni (t + h)f ni (t + h) ,
(2)
148
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
Table 1
Metapopulation simulation model C++ source code. p is the number of patches in the metapopulation. delta*alpha/p is a patch’s adult carrying capacity. R is the gammadistributed per-capita population growth rate in patch i, with expectation, r, standard deviation, sigma, and spatial correlation, rho. gamma(r,sigma,rho) returns
R. ranint(p) returns a uniformly distributed random integer between [0, p). poisson(x) returns a Poisson sample with expectation x. binomial(n,s) returns the
number of survivors from n individuals, where the survival probability is s. Adults and DisProp are vectors that tally adult and dispersed-propagule numbers in each
patch.
Generate & disperse propagules
Standard dispersal
Gravid-female dispersal
Cohort dispersal
Density-dependent recruitment
(Henle et al., 2004). The Henle et al. survey found that densityindependent survival up to a cap was most common (91 case
studies), logistic survival up to a cap was next (17), followed
by Ricker (14), and Beverton–Holt (11). Other investigations
using cap-type models include, Lande (1993), Foley (1994, 1997)
and Hanski et al. (1996). The cap-on-density model, with densityindependent survival up to a cap, is arguably the most biologically
realistic extinction model, among these alternatives. A common
situation when a metapopulation is near extinction is to have
just a few individuals in some patches, where resources are
abundant. If resources are abundant, in the absence of Allee
effects, density should not cause mortality. The cap-on-density
model, with density-independent survival up to a cap, meets the
requirement. By contrast, the Beverton–Holt, Ricker, and logistic
models do not. In fact, the classic models cause self-competition,
reducing the survival probability of a lone individual because of
the 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 is
implemented as follows. If the propagule density in a patch is D,
then the probability that a propagule survives competition to reach
adulthood is min(δ/D, 1), where the population density at carrying
capacity is δ (units: adults/area). In other words, if D ≤ δ then all
propagules in the patch are recruited to the adult population; and
if D > δ then the patch fills to carrying capacity with adults, on
average.
Importantly, the hockey-stick has no Allee effect. If propagules
are dispersed uniformly, the expected number of propagules
surviving recruitment in highly fragmented habitat is the same as
in unfragmented habitat. There is no survival penalty just because
individuals live in small patches.
A simple example demonstrates how lumpy dispersal causes
a reduction in both survival and average density. Two dispersal
mechanisms are considered, but first a few assumptions must be
stated. Consider a two-patch metapopulation where the density
at carrying capacity is δ . Further, let density-dependent survival
be 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 overly
complex, only situations where x > δ are considered. The first
dispersal mechanism, which has the most spatial variance, sends
all of the propagules from both patches to just one patch. If the
post-dispersal density in the occupied patch is D then survival
is δ/D. The second dispersal mechanism, which has no spatial
variance, divides the propagules equally between the patches. If
the total number of propagules is the same as before, then the postdispersal density in both patches is D/2, and survival is 2δ/D. Thus,
uniform dispersal doubles survival. As for average density, under
the variable dispersal mechanism it is δ/2 and under the uniform
dispersal mechanism it is δ . Thus, uniform dispersal doubles the
average population density.
2.2. Dispersal mechanisms
Taxa show great variety in the many ways that propagules are
dispersed to suitable habitat. The adults of some species literally
cast their propagules to the wind, while the females of other
species actively search for a suitable location to place all of their
propagules. And for yet other species, propagules are transported
by ocean currents, with a significant amount of correlation, from
one intertidal location to another. In these examples, dispersal is
structured on the individual, clutch, and patch levels—illustrating
the potential for dispersal structure on various levels of ecological
organization. Given the potential of dispersal to generate spatial
variance, it is important to know how dispersal generates spatial
variance and how much spatial variance to expect from common
dispersal mechanisms.
In important early work on the interaction between dispersal
and localized density-dependence, de Jong (1979) used a collection
of dispersal probability distributions, spanning a range of spatial
variances (i.e., dispersal lumpiness is a tunable parameter). The
mean density of the dispersed propagules is identical for all
distributions and for any number of patches, only the spatial
variance of the dispersed propagule densities changes from one
distribution to another. Here, I take a similar approach.
Uniform dispersal. All patches receive an equal number of dispersers (i.e., propagules). This spatial distribution is deterministically uniform, with zero spatial variance (in practice, there is
some insignificant variance if the propagule number does not divide evenly into the patch number). In a natural system, propagules
that search for low density patches could produce a uniform distribution. Alternatively, as I will show later, uniform dispersal is obtained from standard dispersal if the per-capita population growth
rate is very high.
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
A
B
149
C
Fig. 1. Metapopulation extinction risk versus habitat fragmentation level (p). Metapopulation extinction risk is estimated by repeatedly simulating the model for 256
generations, and counting the number of metapopulation extinctions. Metapopulation extinction occurs when all patches are simultaneously extinct. The x-, y-, and zregions 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 receiving
a disperser (i.e., a propagule), where p is the patch number.
Dispersal is uniform from a probabilistic perspective, as opposed
to 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 patch
number. A gravid-female carries a Poisson distributed number of
propagules.
Cohort dispersal. All patches have probability, 1/p, of receiving a
disperser (i.e., a propagule cohort), where p is the patch number. A
disperser is made up from all of the propagules produced in a patch.
Physical transport or behavior are processes that could produce
cohort dispersal.
Finally, a comment on an alternative dispersal mechanism.
Typically, in work on coexistence in multispecies systems, the
negative binomial distribution is used to model aggregated
dispersal to ephemeral habitat patches. The negative binomial is
obtained when dispersers carry a Poisson distributed number of
propagules, and the expected values of the Poisson distributions,
associated with the dispersers, are gamma distributed (Johnson
et al., 1993, p. 204). The gamma distribution is called the mixing
distribution (Johnson et al., 1993, p. 328). If the habitat patches
are permanent, as they are here, the correct mixing distribution
is the binomial (or multinomial). And the correct dispersal model
is 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 negative
binomial distribution. Here, similarly, gravid-female and cohort
dispersal are the endpoints on an array of parameter values that
tune dispersal lumpiness.
3. Metapopulation extinction risk
The metapopulation extinction risk curves in Fig. 1A–B
are rather interesting because they are U-shaped, rather than
monotonic. The U-shape indicates that mild habitat fragmentation
reduces metapopulation extinction risk, but that severe habitat
fragmentation raises metapopulation extinction risk. Consider
Fig. 1A; starting from a single large patch (p = 1), fragmenting the
patch into a few pieces causes the metapopulation extinction risk
to decline (x-region). Further fragmentation produces a relatively
constant metapopulation extinction risk (y-region). Finally, even
further fragmentation causes the metapopulation extinction risk
to rise (z-region). The curve in Fig. 1B is more or less similar to
Fig. 1A. At first glance, Fig. 1C (note Metapopulation extinction risk
scale change) appears to be quite different, however, the z-region
in Fig. 1C is similar to the z-region in Fig. 1B. In fact, Fig. 1 shows
that 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 controlling the metapopulation extinction risk — one mechanism in control at low fragmentation and the second mechanism in control
at high fragmentation. As I show later, the low-fragmentation
mechanism is the metapopulation rescue effect and the highfragmentation mechanism is the musical chairs effect. To show
how these mechanisms drive metapopulation extinction risk,
some essential tools are developed now. Those wishing to take a
less mathematical path may want to just glance at Sections 4 and
5.
4. Quantifying dispersal lumpiness
The amount of spatial variability produced by dispersal is a consequence of both the dispersal mechanism and the fragmentation
level of the habitat. Independent propagule dispersal produces relatively low spatial variance, while bundled propagule dispersal
produces relatively high spatial variance. An alternative route from
low to high spatial variance is produced by increasing the habitat
fragmentation level, for a given dispersal mechanism.
Both simulation and the moments of appropriate probability
distributions are used to quantify the spatial variance generated by
dispersal in fragmented habitat. The expected propagule number
in a patch after dispersal is, r δα/p, for all of the dispersal
mechanisms.
4.1. Uniform dispersal
Using simulation, Fig. 2A () shows that uniform dispersal
produces zero spatial variance for any patch number. The densities
were generated by having each propagule search for the patch with
the lowest population density, at the time it dispersed.
4.2. Standard dispersal
Using simulation, Fig. 2A () shows spatial variance as a
function of the patch number. By comparison, Fig. 2B () shows
the analytical result. The curves are identical.
The analytical spatial variance assumes the disperser numbers
in the patches are multinomial. However, rather than use the
cumbersome multinomial distribution directly, I take advantage
of a convenient fact. From the perspective of a given patch, a
disperser either landed there or it did not, and so the dispersers
are binomially distributed (i.e., the marginal distribution is
binomial, Johnson et al. (1997, pp. 32–34)). The probability mass
function of the binomial distribution (Johnson et al., 1993) is,
150
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
distribution. However, the disperser is either a gravid-female or
a propagule cohort, rather than a bare propagule. Either a gravidfemale or a cohort carries a Poisson distributed number of propagules. Because the multinomial marginal distribution is binomial
(see above), the probability mass function is that of a binomial mixture of Poisson distributions (Johnson et al., 1993, p. 333),
A
Pr[N = n] =
d X
d
j =0
j
n = 0, 1, 2, . . . ,
B
Fig. 2. Spatial variability of dispersed propagule densities (i.e., lumpiness), as
measured by CV(D), versus habitat fragmentation level (p). The expected number
of dispersed propagules is r δα ; 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 was
simulated 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 for
CV(D) (see text). Symbols same as (A).
Pr[N = n] =
d
n
qn (1 − q)d−n ,
n = 0, 1, 2, . . . , d,
(3)
where N is a binomial random variable, q is the probability a
disperser arrives in the patch and d is the number of dispersers
in 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 = r δα , and, q = 1/p, then, E [N ] = r δα/p,
and, CV(N ) = (p − 1)/(r δα). Further, the coefficient of variation
is the same if the local population state is density, D, rather than a
count, that is, CV(D) = CV(N ).
4.3. Gravid-female and cohort dispersal
Simulation and theory, for both gravid-female and cohort dispersal, produce curves that are in excellent agreement (Fig. 2A–B,
, N).
Both the gravid-female and cohort dispersal mechanisms produce disperser numbers in the patches that follow a multinomial
A
B
qj (1 − q)d−j e−jφ (jφ)n /n!,
(4)
where N is a mixture random variable, q is the probability that a
disperser arrives in the patch and d is the number of dispersers in
the metapopulation. The expectations of the Poisson distributions
are sampled from the random variable, φ V , where φ is a constant
and V is binomially distributed with parameters, d and q. The
mixture distribution has expectation,
dqφ , variance, dqφ + dq(1 −
√
q)φ 2 , 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 distributed
propagule number, with expectation, r. If the dispersed propagule
expectation in a patch is δα r /p, then√δαφ/p = δα r /p, implying
φ = r. Therefore, CV(D) = CV(N ) = (p + rp − r )/(r δα).
Cohort dispersal. For dispersed cohorts, d = p, and, q = 1/p.
Each cohort contains a Poisson distributed propagule number, with
expectation, r δα/p. If the dispersed propagule expectation in a
patch is δα
√r /p, then pφ/p = φ = δα r /p. Therefore, CV(D) =
CV(N ) = (p + r δα − r δα/p)/(r δα).
4.4. Sensitivity analysis
An important question is whether a high per-capita population
growth rate reduces, or eliminates, the spatial variance that is
generated by dispersal. In other words, do the patches become
more uniformly occupied as the per-capita population growth rate
is increased. The spatial coefficient of variation clearly shows that
the answer depends on the dispersal mechanism.
Standard dispersal. Simulations show that the spatial variance
vanishes as the per-capita population growth rate is exponentially
increased (Fig. 3A). Further, letting the per-capita population
growth rate go to infinity in the analytical coefficient of variation
causes spatial variance to go to zero. Thus, if r → ∞ then CV(D) →
0. This limit shows that standard dispersal converges to uniform
dispersal 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 rate
is exponentially increased (Fig. 3B). Further, letting the per-capita
population growth rate go to infinity in the analytical coefficient of
variation causes spatial variance
to approach a positive limit. Thus,
√
if r → ∞ then CV(D) → (p − 1)/(δα).
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 other
parameters. 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.
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
151
Cohort dispersal. Simulations show that high spatial variance
is maintained as the per-capita population growth rate is
exponentially increased (Fig. 3C). Further, letting the per-capita
population growth rate go to infinity in the analytical coefficient of
variation causes spatial variance
to approach a positive limit. Thus,
√
if r → ∞ then CV(D) → 1 − 1/p.
5. The scale transition
A
Chesson (1996) calls ‘‘changes that take place in population
dynamics when the view shifts from one scale in space or time
to another’’ the scale transition. In a metapopulation, important
dynamical properties, such as the average population density
or the average per-capita growth rate, may change in value
as the measurement scale moves from the patch level to the
metapopulation level, or from a single generation to long expanses
of 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 risk
requires some measure of the macro-scale population dynamics
in terms of the local population dynamics. The two macro-scale
metrics that I use are the average population density and the
average per-capita growth rate.
5.1. Average population density
The interaction between spatially localized density-dependent
survival and spatially variable local-population densities may reduce the average population density (de Jong, 1979; Ives and May,
1985; Chesson, 1996, 1998). Here, the average adult population
density 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 propagule density, and after competition, Di (t + 1) is the local adult density. The average adult population density, D(t + 1), is,
D(t + 1) =
p
1X
p i=1
Di (t + 1) =
p
1X
p i=1
Di (t + h)f Di (t + h) ,
(5)
where f is some density-dependent survival function and p is the
patch number. If survival follows the hockey-stick,
D(t + 1) =
p
1X
p i=1
min Di (t + h), δ .
(6)
The next step could be skipped, however, it shows how variable
propagule densities and density-dependent survival act together
to reduce the average adult population density. Using algebra, (6)
yields,
D(t + 1) = min D(t + h), δ − min(B, A),
B=
A=
1
p
1
p
X
Fig. 4. (A) Average per-capita growth rate, b
R, from Eq. (9), versus habitat
fragmentation 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), from
Eq. (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 local
departures from the carrying capacity, δ . B measures the average density-departure of propagule populations below carryingcapacity, while A measures the average density-departure of
propagule populations above carrying capacity. Although the average density-departure is not a true variance, it is close in form,
providing similar insight. Importantly, the metapopulation now
suffers from a possibly devastating reduction in average adult population density because of intense localized competition for resources by propagules.
At this point the post-dispersal propagule densities are
required. de Jong’s (1979) approach uses a dispersal distribution’s
probability mass function to provide the probability of observing
any given population count at time t + h. Using the probabilities
as 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),
B=
(8)
δα/
Xp
(δ − pn/α) Pr[N (t + h) = n],
n=0
(7)
δ − Di (t + h),
Di (t +h)≤δ
X
B
Di (t + h) − δ.
Di (t +h)>δ
In the absence of spatial variance, min(B, A) is zero, and the
average adult population density is just the survival function
applied to the spatially averaged post-dispersal propagule density (i.e., min(D(t + h), δ)). Importantly, this means that in the
absence of spatial variance, density-dependent survival in fragmented habitat is identical to density-dependent survival in nonfragmented habitat. However, if there is spatial variance, min(B, A)
pE [N ]
A=
X
(pn/α − δ) Pr[N (t + h) = n],
n=1+δα/p
where Pr[N (t + h) = n] is the probability that a patch contains
n dispersed propagules. Using the probability mass functions
developed earlier, Fig. 4B shows the average adult population
density 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 accounts for two effects — the effect of dispersal within a generation and the effect of multiplying serial random per-capita growth
152
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
rates across generations. For the metapopulation as a whole, within
a generation, dispersal produces a per-capita growth rate equal to
the spatial arithmetic average of the random per-capita population growth rates in the patches. Over the course of generations the
metapopulation grows as if these spatial arithmetic averages were
multiplied. Further, the average per-capita growth rate over those
generations is the geometric average of the spatial arithmetic averages (Roff, 1974a,b; Strathmann, 1974; Palmer and Strathmann,
1981; Ives et al., 2004). Taking both effects into account, the average per-capita growth rate, b
R, is,
b
R=
"
t
Y
#1/t
R(j)
=
"
p
t
Y
1X
j=1
p i =1
j =1
!#1/t
Ri (j)
,
(9)
random per-capita population growth rates in the patches. Thus,
for all of the dispersal mechanisms investigated here the average
per-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 differing
spatial variabilities, is that all of the dispersal mechanisms are
spatially isotropic. That is, under all dispersal mechanisms the
expected number of propagules dispersed to any given patch is the
same. Because of this assumption the expected adult numbers in
the patches are all the same, too. Therefore, (13) holds for all of the
dispersal mechanisms, and then the rest of the derivation follows.
6. The mechanisms driving metapopulation extinction risk
where Ri (j) is the random per-capita population growth rate in
patch i at time j; R(j) is the spatial arithmetic average of the Ri (j)’s,
over p patches, at time j; and b
R is the geometric average of the spatial arithmetic averages over the time span [1, t ]. Fig. 4A shows
how b
R responds to habitat fragmentation, and to spatial correlation 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,
Dispersal in an increasingly fragmented habitat both raises the
average per-capita growth rate (Fig. 4A) and reduces the average
population density (Fig. 4B). In this section I will show how these
counteracting extinction-forces shape metapopulation extinction
risk.
Ri (t )Ni (t ) propagules,
Metapopulation extinction risk shows a U-shaped pattern in
Fig. 1A–B (, ). The x, y, and z labels denote three important places
on the U where habitat fragmentation has altered the balance of
power between the metapopulation rescue effect and the musical
chairs effect.
The x-region shows that metapopulation extinction risk is
declining as the habitat becomes mildly fragmented. The cause
of the declining risk is the rising average per-capita growth rate
associated 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 average
population 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 of
moderate fragmentation levels. The risk is constant because the
rise in the average per-capita growth rate is nearly over (Fig. 4A,
ρ = .4) and the decline in the average population density is
just beginning (Fig. 4B; , ). The onset of the decline in average
population density is driven by the onset of appreciable dispersal
lumpiness (Fig. 2; , ).
The z-region shows that metapopulation extinction risk rises
as the habitat becomes severely fragmented. The cause of the
rising risk is the declining average population density associated
with dispersal in a severely fragmented habitat (Fig. 4B; , ). In
severely fragmented habitat dispersal is highly lumpy (Fig. 2; , ),
and it is high dispersal lumpiness that directly causes the decline
in the average population density.
p
1X
p j =1
f
Rj (t )Nj (t )
p
1X
p j =1
(10)
dispersed propagules,
!
Rj (t )Nj (t )
p
1X
p j =1
Rj (t )Nj (t ) adults.
(11)
(12)
If dispersal is uniform, then at the beginning of a generation, the
expected adult numbers in the patches are equal, that is, N1 (t ) =
· · · = Np (t ) = N (t ), where N (t ) stands for the common expected
number. Then,
p
1X
p j =1
Rj (t )Nj (t ) =
p
N (t ) X
p
Rj (t ) = N (t )R(t ).
(13)
j=1
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
Y
f N (j)R(j) R(j)
j=1
= N (1)b
Rt
t
Y
f N (j)R(j) ,
(15)
j =1
1/t
where, b
R =
. The product of the f ’s must be
j=1 R(j)
≤1, because they are probabilities. Therefore, metapopulation
persistence requires b
R ≥ 1.
Remarkably, even though the derivation of (9) assumes uniform
dispersal, it can be shown using simulation that b
R is invariant
across all of the dispersal mechanisms used here. An analytical
derivation of b
R for the nonuniform dispersal mechanisms can
be done using the probability mass function approach employed
earlier. Unfortunately, at a key step in the derivation the
multinomial coefficients of a vast number of possible spatial
configurations are required, making the analytical approach less
useful. However, for cases with just a few patches and individuals,
it is clear from the symmetry of the multinomial coefficients
that the nonuniform dispersal mechanisms also produce spatially
averaged growth that is equal to the arithmetic average of the
Qt
6.1. Metapopulation rescue effect versus musical chairs effect
6.2. Effective fragmentation level
Metapopulation extinction risk shows an unusual pattern under
cohort dispersal (Fig. 1C, N; note Metapopulation extinction risk
scale change). At first glance the form of the extinction-risk curves
appears to be novel, when compared to standard or gravid-female
dispersal, with almost all levels of habitat fragmentation producing
near certain extinction. However, the cohort extinction-risk curves
(Fig. 1C, N) are a far left-shifted version of the standard and
gravid-female extinction-risk curves (Fig. 1A–B; , ). Cohort
dispersal left-shifts the extinction-risk curves by generating very
lumpy dispersal at much lower levels of habitat fragmentation
than either standard or gravid-female dispersal do (compare: ,
, N; Fig. 2). Because cohort dispersal is extremely lumpy even
in mildly fragmented habitat, cohort dispersal causes the average
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
population density to decline precipitously when the habitat is
broken into just two fragments (Fig. 4B; N). By contrast, an
equivalent decline in average population density under either
standard or gravid-female dispersal requires that the habitat be
broken into thousands of fragments (compare: , , N; Fig. 4B).
The dashed line in Fig. 2B shows that the three dispersal mechanisms produce identical levels of dispersal lumpiness at three
different levels of habitat fragmentation. From the perspective of
dispersal lumpiness, the effective habitat fragmentation level along
the dashed line is the same for all three dispersal mechanisms,
although the actual habitat fragmentation levels are quite different. Importantly, raising the propagule-bundle size is equivalent to
raising the habitat fragmentation level. Therefore, it is the joint action of propagule-bundle size and habitat fragmentation level that
sets the dispersal lumpiness level, the average population density,
and the metapopulation extinction risk.
153
A
6.3. Turning on lumpy dispersal raises metapopulation extinction risk
Uniform dispersal spreads propagules across habitat fragments
in an even layer, with zero spatial variance (Fig. 2, ). By contrast, standard dispersal spreads propagules across habitat fragments with uniform probability, generating a level of lumpiness
that grows with the level of habitat fragmentation (Fig. 2, ).
The metapopulation extinction-risk curves for these two dispersal
mechanisms, where all parameters are otherwise identical, show
very marked divergence (Fig. 5A–B). The curves are identical at
low habitat fragmentation levels, indicating the metapopulation
extinction risk is the same for both dispersal mechanisms. By contrast, the curves are entirely different at high habitat fragmentation levels, indicating that the metapopulation extinction risk under smooth dispersal is far below the metapopulation extinction
risk under lumpy dispersal. This divergence of risk for the smooth
and lumpy dispersal mechanisms provides a striking example of
the power of the musical chairs effect to raise metapopulation extinction risk.
6.4. Recovering the metapopulation extinction risk of a constant
environment
The convergence of the metapopulation extinction risk curves
at high fragmentation levels in Fig. 5 suggests something
interesting may be happening. In the convergence region the
metapopulation extinction risk in the random environment cases
is approaching the metapopulation extinction risk of the constant
environment case. From the perspective of metapopulation
extinction risk, the random metapopulation environment is
effectively constant, even though the local environments in the
patches are fluctuating intensely. Remarkably, the metapopulation
rescue effect has recovered the metapopulation extinction risk of
a constant environment.
The recovery of the metapopulation extinction risk of a constant
environment is driven by the rise of the average per-capita
growth rate to the constant environment per-capita growth
rate, as the habitat becomes sufficiently fragmented. Given a
sufficiently fragmented habitat, with independent random percapita population growth rates in the fragments (ρ = 0), the
average per-capita growth rate, b
R, rises to the expected value of the
random per-capita population growth rates in the patches, E [R],
(Fig. 4A, ρ = 0), where E [R] = r, the constant environment percapita 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 — the
metapopulation extinction risk is the same at all levels of habitat
fragmentation when dispersal is uniform.
B
Fig. 5. Turning on lumpy dispersal raises metapopulation extinction risk.
Metapopulation extinction risk versus habitat fragmentation level (p). (A) Uniform
dispersal (no lumps). (B) Standard dispersal (lumpy). Bottom curve in (A) or (B)
shows metapopulation extinction risk in a constant environment; all higher curves
are 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. The
x-, w -, and z-regions are discussed in the text.
6.5. Environmental correlation impairs the metapopulation rescue
effect
The metapopulation rescue effect relies on the power of
dispersal to raise the average per-capita growth rate as the habitat
fragmentation level rises. For the metapopulation rescue effect
to properly function there must be some degree of asynchrony
between habitat fragments. Fig. 4A shows the average per-capita
growth rate, b
R, for three levels of environmental correlation,
as functions of habitat fragmentation. If the patch number is
sufficiently high and if the local environments are independent
(ρ = 0), the average per-capita growth rate, b
R, rises by the
maximum amount to E [R], the per-capita growth rate in a constant
environment. By contrast, in a partially correlated environment
(ρ = .4) the rise in b
R falls short of the maximum. In the worst case,
in an exactly correlated environment (ρ = 1) there is no boost
to b
R from dispersal in fragmented habitat — at all levels of habitat
fragmentation the metapopulation grows as if it were a single
population in a random environment. In an exactly correlated
environment there is no metapopulation rescue effect. In related
work, Palmqvist and Lundberg (1998) show that environmental
correlation raises metapopulation extinction risk in models with
local density-dependent dynamics.
6.6. Extinction threshold
Metapopulation extinction is certain if the fraction of suitable
habitat patches drops below a critical value called the ‘‘extinction
threshold’’ (Lande, 1987). In the metapopulation model developed
here, from the perspective of a dispersing propagule, a destination
patch with too many propagules has a degree of unsuitability that
increases with the level of overcrowding there. Raising dispersal
lumpiness is tantamount to reducing the quality of the suitable
habitat patches in the metapopulation. If dispersal is very lumpy,
154
K. Higgins / Theoretical Population Biology 76 (2009) 146–155
the effective fraction of suitable habitat patches drops below the
extinction threshold. The extinction threshold behavior predicted
by Lande is self-evident in the gravid-female and cohort dispersal
z-regions of Fig. 1B–C (, N).
7. Discussion
The metapopulation model developed here brings together
several important mechanisms to investigate metapopulation
extinction risk. The joint action of random environments, densitydependent survival, dispersal, and habitat fragmentation; affect metapopulation extinction risk in ways that might not be
anticipated from the action of these forces in isolation. Two important mechanisms are identified that provide a complete explanation for the metapopulation extinction risk produced by the model.
Metapopulation extinction risk is reduced by dispersal in an increasingly fragmented habitat when there is some degree of asynchrony between local random per-capita growth rates. Dispersal in
an increasingly fragmented habitat raises the average per-capita
growth rate of the metapopulation by reducing the generationto-generation variability of metapopulation growth. Lewontin and
Cohen (1969) showed that such generation-to-generation variability may be devastating to single-population growth; and here
it is no less devastating to metapopulation growth. Metapopulation extinction risk is raised by lumpy dispersal in an increasingly fragmented habitat. Lumpy dispersal reduces the average
population density in fragmented habitat with localized densitydependent competition for resources. Because survival is density
dependent, reduced survival in overcrowded patches is not offset
by improved survival in underpopulated patches. Although both
mechanisms are driven by dispersal, the counteracting mechanisms affect metapopulation extinction risk in opposite fashion,
with the habitat fragmentation level deciding which mechanism
prevails. If habitat fragmentation is mild, dispersal raises the average per-capita growth rate, lowering metapopulation extinction
risk. If habitat fragmentation is severe, dispersal raises the percapita growth rate by the maximum amount, but this time lumpy
dispersal reduces the average population density below the extinction threshold, making metapopulation extinction certain. That the
joint action of dispersal and habitat fragmentation might either
raise or lower metapopulation extinction risk is consistent with
the results from 17 empirical investigations on the effects of habitat fragmentation on biodiversity — more than half showed that
habitat fragmentation had positive effects (Fahrig, 2003).
In a sufficiently large single-population, demographic stochasticity is a relatively weak extinction force, by comparison to environmental stochasticity (Lande, 1993). In the metapopulation
model developed here, in a sufficiently fragmented habitat, the
ordering is reversed, with demographic stochasticity becoming a
stronger extinction force than environmental stochasticity. Demographic stochasticity comes from a surprising source — global
dispersal. Global dispersal of discrete propagules in fragmented
habitat has great potential to generate highly variable propagule
densities, that then interact with localized density-dependence
to generate high systemic risk for the metapopulation. Thus, demographic stochasticity, usually associated with local events in a
small population, here generated by a global process, threatens
global extinction.
That demographic stochasticity might cause systemic effects
in large spatially subdivided populations was shown in important
early work by Chesson (1978, 1981). Chesson found that the joint
action of patchiness, localized variability, and density-dependence
has great potential to thoroughly alter population dynamics at
the system level. More recent work, on the extinction of isolated
local populations, unconnected by dispersal, finds that the joint
action of patchiness, localized variability, and density-dependence
is, 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 to
model habitat fragmentation, given that both features contribute
to dispersal lumpiness. Importantly, dispersal lumpiness may be
intensified by barriers that prevent, or hinder, efficient dispersal
to suitable habitat fragments. The upshot is that dispersal barriers
may cause the effective habitat fragmentation level to be far
greater than the actual habitat fragmentation level. Dispersal
barriers, especially those erected by humans, may make an
otherwise highly resilient metapopulation extremely vulnerable to
extinction.
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 may
have similar potential to reduce the genetic effective size, even
though the biology is quite different. Lumpy dispersal produces
a propagule-bundle rain over all habitat fragments, like an island
model. However, the local populations created by lumpy dispersal
are composed of propagules that are more likely to have a common geographic origin than a random sample from the pool of all
propagules, similar to a stepping-stone model. Thus, lumpy dispersal makes mating between close relatives more likely; and even
though dispersal is global, inbreeding occurs as if dispersal were
more local. Such novel behavior suggests that a metapopulation
genetics model with lumpy dispersal is an island/stepping-stone
hybrid.
The dynamically rich and well-studied wasp–butterfly–plant
system in Southwest Finland (Ehrlich and Hanski, 2004) affords
several opportunities where both the metapopulation rescue effect
and the musical chairs effect are likely to be important drivers
of metapopulation extinction dynamics. The pairwise interactions
between constituent species form a hierarchy with several
levels (Lei et al., 1997; Kuussaari, 1998; Lei and Hanski, 1998). The
butterfly larvae feed on two host plant species that live on about
4000 dry meadows on the Åland islands in the Baltic Sea. In turn,
the butterfly larvae are parasitized by two specialist parasitoid
wasps. Finally, two hyperparasitoid wasps parasitize the two larval
parasitoids. For a given exploiting-species (one of the four wasps or
the butterfly), dispersal over the spatially variable patch-densities
of its host may raise the exploiting-species’ average per-capita
growth rate. The actual rise will be sensitive to the amount
of spatial correlation in the host’s local population densities.
Further, for a given exploiting-species, its average population
density may be reduced by lumpy dispersal. However, dispersal
lumpiness is highly sensitive to search efficiency, a property that
varies markedly between exploiting-species. The great potential
of the Åland system for systemic growth and density effects –
caused by patchiness, localized variability, density-dependence
and dispersal – make Åland an exemplary system for the study of
metapopulation extinction.
Acknowledgments
I thank Oscar Gaggiotti, Ilkka Hanski, Jerry Hilbish, Beth Krizek,
Don Krizek, Simon Levin, Dave Wethey, and the anonymous
reviewers for their generous help. I am grateful to Jinhua Wu; who
did a yeoman’s job on the simulations. This work was supported by
National Science Foundation Grant DEB–0238354 and by start-up
funding from the University of South Carolina.
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