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vol. 160, no. 2 the american naturalist august 2002
A Unifying Framework for Metapopulation Dynamics
Karin C. Harding
1,
*
and John M. McNamara
2,†
1. Department of Marine Ecology, Go¨teborg University, Box 461,
405 30 Go¨teborg, Sweden;
2. School of Mathematics, University of Bristol, University Walk,
Bristol BS8 1TW, United Kingdom
Submitted March 12, 2001; Accepted January 18, 2002
abstract: Many biologically important processes, such as genetic
differentiation, the spread of disease, and population stability, are
affected by the (natural or enforced) subdivision of populationsinto
networks of smaller, partly isolated, subunits. Such “metapopula-
tions” can have extremely complex dynamics. We present a new
general model that uses only two functions to capture, at the meta-
population scale, the main behavior of metapopulations. We show
how complex, structured metapopulation models can be translated
into our generalized framework. The metapopulation dynamics aris-
ing from some important biological processes are illustrated: the
rescue effect, the Allee effect, and what we term the “antirescue
effect.” The antirescue effect captures instances where high migration
rates are deleterious to population persistence, a phenomenon that
has been largely ignored in metapopulation conservation theory.
Management regimes that ignore a significant antirescue effect will
be inadequate and may actually increase extinction risk. Further,
consequences of territoriality and conspecific attraction on meta-
population-level dynamics are investigated. The new, simplified
framework can incorporate knowledge from epidemiology, genetics,
and population biology in a phenomenological way. It opens up new
possibilities to identify and analyze the factors that are important
for the evolution and persistence of the many spatially subdivided
species.
Keywords: metapopulation, rescue effect, multiple equilibria, Allee
effect, antirescue effect.
The spatial structure of a population can profoundly affect
genetic, epidemiological, and population dynamic pro-
cesses within that population. For example, subdivision of
a population can influence gene flow and enhance the rate
of loss of genetic diversity in subpopulations (Lacy 1987;
* Corresponding author; e-mail: kharding@whoi.edu.
†
E-mail: john.mcnamara@bris.ac.uk.
Am. Nat. 2002. Vol. 160, pp. 173–185. 䉷2002 by The University ofChicago.
0003-0147/2002/16002-0003$15.00. All rights reserved.
Pannell and Charlesworth 1999), which sometimes results
in harmful levels of inbreeding (Sacceri et al. 1998). How-
ever, spatial subdivision is also a requirement for beneficial
local adaptations to occur (Rola´n-Alvarez et al. 1997; Joshi
et al. 2001). Within epidemiology, the rate of contact be-
tween subpopulations crucially influences the develop-
ment of a disease outbreak (Grenfell and Harwood 1997).
Population biologists have found a wide variety of cases
where spatial structure influences the stability of single-
and multispecies systems, both in nature and in theoretical
settings (Tilman and Kareiva 1997; Hanski 1999). All the
processes exemplified above also have the potential to af-
fect the viability or extinction risk of single subpopula-
tions. An appreciation of spatial influences on biological
processes has proven important for the understanding of
basic evolutionary and ecological phenomena. This ap-
preciation is also urgently needed if we are to improve the
management of the many endangered species living in
fragmented habitats.
However, the construction and analysis of mathematical
models of spatially structured populations has turned out
to be a formidably complicated task. The processes within
a subpopulation can be as complex as in an ordinary single
population and are, in addition, intricately affected by the
dynamics of neighboring patches via the migration of in-
dividuals. Impressive efforts have been made to model
such “metapopulations,” taking into account the size,
growth rate, and age structure of each subpopulation (Lev-
ins 1969; Hastings and Volin 1989; Allen et al. 1993), patch
quality (Hanski 1994), matrix quality (Vandermeer and
Carvajal 2001), correlated random events (Harrison and
Quinn 1989), and interactions with predators (Taylor
1990; Holt 1997). A few models have even attempted to
predict the dynamics of real-world metapopulations (Han-
ski 1994). These detailed, so-called structured models are
necessary to describe specific populations or specific pro-
cesses. Unfortunately, as the models describe specific sit-
uations and populations with increasing accuracy, they
inevitably grow in complexity and have been found to be
unsuited for establishing general principles concerning
metapopulation dynamics: “This general framework, how-
ever, leads to a class of models that are typically too com-
plex to analyze” (Hastings and Harrison 1994, p. 175).
(With correct figure no 1)
174 The American Naturalist
“Although limited mathematical analysis is possible, … it
is hard to extract many useful ecological results and pre-
dictions without making some further simplifying as-
sumptions” (Hanski 1999, p. 64).
Simplified models are needed to address general ques-
tions in biology that concern the whole metapopulation
scale. However, the simple metapopulation models that
have been available, Levins’s (1969) model and variants
thereof, are too restricted (Gyllenberg et al. 1997). As we
will show in this study, this is because they do not allow
for a range of biological phenomena affecting the dynamics
of metapopulations at the metapopulation level. While a
good general model does not have to include all details
of the real world to give interesting insights, it should be
able to express the dynamics of the system, at the level it
attempts to capture (Caswell 1988). While it has been
argued that the shortcomings of the unstructured Levins
model stem from its many unrealistic assumptions about
the local dynamics of subpopulations (Burgman et al.
1993; Hanski and Zhang 1993), we show that the limi-
tations are instead at the description of the metapopula-
tion-level processes themselves. We use the vital core of
Levins’s model to build a new phenomenological model
that is unstructured but nonetheless captures much more
of the dynamic flexibility of metapopulations.
The new model is used to visualize and classify the
metapopulation dynamics that can arise from different
main types of extinction and colonization processes. We
show how structured models can be incorporated into our
simplified framework, making it easier to understand the
origin of their behavior. Finally, we give an example on
how extinction times of small metapopulations are influ-
enced by the assumed shape of the colonization and ex-
tinction functions.
About Levins’s Model
Richard Levins introduced a new approach to the study
of patchy populations (Levins 1969). In his model he as-
sumes that the environment is composed of Tidentical
habitat patches. Patches are classified as either occupied
or empty; there is no additional local structure. Occupied
patches send out emigrants at rate m, and these migrants
colonize unoccupied patches. Occupied patches go extinct
with the rate e. When the two processes of extinction and
colonization counterbalance there is an equilibrium num-
ber of occupied patches. The characterization of equilibria
is important since natural metapopulations are expected,
and are sometimes also found, to move toward such “equi-
libria of patch occupancy” (Hanski 1999).
However, in Levins’s model, it is assumed that the arrival
of a migrant at an empty patch always turns that patch
into an occupied patch. Thus, the colonization rate of each
patch in Levins’s model is preset; it increases proportion-
ally with the number of patches occupied (N). Conse-
quently, total colonization rate in the whole metapopu-
lation (C) is given by . Further, Levins’sCpmN(1 ⫺N/T)
model assumes the extinction rate per patch to be a con-
stant, independent of the number of surrounding occupied
patches. Thus, the total extinction rate (E) in a Levins-
type metapopulation is preset to just increase linearly as
the number of occupied patches increases: .EpeN
As we will demonstrate, these preset functions (Cand
E) leave out important biological processes that act at the
metapopulation scale and are the reasons why Levins’s
model cannot act as a good phenomenological model for
metapopulations. In order to capture the dynamics of
metapopulations, we must allow for different types of de-
pendencies between migration and colonization rates and
between migration and extinction rates. This is because
per patch colonization rate can differ systematically from
the assumed function of Levins’s model; for example, it
might take several individuals to establish a new subpop-
ulation so that the colonization rate can have a nonlinear
dependence on migration rate (see “Allee Effect” and “The
Effects of Differential Migration”). Also, the influx of mi-
grants can significantly affect the subpopulation extinction
rate. This effect can be positive (e.g., by the demographic
contribution of immigrants to small populations) or neg-
ative (e.g., by the introduction of a lethal disease); see
“Rescue Effect” and “Antirescue Effect.” We develop a
framework that can embrace most of the relevant meta-
population-level processes (i.e., those that affect patch
occupancy).
A Generalized Metapopulation Model
Faithful to the classic metapopulation scenario of Levins
(1969), we let the environment be composed of Tdiscrete
patches, where at any given time each patch is either oc-
cupied or not. Each occupied patch sends out successful
migrants at rate m. (By “successful” migrants, we mean
those that survive to reach another patch.) However, in
contrast to traditional patch-occupancy models, we will
not assume specific shapes of the functions for the rates
at which patches are colonized (C)orgoextinct(E). And,
importantly, we shall allow for the possibility that the ex-
tinction rate is influenced by the rate of immigration. We
define two general functions, the colonization rate per
patch C
patch
(a) and the extinction rate per patch E
patch
(b),
where adenotes the rate of arrival of colonizers onto an
unoccupied patch and bdenotes the rate at which im-
migrants arrive at an occupied patch. We assume thatthese
arrival rates for immigrants are functions a(N) and
b(N) of the number of occupied patches (N). The rate at
Unifying Metapopulation Model 175
which empty patches are colonized in a metapopulation
according to this new generalized framework is given by
C(N)p(T⫺N)C[a(N)]. (1)
total patch
The rate at which occupied patches switch to empty
patches E
total
(N) in the new framework is
E(N)pNE [b(N)]. (2)
total patch
These two functions (eqq. [1], [2]) are the core of the
new flexible model. By allowing for various dependencies
between extinction and colonization on migration we now
can account for most metapopulation-level dynamics. To
find the rate at which the number of occupied patches
changes with time in the metapopulation, we make the
same deterministic approximation to the stochastic change
in Nas is made in Levins’s model. That is, we assume that
the number of patches is large enough to make stochastic
deviations in Ninsignificant. Then the rate of change in
patch occupancy is given by
dN
pC(N)⫺E(N). (3)
total total
dt
The immigration rates a(N) and b(N) could be different
and have a complex dependence on the number of oc-
cupied patches, N. However, for ease of exposition, we
assume that migrants settle at random so that apbp
. This assumption will be relaxed later. SettingmN/T
in equation (1), we obtaina(N)pmN/TC(N)p
total
, where denotes the proportionT(1 ⫺x)C(mx)xpN/T
patch
of occupied patches. Similarly, setting inb(N)pmN/T
equation (2), we obtain . SinceE(N)pTxE (mx)
total patch
, equation (3) now reduces todN/dt pTd x /dt dx /dt p
, wheref(x)
f(x)p(1 ⫺x)C(mx)⫺xE (mx). (4)
patch patch
When extinction and colonization rates counterbalance,
we have an equilibrium proportion of occupied patches
( ). At equilibrium we have , so that satisfies
∗∗
xdx/dt p0x
. If , the equilibrium is stable, and the
∗∗
f(x)p0f(x)!0
metapopulation will deterministically return to the equi-
librium after a small perturbation. If , the equi-
∗
f(x)10
librium is unstable and a small perturbation away from
the equilibrium will either make the metapopulation move
toward another equilibrium of patch occupancy or cause
metapopulation extinction. We will now look at the meta-
population dynamics that arise from several different types
of biologically interesting colonization C
patch
and extinction
functions E
patch
. However, we first look at the traditional
Levins model again. We will repeatedly return to this
model, but as a specific, basic case of the generalized
framework.
Levins’s Functions Rewritten
In Levins’s model the arrival of an immigrant at an un-
occupied patch changes that patch into an occupied patch.
This can be written as in our generalizedC(a)pKa
patch
model, where the constant Kis the proportion of immi-
grants arriving at an unoccupied patch that succeed in
establishing an occupied patch. The total colonization rate
C
total
(N) in the metapopulation thus satisfies
C(N)NN
total
p1⫺Km p(1 ⫺x)Kmx.(5)
()
TTT
Levins’s model assumes the patch-specific extinction
rate to be , where eis a constant, which givesE(b)pe
patch
E(N)eN
total
ppxe.(6)
TT
Equation (4) becomes
f(x)p(1 ⫺x)Kmx ⫺xe px(Km ⫺e⫺Kmx). (7)
Thus, the preset shapes of the functions limit Levins’s
model to the single equilibrium . Because
∗
xp1⫺e/Km
, the equilibrium is stable. The well-known dy-
∗
f(x)!0
namics of Levins’s model is visualized in figure 1. An
equilibrium point occurs where the colonization rate
equals the extinction rate (fig. 1A). It can be seen that for
any given value of the migration rate (m) the two functions
intersect only once (fig. 1B). A higher migration rate al-
ways implies a higher patch occupancy (N/T) at equilib-
rium in Levins’s metapopulation scenario. Below we will
illustrate what happens with the dynamics of metapopu-
lations for a range of examples when migration has a
different influence on subpopulation extinction and
establishment.
Rescue Effect
The demographic contribution of migrants to small dwin-
dling subpopulations may reduce subpopulation extinc-
tion risk—the “rescue effect” (Brown and Kodric-Brown
1977). Logically, if all subpopulations are of the same size
and have the same migration rate (as assumed in Levins’s
model), there will be no rescue effect, since subpopulations
will lose as many (or more) migrants as they gain. There-
fore, the rescue effect has primarily been included in more
complex models (Gyllenberg et al. 1997; but see Hanski
176 The American Naturalist
Figure 1: Metapopulation dynamics for Levins’s classic metapopulation
model showing the dependence of the colonization rate (C) and extinc-
tion rate (E) on the proportion of occupied patches (N/T). The pro-
portion of occupied patches is in equilibrium when , that is, whenCpE
the colonization and extinction function intersect. When the migration
parameter mis small, the colonization and extinction function do not
intersect (except at 0) and there is no equilibrium at whichthe population
is not extinct. When the migration parameter mis sufficiently large, they
intersect once and there is a unique stable equilibrium. A,C, and E
illustrated for . B, Colonization rate (white surface) and extinc-mp0.5
tion rate (dark surface) as functions of the proportion of occupied patches
(N/T) for a range of migration parameter m. Equations (5) and (6) with
and . (Concerning the range of migration rates [m] thatep0.2 Kp1.0
are illustrated in figs. 1–6, the absolute value of mcan be rescaled by
rescaling the time, so it has no significance in the figures. In contrast,
is dimensionless, and its absolute value is critical in determiningKm/e
metapopulation properties.)
Figure 2: Metapopulation with a general rescue effect; that is, the per
patch extinction risk decreases with N/Tand m.A, For some value of
the migration parameter (m) a second equilibrium, which is unstable,
occurs at the left side of the “hill,” here illustrated for . B,mp0.37
Colonization rate (white surface) and extinction rate (black surface) for
a range of the migration parameter m. Colonization rate as in Levins’s
model (eq. [5] with ). Extinction rate given by equation (8) withKp1.0
and .Ap10 ep0.5
1982; Gotelli and Kelly 1993). In our framework, the rescue
effect has a broader definition than just the effect of demo-
graphic enhancement. Here a decreasing extinction risk
with higher immigration rate (b)—regardless of the un-
derlying mechanism of this relationship—will define a res-
cue effect. For example, immigration may reduce local
extinction risk by reducing inbreeding (Sacceri et al. 1998)
or dampening local stochastic fluctuations (Allen et al.
1993; Burgman et al. 1993). What matters at the meta-
population level is that the net effect of increased migration
is a reduction of the local extinction rate. Different bio-
logical processes can be expected to give rise to different
shapes of the extinction function, and these shapes could
be classified. The shape of the extinction function is crucial
to metapopulation dynamics, as we now illustrate. If im-
migration reduces the risk of inbreeding, the generalized
rescue effect should be most pronounced initially, as a few
migrants per generation are sufficient to genetically ho-
mogenize a metapopulation (Hastings and Harrison 1994).
To illustrate this type of mechanism, we assume the col-
onization function of Levins (eq. [5]) but take
e
E(b)p(8)
patch
1⫹Ab
(where Ais a positive constant), so that bhas a strong
positive effect at low values. The result of this generalized
rescue effect is to bend the extinction planes in figures 1A
and 1Buntil those in figures 2Aand 2Bare obtained. If
a stable metapopulation size is possible in Levins’s model
(the current model with ), then including the rescueAp0
effect ( ) simply increases the stable size. However, ifA10
Levins’s model predicts the metapopulation would go ex-
tinct ( ), a stable metapopulation size can still existKm !e
when a rescue effect is present. And, interestingly, if there
is such a stable size, then it can be shown that there must
be a second, smaller, unstable metapopulation size as well
(fig. 2A). This phenomenon of multiple equilibria in patch
correct
figure
Unifying Metapopulation Model 177
Figure 3: The dynamics for a metapopulation with another type of rescue
effect, which can have three equilibria. The equilibrium in the middle is
unstable. A, Cross section of Bat . B, Colonization rate (whitem p0.95
surface) and extinction rate (black surface) for a range of the migration
parameter m. Colonization rate as in Levins’s model (eq. [5] with
). Extinction rate given by equation (9) with andKp1.0 Ap8ep
.0.7
Box 1: Increased Migrations Can Increase
Extinction Risk
A. The periwinkle. The periwinkle, Littorina saxatilis, has dif-
ferent ecotypes living in different microhabitats, just a few meters
apart. Attacks from crabs favor large, ridged shells in the upper
shore, whereas heavy wave action promotes small, smooth shells
at the lower shore. The two morphs and their hybrid survive
equally well along a narrow zone of intermediate habitat (Rola´n-
Alvarez et al. 1997). The parental morphs as well as hybrids have
very low survival if transplanted to a nonnatal microhabitat. In-
terestingly, gene flow between the two morphs is strongly reduced
by the combination of nonpelagic larvae and assortative matings
(Rola´n-Alvarez et al. 1997). Thus, a low migration rate and as-
sortative mating strongly promote the locally adapted genotypes
that are crucial to periwinkle survival in different microhabitats.
B. A New Zealand weevil. A metapopulation of the weevil Had-
ramphus spinipennis inhabits steep, partly isolated creeks along
the shores of two islands in New Zealand. The subpopulations
of weevils overexploit the host plant and have frequent, asyn-
chronous local extinctions (Scho¨ps 1999). The distribution area
of the host plant was dramatically increased after a deforestation
of the islands. The increased connectivity between subpopulations
increased the local extinction rates. Future survival of H. spini-
pennis depends on a maintained low connectivity between sub-
populations (Scho¨ps 1999).
occupancy has earlier been described in more complex
models only (Gyllenberg et al. 1997).
Metapopulation dynamics depend crucially on the form
of the relationship between the immigration rate and the
reduction in extinction risk. For example, a linear de-
creasing extinction risk, , has only oneE(b)pe⫺Ab
patch
equilibrium. In contrast, for threshold populations, ex-
tinction rate may be little affected by immigration rate
while this rate is small but may drop rapidly once a thresh-
old immigration rate is reached (May 1977). To illustrate
this mechanism, consider this case:
e
E(b)p.(9)
patch 4
1⫹Ab
For this function the metapopulation has three equilibria
for some values of eand m(fig. 3). When three equilibria
occur, two stable equilibria are separated by an unstable
equilibrium. For low patch occupancy the rescue effect is
weak and the model is similar to Levins’s model. Thus,
when eis slightly smaller than Km, there is a stable equi-
librium at low patch occupancy. As patch occupancy in-
creases above a threshold, the rescue effect takes hold and
the extinction rate plummets below the colonization rate,
resulting in one unstable and a second stable equilibrium
at higher patch occupancy.
Antirescue Effect
The literature on metapopulations has largely ignored det-
rimental effects of immigration that are well known from
epidemiology and genetics. An increased extinction rate
with increasing immigration rate, the “antirescue effect,”
might be due to immigrants carrying parasites or diseases
(Grenfell et al. 1995; Grenfell and Harwood 1997) or gene
flow reducing local adaptation (Hastings and Harrison
1994; Rola´n-Alvarez et al. 1997; Joshi et al. 2001; see box
1A). A high migration rate can also cause synchronous
fluctuations in local population sizes, which increases the
extinction risk for the whole metapopulation (Allen et al.
1993; Burgman et al. 1993; Scho¨ps 1999; box 1B). Al-
though our framework cannot explicitly handle these pro-
cesses, it can capture the main effect at the metapopulation
level by allowing local extinction risk to increase with im-
migration rate (fig. 4). Consider a case where the colo-
nization rate is as in Levins’s model (eq. [5]) but the per
patch extinction function is increasing in b:
178 The American Naturalist
Figure 4: A metapopulation with an antirescue effect. A, The hampering
effect of an increased immigration rate leads to increased extinction risk
as compared to the extinction risk assumed by Levins’s model (dashed
line); . B, Colonization rate (white surface) and extinction ratemp1.1
(black surface) for a range of the migration parameter m. Colonization
rate as in Levins’s model (eq. [5], with ). Extinction rate givenKp1.0
by ; .
22
E(N)/Tpe(N/T)[1 ⫹m(N/T)] ep0.3
total
Figure 5: A metapopulation with patch colonization subject to an Allee
effect. The bell-shaped colonization function indicates poor colonization
success at small N/T. Unstable equilibrium points occur at the left side
of the “hill.” A, Cross section of Bat . B, Colonization rate (whitem p0.6
surface) and extinction rate (black surface) for a range of the migra-
tion parameter m. Colonization rate given by equation (1), where
and . Extinction rate as in Levins’s model
22 2
C(a)pa/(a⫹y)yp0.6
patch
(eq. [6], with ).ep0.12
E(b)ph(b), (10)
patch
where . This gives
h(b)10
f(x)px[(1 ⫺x)Km ⫺h(mx)]. (11)
It is straightforward to show that if , then
∗
f(x)p0
. Thus, there can be only one equilibrium point
∗
f(x)!0
and it is stable. The equilibrium patch number is
∗
N
smaller than in Levins’s model with extinction rate ep
(fig. 4A). One consequence of the antirescue effect,h(0)
opposite to the earlier investigated scenarios, is that in-
creased migration will not always lead to successively larger
(fig. 4B). In particular, if h(b)/bis increasing for large
∗
N
b(e.g., ), then will decrease for large
2∗
h(b)pe[1 ⫹b]N
m(but never reach 0). This might have important bio-
logical consequences; for example, conservation efforts
such as increasing the connectivity between patches or
artificially colonizing empty patches in a threatened meta-
population can be detrimental if the antirescue effect is
strong.
Allee Effect
In Levins’s model the arrival of an immigrant at an un-
occupied patch changes that patch into an occupied patch.
This is generalized to in our model. How-C(a)pKa
patch
ever, colonization rate need not be proportional to im-
migration rate. For example, it may be difficult initially
to establish subpopulations because of demographic, sto-
chastic, or genetic complications at low population num-
bers—the “Allee effect” (Allee 1931). Consider the S-
shaped colonization function ,
22 2
C(a)pa/(a⫹y)
patch
where yis a constant (as earlier used in a structured model
by Hanski [1994]) and . At the metapopu-E(b)pe
patch
lation level, the most important consequence of this
change in the colonization function is that there can be
two equilibria of patch occupancy (fig. 5). If patch oc-
Unifying Metapopulation Model 179
cupancy is below , extinction rate exceeds colonization
∗
N
L
and the population rapidly goes extinct. Above and
∗
N
L
below the higher equilibrium colonization rate exceeds
∗
N
H
extinction rate, and the patch occupancy will increase until
is reached (fig. 5A). Other similar S-shaped coloni-
∗
N
H
zation functions give the same qualitative results.
The Effects of Differential Migration
So far we have assumed that migrants settle on a patch
chosen at random. Migrants may, however, avoid patches
that are already occupied in order to avoid competition
for territories or food. Alternatively, migrants seeking a
mate, or those at risk from predation while alone, may try
to settle on already occupied patches. When there is such
differential migration, the rate of arrival of potential col-
onizers is no longer given by , and this willa(N)pmN/T
change how colonization rate depends on N. To investigate
the effect of differential migration, we take the per patch
colonization and extinction rates to be andC(a)pKa
patch
, respectively, as in Levins’s model. Note thisE(b)pe
patch
means that does not depend on NwhenC[a(N)]/ N
patch
.a(N)pmN/T
Consider first the extreme case when migrants avoid
occupied patches and each settles on an empty patch cho-
sen at random. Then and hencea(N)pmN/(T⫺N)
C(N)
total
pKmx.(12)
T
If , then the metapopulation goes extinct as in Lev-Km !e
ins’s model. If , then at the unique equilibrium allKm 1e
patches are occupied. In reality, differential migration is
not likely to be this extreme, particularly since not all
migrants will be able to find an unoccupied patch when
these are rare. Different degrees of this “repellent behavior”
will produce different colonization functions, but all have
steeper initial increase and result in higher equilibrium
patch occupancies (N/T) than the symmetrical coloniza-
tion function of Levins’s model.
Now suppose that migrants seek occupied patches. In
the extreme case in which there is no immigration to un-
occupied patches, there is no colonization, and the pop-
ulation will go extinct. Suppose instead that migrants settle
on an unoccupied patch if they do not soon encounter an
occupied patch. Then a(N)/Nwill be a decreasing function
of N. Thus, the function C
patch
[a(N)]/Nwill be decreasing
in N. This is in contrast to models in which there is
an Allee effect and no differential migration, where
C
patch
[a(N)]/Nis an increasing function of N. In this sense
there is an “anti-Allee effect” when migrants seek occupied
patches.
We can also consider the effect of differential migration
on the extinction rate. When migration rate affects per-
patch extinction rate, as happens when there is a rescue
or antirescue effect, differential migration will interactwith
these effects to distort the extinction function E
total
(N). For
example, suppose that the patch extinction rate increases
with immigration rate as a result of the spread of disease.
Then extinction rate will be lowest when few patches are
occupied. Suppose that migrants also try to avoid occupied
patches. Since they will be most successful at doing so
when few patches are occupied, the extinction rate will be
further reduced when few patches are occupied but could
be little affected when most patches are occupied. Thus,
in this example, the differential migration would accen-
tuate the antirescue effect in the metapopulation even
more.
Translating Structured Models
Metapopulation models in which local populations have
structure appear fundamentally more complex than our
generalized metapopulation model. However, as the fol-
lowing example shows, they can be translated into our
much simpler framework while still preserving the im-
portant metapopulation-scale features.
Let us first consider a simple structured metapopulation
model that has been developed to study metapopulations
where immigration positively affects local populations, in-
creasing population density and reducing population ex-
tinction risk. Thus, this is a model designed to capture
the rescue effect (Hanski 1985; see also Hanski and Gyl-
lenberg 1993; Hanski 1999). Assume that patches are in
one of three states: empty (state 0), low population (state
1), or high population (state 2). A low-population patch
transforms into a high-population patch at rate r
1
(a),
where ais the rate at which it receives immigrants. A
high-population patch changes to a low-population patch
at rate r
2
(a). We assume that r
1
(a) is an increasing function
of aand r
2
(a) is a decreasing function. A patch in state
i( ) sends out migrants at rates m
i
. We assume thatip1, 2
. A patch in state i( ) changes into anm!mip1, 2
12
empty patch at rate e
i
. We assume . An empty patche1e
12
changes into a low-population patch at rate Ka.
We now assume that extinction and colonization rates,
e
1
,e
2
, and Ka, are small compared with the rates r
1
and
r
2
. (Later we comment on the case when this approxi-
mation does not hold.) Then the proportion of occupied
patches that are of each type can be assumed to equilibrate
before there is a change in the number of occupied patches
due to a colonization or extinction event. Specifically, let
there be Noccupied patches, of which a proportion rare
high population. Then the rate at which patches receive
immigrants is
180 The American Naturalist
Figure 6: An example of how the dynamics of a three-state model (with
empty, small, and large populations in a patch) can be represented by
the two functions of the generalized framework (eqq. [15], [16]). Kp
,, ,, ,.A, Cross section of1mp0ep0.03 ep0rp1⫹10arp4
11 21 2
Bat . B, Colonization rate (white surface) and extinction ratemp0.9
2
(black surface) for a range of the migration parameter . Thempm
2
unstable equilibrium point at low patch occupancy is due to a rescue
effect and a pronounced Allee effect.
N[(1 ⫺r)m⫹rm]
12
ap.(13)
T
Thus, ignoring colonization and extinction events, at equi-
librium the rates at which high- and low-population
patches convert to one another are equal,
Nrr(a)pN(1 ⫺r)r(a). (14)
21
Eliminating afrom equations (13) and (14) gives the equi-
librium proportion of occupied patches that are high pop-
ulation, r(N). Our assumptions about the dependence of
r
1
(a) and r
2
(a)ona, and that , guarantee thatm!m
12
r(N) is an increasing function N. The total patch extinc-
tion rate in the metapopulation is then
E(N)pN{[1 ⫺r(N)]e⫹r(N)e}
total 1 2
pN[e⫹r(N)(e⫺e)], (15)
121
and the total colonization rate is
{[1 ⫺r(N)]m⫹r(N)m}
12
C(N)p(T⫺N)KN
total
T
[m⫹r(N)(m⫺m)]
121
p(T⫺N)KN .(16)
T
These two functions are illustrated in figure 6. In this
model, colonization of empty patches is as in Levins’s
model, and the extinction rate of a patch of a given type
does not depend on the immigration rate into that patch.
Thus, at the microscopic level there is no rescue effect or
Allee effect. However, at the macroscopic level the model
exhibits both effects (fig. 6). To see this, first focus on the
average per patch extinction rate:
E(N)
total
.(17)
N
This is a constant function of Nin Levins’s model. It
decreases as Nincreases when there is a rescue effect, and
it increases with Nwhen there is an antirescue effect. In
our structured model r(N) is an increasing function of N.
Thus, by equation (15), E
total
(N)/Nis decreasing in N
(since it is assumed that ). In other words, at thee!e
21
macroscopic level, our structured model exhibits a gen-
eralized rescue effect. Now consider the function
C(N)
total
.(18)
(T⫺N)N
In Levins’s model this is constant, whereas it is increasing
in Nwhen there is an Allee effect. In our structured model,
equation (16) shows that the function (18) is increasing
in N(since it is assumed that ). This is synony-m!m
12
mous with a pronounced Allee effect at the macroscopic
level (cf. figs. 5A,6A). The tendency of this model to
produce two equilibria thus stems from generalized rescue
and Allee effects. The dynamics of this type of model have
earlier been claimed to stem from a rescue effect only
(Hanski 1985, 1999; Hanski and Gyllenberg 1993). Thus,
the simplified framework presented here can be used to
avoid confusion about the origin of the dynamics of struc-
tured metapopulation models.
The above method, by which a structured metapopu-
lation leads to an unstructured analogue with the same
metapopulation-level dynamics, is easily generalized (see
next paragraph). To make this translation it is assumed
that colonization and extinction occur on a slower time-
scale than local subpopulation dynamics. When this time-
scale assumption is not met the detailed population dy-
Unifying Metapopulation Model 181
namics of the structured metapopulation model and our
unstructured analogue will no longer agree. However, if
the goal is to determine the equilibria of a structured meta-
population, the approach outlined above is still valid. This
is because, at an equilibrium of a metapopulation, time-
scales are not important since the system is stationary.
Thus, the unstructured analogue, which assumes fast local
dynamics, has the same equilibria as the original structured
metapopulations with slow local dynamics.
More Complex Structured Models
The above approach, in which a three-state model is trans-
lated into an unstructured model, generalizes to more
complex models as follows. Suppose that there are a total
of Tpatches and that the population on a patch may be
of the types , where type 0 correspondsI⫹1 0,1,2, …,I
to the patch being empty. The other states can describe
different patch sizes or actual local population sizes. As in
our example, we assume that patch dynamics is fast relative
to colonization and extinction rates. The number Nof
occupied patches then determines the proportion r
i
(N)
that are type i. Let a type ipopulation contribute migrants
to a pool at rate m
i
. Then the total rate at which migrants
enter the pool is Nm(N), where
I
m(N)pr(N)m.(19)
冘
ii
ip1
If pool members settle on patches at random, the rate at
which unoccupied patches receive immigrants is ap
. Assuming that the colonization rate of an emptym(N)N/T
patch is C
patch
(a), we have
C(N)
total
p(1 ⫺x)C[xm(N)], (20)
patch
T
where . Let a type ipopulation ( )xpN/Tip1, … , I
become extinct at rate e
i
. Then the extinction rate is
, whereE(N)pNE (N)
total patch
I
E(N)pr(N)e.(21)
冘
patch ii
ip1
Thus, a structured model (with constant extinction prob-
ability for given structure) can be replaced by the gener-
alized Levins-type model with no structure but a migra-
tion-rate-dependent extinction risk.
Finite Patch Numbers
So far we have assumed that the total number of patches
(T) is large. It has also been implicitly assumed that col-
onization and extinction events at one location occur in-
dependently of what occurs at other patch locations (given
the current N). It is then reasonable to treat colonization
and extinction events as occurring at a smooth rate. Meta-
population dynamics are consequently deterministic, and
a metapopulation that reaches a stable equilibrium will
remain there indefinitely. Conservation theory is, however,
often concerned with the vulnerability to extinction of
metapopulations that are subject to stochastic effects (Til-
man et al. 1994; Hill and Caswell 2001). The details of
how stochasticity operates are then crucial to questions of
metapopulation persistence (Lande et al. 1998). For ex-
ample, there might be large-scale fluctuations in weather
conditions that affect all patches in a highly correlated way
(Harrison and Quinn 1989). Alternatively, patches could
be independent as in our deterministic model, with the
stochasticity due to the low number of patches (Hanski
1999; Hill and Caswell 2001). We do not attempt a full
analysis of the effects of stochasticity in this article. Instead,
we illustrate the effects of stochasticity in this latter
alternative.
When Tis low, the actual events of extinction and col-
onization become important. Suppose Npatches are
currently occupied. Before, the functions C
total
(N) and
E
total
(N) were interpreted as rates. Now we assume that
the time to wait for the next patch colonization (given no
extinction occurs before then) is an exponentially distrib-
uted random variable with parameter C
total
(N). Similarly,
the time to wait for the next extinction (given no colo-
nization occurs before then) is an exponentially distributed
random variable with parameter E
total
(N). These two ran-
dom variables are assumed to be independent. Under these
assumptions the time for the number of occupied patches
to change has an exponential distribution with mean
. After the change, the number
⫺1
t(N)p[C(N)⫹E(N)]
of occupied patches is with probabilityN⫹1p(N)p
and is with probabilityC(N)/[C(N)⫹E(N)] N⫺11⫺
.p(N)
Figure 7 illustrates the use of this stochastic version of
our generalized model. The figure shows how the mean
time to population extinction depends on the total number
of patches in the environment. In each of the four cases
shown, we have kept the colonization function as in Lev-
ins’s model. The extinction functions presented are for
two different cases of the rescue effect and a case with the
antirescue effect, together with the baseline Levins case for
comparison. When there is an antirescue effect, the mean
time to extinction is only substantially reduced when the
detrimental effect of increased immigration is very strong
182 The American Naturalist
Figure 7: Mean time to extinction for the whole metapopulation as a
function of the number of available habitat patches, for a stochastic
version of the generalized metapopulation model. Initially all patches are
occupied. Extinction times for the following four different extinction rate
functions are illustrated: Levins’s extinction rate (eq. [6]); the rescue effect
given by equation (8) with ; the rescue effect given by equationAp0.25
(9) with ; the antirescue effect given by .
2
Ap10 E(b)pe(1 ⫹5b)
patch
In all cases, and the colonization rate is given by equation (5),ep0.71
with and .Kp1.0 mp0.9
(as in the case shown). This is because the antirescue effect
is strongest when there are many patches occupied, and
extinctions matter least. When the per patch extinction
rate drops rapidly once immigration rates reach a thresh-
old (eq. [9]), there is a stable equilibrium comparable with
that in the Levins case and a second stable equilibrium at
a higher proportion of occupied patches. As a result, the
mean time to extinction is very much above that for Lev-
ins’s model, especially when the total number of patches
is small. In contrast, the advantage of a rescue effect of
the form given by equation (8) has its strongest effect when
the total number of patches is large (fig. 7).
Discussion
By stochastic necessity, all populations will eventually go
extinct, and this is true in particular for small populations.
The persistence of a metapopulation that is divided into
a network of smaller subunits is dependent not only on
the local development within single subpopulations but
also on the overall picture of how many new habitat
patches are colonized each year, compared with the num-
ber of subpopulations that go extinct. The metapopulation
concept introduced by Levins acknowledges this fact by
focusing solely on the rates of colonizations and extinc-
tions (Levins 1969). However, Levins assumed a fixed
shape of the colonization function and also assumed ex-
tinction rate to be a per patch constant. Thereby, Levins’s
model became unable to express the effects of many bi-
ological processes that affect metapopulation dynamics.
Many processes influencing metapopulation dynamics
have their origin in local processes of subpopulations and
are linked to migration. We have mentioned processes such
as local genetic adaptations, inbreeding, demographic con-
tributions, and epidemics. They all act at the local scale,
are mediated via migrations, and can have a profound
impact on the whole metapopulation (its patch occupancy
and/or its persistence). As is evident from the examples
above, the expected influence from migration on subpop-
ulation persistence and metapopulation dynamics can be
very different among metapopulations. The new frame-
work presented here allows for any functional relationship
between migrations and colonization rates and between
migrations and extinction rates. Therefore, although the
new model does not explicitly describe processes operating
at the subpopulation scale, it captures their metapopula-
tion-level effects.
We have sampled the literature for examples of different
functional relationships for Cand Eon migration rate.
With the appropriate choice of functions, the new model
can incorporate the effects of different colonization pat-
terns. Some populations have problems establishing col-
onies at low population sizes and are increasingly suc-
cessful at higher immigration rates. This is allowed for by
the hill-like colonization function in the Allee effect (Allee
1931; Hanski 1994; fig. 5). Also, differential migration be-
havior of individuals can lead to an Allee effect at the
metapopulation level (if individuals prefer empty patches)
or to an anti-Allee effect (if individuals prefer already oc-
cupied patches). Further, by allowing the extinction func-
tion to vary with migration rate the new model describes
how immigration affects existing subpopulations. The ex-
tinction function can describe beneficial effects, such as
demographic contribution of immigrants and prevention
of inbreeding via gene flow with different types of rescue
effects (figs. 2, 3). However, the rate of extinctions can
increase with increased migration rates. Examples of del-
eterious effects of immigration are the spread of disease
and the loss of local adaptation via gene flow. We term
the general phenomenon of harmful effects of increased
immigration on local extinction risk the antirescue effect
(fig. 4).
The actual functions given in this article are to be re-
garded as examples of “main type functions” used to il-
lustrate the phenomenology of how different processes af-
fect overall metapopulation dynamics. For implementation
on specific questions, the appropriate Cand Efunctions
must be found. After mean extinction and colonization
rates have been derived from a natural population, or from
a complex model, our model can describe the meta-
Unifying Metapopulation Model 183
population-scale dynamics in a transparent and easily an-
alyzed way. This makes it possible to classify and compare
the main behaviors of completely different complex
systems.
So what can we expect the Cand Efunctions to be like
in real-world metapopulations? We believe it can be useful
to consider how the metapopulation has been formed.
Many populations live on “naturally” patchy habitats.
Habitats can be networks of rock pools, islands, the in-
testines of birds, or, as a more unusual example, thesunken
carcasses of whales (T. Dahlgren, unpublished manu-
script). We suggest that species evolved in such systems
can be expected to have developed optimal dispersal strat-
egies related to the benefits and risks of migrations in their
environments. Another type of patchy population is ar-
tificially created. Modern farming and forestry have
changed the main vegetation of entire landscapes, leaving
only scattered patches of the once dominant vegetation
types left. Species that have inhabited larger, unbroken
habitats for perhaps thousands of generations might not
have optimal dispersal strategies to cope with the new
constraints imposed by a sudden fragmentation of their
habitats.
The focus of metapopulation research has been directed
toward endangered species, living on small remnants of
their natural habitat. Those populations are likely to ben-
efit from increased migration rates to counteract their im-
posed isolation. Consequently, a rescue effect is included
in most metapopulation models investigating conservation
issues (such as habitat loss). However, it is important to
rule out other functional relationships since high migra-
tion rates can be detrimental for some populations (e.g.,
by inducing synchronous fluctuations in local populations
[Vandermeer and Carvajal 2001] or by loss of genetic di-
versity [Lacy 1987; Pannell and Charlesworth 1999]). The
antirescue effect highlights this possibility in a phenom-
enological way (fig. 4). Two examples where extinction
risk increases as migration rate increases are the periwinkle
(Rola´n-Alvarez et al. 1997; box 1A) and a New Zealand
weevil (Scho¨ps 1999; box 1B). Studies of metapopulations
with antirescue effects can give insights into the evolution
of nondispersal strategies, for example, site fidelity, wing-
lessness, and territoriality. The awareness of how coloni-
zation and extinction functions can differ from what is
mostly assumed in models of metapopulations thus also
has practical significance. The successful management
strategy of a metapopulation with an antirescue effect will
differ dramatically from the common scenario. One illus-
tration of this is how the expected mean time to extinction
will change, depending on the assumed shapes of extinc-
tion and colonization functions, as the number of habitat
patches decreases (fig. 7). We want to underline the ne-
cessity to develop conservation theory for several different
types of metapopulations and to find the relevant colo-
nization and extinction functions for real-world meta-
population before implementation.
Earlier extensions of Levins’s model that tried to make
it more realistic added a linear rescue effect to Levins’s
model to take into account demographic contribution
from migration (Hanski 1982, 1985). Also, Gotelli and
Kelly (1993) suggested that Cand Ecould be any function
of N; however, they only explored the linear rescue effect.
The linear rescue effect gives only one stable equilibrium
of patch occupancy. The dynamic limitations of Levins’s
model and these extensions with linear rescue effect were
believed to be an inherent property of all unstructured
metapopulation models: “The most fundamental differ-
ence between the unstructured and structured metapop-
ulation models is that with the structured models there
can be multiple equilibria” (Gyllenberg et al. 1997, p. 107;
see also Hanski and Gyllenberg 1993). Consequently, the
use of simple Levins-type models in analyzing general
questions such as consequences of patch destruction on
metapopulation persistence (Tilman et al. 1994) has been
criticized (Gyllenberg et al. 1997). We show that a simple
model can express most of the complex behaviors de-
scribed for structured metapopulation models by varying
the different functional relationships between migration,
colonization, and extinction rates. Although there have
previously been specific simple extensions to Levins’s
model, there has been no attempt to systematically inves-
tigate how the form of the colonization and extinction
functions affects metapopulation dynamics. Our article ad-
vocates such a study.
For some studies, the desired general Cand Efunctions
for a specific biological process can be found by first build-
ing a structured model, which then is reduced to our sim-
plified model. The simplified model can capture the es-
sential detail, at the metapopulation level, of the structured
metapopulation model. Once such a simplified model has
been constructed, the dependence of the colonization and
extinction functions on the number of occupied patches
(cf. formulas [17] and [18]) reveals the metapopulation-
scale properties of the original structured model. For ex-
ample, it can be seen whether there is an implicit gener-
alized Allee effect or a generalized rescue effect. Note that
our definition of these effects differs from those of previous
authors (Brown and Kodric-Brown 1977; Hanski and Gyl-
lenberg 1993; Lande et al. 1998). Our definition captures
the important features at the metapopulation scale rather
than at the smaller subpopulation scale. Some structured
models with complex local dynamics (e.g., due to infec-
tious diseases or predator-prey interactions) can exhibit
limit cycles and chaos at the metapopulation level (May
1977; Hassell and Wilson 1997). We propose such complex
dynamics should be captured by the introduction of time
184 The American Naturalist
lags into our unstructured analogue. Studies that address
specific processes (such as influence from explicit spatial
location, time since colonization, and genotype frequen-
cies), must obviously utilize more complex models allow-
ing for these parameters to be estimated. The generalized
model is designed to capture processes that leave a sig-
nature at the metapopulation scale.
The generalized model presented here can be used as a
phenomenological model in the study of general aspects
of metapopulation structure. Interesting areas for theo-
retical applications are environmental stochasticity and ex-
tinction risks for metapopulations with different types of
colonization and extinction functions, evolution of opti-
mal migration rates, time-lagged responses in patch oc-
cupancy, and investigation of the implicit assumptions of
more complex models.
Acknowledgments
We wish to thank the following persons for valuable com-
ments: I. Hanski, T. Ha¨rko¨ nen, R. Holt, A. Houston, H.
Kokko, E. Ranta, A. Sih, and D. Wahlstedt. K.C.H. was
supported by the Oscar and Lili Lamm Research
Foundation.
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Associate Editor: Per Lundberg
