Biological Journal of fhe Linnean Sociep (1991), 42: 57-71. With 3 figures
Structured models of metapopulation dynamics
ALAN HASTINGS
Division of Environmental Studies and Institute of Theoretical Dynamics,
University of California, Davis, California 95616, U.S.A.
I develop models of metapopulation dynamics that describe changes in the numbers of individuals
within patches. These models are analogous to structured population models, with patches playing
the role of individuals. Single species models which do not include the effect of immigration on local
population dynamics of occupied patches typically lead to a unique equilibrium. 'I'he models can be
used to study the distributions of numbers of individuals among patches, showing that both
metapopulations with local outbreaks and metapopulations without outbreaks can occur in systems
with no underlying environmental variability. Distributions of local population sizes (in occupied
patches) can vary independently of the total population size, so both patterns of distributions of local
population sizes are compatible with either rare or common species. Models which include the effect
of immigration on local population dynamics can lead to two positive equilibria, one stable and one
unstahlr, thr latter representing a threshold between regional extinction and persistence.
KEY WORDS:-Metapopulations
-
structured models
-
stability
-
extinction - population size.
CONTENTS
Introduction . . . . . . . . . . . . .
Modelling approach . . . . . . . . . . .
A general model for a single species metapopulation
. . .
Models with total disasters only . . . . . . .
Numerical results for models with total disasters only . .
Models which include disasters that do not lead to extinction
A simpler model . . . . . . . . . . . .
Model formulation
. . . . . . . . . .
Equilibria and stability . . . . . . . . .
Discussion . . . . . . . . . . . . . .
Acknowledgements
. . . . . . . . . . .
References
. . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . .
The threc-statr model as a special case of the complex model
The three-state model . . . . . . . . . .
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INTRODUCTION
The importance of spatial structure in the dynamics of natural populations has
become accepted by most population biologists (Kareiva, 1986). In particular,
the role of spatial distribution of individuals and species has become central to
0024-4066/91/010057 + I5 SOS.OO/O
57
0 1991 The Linncan Society of London
58
A. HASTINGS
questions in conservation biology and many approaches to understanding
population dynamics. Moving beyond the simple recognition that spatial
structure is important to a formal model or to an experimental or field
investigation requires a careful assessment of the spatial and temporal scales over
which the important biological processes of population dynamics take place
(Hastings, 1990a). Broadly speaking, one must determine the spatial and
temporal scales of movement, and the spatial or temporal scales of local
population dynamics. Implicit in this determination of scales are assumptions
about the role of stochasticity in population dynamics. There are several ways
one can include spatial factors in the study of the dynamics of biological
populations (Hastings, 1990a), but here I will restrict attention to the metapopulation approach.
The term metapopulation has been used to refer to populations with frequent
local extinctions, in which the interplay between dispersal and local dynamics
becomes a key to understanding the overall population dynamics. Stochasticity
is important at the level of the local populations. The only tractable approach to
formulating models which include local extinctions as a stochastic process is to
formulate a model in terms of numbers of occupied patches. Most models based
on this approach have considered only the presence or absence of species, as
reviewed by Hanski (1991). Yet, there are many questions for which knowledge
of the numbers within a patch provide vital information. A complete
understanding to the relationship between local stochasticity in population
dynamics and metapopulation persistence will require consideration of numbers
within local populations. The ‘rescue effect’ (Hanski, 1982, 1991), which
considers the role of immigrants into occupied patches, depends on information
about numbers within local populations. Better understanding of the
relationship between persistence and stability in metapopulations (e.g. Murdoch
& Oaten, 1989) also requires knowledge of the numbers within populations.
Here, a stable metapopulation is one which returns to an equilibrium when
perturbed.
I have begun, in collaboration with C. Wolin, investigations of metapopulation models which include numbers within populations (Hastings &
Wolin, 1989; Hastings, 1990a, b; Hastings & Wolin, unpublished), and such
models have also been formulated by Metz & Diekmann (1986). In the present
paper, I will present the model for a single species in a patchy environment
developed in Hastings & Wolin (1989, unpublished) and Hastings ( 1990b),
which includes the effects of local ‘disasters’ on population dynamics. In general,
changes in the numbers of individuals in a local subpopulation result from local
population growth, immigration, emigration and disasters. Here, I separate out
disasters from other aspects of local population growth because they can occur
on a faster time scale. I then present some numerical results for a particular
version of this model which show a variety of different patterns consistent with
natural metapopulations (Hanski, 1991 ; Harrison, 1991) .
It is difficult to analyse these models when immigration into occupied patches
is an important factor. Thus, I will therefore present an alternative simpler
model, developed in Hanski (1985), which can be solved more easily and can be
used to address questions of the importance of immigration to existing
populations in metapopulation dynamics. I will also consider explicitly the
relationship between these simpler models and the more detailed ones, showing
SI‘RUCI’URED MODELS
59
in the Appendix that with appropriate parameter choices, the simpler model can
be derived as a special case of the more detailed model.
Many natural populations well described by these metapopulation models are
described in the contribution by Hanski (1991). One example of a population for
which this modelling approach is appropriate are the aphids on fireweeds
studied by Addicott (1978).
MODELLING APPROACH
Before considering explicitly the assumptions and the models, I will discuss
heuristically the motivation behind structured metapopulation models which
include numbers of individuals within populations. All ‘patch’ models of metapopulations are formally analogous to traditional models for population
dynamics in ecology, with the populations playing the role of individuals. The
simple models for mctapopulations rcvicwed by Hanski (1991) make the
simplifying assumption that all patches within a class are identical, as the
simplest models of population dynamics, such as the logistic model, assume that
all individuals within a species are identical.
Quite early in the development of theoretical ecology, the role of age
differences among individuals within populations in determining population
dynamics was recognized. The early work of Leslie (1945) on discrete time
models with discrete age classes and the corresponding work of Lotka (1922),
McKendrick ( 1926) and Von Foerster ( 1959) on the analogous continuous time
models, showed that age structure could have an important effect on population
dynamics. More recently, work on ‘structured population models’ has been
extended in a variety of ways to include both additional or alternate structuring
variables, such as size or physiological state, as well as the effects of density
dependence. The mathematical aspects of these models is reviewed in Metz &
Diekmann (1986) and a number of biological examples are discussed in
Ebenman & Persson (1988).
Note that several modelling approaches have been used to formulate these
demographic models for individuals. Descriptions have been formulated in either
continuous or discrete time, and with classification of individuals into either a
finite number of age, size, and/or physiological stage classes or into an infinite
number of different classes using continuous variables.
Structured models for metapopulations can be formulated using an approach
analogous to that used in formulating structured models for single populations,
with the ‘patch’ playing the role of the individual. The metapopulation models
are described in terms of the demography of patches, where patches are classified
by the numbers of individuals of different species, and perhaps, as well, by
underlying environmental variables. Changes in the state of a patch correspond
to changes in population size(s). As noted below, this framework places
restrictions on the kinds of dispersal that can be modelled.
A GENERAL MODEL FOR A SINGLE SPECIES METAPOPULATION
The simplest version of a patch model describes the dynamics of a single
species. The assumptions and approach used in the following model are
analogous to those in more complex models. I will both describe the assumptions
60
A. HAS'I'INGS
I use here to produce a tractable model and indicate other possibilities.
1. The number of patches is assumed to be so large that a deterministic model
is appropriate. Note that the model is deterministic only on the scale of large
numbers of patches, and not at the scale of a single patch. At the latter scale, the
dynamics have a stochastic component. For models where the number of patches
is small, see Chesson (1982).
2. All patches are assumed to be equally accessible from all other patches. No
effect of spatial arrangement is considered. T o go beyond this assumption, and
still allow the kind of within-patch stochasticity implicit in these models, it
appears necessary to use computer simulations, although results from percolation
theory (e.g. Durrett, 1988) may provide guidance in some simple cases.
3. One needs to describe population dynamics within populations. I make
several assumptions about this process. I assume that the initial propagule size is
always the same. Dynamics within a population is a mixture of deterministic
processes, involving population growth, and stochastic processes, involving
immigration and disasters. I incorporate stochasticity into the local dynamics by
allowing for the possibility of local disasters, changes in the population size
within a patch that occurs on a fast time scale relative to other population
processes. These disasters, which can be of varying magnitude, could be due to
environmental changes, or the arrival of a forager, parasitoid or predator. Rapid
increases in local population size as a result of unusually favourable
environmental conditions could be modelled in the same way.
4. In the first models I develop, I assume that local populations growth
always increase monotonically from the initial propagule size, possibly
approaching a limiting value. The logistic model is one example of this kind of
population growth.
5. I assume that the environment is the same in all patches. By this I do not
mean that the environment is the same in all patches at all times, but merely
that the stochastic process describing environmental effects is the same in all
patches. It is relatively straightforward to formulate models which incorporate
environmental variability by the use of an additional structuring variable.
6. I assume that different patches are affected independently by the stochastic
processes.
Models with total disasters only
The simplest model is a single species model in which all the disasters are total,
leading to local extinction (Hastings & Wolin, 1989). In writing the equations
for these models it is simpler to change the underlying variable in the model from
the fraction of patches with a given number of individuals to the fraction of
patches of a given age since age changes at the same rate as time. In the simplest
single species model, age of a local population is simply the time since
colonization. Although ages of populations are used in the formulation of the
model to simplify the equations, results can be presented in terms of population
sizes, as is done below. If population growth within a population is assumed to
be deterministic except for the possibility of randomly distributed disasters,
which remove all individuals within a population, and the initial propagule sizes
are all the same, then there is a fixed relationship between population age and
the numbers within a population. Thus, a simple model can be formulated for
STRUCTURED MODELS
61
the dynamics of populations of different ages, and well-developed techniques
from age and density-dependent demography (e.g. Metz & Diekmann, 1986)
can be used to analyse the model.
This approach was used in Hastings & Wolin (1989) to arrive a t the model
where p ( t , a ) is a density function for the fraction of patches of age a at time 1.
The probability of extinction in a patch of age a is p ( a , p ( t , * ) , t ) . T o complete
the model, one needs a description of the process of a colonization of empty
patches:
where the function M gives the rate of colonization of empty patches per
population of age a, which may depend on the state of the whole metapopulation. The major result of Hastings & Wolin (1989) was that if the
probability of population extinction is a declining function of population size
(age), then the regional outcome is a stable equilibrium.
This model can be used to make predictions of distributions of numbers of
individuals within populations. After specifying a function describing the
dynamics within a single patch, and the extinction probabilities and the rate of
colonization of empty patches, one can determine the equilibrium distribution of
numbers within patches. For example, Hastings & Wolin (1989) consider the
case where growth within patches is exponential (at a rate r ) . Extinction is
assumed to be the result of either demographic or environmental factors. In
small populations, less than N individuals, the extinction rate is assumed to be
p,, while in larger populations it is assumed to be p2, with p , > p s . The
equilibrium density of population sizes is found to be:
for n < N and
p^(n) = C[n/n(O)]-rc'ir
p^( n) = C"/n
(3)
(0)]pr - a"" [n/n(0)3 -W/'
(4)
for n > N . Here, n ( 0 ) is the initial propagule size, and C is a constant depending
on the other parameters in the model. As noted in Hastings & Wolin (1989), this
formula implies that the frequency of populations of a given size decreases with
increasing size-there are fewer patches with large numbers of individuals in
them, even though the extinction rate decreases with population size. Since the
extinction rate decreases with size, larger patches turn over less frequently. This
is consistent with many of the systems reviewed by Hanski (1991) and Harrison
(1991), and the system of spiders in the Bahamas studied by Schoener & Spiller
(1987), even though the natural systems generally include the effects of
underlying variability in the habitat in different patches.
Numerical results for models with total disasters
onb
To illustrate that even the simple model just developed can produce some of
the general patterns of distributions of population sizes observed in meta-
62
A. HAS’HNCS
populations, I will present some numerical results based on equations (3) and (4)
for the density function for the fraction of populations with n individuals. For
selected values of the parameters, I plot the distribution function
for the fraction of populations with n individuals. This is equivalent to plotting
the fraction of populations with n or fewer individuals as a function of n. Note
that in doing this, I am presenting results based only on occupied patches, which
is appropriate when comparing to observations of natural metapopulations. I
will also plot the distribution function for the fraction of the total number of
individuals in populations with n individuals
xi(.)
dx
x i ( x ) dx
which is equivalent to plotting the fraction of all individuals that are in
populations with n or fewer individuals.
A range of outcomes is possible, as the parameters are varied, as shown in
Fig. 1. One important observation is that both (5) and (6) are independent of
any assumptions about the rate of colonization of empty patches. Thus, the
distribution of population sizes (in occupied patches) and the total number of
populations (occupied patches) can be varied independently in the model.
Overall rarity and overall commonness are both compatible with any
distribution of population sizes that can be produced in the model.
Here, I concentrate on numerical solutions, although both (5) and (6) can be
computed explicitly. Observe, however, that unless p2 > 2r, the integral in the
denominator of (6) does not converge since the number of individuals in large
populations becomes unbounded. This is biologically unreasonable, and thus I
restrict attention to the case where p2 > 2r. More generally, the nth moment of
i(n) is unbounded unless p2 > (n- 1)r.
The outcome of the model can be large local outbreaks in population
numbers. As expected from the observations of the previous paragraph, if p2- 2 r
is small, most individuals are found in large populations, even though the
number of large populations is small, as illustrated in Fig. 1. These local
outbreaks are not the result of any underlying environmental differences, but
merely the result of random processes.
The model can also lead to a metapopulation with very few outbreaks. In
what may at first seem counterintuitive, a metapopulation structure with very
few large populations results when the extinction rate in small populations is
very high. As shown in Fig. 1, in this case, most of the individuals will be found
c
0
I-
....................
C
0.2
0
I
I
I
20
40
60
I
80
I
100
D
,,
I
0
20
I
40
I
60
I
80
;1
Population size
Figure 1. Equilibrium behaviour of the metapopulation model as computed in equations (3)-(6). In
each panel, the upper, solid line represents the fraction of populations with fewer than n individuals,
and thr lower, dashed line rcprcsents the fraction of individuals in populations with fcwcr than n
individuals. In each panel, the initial propagule size n(0)is 10 individuals, the growth rate of local
populations is 0.5 per time unit, and there is a threshold population size of 20. In A the extinction
rates below and above the threshold are 2.0 and 1.5 per unit time, respectively. In B the extinction
rates below and above the threshold are 2.0 and 1.2 per unit time, respectively. I n C the extinction
rates below and above the threshold are 5.0 and 1.2 per unit time, respectively. In D the extinction
rates below and above the threshold are 2.0 and I . I per unit time, respectively. Note that the time
scalp is arbitrary, and immigration rates affect only total numbers of populations and not
distributions of population sizes. Panels B and D are metapopulations where there arc large local
outbreaks, and panels A and C have primarily small populations.
in small populations. This is the consequence of an extinction rate that is high
enough in small populations so that very few populations become large.
Models which include disasters that do not lead to extinction
In a more realistic model, one would need to extend the role of stochasticity to
allow random events besides those which remove all individuals within a local
population. A model which incorporates the possibility of disasters that remove
only some of the individuals within the local population has been formulated and
analysed by Hastings (1990b) and Hastings & Wolin (unpublished). In this case,
we use the approach described above of considering the ‘age’ of a population as
the underlying independent variable, but redefine ‘age’. Now, ‘age’ is not taken
as the time since colonization, but the time it would have taken the population
within a patch to grow from initial propagule size to reach the current size given
that no disasters had occurred. As above, the results of the model can be phrased
in terms of the biologically measurable variable, population size.
Under the assumptions about local population growth given above, this device
64
A. HASTINGS
provides a unique way to assign ages and sizes to populations. Let p ( l , a ) be the
fraction of patches with a population of ‘age’ a,
p ( a , P ( t , * 1) = CL,(a,P(4 ’ ))+CLp(o,P(4 -1)
be the rate of all disasters in a population of age a, with pl the rate of ‘total’
disasters, and p,, the rate of partial disasters, which remove only some proportion
of the individuals. Partial disasters need not be of a fixed size, so let y ( 6 , a ) be the
probability that a disaster in a population of ‘age’ b leads to a patch of ‘age’ a.
The rate at which empty patches are colonized is:
Here, M(a, s(P(1, - ) ) ) is the rate at which empty patches are colonized due to
colonists from a population of ‘age’ a when the total ‘size’ of the metapopulation
is given by s ( t ) .
The dynamics within patches are thus governed by two features: local ageing
and disasters. These two events are incorporated within the following equation:
The second term on the right-hand side of this equation represents the effect of
partial disasters. I have found an explicit method for determining the unique
equilibrium for this model, under the assumption that there is no effect of other
populations on the local dynamics of a given population (Hastings, 1990b). This
assumption is equivalent to assuming that pl and pp.are independent ofp(l . ). As
in the model where all disasters lead to local extinction, if the probability of
extinction of a population is a declining function of the actual time since
colonization, the overall equilibrium is stable. However, there are also cases
where cyclic behaviour in total population numbers is possible. This occurs when
the disaster rate increases sharply after the local population reaches a threshold
size.
A SIMPLER MODEL
There are both advantages and drawbacks to the general model presented in
the previous section. As I noted, analysis can be difficult and, in attempting to
either fit the model to data, or just to get a n idea of reasonable parameter values,
one is faced with the problem of providing a large number of functions, for
example the disaster probabilities as a function of local population sizes. An
alternative approach to the one just outlined would be to classify patches into a
large number of distinct states, based on the numbers and kinds of species
present. Thus, a state for the metapopulation(s) would be a vector giving the
fraction of patches in these different states, rather than a distribution function.
One can then prescribe transition probabilities among the various states, and
from this determine the behaviour of the metapopulation. This approach both
provides an alternative way of incorporating stochasticity and a recipe for
performing numerical simulations. Carole Wolin and I (unpublished) are using
STRUCTURED MODELS
65
this approach to investigate host-parasitoid models. Moreover, as I show in the
Appendix, this simpler model can be viewed as a special case of the more
detailed model with appropriate choices of function, in the case where disasters
include the possibility of jumps upward in local population size, as discussed in
assumption 3 above.
Here, I will indicate how this approach can be used to investigate the
immigration in metapopulation dynamics, within the context of a very simple
model. Rather than use a large number of states for the patches, I will describe a
model due to Hanski (1985), which merely extends the simple model with two
states, empty and occupied (Hanski, 1991), to one with three states: empty, low
population level, and high population level. Even this simple extension allows
one to include the role of immigration on population dynamics in occupied
patches in a mechanistic way. A diagrammatic presentation of the model is given
in Fig. 2.
Model formulation
The variables in the model are pe, the fraction of empty patches, PI, the
fraction of patches with a low population level, and Ph, the fraction of patches
with a high population level, following Hanski (1985). Since these are fractions,
+PI + P h
(9)
and only two of the variables are independent. The total fraction of occupied
patches is
Pt
=
>
PO= P I + P h -
(10)
The equations for the model will be defined if rates are described for the arrows
describing the different transitions allowed as illustrated in Fig. 2. In this
diagram I extend Hanski’s (1985) model slightly by explicitly including the
possibility of a transition not included in Hanski’s original formulation, from
high population level to empty patch, which might result from environmental
factors. The colonization rate of empty patches will be assumed to be a product
of the fraction of empty patches and a linear function of the fractions of low and
high level patches:
mdh),
(11)
where m, and m h are constants. This corresponds to an assumption that patches
produce propagules at a rate proportional to local population size, and that
these propagules are dispersed independently.
The rate at which individual low level populations become high level
populations depends on two biological processes: local population growth and
immigration. Thus, the rate at which this transition takes place will be:
Pe(m&l+
PI ( a+ ‘lP1-k
(12)
where c1 is the rate due to local population growth in low level populations, and
a, and a,, measure the contribution due to immigration from low and high level
populations into low level populations, respectively.
The remaining transitions are all density independent, not depending on the
states of other patches. The rate at which low level patches go extinct is elpI.The
‘hph)
7
A. HASTINGS
66
I
I
I
I
I
I
\
\
\
\
\
\
Figure 2. An illustration of the three possible states for a patch in the three-state model introduced
by Hanski (1985). Arrows represent possible transitions among different states, with a dashed line
representing a transition which may not be possible in some circumstances.
rate at which ‘disasters’ cause high level patches to become low level ones is &,,
and the rate at which high level patches go extinct is e&,. If the ultimate cause
of extinction were always due to demographic factors, then there would be no
extinctions of high level patches, and eh would be zero.
Coupling all these transition rates leads to the following system of equations
describing the dynamics of patches.
Even this model is algebraically quite complex. To illustrate many of the
important biological principles, it is both sufficient and simpler to assume that
there is no effect of emigration from low level patches, so
m,= El = 0.
(16)
As noted above, the variable p,, the fraction of occupied patches is a useful
STRUCTURED MODELS
67
choice, and this variable plus fib provide a complete description of the dynamics:
The model is now reasonably easy to analyse.
Equilibria and stability
I will concentrate on the equilibrium behaviour of this system. First, when are
there two positive equilibria for this system? I will present these results in a
slightly different fashion than Hanski (1985). Note that from equation (17), one
sees that any positive equilibrium automatically must satisfy pn < 1. At
equilibrium, equation (18) implies that
Thus, knowing P h at equilibrium, one can find po from (19). Thus, in what
follows I concentrate on determining fib at equilibrium. In the Appendix (see
also Hanski, 1985) I show that the equilibrium solutions and stability are, as a
function of el, as graphed in Fig. 2. There is a critical value of the extinction rate
el
,
el* =
amh- aeh
eh+fi
'
above which the equilibrium with the species absent is stable, and below which
the equilibrium with the species absent is unstable. If
then for
there are two positive equilibria, where el** is a complicated function of the other
parameters, as in Fig. 3A. The positive equilibrium with larger overall
population size is stable, whilst the one with smaller overall population size is
unstable. In the case where there is no extinction from high level patches, eh = 0,
the formula for e;"* simplifies to:
If, on the other hand, (21) does not hold, then there is only one positive
equilibrium which is always stable, as in Fig. 3B.
Thus, the formula (21 ) is critical to understanding the equilibrium behaviour
of the metapopulation. Not surprisingly, a large value of ah, the effect of
immigration on the transition from low level to high level populations, and a
small value for eh, the extinction rate of high level populations, are required for
68
A. HASTINGS
0
Figure 3. Representation of possible equilibrium behaviour for the three-state modcl, in terms of the
equilibrium value for high level populations as a function of the extinction rate in low level
populations. Solid lines represent stable equilibria, dashed lines unstable equilibria. In A I W O
positive equilibria are possible for ey < el < e r * , while in B only one positive equilibrium is possible,
as determined by equation (21). Parameters are defined in the text.
there to be a threshold and two equilibria. I n the case where there is no
extinction of high level populations, eh = 0, the formula (21) simplifies to:
ah > a+p.
(24)
This formula basically says that the effect of immigrants on the transition from
low level patches to high level patches must be strong relative to the effect of
local population growth for the-model to have two positive equilibria (Hanski,
1985).
DISCUSSION
The overall theme that emerges from the metapopulation models discussed
here is that stochasticity, extinction or unstable dynamics at the level of local
populations, may be consistent with stability, persistence or coexistence at the
level of the metapopulation. The consequences of the structure in the metapopulation depend on the questions asked. For the models considered here, if
there is no effect of immigration on numbers within occupied patches, the
systems have a unique equilibrium. A consideration of the numbers within
populations is nonetheless important in understanding the role played by local
extinctions and disasters coupled with colonization of empty patches in
generating local outbreaks in numbers, as observed in many arthropod
populations.
Even the very simple model analysed numerically here can reproduce local
outbreaks, as well as other patterns of distributions of sizes of local populations
consistent with observations of metapopulations, without any underlying
environmental variability. Both cases where most individuals are found in a few
very large populations, and cases where most individuals are found in small
populations can be produced. Another striking feature in this simple model is
that the formula for the distribution of population sizes (in occupied patches)
S’I’RUClURED MODELS
69
does not depend on the rate of colonization of empty patches, which determines
the total fraction of patches which are occupied. Thus the model outcome can be
a rare species with large local outbreaks, a rare species that is always found in
small populations, a common species that has large local outbreaks, or a
common species that does not have outbreaks.
In a simple three-state model (developed by Hanski, 1985) which explicitly
includes the effect of immigration on local population dynamics, there is a
possibility of multiple equilibria, with both overall extinction and persistence
locally stable. The three-state model provides an explicit description of the range
in parameter space where two equilibria and a threshold are possible. If, in a
model with three states, two equilibria are possible, then clearly more complex
models may have many equilibria. Even for the three-state model, if
immigration from low level patches is allowed to affect local population
dynamics, three positive equilibria are possible. Since the three-state model is a
special case of the more complete structured model presented here, the model
which includes the possibility of local disasters and the effects of immigration on
numbers within populations is likely to have extraordinarily complex
equilibrium behaviour. Development of more complex structured models must
necessarily be done in consort with consideration of explicit experimental or
observational systems, such as the system of aphids on fireweed studied by
Addicott ( 1978).
One interesting feature concerning the threshold effect with important
implications for conservation biology is apparent from Fig. 3A. If e, is just slightly
smaller than e:*, there can be a stable equilibrium with the regional population
at a level far from zero. However, a small increase in the extinction rate of low
level populations, e , , will cause the system to move into a range of parameter
space where the only outcome is regional extinction.
Although the structured metapopulation models developed here show great
promise for developing a theory for understanding natural metapopulations, it is
also important to be aware of other important features that are not included. I
have considered only cases where all patches are equally accessible from all other
patches. Other cases, where short range dispersal predominates, such as plants
with only short-range seed dispersal, must be dealt with in a very different
fashion, perhaps through ideas from percolation theory (Durrett, 1988). I have
not considered a metapopulation with only a small number of local populations
(but see Hanski, 1991). This case would be important for understanding
problems in conservation biology (e.g. Quinn & Hastings, 1987). I have not
considered metapopulations where the underlying habitats vary among the
patches. All of these omitted features are likely to have important biological
consequences, and suggest avenues for future research.
AC:KNOClrL~DCEI1.IE~’l’S
This research was supported by grant DE-FG03-89ER60886/AOOO from the
Ecological Research Division, Office of Health and Environmental Research,
US.Department of Energy. This support does not constitute an endorsement of
the views expressed here. I gratefully acknowledge helpful discussions on many
of these questions with Carole Wolin, comments on an earlier draft by Elaine
70
A. HASTINGS
Fingerett and Carole Hom, and extensive helpful comments on earlier versions
by Ilkka Hanski.
REFERENCES
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DURRETT, R., 1988. Lecture Notes on Particle Systems and Percolation. New York: Springer-Verlag.
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HANSKI, I., 1982. Dynamics of regional distribution: the core and satellite species hypothesis. Oikos, 38:
2 10-22 I .
HANSKI, I., 1985. Single-species spatial dynamics may contribute to long-term rarity and commonness.
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HANSKI, I., 1991. Single-species metapopulation dynamics: concepts, models and observations. Biological
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QUINN, J. F. & HASTINGS, A,, 1987. Extinction in subdivided habitats. Conservation Biology, 1: 198-208.
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APPENDIX
Here I present some of the mathematical formalism which arises in the study of the three-state model and its
relationship to the more complex model presented here.
The three-state model as a special case of the complex model
Here I exhibit the function choires for which the model ( I ) and (2) reduces to the three-state model given in
equations (13)-( 15). First, identify as low level populations those smaller than some fixed size, and as high level
populations those larger than this fixed size:
“41)
roo
71
STRUCTURED MODELS
where the constant a, may have to be chosen large enough, as indicated below. Next, choose the extinction rates
so they correspond in the two models:
The choice of the rate of partial disasters requires more care. In this case, partial disasters in small populations
will actually not be disasters, but sudden upward jumps in population size, as discussed in assumption 3 in the
text. Also, to ensure that ppalways is positive, one must choose a, above large enough. Use the following choice:
Then pick
S(a-a,)
y(b'a) = {S(o)
for b < a,
for b 2 a,,
where S ( a ) is the function which is zero, everyone except at 0 and has integral one.
Finally, let
('46)
and
A straightforward computation shows that with these choices the more complex model reduces to the threestate model.
The three-state model
Here I will consider the form and stability of the equilibria. Substituting (19) into (17) and simplifying
implies that either
p,
= ph = 0,
('48)
or that phsatisfies the quadratic equation
O = a,m,,p:
+ (ehmh+/3mh-ahmh
+am, +a,e,)p,-amh+
ehe,+~e,+aeh.
('49)
Although this is a quadratic equation and therefore one can write down a formula for the solutions, it is more
informative to proceed in an indirect fashion, letting e, play the role of a bifurcation parameter, studying the
brhaviour of solutions as a function of el. More precisely, 1 will focus on the graph of the solutions Ph as a
function of e,, which is easy to obtain since (A9) is easily solved for el:
Note that (A10) implies that for any value of c, there are at most two positive equilibria. (However, if one
does allow for the effect of emigration from low level patches, then the equation for the non-trivial equilibria
becomes a cubic, and three solutions are possible.) Equation (20) follows from a straightforward stability
analysis of the equilibrium with ph= 0. Rather than considering the quadratic equation, I distinguish between
the two cases in Fig. 2 by computing the derivative ofe, with respect top,, evaluated atp, = 0, from (A10). To
find ey*, 1 find the maximum of el, as a function ofp,, from (AIO).
Standard arguments from bifucation theory (e.g. Iooss & Joseph, 1980) show that the stability of the
equilibria is as illustrated in Fig. 3, provided that there are never any purely imaginary eigenvalues at the nontrivial equilibrium. This will be the case if the trace of the Jacobian matix evaluated at the equilibrium is
always negative. The trace of the Jacobian matrix is
It is straightforward to show that this trace is zero whenp, is zero, and that the denominator in (A1I ) is always
positive. It is also easy to compute that both the first and second derivatives with respect toph of the numerator
in (A1 1 ) are negative. Consequently, the trace of the Jacobian is always negative. Therefore, by Bendixson's
criterion, there are never any limit cycles (Lefschetz, 1963). Thus, all solutions will always approach one of the
equilibria.