Doebeli, M., Killingback, T. Metapopulation Dynamics with quasi-local competition. 2003. Theor. Pop. Biol. 64, 397-416.

Doebeli, M., Killingback, T. Metapopulation Dynamics with quasi-local competition. 2003. Theor. Pop. Biol. 64, 397-416.
ARTICLE IN PRESS
Theoretical Population Biology 64 (2003) 397–416
http://www.elsevier.com/locate/ytpbi
Metapopulation dynamics with quasi-local competition
Michael Doebelia,b, and Timothy Killingbackc
a
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
b
Department of Zoology, University of British Columbia, Vancouver, British Columbia, Canada
c
Ecology and Evolution, ETH Zurich, 8092 Zurich, Switzerland
Received 6 July 2001
Abstract
Stepping-stone models for the ecological dynamics of metapopulations are often used to address general questions about the
effects of spatial structure on the nature and complexity of population fluctuations. Such models describe an ensemble of local and
spatially isolated habitat patches that are connected through dispersal. Reproduction and hence the dynamics in a given local
population depend on the density of that local population, and a fraction of every local population disperses to neighboring patches.
In such models, interesting dynamic phenomena, e.g. the persistence of locally unstable predator–prey interactions, are only
observed if the local dynamics in an isolated patch exhibit non-equilibrium behavior. Therefore, the scope of these models is limited.
Here we extend these models by making the biologically plausible assumption that reproductive success in a given local habitat not
only depends on the density of the local population living in that habitat, but also on the densities of neighboring local populations.
This would occur if competition for resources occurs between neighboring populations, e.g. due to foraging in neighboring habitats.
With this assumption of quasi-local competition the dynamics of the model change completely. The main difference is that even if
the dynamics of the local populations have a stable equilibrium in isolation, the spatially uniform equilibrium in which all local
populations are at their carrying capacity becomes unstable if the strength of quasi-local competition reaches a critical level, which
can be calculated analytically. In this case the metapopulation reaches a new stable state, which is, however, not spatially uniform
anymore and instead results in an irregular spatial pattern of local population abundance. For large metapopulations, a huge
number of different, spatially non-uniform equilibrium states coexist as attractors of the metapopulation dynamics, so that the final
state of the system depends critically on the initial conditions. The existence of a large number of attractors has important
consequences when environmental noise is introduced into the model. Then the metapopulation performs a random walk in the
space of all attractors. This leads to large and complicated population fluctuations whose power spectrum obeys a red-shifted power
law. Our theory reiterates the potential importance of spatial structure for ecological processes and proposes new mechanisms for
the emergence of non-uniform spatial patterns of abundance and for the persistence of complicated temporal population
fluctuations.
r 2003 Elsevier Inc. All rights reserved.
Keywords: Population dynamics; Spatial structure; Metapopulations; Local interactions; Theory; Attractor; Power spectrum; Red shift; Power law;
Stochastic models; Environmental perturbation; Spatial pattern formation
1. Introduction
The past decade has seen an upsurge of interest in the
role of spatial structure on ecological and evolutionary
processes (e.g. Gilpin and Hanski, 1997; Bascompte and
Sole, 1998; Dieckmann et al., 2000). Models for spatially
structured populations have a long tradition in population genetics and in ecology (e.g. Wright, 1943; Levene,
Corresponding author.
E-mail addresses: [email protected] (M. Doebeli),
[email protected] (T. Killingback).
0040-5809/$ - see front matter r 2003 Elsevier Inc. All rights reserved.
doi:10.1016/S0040-5809(03)00106-0
1953; Kimura and Weiss, 1964), but such models tend to
quickly become analytically intractable, and advances in
computational power have greatly facilitated the numerical investigation of spatial models in recent years.
In ecological theory, the principal interest in spatial
structure concerns its effects on the nature of population
fluctuations and on geographical patterns of population
abundance. For example, starting with Levins’ classic
metapopulation model (Levins, 1970), the study of the
effects of spatial structure on processes of extinction has
been a recurrent theme, and it has for instance been
shown that spatial pattern formation in the form of
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spiral waves of population abundance can prevent
extinction in intrinsically unstable predator–prey systems (Hassell et al., 1991, 1994; Comins et al., 1992).
One of the main questions in this context is whether
population dynamics tend to become simpler, and hence
population fluctuations decrease, when ecological interactions are localized and migration induces mixing
between local populations. A common model used to
investigate this question is an ecological version of
Kimura’s stepping stone model (Kimura, 1953), in
which one envisages a metapopulation consisting of a
number of local populations occupying local habitat
patches that are arranged on a spatial grid and that are
coupled by migration between habitats. In each habitat
patch the dynamics of the local populations is given by
difference equations that describe population growth or
decline resulting from the local ecological interactions.
After the local dynamics have taken place in each
generation, migration to neighboring patches occurs,
typically in the form of passive dispersal, in which a
constant fraction of the local population is distributed
evenly among the surrounding habitat patches. Such
spatially and temporally discrete dynamical systems
have been studied extensively in the physics literature
(e.g. Kaneko, 1998), and have also attracted considerable attention in theoretical ecology (e.g. Hastings, 1993;
Gyllenberg et al., 1993; Lloyd, 1995; Doebeli, 1995;
Ruxton, 1996; Scheuring and Janosi, 1996; Holt and
McPeek, 1996; Earn et al., 2000). One common message
emerging from this large body of work is that spatial
structure typically tends to simplify population dynamics by inducing coherent spatial patterns in the
metapopulation. For example, even if the local populations would exhibit chaotic dynamics in isolation, the
coupling between them induced by dispersal can lead to
spatially and temporally regular patterns of abundance
(Kaneko, 1998). In particular, in a metapopulation
consisting of two chaotic local populations, coupling
through dispersal can induce a spatially inhomogeneous
but temporally stable equilibrium for the metapopulation dynamics (Doebeli, 1995). Further simplifying
effects can be observed if dispersal occurs over larger
distances than just to neighboring habitat patches
(Doebeli and Ruxton, 1998), or if dispersal is conditional and depends on the local population size in a
habitat (Ruxton, 1996; Scheuring and Janosi, 1996;
Doebeli and Ruxton, 1998).
However, to observe these effects in single-species
models one always needs to assume very complicated
local dynamics and hence strong density dependence. In
fact, if one assumes that the local populations exhibit
stable equilibrium dynamics in isolation, and that all
habitat patches are identical (i.e. that the demographic
parameters are the same in each patch) then one can
show that passive dispersal cannot induce any change in
the dynamics: there is always a spatially homogeneous
and temporally constant equilibrium for the metapopulation in which each local population is at the
equilibrium size it would have in isolation (Rohani
et al., 1996). Therefore, these types of models for spatial
structure only yield interesting results if some form of
dynamical instability resulting form the local ecological
interactions in each patch is assumed.
In situations in which two or more species are present
it is possible, in continuous time systems, to have stable
spatially inhomogeneous configuration even with stable
local dynamics via the Turing mechanism (Turing,
1952). For continuous-time predator–prey systems such
diffusion driven instability seems to occur only under
rather restrictive conditions (Levin and Segel, 1976;
Mimura and Murray, 1978). Spatially structured interspecific competition models, either in discrete time
(Karlin and McGregor, 1972) or in continuous time
(Levin, 1974, 1979), can also result in stable spatially
inhomogeneous population distributions. It is also
possible to obtain stable inhomogeneous spatial distributions in discrete-time predator–prey systems in
which the local dynamics has a spiral instability (Hassell
et al., 1991, 1994; Comins et al., 1992; White et al.,
1996), in age-structured single-species models in which
different age classes have different dispersal parameters
(Hastings, 1992), and in invasion models which incorporate an Allee effect (Keitt et al., 2001). These models
are fundamentally different from the type of model
introduced in this paper in that in the former, all nondispersive ecological interactions are confined to take
place within each habitat patch, with dispersal being the
only coupling between the patches.
The novelty of the approach taken here is that we
consider the coupling between neighboring habitat
patches to include other ecological interactions in
addition to simple dispersal. In this approach we also
assume that each local population is described by a
difference equation, but we assume that the coupling
occurs primarily through ecological interactions, such as
competition, rather than through dispersal. Thus, we
assume that the reproductive dynamics of a local
population in a given patch depends not only on the
population density in that patch, but also to a certain
extent on the population densities in the neighboring
patches. This is a reasonable assumption for situations
in which reproduction and hence population dynamics
takes place in a habitat patch whose resources are also
used, e.g. through foraging, by individuals that live and
reproduce in neighboring patches. Then the depletion of
resources in a given habitat patch occurs not only due to
the individuals living and reproducing in that patch, but
also due to individuals living in a certain neighborhood
of that patch. In other words, competition for resources
is not strictly local, but quasi-local: the strength of
competition impinging on reproduction of an individual
in a given patch depends not only on the density of the
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M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
population inhabiting that patch, but also on the
population densities of neighboring patches. Such an
assumption is commonly used for individual-based
models for spatially structured populations (e.g. Neuhauser and Pacala, 1999; Sasaki, 1997). In those models
it is assumed that the strength of the ecological
interactions impinging on an individual is a function
of the number of neighboring individuals as well as of
their spatial distance, with more distant individuals
having a smaller impact. However, to our knowledge
this from of coupling has never been used for spatially
discrete metapopulations consisting of a number of local
habitat patches.
In contrast to the case where the coupling is due to
dispersal alone, coupling through quasi-local competition can lead to very interesting dynamical phenomena
even if the local population dynamics exhibit a stable
equilibrium in isolation, as we will show below. If the
dynamics of a local population in isolation exhibit a
stable equilibrium, then it is still true that the dynamics
of the whole metapopulation will always converge to a
temporally constant equilibrium state. However, despite
the fact that all habitat patches are ecologically
identical, this equilibrium state is typically spatially
inhomogeneous, with some habitat patches having high
population densities and others having low densities.
Thus, coupling through quasi-local ecological interactions can lead to spatial pattern formation even if the
habitat quality is spatially homogenous throughout the
metapopulation and if the local population dynamics
exhibit a stable equilibrium. This ecological mechanism
for pattern formation seems to have been overlooked in
the past.
In fact, spatial pattern formation due to quasi-local
interactions may be relevant for many spatially structured ecological systems in nature, for which the
assumptions of local population dynamics but quasilocal competition dynamics are plausibly satisfied. For
example, many territorial birds, such as Dunnocks,
occupy spatial regions within which they carry out most
of their foraging, but different territories often overlap
(Davies, 1992), suggesting the possibility of quasi-local
competition between populations in neighboring territories. Also, in social insects such as ants the colonies
occupy territories and regularly engage in competitive
activities with neighboring colonies, e.g. robbing of food
(Wilson, 1971; Holldobler and Wilson, 1990). Finally,
quasi-local competition for water appears to be important for many plant populations, simply because the
effect of a plant taking up water locally is to remove
water from the surrounding regions. Thus quasi-local
competition can result from resource diffusion. Interestingly, spatial pattern formation in plant populations
is found in many semi-arid regions (White, 1970; Belsky,
1986; Klausmeier, 1999), in which quasi-local competition for resources would appear to be particularly
399
strong. These patterns are in qualitative agreement with
the results obtained below from our models. Therefore,
the basic mechanisms of spatial pattern formation due
to quasi-local interactions introduced in this paper may
well be relevant for explaining spatial patterns in
ecological systems.
Our models not only exhibit spatially inhomogeneous
equilibria, but for a given set of model parameters the
equilibrium spatial pattern is not unique and depends
critically on the initial conditions. In fact, even for
metapopulations consisting of a moderate number of
local populations the number of different attractors, i.e.
different spatially inhomogeneous equilibrium states,
can be very large. The existence of a multitude of
attractors has important consequences for the population dynamics when a small amount of noise is
introduced in the system. With many attractors each
one of them tends to have a small basin of attraction, i.e.
the set of all initial conditions from which the given
attractor is reached is typically small. In the presence of
environmental noise, the metapopulation therefore
continually jumps between the basins of attraction of
different attractors and thus performs a random walk in
the set of all attractors. This random walk results in
persistent population fluctuations of the local population densities as well as of the density of the whole
metapopulation. Since the fluctuations are the result of a
random walk, their power spectrum is red-shifted and
obeys a power law, in which long-term fluctuations have
a much larger amplitude than short-term fluctuations.
Such red-shifted power spectra are well-known from
empirical analyses of population time series (Diamond
and May, 1977; Pimm and Redfearn, 1988; Arino and
Pimm, 1995). Our results indicate a potential new
explanation for red-shifted ecological time series that
require neither strong density-dependence inducing
dynamic instabilities nor a red shift in the demographic
noise. Instead, a repeated sequence of convergence
towards different equilibrium attractors and noiseinduced dislocation from these equilibria can produce
red-shifted time series for population abundances. Thus,
despite its inherent simplicity due to non-overcompensating density dependence, our model exhibits very rich
dynamical behavior.
2. Model and results
2.1. Spatially one-dimensional metapopulations:
chains of local habitat patches
For ease of presentation we first consider the case
where the metapopulation extends along one spatial
dimension and therefore consists of a chain of m local
populations Ni ; i ¼ 1; y; m: Such a one-dimensional
case could arise, for example, if the habitats lie along a
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coastline or along the shore of a lake or pond. In each
habitat patch the dynamics of the local populations is
assumed to be described by a difference equation of the
form
Ni0 ðtÞ ¼ Ni ðtÞ f ðNi1 ðtÞ; Ni ðtÞ; Niþ1 ðtÞÞ;
ð1Þ
where Nj ðtÞ is the local population size in patch j at the
start of year t; and where Ni0 ðtÞ is the local population
size in patch i after reproduction, but before dispersal.
Thus we assume that the per capita number of
descendants, f ; in a patch is a function of the density
of the local population in that patch as well as of the
densities of the local populations in the two neighboring
patches, e.g. due to foraging or interference of individuals from the neighboring patches.
In this paper we will always assume that all the local
habitat patches are ecologically identical, so that the
demographic parameters influencing the density-dependent reproductive output are the same in all patches.
Moreover, we assume that the per capita reproductive
output f is of Beverton–Holt type (Yodzis, 1989):
f ðNi1 ðtÞ; Ni ðtÞ; Niþ1 ðtÞÞ
l
¼
:
1 þ a½Ni ðtÞ þ afNi1 ðtÞ þ Niþ1 ðtÞg
ð2Þ
Thus, compared to individuals living in patch i;
individuals from the two neighboring patches have a
relative competitive impact a on reproduction in patch i:
If we define
ei ðtÞ ¼ Ni ðtÞ þ aðNi1 ðtÞ þ Niþ1 ðtÞÞ;
N
ð3Þ
ei ðtÞ is the effective density that impinges on
then N
reproduction in patch i; and the per capita reproductive
output in patch i can be written as
ei ðtÞÞ ¼
f ðNi1 ðtÞ; Ni ðtÞ; Niþ1 ðtÞÞ ¼ f ðN
l
ei ðtÞ
1 þ aN
:
ð4Þ
The parameter l in Eq. (4) describes the maximal per
capita reproductive output attained under ideal conditions, i.e. in the absence of competition, and the
parameter a is a measure of the impact of individuals
in patch i on reproduction in patch i: As mentioned
above, we assume that these parameters are the same in
all patches, hence all local habitats are ecologically
identical, and any difference between patches resulting
from a non-uniform distribution of local population
abundances must be due to internal dynamical properties of the model.
To complete the description of the metapopulation
dynamics, we assume that in each year reproduction is
followed by passive dispersal of a constant fraction of
the local populations, which is distributed evenly
between the two neighboring patches. Therefore, the
population density in patch i at the start of year t þ 1 is
given by
d 0
d 0
ðtÞ þ Niþ1
ðtÞ:
ð5Þ
Ni ðt þ 1Þ ¼ ð1 dÞNi0 ðtÞ þ Ni1
2
2
Here the population sizes Nj0 ðtÞ after reproduction in
year t; but before dispersal, are given by Eqs. (1) and (2),
and d is the fraction of dispersers, which we assume to
be the same in each patch.
We note that the Beverton–Holt equation (4) is one of
the standard difference equations used to describe the
dynamics of discrete-time ecological systems (Skellam,
1951; Leslie, 1957; Utida, 1967) and is a special case of
other well-known difference equations employed in
ecology (Hassell, 1974; Hassell et al., 1976; Maynard
Smith and Slatkin, 1973). The Beverton–Holt equation
is particularly well-suited to our purpose in this paper
since, provided l is greater than 1, a local population in
isolation whose dynamics is described by Eq. (4) will
always converge monotonically towards a globally
stable equilibrium K defined by f ðKÞ ¼ 1; i.e. towards
the carrying capacity K ¼ l1
a : In fact, it is easy to see
that for any Nð0Þ40; the general solution of the (local)
Beverton–Holt difference equation is given by
lt
NðtÞ ¼ aðlt 1Þ
;
1
þ
Nð0Þ
l1
ð6Þ
which shows that NðtÞ-0 as t-N for lo1; and
NðtÞ- l1
a as t-N for l41: Consequently, we can be
assured that any dynamic complexity exhibited by the
metapopulation dynamics (5) results from the effects of
spatial structure and not from intrinsically complex
local dynamics.
Now that we have formally defined the model that we
will study it may be helpful to explain how the quasilocal competitive interactions can be interpreted in
terms of foraging. Note that this interpretation of the
parameter a is of course not the only possible one and is
meant as an illustrative example of quasi-local ecological interactions. Recall that the parameter a measures
the competitive impact of individuals from the neighboring patches on reproduction in a given patch, relative
to the impact of individuals living in that patch. To
illustrate how this parameter can be related to quasilocal foraging outside the natal patch, let p be the
fraction of time an individual born in a given patch i
spends, on average, foraging in neighboring patches
before it returns to the natal patch i to reproduce.
Alternatively, p could be interpreted as the fraction of
resources that individuals born and reproducing in a
given patch acquire from outside that patch. Given p;
the amount of resource depletion in patch i that is due to
individuals born in patch i is determined by the scaled
population size ð1 pÞNi ðtÞ: As mentioned we assume
that individuals always reproduce in their natal patch. If
we further assume that reproduction in patch i is
described by a Beverton–Holt equation, and if we
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neglect for a moment foraging in patch i by individuals
born in neighboring patches, then the equation for
reproduction in patch i would have the form
Ni0 ðtÞ ¼
lNi ðtÞ
;
1þ
pÞNi ðtÞ
a0 ð1
ð7Þ
where a0 is a parameter that scales resource depletion
due to the effective natal population ð1 pÞNi ðtÞ: Note
that this equation incorporates a benefit to foraging in
neighboring patches because due to foraging outside the
natal patch the effective natal population that is
depleting resources is smaller than the total population
size in patch i; i.e. because ð1 pÞNi ðtÞoNi ðtÞ: Of
course, we now have to incorporate the depletion of
resources in patch i that is due to foraging by individuals
from neighboring patches. Clearly, given p; which is
assumed to be the same for all patches, the effective
population size of individuals from the neighboring
patch i 1 that forage in patch i is pNi ðtÞ=2; and
similarly for the neighboring patch i þ 1: Therefore,
when foraging by individuals from neighboring patches
is included, the equation for reproduction in patch i
becomes
Ni0 ðtÞ ¼
lNi ðtÞ
: ð8Þ
1 þ a0 ½ð1 pÞNi ðtÞ þ p2fNi1 ðtÞ þ Niþ1 ðtÞg
If we now set a ¼ a0 ð1 pÞ; this equation takes the form
of Eq. (2), i.e.
Ni0 ðtÞ ¼
lNi ðtÞ
;
1 þ a½Ni ðtÞ þ aðNi1 ðtÞ þ Niþ1 ðtÞÞ
ð9Þ
with
a¼
p
:
2ð1 pÞ
ð10Þ
Note that with this interpretation of the model, in which
quasi-local competitive interactions measured by a are
due to quasi-local foraging described by p; a412 if and
only if p412: Thus, the combined relative impact of
individuals from the two neighboring patches exceeds 1
if the fraction of resources acquired outside the natal
patch is bigger than 50%. Note also that a is a nonlinear function of p; so that, e.g. a ¼ 2 for p ¼ 0:8: Since
foraging outside the natal patch reduces resource
depletion in the natal patch and is thus advantageous
from the point of view of the individuals living and
reproducing in that patch, it would therefore appear to
be justified to consider a large range of a-values as
biologically realistic. This should be kept in mind during
the subsequent analysis.
Before describing the stability and dynamics in the
general case it is instructive to consider the simple
case of two patches with no dispersal. If we denote
the population densities in the two patches by NðtÞ
and MðtÞ; respectively, then system (1) and (2) takes
401
the form
lNðtÞ
;
1 þ a½NðtÞ þ aMðtÞ
lMðtÞ
Mðt þ 1Þ ¼
:
1 þ a½MðtÞ þ aNðtÞ
Nðt þ 1Þ ¼
ð11Þ
If the parameter a is interpreted in terms of foraging as
above, so that quasi-local competition is due to
individuals in both patches acquiring a fraction p of
their resources in the other patch, then similar arguments as above show that in this case a ¼ p=ð1 pÞ: In
particular, we have a41 if p412; i.e. in each habitat the
relative competitive impact of individuals from the other
habitat is larger than 1 if the fraction of time spent
foraging outside the natal patch is larger than 12: Note
that system (10) is formally equivalent to a discrete-time
version of the standard Lotka–Volterra equations for
competition between two species. Therefore, the standard analysis of the Lotka–Volterra system directly
applies. Accordingly, the dynamical system (10) has four
equilibria, with the trivial equilibrium N ¼ M ¼ 0 only
being stable if lo1; a case we will not consider further.
e ; 0Þ
In addition, there are two asymmetric equilibria ðN
e
e
e
and ð0; M Þ; where N ¼ M ¼ ðl 1Þ=a; and a symmetric
equilibrium ðN ; M Þ; where N ¼ M ¼ ðl 1Þ=½að1 þ
aÞ: It is well known that this symmetric equilibrium is
stable, and the two asymmetric equilibria are unstable, if
and only if ao1 (i.e. if and only if po12). For a41; i.e.
for p412; the symmetric equilibrium is destabilized, and
the two asymmetric equilibria both become locally
stable. Thus, if ao1 any initial configuration will
converge to the stable symmetric equilibrium
ðN ; M Þ: For a41; however, symmetry breaking
occurs and any initial configuration will converge either
e ; 0Þ or to ð0; M
e Þ: In particular, for a41 there are
to ðN
two attractors, with the corresponding basins of
attraction being separated by the diagonal N ¼ M:
These features, the existence of a unique stable
equilibrium for a below a certain critical value with
destabilization of this equilibrium and the appearance of
multiple attractors for larger a values, which appear
here in prototype form, are also present in the general
case in both one and two spatial dimensions. The most
significant difference between the two-patch case and the
general case is that, after symmetry breaking has
occurred, in the former there are only two attractors
while in the latter there are typically a very large number
of attractors (which increases rapidly with the number
of patches).
We now turn to the analysis of the dynamical
properties of the model in the general case. Eqs. (1)
and (5) fully describe the dynamics of a metapopulation with quasi-local interactions and dispersal, except
for the fact that we have not yet specified how to
deal with the marginal populations N1 and Nm in the
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402
one-dimensional chain of local populations. The model
that is most easily tractable analytically is obtained by
assuming that the two marginal populations are actually
neighbors, i.e. one assumes periodic boundary conditions, so that the chain of habitat patches is actually a
ring. In case of a one-dimensional habitat distribution it
is biologically perfectly plausible to consider periodic
boundary conditions, for example habitat patches lying
around the shore of a lake or pond might have this
property, as well as absorbing or reflecting boundary
conditions which might be plausible in other situations,
such as for habitat patches lying along a coastline. In
view of the analytic tractability of periodic boundary
condition, and their biological plausibility in certain
one-dimensional situations, we will focus on this case in
this section. In the next section, when we discuss twodimensional habitat distributions for which periodic
boundary conditions are clearly less biologically plausible, we will still consider periodic boundary conditions
for reasons of analytical tractability, but we will also
present numerical results for absorbing boundary
conditions. In general the dynamical behavior of the
models with different boundary conditions is qualitatively very similar and the same ecologically relevant
features are present in all cases.
With periodic boundary conditions when N1 and Nm
occupy neighboring patches, the index i 1 in Eqs. (1)–
(5) has to be substituted by m whenever the index i ¼ 1
on the left-hand side of these equations. Similarly, the
index i þ 1 must be substituted by the index 1 in
Eqs. (1)–(5) whenever i ¼ m on the left-hand side of
these equations. With periodic boundaries the conditions in all patches are exactly the same, because there
are no marginal patches that have to be treated
separately. As a consequence, the metapopulation
always has a unique spatially uniform equilibrium of
local population abundances, which can be found by
noting that if N is the size of each local population at
such a uniform equilibrium, then the effective density in
each patch is given by
e ¼ N þ 2aN :
N
ð12Þ
Since dispersal is passive a homogenous equilibrium
must be characterized by the fact that the per capita
e can be
reproductive output in each patch is 1, hence N
e
found by solving the equation f ðN Þ ¼ 1: Therefore,
e ¼ ð1 þ 2aÞN ¼ l1; and hence
N
a
N ¼
l1
:
að1 þ 2aÞ
ð13Þ
l1
Substituting N ¼ að1þ2aÞ
into Eqs. (1)–(5) shows that the
metapopulation state in which each local population has
size N is indeed an equilibrium. The uniqueness of this
homogenous equilibrium follows from the fact that the
equations which define the equilibrium are linear.
It is well known that for metapopulations consisting
of a number of identical single-species difference
equations that are coupled only by passive dispersal,
but not by any quasi-local ecological interactions, the
stability of the unique spatially homogenous equilibrium
is identical to the stability of the equilibrium of the local
difference equation. More precisely, the spatially uniform metapopulation equilibrium state in which each
local population is at the equilibrium of the local
difference equation is stable if and only if this
equilibrium is stable for the difference equation when
considered in isolation, i.e. if and only if the difference
equation taken in isolation does not exhibit periodic or
chaotic fluctuations (Rohani et al., 1996).
In our metapopulation model, the coupling between
local populations occurs both through passive dispersal
and through quasi-local competition. However, the case
in which coupling is by dispersal only is of course
obtained by setting the interaction parameter a ¼ 0 in
Eq. (2). In this case, the local population size at the
spatially uniform equilibrium is N ¼ l1
a ; i.e. equal to
the carrying capacity K of the local habitat patches.
Thus, since K is always a stable equilibrium for the
Beverton–Holt difference equation, the spatially uniform metapopulation equilibrium is stable if a ¼ 0;
independent of the dispersal rate d: We will now show
that increasing the strength of the quasi-local interactions by increasing the parameter a can destabilize the
spatially uniform equilibrium
For the stability analysis we write the dynamics of the
metapopulation in vector form as
N1 ðt þ 1Þ ¼ g1 ðN1 ðtÞ; y; Nm ðtÞÞ
^
Ni ðt þ 1Þ ¼ gi ðN1 ðtÞ; y; Nm ðtÞÞ
^
Nm ðt þ 1Þ ¼ gm ðN1 ðtÞ; y; Nm ðtÞÞ;
ð14Þ
where the functions gi are obtained by combining
Eqs. (1), (2) and (5) as
gi ðN1 ðtÞ; y; Nm ðtÞÞ
lNi ðtÞ
1 þ a½Ni ðtÞ þ aðNi1 ðtÞ þ Niþ1 ðtÞÞ
d
lNi1 ðtÞ
þ
2 1 þ a½Ni1 ðtÞ þ aðNi2 ðtÞ þ Ni ðtÞÞ
d
lNiþ1 ðtÞ
þ
:
2 1 þ a½Niþ1 ðtÞ þ aðNi ðtÞ þ Niþ2 ðtÞÞ
¼ ð1 dÞ
ð15Þ
In these equations the various indices have to be
adjusted to take the periodic boundary conditions into
account. Expression (13) makes it clear that due to the
quasi-local interactions and dispersal the dynamics of
each local population Ni actually depends on its two
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M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
nearest neighbors on either side, i.e. on Ni2 ; Ni1 ; Niþ1 ;
and Niþ2 (as well as on Ni itself).
To determine the stability of the uniform equilibrium
l1
given by ðN1 ; y; Nm Þ ¼ ðN ; y; N Þ with N ¼ að1þ2aÞ
;
we have to consider the Jacobian matrix J whose ij-th
entry is the derivative of the function gi with respect to
Nj evaluated at the equilibrium:
@gi J¼
ðN ; y; N Þ
:
ð16Þ
@Nj
i;j¼1;y;m
Taking into account that the local population size is
l1
N ¼ að1þ2aÞ
at the uniform equilibrium, one calculates
that
@gi u¼
@Ni l1 1
l1 a
;
¼ ð1 dÞ 1 d
l 1 þ 2a
l 1 þ 2a
@gi @gi v¼
¼
@Ni1 @Niþ1 d
l1 1
l1 a
;
¼ 1
ð1 dÞ
2
l 1 þ 2a
l 1 þ 2a
@gi @gi dl1 a
ð17Þ
w¼
¼
¼
2 l 1 þ 2a
@Ni2 @Niþ2 (where the notation j means evaluated at the [email protected]
brium N ). We already know that @N
¼ 0 if jai 2;
j
i 1; i; i þ 1; i þ 2; and therefore the Jacobian matrix at
the uniform equilibrium has the form
0
u
Bv
B
B
Bw
B
B
B
B
J¼B
B0
B
B
B
B0
B
B
@w
v
v
u
w
v
0
w
0
v
u
v
w
y
0
w
v
y
0
w
y
0
0
y
y
y
0
&
u
&
0
w
0
y
v
w
0
y
w
0
v
w
0
u
v
w
v
u
v
1
v
wC
C
C
0C
C
C
C
C
0C
C
C
C
C
wC
C
C
vA
u
ð18Þ
This is a circulant matrix, i.e. a matrix in which all rows
are identical, except for a cyclic shift by one element that
is applied to each row of the matrix to obtain the row
below. The circulant property is simply a reflection of
the periodic boundary conditions that we assumed for
the metapopulation.
To determine the stability of the uniform equilibrium
we have to calculate the dominant eigenvalue of the
matrix J; i.e. the eigenvalue with largest absolute value.
The equilibrium is stable if and only if the dominant
eigenvalue has absolute value o1: It is well known
(Davis, 1979) how to calculate all the eigenvalues of
circulant matrices. Suppose first that d ¼ 0; i.e. that
403
there is no dispersal, and that the number of patches m
in the metapopulation, and hence the size of the matrix
J; is even. If d ¼ 0; then w ¼ 0 in Eq. (15), so that each
row of J has three adjacent entries. Because m is even,
the dominant eigenvalue of J is then given by the sum
over the absolute values of the three entries in a row
(Davis, 1979). For d ¼ 0 we have u40 and vo0 in
Eq. (15), because l41 and a40 by assumption. Therefore, the dominant eigenvalue z of J is given by
z ¼ u 2v
l1 1
l1 a
þ2
¼ 1
l 1 þ 2a
l 1 þ 2a
4la 2a þ 1
:
¼
2la þ l
ð19Þ
Therefore, z will be larger than 1 if 4la 2a þ 142la þ
l; i.e. if 1 2a4lð1 2aÞ: This occurs as soon as
1
a4 :
2
ð20Þ
In other words, if there is no dispersal, quasi-local
competition destabilizes the spatially uniform equilibrium if the competitive impact on reproduction in a
given patch from an individual from a neighboring
patch exceeds one half of the impact of an individual
from the given patch itself. This result holds for an even
number m of patches, but one can show that it is also
true for odd m as long as m is not very small. Moreover,
qualitatively similar results hold if dispersal rates are
non-zero. In fact, dispersal has a stabilizing effect, so
that the strength of quasi-local competition needed to
destabilize the uniform equilibrium is larger than 12 if
dispersal rates are non-zero. Nevertheless, with non-zero
dispersal the equilibrium still becomes unstable if a is
large enough. More precisely, it can be shown by a
detailed analysis of the Jacobian matrix J that the
critical a-value a ðdÞ needed for destabilization of the
spatially uniform equilibrium in a chain of habitat
patches with dispersal rate d is given by
a ðdÞ ¼
l 1 þ 2d
;
2dð1 3lÞ þ 2ðl 1Þ
ð21Þ
where l is the basic growth rate in the Beverton–Holt
model (4). This expression holds as long as dispersal
rates are not too large. In Fig. 1 the critical a-values are
plotted as a function of dispersal rates d:
Thus we have obtained our first main result: in a
metapopulation consisting of a chain of identical habitat
patches each of which is coupled to its two nearest
neighbors by quasi-local competition and dispersal, the
spatially uniform equilibrium state becomes unstable if
quasi-local competition is strong enough. Similar results
also hold when different boundary conditions are used,
although the systems then tend to become analytically
intractable. For example, in a metapopulation with
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404
Clacitir α
1.5
1
0.5
0
0.05
0.1
0.15
0.2
Dispersal rate d
Fig. 1. Strength of quasi-local competition destabilizing the spatially
uniform steady state in a chain of local habitats as a function of
dispersal rates. The critical a-value above which the uniform
equilibrium becomes unstable is given by Eq. (15) and shown here
for l ¼ expð2:3Þ as a function of the dispersal rate d:
absorbing boundary conditions there is always an
equilibrium state that corresponds to the uniform
equilibrium for periodic boundary conditions. However,
due to the absorbing boundaries this equilibrium is not
spatially uniform anymore. Nevertheless, it is still true
that while dispersal alone does not destabilize this
equilibrium, the equilibrium becomes unstable if quasilocal competition is strong enough.
What we have seen so far is that quasilocal competition can break the symmetry in an intrinsically homogenous metapopulation by destabilizing
the spatially uniform equilibrium. The question we
now address is: what are the ecological consequences
of this broken symmetry, i.e. what is the dynamics
of the metapopulation when the uniform equilibrium is
unstable?
To study the dynamics of spatially non-uniform
patterns of population abundance we have used
extensive numerical simulation, which revealed two
main results. First, the metapopulation always converges to an equilibrium state and never exhibits
periodic or chaotic dynamics. Thus, the intrinsic
stability provided by the local Beverton–Holt difference
equation always ensures convergence to a necessarily
spatially non-uniform stable equilibrium if the uniform
equilibrium is unstable. The second result, however, is
that for large metapopulations there are a large number
of different spatially non-uniform equilibrium states
which are locally stable for the metapopulation
dynamics. The different stable equilibria tend to have
small basins of attraction so that the final equilibrium
pattern of local abundances that is attained by the
metapopulation depends critically on the initial distribution of local population sizes in the various habitat
patches. Fig. 2 shows examples of different equilibrium
configurations to which the metapopulation converges
depending on the initial conditions. One source for the
plethora of different stable equilibrium states is that if
an equilibrium is spatially non-uniform one obtains a
new equilibrium by shifting the given one along the
chain of local populations. For example, if there is no
dispersal, and if the number of local populations m is
even, then one obvious non-uniform equilibrium configuration of local abundances is to have local population sizes of 0 and K; the local carrying capacity,
alternate across the metapopulation. This can obviously
be done in two different ways (depending on whether the
even numbered local populations are at 0 or at K), and
hence this spatially non-uniform equilibrium corresponds to two different equilibrium states of the
metapopulation. Note that the difference between the
two equilibria is biologically not trivial since they
induce very different population sizes in a given habitat
patch.
Most non-uniform equilibrium configurations are
much less symmetrical than the one alternating between
0 and K just described, and therefore most non-uniform
equilibria give rise to many other equilibria by shifting
them across the metapopulation. This is particularly
true in large metapopulations consisting of many
patches. In fact, the number of different stable equilibria
can be very large. This is illustrated in Fig. 3, where we
have plotted the cumulative number of different
equilibrium states attained as a function of the number
of trials in which the metapopulation dynamics was
started using different initial conditions. Fig. 3 illustrates that the number of different equilibrium attractors
can become very large if the metapopulation consists of
many patches. In the next section we will see that this
result is also true for metapopulations that extend over
two spatial directions and consist of a two-dimensional
grid of local populations. In the subsequent section we
then discuss the significance of the existence of a
multitude of attractors for the ecological dynamics in
metapopulations that are subject to environmental
noise.
2.2. Spatially two-dimensional metapopulations:
grids of local habitat patches
Here we extend the result of the previous section to
metapopulations in which the local populations occupy
a two-dimensional lattice of habitat patches. We
assume, as before, that quasi-local competition and
dispersal occur between neighboring patches, and we
thus have to specify what constitutes a neighborhood of
a given local population. Typical choices for neighborhoods consist of taking either the four or the eight
patches that are closest to a given site on the lattice. In
addition, we again have to specify boundary conditions,
which can for example be periodic or absorbing.
However, our extensive numerical simulations showed
ARTICLE IN PRESS
0.8
0.8
0.7
0.7
Local population size
Local population size
M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
2
4
6
8
10 12 14 16 18 20
0
2
4
Local population number
6
8
10 12 14 16 18 20
Local population number
0.8
0.8
0.7
0.7
Local population size
Local population size
405
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
2
4
6 8 10 12 14 16 18 20
Local population number
0
2
4
6
8 10 12 14 16 18 20
Local population number
Fig. 2. Different spatially non-uniform equilibrium attractors for a chain of 20 local habitats in which the strength of quasi-local competition is
strong enough to destabilize the spatially uniform equilibrium. The top left panel shows the alternating steady state described in the text. The
remaining panels show attractors with a less regular pattern of local population abundance. The two lower panels show attractors that are obtained
from each other by a permutation of the local habitats. Note that despite this permutational symmetry the steady state of a given local population
may be very different in the two attractors. Parameter values are l ¼ exp ð2:3Þ; a ¼ 0:9; and d ¼ 0:05:
that all these choices have little qualitative effect on the
results reported below.
We again assume that all habitat patches are
ecologically identical, and that the local ecological
dynamics in each patch are described by a Beverton–
Holt equation f : Thus, if Nij ðtÞ is the population size in
the patch with coordinates ði; jÞ on the metapopulation
lattice (i; j ¼ 1; y; n; where n2 is the size of the square
lattice), then we have
eij ðtÞÞ
Nij0 ðtÞ ¼ Nij ðtÞ f ðN
¼ Nij ðtÞ
l
;
eij ðtÞ
1 þ aN
ð22Þ
where Nij0 ðtÞ is the population size in patch ði; jÞ
after reproduction, but before dispersal in year t; f is
eij ðtÞ is the
the per capita reproductive output, and N
effective population density impinging on reproduction
in patch ði; jÞ in year t: In analogy to the onedimensional metapopulation of the previous section we
assume that the effective population size is a function of
the population sizes in patch ði; jÞ itself and of the
population sizes in neighboring patches. For example, if
the four nearest neighbors constitute the neighborhood
of a local population, then
eij ðtÞ ¼ Nij ðtÞ þ aðNi1j ðtÞ þ Niþ1j ðtÞ þ Nij1 ðtÞ
N
þ Nijþ1 ðtÞÞ:
ð23Þ
Following the same arguments as in the case of a onedimensional chain of habitats, it is easy to see that if the
quasi-local ecological interactions occur because individuals in a given patch forage a fraction p of the time in
neighboring patches, then the parameter a is given by
p
;
ð24Þ
a¼
kð1 pÞ
where k is the number of neighboring patches involved.
Since foraging outside a given patch is advantageous for
the individuals reproducing in that patch, it follows again
that a large range of a-values is biologically feasible.
In addition to quasi-local interactions we again
assume that passive dispersal from and to nearest
neighbors completes the dynamics in a given year. For
example, with four nearest neighbors we have
d
0
0
Nij ðt þ 1Þ ¼ ð1 dÞNij0 ðtÞ þ ðNi1j
ðtÞ þ Niþ1j
ðtÞ
4
0
0
ðtÞ þ Nijþ1
ðtÞÞ:
þ Nij1
ð25Þ
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M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
406
Fig. 3. Cumulative number of different spatially non-uniform
attractors in chains of local populations of different length. In each
case, the metapopulation dynamics were run 30,000 times, each time
from different, randomly chosen starting conditions, until an
equilibrium state was reached. The program then counted the number
of different equilibria attained during the course of the 30,000 trials. (a)
With m ¼ 20 local habitats, the cumulative number of different
attractors saturates at 62. (b) With m ¼ 50 and 100 local habitats, the
cumulative number does not saturate and instead increases steadily.
The increase is steeper with more habitats. This indicates that the
number of different attractors would further increase with larger
numbers of trials, suggesting the presence of a huge total number of
different attractors. Parameter values used were l ¼ exp ð2:3Þ; a ¼ 0:7
and d ¼ 0:07:
Eqs. (22) and (24) have to be appropriately adjusted for
marginal patches and if different neighborhoods are
used. With periodic boundary conditions there is again a
spatially uniform equilibrium state in which all local
populations have size N ; which can be obtained by
eij Þ ¼ 1: For example, with
solving the equation f ðN
neighborhoods consisting of the four nearest neighbors
eij ¼ N þ 4aN at the equilibrium, hence
we have N
l
solving
¼ 1 yields
eij ðtÞ
1þaN
N ¼
l1
:
að1 þ 4aÞ
ð26Þ
If there is no quasi-local competition, i.e. if a ¼ 0; then
N ¼ K; the carrying capacity of the Beverton–Holt
equation, and the local populations are only coupled by
dispersal. In this case it is again true that the stability of
the spatially uniform equilibrium is exactly the same as
the stability of the equilibrium K for the dynamics
of an isolated local population (Jansen and Lloyd,
2000). In particular, in the case considered here the
uniform equilibrium is always stable, because K is a
stable equilibrium for the Beverton–Holt dynamics.
However, quasi-local competition can again destabilize
this equilibrium.
This can be seen by noting that one can approximate
the spatially two-dimensional metapopulation by a
spatially one-dimensional description if one numbers
the patches consecutively by going through the rows of
the lattice one after the other. In a square lattice with n
patches in each of n rows, the patch at position ði; jÞ is
then assigned the number in þ j: In this way one obtains
a spatially one-dimensional metapopulation consisting
of a chain of habitat patches as used in the previous
section, except that now the coupling between different
patches in this chain does not only occur between
nearest neighbors, but also between distant patches. For
example, if in a square lattice, Nij receives competition
from Ni1j ; Niþ1j ; Nij1 and Nijþ1 ; then the local
population Ninþj in the corresponding one-dimensional
chain of habitats receives competition from its two
immediate neighbors Ninþj1 and Ninþjþ1 ; as well as from
Nði1Þnþj and from Nðiþ1Þnþj ; i.e. from populations that
are n steps away on either side in the chain. Therefore,
the analysis of the two-dimensional lattice case reduces
to an analysis of a one-dimensional chain of habitats
with a more general form of couplings than considered
in the previous section. Similar remarks apply for other
choices of neighborhoods on the square lattice.
It must be noted that this one-dimensional description
is not a completely faithful representation of the twodimensional metapopulation. For example, in a square
lattice of size m ¼ n2 with periodic boundary conditions
and local neighborhoods consisting of four neighboring
patches, every patch with index ð1; jÞ or ðn; jÞ has two
neighbors on the same row as the patch itself, but to
obtain the one-dimensional description one of these
neighbors is replaced by a corresponding patch on an
adjacent row. However, for our results this discrepancy
only has visible effects for metapopulations with a very
small number of patches. We also note that it is
straightforward to show that the Jacobian of the twodimensional model is exactly given by a block circulant
matrix. We have chosen to use the approximate
representation given above rather than the exact block
circulant description as the former allows us to reduce
the calculation to the one-dimensional case considered
earlier. Furthermore, the technique of reducing the
stability analysis of a two-dimensional system to a onedimensional one seems to be of some intrinsic interest.
For the stability analysis the reduction to a onedimensional chain of patches implies that the Jacobian
matrix J at the spatially uniform equilibrium in which
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M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
all local populations have size N has the same form as
(17), except that now not only entries adjacent to the
diagonal entries are non-zero, but also entries that are
some distances away, reflecting the fact that in the
reduced descriptions interactions occur between distant
local populations. For example, when considering
interactions with the four nearest neighbors on a square
lattice with d ¼ 0; so that there is no dispersal, and with
periodic boundaries, every local population interacts,
through quasi-local competition, with four other
patches in the reduced one-dimensional description:
two of these are its adjacent neighbors, and two are n
steps away on either side in the chain. (Note that if
dispersal is non-zero, then interactions occurs with more
local populations, some of which are even further away
from the focal patch in the chain.) Therefore, the
derivative of the function gi describing the population
dynamics of a patch through
Ni ðt þ 1Þ ¼ gi ðN1 ðtÞ; y; Nm ðtÞÞ;
ð27Þ
2
(where m ¼ n is the size of the lattice, and hence the
length of the corresponding one-dimensional chain) is
non-zero only with respect to Nin ; Ni1 ; Ni ; Niþ1 ; and
Niþn ; and at the uniform equilibrium N we have
@gi l1 1
;
u¼
¼1
l 1 þ 4a
@Ni @gi @gi @gi @gi v¼
¼
¼
¼
@Nin @Ni1 @Niþ1 @Niþn l1 a
:
ð28Þ
¼
l 1 þ 4a
Therefore, the Jacobian matrix J at the uniform
equilibrium has the form
0
B
J ¼ @0
y 0
v |ffl{zffl}
0y0
n1
y
v u
y
1
v |ffl{zffl}
0y0
n1
v 0
y
0C
A:
407
Thus, in the absence of dispersal the uniform equilibrium becomes unstable when the relative competitive
impact of individuals from neighboring patches on the
lattice exceed 14 that from individuals living in the same
patch. More generally, for small non-zero dispersal rates
d a detailed analysis of the Jacobian matrix J yields the
following formula for the critical a-value a ðdÞ needed
for destabilization of the spatially uniform equilibrium
in a square metapopulation with periodic boundary
conditions
a ðdÞ ¼
l 1 þ 2d
:
2d½1 lðk þ 1Þ þ kðl 1Þ
ð32Þ
Here k is the number of neighbors in the local
neighborhood, i.e. k ¼ 4 or k ¼ 8: Note that for k ¼ 2
Eq. (31) reduces to the corresponding Eq. (20) for chains
of habitat patches. It follows from Eq. (31) that
dispersal has a stabilizing effect, but that quasi-local
competition can destabilize the spatially uniform equilibrium also for non-zero dispersal rates, and that the
more interacting neighboring populations there are, the
more likely it is that the symmetry of the uniform
equilibrium will be broken. This is illustrated in Fig. 4.
Similar results can be obtained for other types of
boundary conditions, although in such cases the
corresponding equilibrium that is destabilized by
quasi-local competition is not strictly spatially uniform
anymore, because the marginal patches are intrinsically different from all the other patches in the
metapopulation.
The instability of the uniform equilibrium again raises
the question about the ensuing dynamics. The results for
two-dimensional lattice metapopulations are qualitatively similar to those of spatially one-dimensional
metapopulations described in the previous section: the
lattice metapopulations always converge to a spatially
ð29Þ
0.5
Critical α
With periodic boundaries J is again a circulant matrix,
so that all the rows have the same form as the one
indicated in (28), except for a cyclic shift depending on
the position of the focal patch. In particular, all rows
have the same non-zero entries. If m ¼ n2 is even, it
follows as in the previous section that the dominant
eigenvalue of J is simply the sum of all the absolute
values of the entries in a given row of J: Since u40 and
vo0 because l41 and a40; the dominant eigenvalue z
of J is then
k=4
0.25
k=8
0.125
0
z ¼ u 4v;
8la 4a þ 1
¼
:
4la þ l
It easily follows that z41 if
ð30Þ
1
a4 :
4
ð31Þ
0.05
0.1
0.15
0.2
Dispersal rate d
Fig. 4. Strength of quasi-local competition destabilizing the spatially
uniform steady state in a square lattice of local habitats as a function
of dispersal rates. The critical a-value above which the uniform
equilibrium becomes unstable is given by Eq. (25) and shown here for
l ¼ exp ð2:3Þ as a function of the dispersal rate d for two different
neighborhood sizes, k ¼ 4 and 8.
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408
0.3
0.2
0.1
7
Local population size
Local population size
0.3
0.1
7
4
0
1
2
3
4
5
6
4
0
1
1
2
3
7
0.3
4
5
6
1
7
0.3
0.2
0.1
7
4
0
1
2
3
4
5
6
1
7
Local population size
Local population size
0.2
0.2
0.1
7
4
0
1
2
3
4
5
6
1
7
Fig. 5. Different spatially non-uniform equilibrium attractors for a square lattice of 7 7 local habitats in which the strength of quasi-local
competition is strong enough to destabilize the spatially uniform equilibrium. All attractors have irregular patterns of local population abundance.
The two top panels show attractors that are obtained from each other by a permutation of the local habitats. Note that despite this permutational
symmetry the steady state of a given local population may be very different in the two attractors. Parameter values were l ¼ exp ð2:3Þ; a ¼ 0:7 and
d ¼ 0:05:
non-uniform equilibrium and never exhibit non-equilibrium dynamics. However, there is typically a large
number of different equilibrium states to which the
metapopulation may converge depending on initial
conditions. Examples of possible configurations of local
population abundances at non-uniform equilibria are
shown in Fig. 5.
In Fig. 6 we have plotted the numbers of different
ecological attractors reached in 10,000 trials started
from different initial conditions as a function of the
strength of quasi-local competition a: This is shown for
different boundary conditions in the metapopulation
and for different dispersal rates. The results indicate that
there can be huge numbers of different attractors. For
example, with zero dispersal and a ¼ 0:6; which
corresponds to spending approximately 55% of the
time foraging in neighboring patches, we get more than
5000 different attractors out of 10,000 trials, which
suggests that the total number of different attractors
(which could be obtained only from a much larger
number of trials) is indeed exceedingly large. In Fig. 6
we also illustrate the fact that the occurrence of large
numbers of different attractors is not an artefact of
periodic boundary conditions in the metapopulation.
Due to the decreased symmetry in the system the
number of attractors tends to be lower in metapopulations with absorbing boundaries than in metapopulations with periodic boundaries. However, with
absorbing boundaries the number of attractors also
becomes very large even for moderate values of a as the
system size increases. Fig. 6 also shows that dispersal
tends to have a stabilizing effect in the sense that for
higher dispersal rates the number of attractors increases
more slowly with a:
We have presented above detailed results for the
Beverton–Holt model with quasi-local competition, and
it may be appropriate here to describe briefly the results
we obtained for other models with quasi-local competition. We have studied discrete-time models with quasilocal competition in which the reproductive output f is
of Ricker (1954) or Bellows (1981) type, rather than of
Beverton–Holt type. For parameter values which yield
stable local dynamics the results we obtain are exactly
analogous to those obtained using the Beverton–Holt
models, namely: (i) for any values of a and d the system
converges to a stable equilibrium; (ii) for d ¼ 0 and ao12;
in one dimension, or ao14 in two dimensions with four
neighbors, or ao18 in two dimensions with eight
ARTICLE IN PRESS
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periodic 7x7 lattice, d=0
periodic 7x7 lattice, d=0.1
absorbing 10x10 lattice, d=0
Cumulative number of different
attractors after 10'000 trials
absorbing 7x7 lattice, d=0
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
strength of non-local competition α
Fig. 6. Cumulative numbers of different spatially non-uniform
attractors as a function of the strength of quasi-local competition a:
For equidistant values of a (increments of 0.05) the metapopulation
dynamics were run 10,000 times, each time from different, randomly
chosen starting conditions, until an equilibrium state was reached. For
each value of a the simulation counted the number of different
equilibria attained during the 10,000 trials. This was done for lattices
with different boundary conditions and for different dispersal rates.
With absorbing boundaries the number of attractors is generally
smaller than with periodic boundary conditions (cf. filled triangles and
circles). However, even with absorbing boundaries the number of
attractors becomes very large as the lattice size is increased (open
squares). In general, the number of attractors decreases as dispersal
rates increase (cf. filled circles and squares), but even for relatively high
dispersal rates the number of attractors becomes very large for high
intensities of quasi-local competition. The parameter l ¼ exp ð2:3Þ in
all simulations.
neighbors, the system converges to a unique homogeneous stable equilibrium; (iii) if a is sufficiently large
to break the homogeneous equilibrium then there exist,
in general, a large number of stable inhomogeneous
equilibria. Thus the features we have described above
for the Beverton–Holt model with quasi-local competition seem to be quite general features of any singlespecies difference equation with quasi-local competition.
We have also considered continuous-time singlespecies ecological models with quasi-local competition.
The simplest example of such a model, which describes a
metapopulation in one dimension, is the following
system of Lotka–Volterra type. If we let Xi ðtÞ denote
the population size in patch i as a function of time t;
then the dynamics of the corresponding metapopulation
is given by the following system of ordinary differential
equations:
dXi
¼ rXi ½1 aðaXi1 þ Xi þ aXiþ1 Þ:
dt
ð33Þ
This equation has obvious analogs in two dimensions
with four and eight neighbors. Here we have neglected
409
dispersal, however, this can easily be included by adding
a suitable dispersal term. The stability analysis of this
system (without dispersal) can be carried out in
complete analogy with that given above for the
discrete-time model. We find (in one dimension with
periodic boundaries) that for ao12 there is a unique
spatially homogeneous stable equilibrium. For a412 this
homogenous equilibrium becomes unstable and the
system always converges to an inhomogeneous stable
equilibrium. Numerical integration of the system (32)
shows that for a412 there are, in general, for both
periodic and absorbing boundary conditions, many
distinct inhomogeneous stable equilibria. The obvious
analogs of these results hold in two dimensions. That is,
we find that points (i)–(iii) above hold for this
continuous time system with quasi-local competition.
This does not come as a surprise, since the discrete-time
Beverton–Holt equation that we used for most of our
analysis is homologous, i.e. dynamically equivalent, to
the continuous time logistic equation on which system
(32) is based (Yodzis, 1989). Hence, it appears that these
features are very general properties of single-species
population models, in both discrete and continuous
time, with quasi-local competition.
The central theme of this paper has been that
quasi-local competition can lead to the destabilization
of the homogenous equilibrium and the appearance
of a multitude of inhomogeneous stable equilibria.
This phenomenon can have important consequences
for the population dynamics of the system in the
presence of noise. When there are many different
attractors the basins of attraction of each one of them
is likely to be small. In this case, environmental
stochasticity can easily perturb the metapopulation
from one basin of attraction to another. As a
consequence, the dynamics of the metapopulation will
continually shift between convergence to different
equilibrium attractors and hence exhibit persistent
fluctuations. Therefore, small amounts of environmental stochasticity can have profound effects on the
dynamics of metapopulations with quasi-local competition. In the next section we explore in more detail the
nature of the population fluctuations that environmental noise induces.
2.3. Random walks in attractor space
We introduce environmental stochasticity into the
local population dynamics given by Eq. (4) and by
Eq. (21) for the spatially one and two-dimensional
metapopulations, respectively, by assuming that the
deterministic per capita reproductive output described
by these equations is multiplied by a random variable
with mean 1. Thus, in a two-dimensional lattice
metapopulation the per capita reproductive output in
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M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
a local population Nij in year t is
l
eij ðtÞÞ ¼
exp ðxij ðtÞÞ;
f ðN
eij ðtÞ
1 þ aN
ð34Þ
eij ðtÞ is, as before, the effective density impinging
where N
on reproduction in year t in the patch considered, and
where xij ðtÞ is a random number that is drawn from a
Gaussian distribution with mean 0 and variance s
independently for each patch in each year. This form of
incorporating noise into the model ensures that biologically meaningless, negative noise terms to do not occur
as multipliers of the per capita reproductive output.
However, for small values of the parameter s; as we will
use below, the noise multipliers are still approximately
normally distributed with mean 1 and variance s:
If the deterministic Eq. (21) is replaced by its
stochastic version (33) in lattice metapopulations in
which there is no quasi-local competition (i.e. in which
a ¼ 0; so that the only coupling between local populations occurs through dispersal) then the uniform
equilibrium that is the only attractor in the corresponding deterministic models is constantly perturbed by the
environmental stochastic fluctuations. As a consequence
the local population sizes as well as the total metapopulation undergo persistent fluctuations, but these
fluctuations simply reflect repeated returns to the unique
uniform equilibrium after each perturbation. As a
consequence, the size of the population fluctuations is
comparable to the size of the environmental perturbations. In particular, local fluctuations are small when the
environmental perturbations are small, and the fluctuations at the scale of the whole metapopulation are even
smaller because fluctuations on the local scale tend to
average out on the global scale.
A very different picture emerges in metapopulations
in which local populations are coupled through quasilocal competition, and in which the strength of the
quasi-local competition a is large enough to destabilize
the uniform metapopulation equilibrium. In this case
there are typically very many different, spatially nonuniform equilibrium attractors for the metapopulation
dynamics, as we have shown in the previous sections. It
is helpful to view each attractor as a point in m-space,
where the dimension m is the number of habitat patches
in the metapopulation. For example, a uniform equilibrium would then correspond to a point on the diagonal
in m-space. Each attractor has associated with it its
basin of attraction, and m-space is divided into mutually
disjoint basins of attraction. Since there are typically
many attractors, there are also very many basins of
attraction, and each one of them is typically small.
It is now easy to imagine what happens when the
system is constantly perturbed by small amounts of
environmental noise, as described by Eq. (33). While the
metapopulation is converging to one of the many
attractors an environmental perturbation can easily
throw the system off its path and into the basin of
attraction of a different attractor, to which it starts to
converge. Since the new attractor may correspond to a
very different distribution of local population sizes, the
corresponding change in population size can be large,
both on the local and on the global scale. Repeated
operation of this process can therefore lead to substantial and persistent fluctuations in both local and
total population size. These fluctuations are thus due to
an interaction between environmental stochasticity and
the deterministic dynamical properties of the metapopulation. In the form of a multitude of different
equilibrium attractors the deterministic metapopulation
dynamics provides the raw material which allows
environmental noise to induce a random walk in which
the system constantly jumps from one attractor to
another. In other words, due to environmental noise, the
system performs a random walk in the space of all
attractors.
This random walk in attractor space is reflected in the
dynamics of the local populations comprising the
metapopulation, as is shown in Fig. 7a and b. The local
population sizes undergo large fluctuations with a
distinct ‘random walk’ pattern. Recall that in the purely
deterministic case the equilibrium to which the metapopulations converge depends sensitively on the initial
configuration of local population abundances. In
stochastic metapopulations, this dependence on initial
conditions results in a dependence of the fluctuating
population trajectories on the initial conditions. Thus,
the random walk dynamics of local populations can
depend critically on the initial distribution of local
population sizes. To illustrate this, panels a and b in Fig.
7 show the dynamics of a particular local population in
a metapopulation undergoing a random walk in
attractor space. The series of randomly generated
environmental perturbations is exactly the same in the
two cases, and the only difference between the panels is
that the metapopulation dynamics was started from
different initial conditions. Clearly, for extended periods
of time the local population trajectories may follow very
different paths for different initial conditions. This
phenomenon is reminiscent of the sensitive dependence
on initial conditions of chaotic systems, but occurs here
not due to chaos in the underlying deterministic system,
but due to the existence of many different attractors.
Fluctuations in the total population size of the
metapopulation of which the local population shown
in Fig. 7a is a part are shown in Fig. 7c. Here the
random walk properties are less apparent, because local
fluctuations tend to average out on the scale of the total
metapopulation. Nevertheless, the time series still has
some attributes of a random walk (see below), in
contrast to the time series of population fluctuations in
metapopulations in which there is no quasi-local
competition ða ¼ 0Þ; and in which environmental noise
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0.55
Local population size
Local population size
0.55
0.45
0.35
0.25
150
411
250
(a)
350
450
0.45
0.35
0.25
150
550
250
350
(b)
generation
450
550
450
550
generation
22.5
22
150
(c)
Local population size
Total population size
23
250
350
450
1.15
1.05
0.95
0.85
150
550
250
350
generation
(d)
generation
Local population size
0.55
0.45
0.35
0.25
150
250
(e)
350
450
550
generation
Fig. 7. Time series for local and total population sizes in stochastic metapopulations. (a) Time series of a local population that is part of a 10 10
square lattice metapopulation with periodic boundaries that performs a random walk in attractor space due to environmental perturbations. The
local population undergoes large fluctuations with a distinct pattern reflecting the random walk in attractor space. (b) Same as (a) except that a
different initial configuration of local population abundance was used to simulate the dynamics, for which exactly the same random environmental
perturbations were used as in (a). This shows the sensitive dependence of the trajectories on the initial conditions in the stochastic systems. (c) The
corresponding time series of the total population size of the metapopulation. (d) Time series of a local population in a metapopulation in which there
is no quasi-local competition ða ¼ 0Þ: In this case the local populations fluctuate around the steady state corresponding to the spatially uniform
metapopulation equilibrium, and the local fluctuations have the same magnitude and frequencies as the environmental perturbations. (e) Same as (a)
but with absorbing rather than periodic boundary conditions for the metapopulation. This shows that random walk fluctuations in local population
size are not an artefact of periodic boundary conditions. Parameter values were l ¼ exp ð2:3Þ; a ¼ 0:5 and d ¼ 0:05; except for (d) in which a ¼ 0:
For the environmental noise, a normal distribution with variance s ¼ 0:05 was used in Eq. (25).
therefore only induces perturbations around the spatially uniform attractor. Fig. 7d shows the fluctuations
of a local population in a stochastic metapopulation
without quasi-local competition. The local fluctuations
simply track the environmental noise around the
spatially uniform steady state, and the qualitative
difference to the dynamics shown in Fig. 7a and b is
apparent. Finally, Fig. 7e illustrates that random walks
of local population densities are not artefacts of periodic
boundary conditions in the metapopulation. In Fig. 7e
everything was exactly the same as in Fig. 7a, except that
absorbing rather than periodic boundary conditions
were used. Again, the local populations undergo large
fluctuations in a distinct random walk pattern.
In general, if a dynamical system performs a random
walk in its state space it tends to explore more and more
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412
basin of attraction can have very irregular boundaries.
Despite these complications one might expect that a
random walk in the set of attractors leads to fluctuations
whose power spectrum obeys a power law. Fig. 8a shows
the power spectrum for the time series of the local
population size whose dynamics are shown in Fig. 7a.
Clearly, the power spectrum obeys a power law with
distinctly negative exponent, showing a prominent red
shift in the local fluctuations. This red shift is due to
intermittent switches between convergence to different
attractors, switches that are also apparent in the
corresponding time series (Fig. 7a). Power spectra
obeying power laws with negative exponents can be
found for local populations in the stochastic metapopulation models whenever the underlying deterministic model
has many different equilibrium attrcators.
The corresponding power spectrum for the time series
of the total population size of the metapopulation (time
series shown in Fig. 7c) is shown in Fig. 8b. This
spectrum also appears to follow a power law with
negative exponent. However, the red shift is less
apparent than in the local populations, and the
-3.5
-3.5
-4.5
-4.5
Log (power)
Log (power)
remote regions of this space as time progresses. As a
consequence, the variation in the states attained by the
system increases with the time window over which the
states are sampled. Thus, in a system performing a random
walk long-term fluctuations are larger than short-term
fluctuations. As a consequence the power spectrum of a
time series of the system’s state tends to be red-shifted, i.e.
it tends to have more power at low frequencies. In fact, the
power spectrum of random walks typically obey power
laws with negative exponents, so that the logarithm of the
power is a linear function with negative slope of the
logarithm of the frequency. For example, random walk
models of Brownian motion yield a power law with
exponent 2 (Hastings and Sugihara, 1993).
In the present case, in which environmental noise is
superimposed on a deterministic framework consisting of
many different basins of attraction, stochastic displacements and deterministic convergence to attractors in state
space interact in a way that is dictated by the geometry of
the basins of attraction, which may be very complicated.
For example, the overall size of the basins of attraction
can be different for different attractors, and any given
-5.5
-6.5
-6.5
-7.5
-7.5
-6
-5
-4
-3
-2
Log (frequency)
-1
0
-6
-5
-4
-6
-5
-4
(b)
-3.5
-3.5
-4.5
-4.5
Log (power)
Log (power)
(a)
-5.5
-3
-2
Log (frequency)
-1
0
-1
0
-5.5
-6.5
-6.5
-7.5
-7.5
-6
(c)
-5.5
-5
-4
-3
-2
Log (frequency)
-1
0
(d)
-3
-2
Log (frequency)
Fig. 8. Power spectra of the time series of local and total population sizes. (a) Power spectrum of the local population shown in Fig. 7a. The power
spectrum is red-shifted and obeys a power law with negative exponent of approximately 0.62. (b) Power spectrum of the total population size of the
metapopulation shown in Fig. 7c. The power spectrum is slightly red-shifted and obeys a power law with negative exponent of ca. 0.13. (c) Power
spectrum of the local population shown in Fig. 7d. The power spectrum reflects the white spectrum of the environmental noise. (d) Power spectrum of
the local population shown in Fig. 7e. The local fluctuations are again governed by a power law, with a negative exponent of ca. 0.58. For all panels
power spectra were calculated as average of the spectra of 200 time series of length 1024 time steps obtained consecutively in a single run of the
corresponding metapopulation dynamics.
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corresponding exponent is closer to 0. Thus, despite its
red shift the spectrum of the total population is similar
to that of white noise, because the local fluctuations tend
to average out on the global scale, so that the global
fluctuations reflect the white power spectrum assumed
for the environmental noise. The white fluctuations of
the environmental noise are even better matched in
stochastic metapopulations without quasi-local competition, in which environmental perturbations induce
fluctuations around the spatially uniform equilibrium
attractor. This can be seen in Fig. 8c, which shows the
power spectrum of the time series of the local population
shown in Fig. 7d. In this power spectrum very short
frequencies still have slightly lower power than expected
for white noise, which is likely due to the interaction
between the white environmental noise and the nonlinear Beverton–Holt dynamics of the local populations
(for an interesting treatment of power spectra resulting
from the interaction between environmental perturbations and non-linear deterministic dynamics see Kaitala
et al., 1997). Finally, Fig. 8d shows the power spectrum
of the time series of the local population shown in Fig.
7e, for which a metapopulation with absorbing rather
than periodic boundaries was used. Again, the power
spectrum obeys a power law with distinctly negative
exponent, showing that the power law behavior is not an
artefact of periodic boundary conditions, and that
quasi-local competition can also lead to red-shifted
population fluctuations in metapopulations with absorbing boundaries.
3. Discussion
In this paper we have presented a new way of
modeling metapopulations consisting of a number of
habitat patches occupied by local populations with
discrete generations. Traditional models of such metapopulations assume that the ecological coupling between local populations occurs only through migration
of individuals between local habitats. The novelty of our
approach consists of assuming that coupling of local
populations also occurs through quasi-local ecological
interactions. For example, in the particular model
considered here we assumed that competition for
resources occurs not only between individuals with a
given patch, but also to some extent between individuals
reproducing in different (but neighboring) patches, an
assumption that seems reasonable in many circumstances in which individuals leave their home territory in
search of food.
Formally, single-species models with discrete time and
space consist of a number of coupled difference
equations and are examples of ‘coupled map lattices’.
Coupled map lattices in which all the local difference
equations are identical and in which the coupling occurs
413
through passive dispersal have been studied extensively
(Kaneko, 1998). In this case, if the local difference
equations exhibit stable equilibrium dynamics, then the
total metapopulation also has a unique stable equilibrium, and nothing interesting happens unless the local
dynamics are complicated when considered in isolation.
We have shown that the situation changes completely
when the coupling between a number of identical
difference equations occurs through quasi-local competition between individuals from neighboring habitat
patches. In this case, the spatially uniform metapopulation equilibrium, in which all local populations have the
same density, can become unstable even if the local
difference equations exhibit a stable equilibrium in
isolation.
Our results are threefold. First, destabilization of the
spatially uniform metapopulation equilibrium occurs as
soon as the quasi-local competition is strong enough.
Second, if the uniform equilibrium is unstable, then the
metapopulation always converges to a spatially nonuniform equilibrium distribution of population abundances, and the number of different non-uniform
equilibrium attractors can be exceedingly large. Third,
even small amounts of environmental noise can induce a
random walk in the set of all attractors, thus leading to
persistent population fluctuations with a red-shifted
power spectrum. All these results are qualitatively
independent of the spatial dimension of the metapopulation, the boundary conditions assumed, and the
neighborhoods chosen for the quasi-local interactions.
It is not hard to find examples of spatially structured
ecological systems for which the assumptions of local
population dynamics but quasi-local competition dynamics are plausibly satisfied. Different systems illustrate the various ways in which the basic scenario of
quasi-local competition may be realized in detail. Birds
provide one class of examples of this type of ecological
behavior. Dunnocks, for instance, occupy territorial
regions within which they carry out most of their
foraging. However, detailed study of dunnock territories
has shown that different territories often overlap
(Davies, 1992), suggesting the strong possibility of
competition between populations in neighboring territories. Insects provide another possible class of examples. In social insects such as ants the colonies occupy
territories and regularly engage in competitive activities
with neighboring colonies, which often extends to
robbing neighboring nests of food (Wilson, 1971;
Holldobler and Wilson, 1990). Plants provide a particularly interesting class of systems in which quasi-local
competition appears to be occurring. Since water is a
diffusible resource it follows that the effect of a plant
taking up water locally is to remove water from the
surrounding regions. Thus the greater the rate of water
uptake by a given plant, the lower the quantity of water
available for neighboring plants. Therefore, in such
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cases, even though the resource uptake is local, quasilocal competition results from the diffusion of the
resource. The strength of quasi-local competition is
determined by the rate of water uptake relative to the
average amount of water per unit volume in the soil.
Hence, quasi-local competition should be stronger in
semi-arid regions. According to our model of metapopulation dynamics with quasi-local competition we
would expect spatial pattern formation to occur under
conditions of strong quasi-local competition. Interestingly, vegetation patterns consisting of stationary
irregular mosaics are found in many semi-arid regions
(White, 1970; Belsky, 1986; Klausmeier, 1999). These
observed patterns are in good qualitative agreement
with the results obtained from our model. It should be
noted, however, that most natural populations do of
course not exist in isolation and instead interact with
populations of other species, e.g. through competition
or predation. This is for example true for most of the
plant populations in semi-arid regions mentioned above.
Thus, the single-species models presented in this paper
are but a first step in the study of the role of quasi-local
interactions for the dynamics of spatially structured
ecosystems. Nevertheless, we expect that the basic
mechanism that quasi-local ecological interactions can
induce spatial pattern formation will also be present in
systems comprising interspecific interactions of various
types. In this sense we expect our findings to be robust,
an expectation that should be corroborated by future
research.
Thus, our findings may shed new light on the main
ecological question of what causes the spatial and
temporal patterns of population abundance observed
in the real world. Spatial pattern formation in population abundance is a classic theme in metapopulation
theory, and apart from the rather trivial observation
that spatial heterogeneity in productivity can induce a
corresponding heterogeneity in population abundance,
the main theoretical explanations for ecological pattern
formation come from reaction diffusion models (e.g.
Holmes et al., 1994). Stable spatially inhomogeneous
population distributions have also been found in
spatially structured models for interspecific competition,
both in discrete time (Karlin and McGregor, 1972) and
in continuous time (Levin, 1974, 1979). In all these
models spatially inhomogeneous equilibrium distributions are more likely to occur when dispersal rates are
low, which is in accordance with our results. However,
these models are limited in the sense that spatial
coupling only occurs through dispersal, and that they
always require interactions between two or more species
in order to produce non-uniform patterns of abundance.
In one of the rare previous single-species models for
spatial pattern formation Sasaki (1997) has shown in a
model with continuous space that quasi-local competition within a single species can produce spatially non-
uniform equilibria if the average interaction radius is
sufficiently different from the average dispersal distance.
However, in contrast to the theory presented here his
model does not yield a multitude of different attractors,
and instead the population always converges to a given
spatial wave of population abundance. In our models it
is also true that quasi-local competition should be
relatively strong while dispersal rates should be relatively small for the uniform equilibrium to become
unstable. However, once this equilibrium is unstable
there exist a multitude of potentially very irregular
spatial patterns of abundance to which the metapopulation can converge (cf. Figs. 2 and 5). Therefore, even in
the absence of any intrinsic differences between habitats
quasi-local ecological interactions have the potential to
induce complicated patterns of abundance, the details of
which may vary significantly between different metapopulations not because of different ecological conditions,
but because different metapopulations are likely to be
on different trajectories and hence to converge to
different equilibrium attrcators. Thus, quasi-local ecological interactions may be very important in determining the spatial patterns of population abundance.
Traditionally, there are two main explanations for
temporal patterns of population abundance. In the first
scenario population fluctuations show a random pattern
and are mainly due to stochastic perturbations from a
deterministic equilibrium. In the second scenario,
population fluctuations are due to overcompensating
ecological interactions that lead to complex deterministic dynamics. Our results suggest a third alternative, in
which complex population fluctuations arise due to an
interaction between environmental noise and deterministic population dynamics. Interactions between stochastic and deterministic mechanisms in ecological
systems have been studied before (e.g. Rand and Wilson,
1991; Petchey et al., 1997), however, the novel aspect of
our scenario is that large population fluctuations occur
even if the environmental noise is small and despite the
fact that the deterministic dynamics always exhibit a
stable equilibrium. Because the deterministic system has
many different equilibrium attractors environmental
noise induces a random walk between different basins
of attraction and leads to red-shifted population time
series, in which the variance in population size increases
with the length of the time period over which population
sizes are sampled.
Red-shifted population time series have attracted
considerable attention, because even though red shifts
seem to be common in time series of natural populations
(Pimm and Redfearn, 1988; Arino and Pimm, 1995), it
appears to be difficult to account for them in ecological
models (Cohen, 1995, but see Blarer and Doebeli, 1996).
For example, chaotic dynamics never yield a power law
with negative exponent over the whole range of
frequencies. In contrast, the power spectrum of time
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M. Doebeli, T. Killingback / Theoretical Population Biology 64 (2003) 397–416
series obtained from the interaction of stochastic and
deterministic processes in our models can show distinct
power law behavior with negative exponents (Fig. 8),
thus suggesting a new mechanism for generating redshifted power laws in ecological time series.
In sum, the theory presented here reiterates the
potential importance of spatial structure for ecological
processes and proposes new mechanisms for the
emergence of non-uniform spatial patterns of abundance and for the persistence of complicated temporal
population fluctuations. The concept of quasi-local
ecological interactions lies at the basis of this theory,
and it is easy to envisage a number of potentially
interesting extensions of the theory in which the
inclusion of quasi-local interactions could lead to new
insights. Two extension that appear to be particularly
interesting consist of considering quasi-local interactions in ecological contexts other than competition, e.g.
in metapopulation models for predator–prey interactions, and of considering quasi-local interactions in
metapopulation models in which the local deterministic
dynamics can be complicated even when habitat patches
are considered in isolation. In such models it will be
interesting to see how dispersal and quasi-local competition interact to determine the spatio-temporal dynamics
of the system. These investigations will likely lead to
further advances in understanding the role of spatial
structure for ecological processes.
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