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Journal of
Ecology 2002
90, 616 – 626
Spatially realistic plant metapopulation models and
the colonization–competition trade-off
Blackwell Science, Ltd
STEVEN I. HIGGINS 1,2 and MICHAEL L. CAIN 3
1
Umweltfoschungszentrum, Leipzig-Halle, Sektion Ökosystemanalyse, Permoserstrasse 15, 04318 Leipzig, Germany,
National Botanical Institute, Private Bag X7, Claremont 7735, South Africa, and 3Department of Applied Biology,
Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA
2
Summary
1 Results from patch-occupancy metapopulation models indicate that a trade-off
between competitive and colonization abilities is necessary for species to coexist in
patchy environments. However, such models are often based on unrealistic ecological
assumptions, such as global dispersal and no local population dynamics.
2 We develop a plant metapopulation model that allows us to sequentially relax unrealistic assumptions about dispersal and local population interactions. We use our model
to examine the extent to which the conclusions of analytical, patch-based coexistence
models depend on their assumptions.
3 We found that the need for an inferior competitor to be a superior colonizer was
reduced by using a dispersal kernel to distribute seeds, and totally removed by relaxing
the assumption of instantaneous competitive displacement. These results hold for
annual or perennial plants with or without seedbanks, for global, local and stratified
dispersal, for delayed competitive exclusion, two-way competition and neighbourhood competition, and for landscapes where habitat patches are either adjacent or
disjunct.
4 We conclude that the results of patch-occupancy metapopulation models differ
greatly from results of models that incorporate more realistic assumptions about
dispersal and local population dynamics. As a result it may be premature to base
biodiversity management and policy on the results of patch-occupancy models.
Key-words: dispersal, long-distance seed dispersal, local population dynamics, competition, landscape connectivity
Journal of Ecology (2002) 90, 616–626
Introduction
A major goal of ecology is to understand factors that
determine the diversity of species within communities.
For sessile organisms, a trade-off between competitive
ability and colonization ability has long been suggested
as a mechanism that promotes the coexistence of species
in patchy environments (Skellam 1951; Hutchinson 1951;
Levins & Culver 1971). Among such competitioncolonization trade-off models, the hierarchical, patchoccupancy model developed by Hastings (1980), Nee &
May (1992), Tilman (1994) and others (hereafter referred
to as the occupancy model) has been particularly influential (Kareiva & Wennergren 1995; McCarthy et al. 1997;
© 2002 British
Ecological Society
Correspondence: Steven I. Higgins, Umweltfoschungszentrum,
Leipzig-Halle, Sektion Ökosystemanalyse, Permoserstrasse
15, 04318 Leipzig, Germany (tel. 49 341235 2479; fax
49 341235 3500; e-mail [email protected]).
Eriksson & Kiviniemi 1999; Lindenmayer 1999). Insights
from competition-colonization models have influenced
how ecologists view a wide range of theoretical and
conservation issues, including species coexistence, limiting similarity, species packing, successional dynamics
and the loss of species due to habitat destruction.
The occupancy model follows the dynamics with
which two or more competitive species occupy a large
set of patches. Two important assumptions are made
that allow the development of a simple and elegant
model. First, it is assumed that the environment is
homogenous and that there is complete mixing of the
populations. This assumption allows the spatial
arrangement of patches to be ignored. Because this
assumption also means that dispersal is equally likely
among patches, we refer to it as the assumption of global dispersal. Secondly, it is assumed that within-patch
population dynamics do not influence the outcome of
metapopulation dynamics. Specifically, this assumption
means that competition is absolute and instantaneous,
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© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
that is, inferior competitors cannot persist, reproduce
or establish in the presence of superior competitors.
Equilibrium solutions of the occupancy model reveal
that an inferior competitor can coexist with a superior
competitor, provided that ci > c 2s /e (where ci and cs are
the colonization rates of the inferior and superior species and e is the rate at which all occupants are removed
from a patch (Hastings 1980; Tilman 1994).
Results from occupancy models indicate that a
trade-off between colonization and competition is
necessary for species coexistence, but are such trade-offs
found in natural systems? Several observations support
the existence of colonization-competition trade-offs in
plants. Trade-offs have been found between allocation
to roots (leading to high competitive ability) and allocation to reproduction (leading to high colonization
ability) (Tilman 1990; Tilman & Wedin 1991a,b). Other
evidence comes from correlations with seed size: smallseeded species tend to produce smaller seedlings and
have lower establishment success (Westoby et al. 1996),
and they tend to be competitively inferior to larger
seeded species (Rees 1995). Small seeds also have low
descent velocities and therefore may have better dispersal abilities (Okubo & Levin 1989), and small-seeded
species tend to produce more seeds (Jakobsson &
Eriksson 2000) and therefore may be better colonizers.
However, it should be noted that many vertebratedispersed seeds are widely dispersed despite being large.
Seed mass and seed morphology also often correlate
poorly with a species’ long-distance dispersal ability
(Webb 1986; Wilkinson 1997), the primary determinant
of colonization ability (Clark 1998; Cain et al. 1998;
Higgins & Richardson 1999). Moreover, competitive
ability as a seedling may have little bearing on adult
competitive ability (Saverimuttu & Westoby 1996). Indeed,
empirical studies suggest that colonization-competition
trade-offs are imperfect or absent (Banks 1997).
Overall therefore we do not find the empirical evidence for colonization-competition trade-offs compelling. In a modelling context, this leads us to ask, under
which conditions is a trade-off between colonization
and competitive ability necessary for coexistence? We
address this question by developing a discrete analogue
of the occupancy model and then sequentially relaxing
its assumptions; this procedure allows us to examine
whether predictions from occupancy models hold when
we no longer assume global dispersal and instantaneous competition. Relaxing these assumptions opens a
Pandora’s box of potentially overwhelming biological
detail. Notably, when we no longer assume global dispersal we need to specify whether patches are adjacent
or disjunct and specify a kernel that distributes seeds
between patches. The dispersal kernel could either
restrict seeds to the local environment or allow some longdistance dispersal (a stratified dispersal kernel; Cain
et al. 2000; Nathan & Muller-Landau 2000). Similarly,
relaxing the assumption of no within-patch population
dynamics forces us to specify which demographic processes (seed production, seed dormancy, recruitment,
mortality) operate within the patch, and how they are
influenced by competition and the size of the neighbourhood over which the competitive interactions operate.
Consideration of within-patch dynamics also means that
we need to separate the effects of catastrophe (‘patch
mortality’) from longevity (‘individual mortality’) on
mortality rates. In other words, we are forced to be
explicit about whether the patch is occupied by a population of plants or by an individual plant. In the former
case the model may be thought of as a metapopulation
model, in the latter case as an individual-based model.
Many studies have relaxed some of the assumptions
of the occupancy model. In a study that relaxed global
dispersal to neighbourhood dispersal (dispersal only to
neighbouring cells in a discrete lattice) and immediate
competitive exclusion to delayed competitive exclusion, Tilman et al. (1997) found the patch occupancy
model’s predictions regarding the sequence of species
extinction following habitat loss to be robust. Neuhauser (1998) also examined the sequence of species extinctions following habitat loss, and found that
the occupancy model’s predictions were robust for
most patterns of habitat loss investigated. Adler &
Mosquera (2000) show that the nature of competitive
advantage assumed in the occupancy model is sufficient to predict coexistence. Furthermore, Yu & Wilson
(2001) relaxed the assumption of absolute and instantaneous competitive exclusion, and found that the
competition-colonization trade-off was insufficient to
generate coexistence. Bolker & Pacala (1999) provide
another perspective; they explored the effects of local
dispersal kernels and local competitive neighbourhoods in homogenous spatial environments and showed
that a trade-off between colonization and competition
was not needed for coexistence. Similarly, Holmes &
Wilson (1998) showed that a colonization-competition
trade-off was not necessary for coexistence when the
inferior competitor dispersed globally and the superior
competitor was rare and dispersed locally.
Collectively these seemingly conflicting results suggest that the predictions of the occupancy model may
not be robust. Our aim is to examine the consequences
of assumptions made regarding plant life-history, competition, dispersal and patch arrangement on the prediction of the occupancy model that competition-colonization
trade-offs are necessary for species coexistence in patchy
landscapes. In our analysis we investigate model behaviour over a broad range of biologically realistic cases
by varying the strength and symmetry of competition,
the neighbourhood over which competition occurs, the
arrangement of patches, the type of dispersal kernel,
and by considering different plant life-history types.
Methods
 
The metapopulation model that we develop includes
the dynamics of within-patch demographic processes,
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Fig. 1 The demographic processes, landscape and patch structures considered in the model. The model considers two landscape
types, one where patches are adjacent and one where patches are disjunct. Patches can either be mixed or neighbourhood patches.
In mixed patches competitive effects are averaged over the entire patch. In neighbourhood patches the competitive influence of
plants within a patch are restricted to neighbourhoods.
© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
the stratified nature of dispersal, catastrophic disturbance, the spatial location of each patch, and the plant
life-history type. We use our model to explore the
conditions under which an inferior competitor can
coexist with a superior competitor when the assumptions of the patch-occupancy model are relaxed in a
sequential manner. Although the model can simulate
the ecological processes listed above it can also be
parameterized to represent a discrete analogue of the
analytical occupancy model. This allows us to systematically add detail to the model and thereby explore
the consequences of two fundamental assumptions of
the occupancy model (global dispersal and no local
population dynamics).
The model is a coupled map lattice model, that is, it
considers the dynamics of a series of spatially linked
populations (Fig. 1). The model landscape is a twodimensional grid of d × d m cells, each cell is potentially
a patch of habitat that can be occupied by K plants of
each of two competing species (Nx,Ny). The size of the
competitive neighbourhood (w) relative to the patch
sizes (d ) defines the level of competitive mixing in each
patch. In the simulations below we consider two levels
of competitive mixing (Fig. 1). In mixed patches (w = d )
competition is between all individuals in the patch, whereas
in neighbourhood patches (w < d ) competition is only
between individuals in the local neighbourhood. By
varying the size of the landscape (h) we can simulate
different interpatch distances (we use 100 patches for
all simulations). When h is appropriately small a system
of adjacent patches is represented, when h is large
a system of disjunct patches is represented (Fig. 1).
When patches are disjunct they are arranged at random
locations in the landscape. The boundaries of the landscape are assumed to be periodic.
Each iteration (1 year) of the model sequentially
simulates catastrophes, seed production, seed dispersal,
adult mortality, germination and establishment, and
seed decay (Fig. 1). The sequence of simulating these
processes is important: simulating catastrophes before
seed production ensures that local extinction can
occur, while simulating mortality before recruitment,
but after seed production and dispersal, ensures that
short-lived plants produce seeds before dying and that
space is released by death for new recruitment. Where
appropriate we denote these different time intervals as:
(t), current year; (t + ∆1), after catastrophe; (t + ∆2),
after mortality; and (t + 1), next year.
Catastrophes are modelled as:
Nxj (t + ∆1) = Nxj (t) − ej Nxj (t),
eqn 1
Here Nxj (t + ∆1) is the size of the population in the j th
patch of species x after the catastrophe (i.e. at time t
+ ∆1). The frequency of catastrophe (e) defines whether
a catastrophe occurs; ej is the outcome of a Bernoulli
trial for patch j. We assume that catastrophes affect
species equally and patches independently.
The seed production of the j th patch of species x (SPxj) is

β N (t + ∆1) 
SPxj = rx N xj (t + ∆1) 1 − y yj
,
K


eqn 2
where rx is the fecundity of species x, K is the carrying
capacity and βy Nyj (t + ∆1) is the effect of interspecific
density on seed production. The term in brackets in
equation 2 is constrained to be positive. We assume
that intraspecific competition does not influence seed
production, this allows a population at carrying
capacity (K) to produce the maximum number of seeds.
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This assumption is consistent with the occupancy model
and seems appropriate for plants, as seed production is
not prevented in closed monospecific stands even though
in such stands recruitment may be non-existent in the
absence of adult mortality. Effects of intraspecific competition are included at the recruitment stage (equation
5). To simulate the occupancy model’s assumption that
the inferior competitor cannot reproduce in patches
occupied by the superior competitor we set K = 1, and
βy = 1 if species y is the superior competitor and βx = 0
if species x is the inferior competitor.
For simulations where we assume global dispersal,
every patch has an equal probability of receiving seeds
and all seeds land in a patch; this assumption matches
the assumption of the occupancy model. For kernel
dispersal we use a mixture of exponential distributions
to describe the distance (x, metres) that seeds are dispersed from each source habitat as
f ( x ) = p1
 x
 x
1
1
exp  −  + p2 exp  −  ,
λ1
λ
λ
 1
 λ2 
2
eqn 3
where p1 is the proportion of seeds that are dispersed
short distances, λ1 is the mean of short-distance dispersal (metres), p2 is the proportion of seeds dispersed long
distances (where p2 = 1 – p1), and λ2 is the mean of longdistance dispersal (metres). For local dispersal we set p1
to 1 (and p2 to 0). Dispersal is implemented by using a
distance (randomly drawn from the above distribution)
and a random direction to calculate the position of
each dispersed seed in the two-dimensional landscape.
All seeds are dispersed from the centre of the competitive neighbourhood (w, Fig. 1). Seeds that land in
habitable patches are added to the seedbank (SBxj)
and are available for recruitment.
Mortality is simulated before recruitment, this allows
space vacated by mortality to be used for recruitment.
The number of adults surviving to time t + ∆2,
Nxj (t + ∆2), is

γ N (t + ∆1) 
N xj (t + ∆ 2) = s x N xj (t + ∆1) 1 − y yj
 , eqn 4
K


© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
where sx is the survivorship rate of species x, γy is the
effect of species y on the survivorship of species x, K is
the carrying capacity and Nxj (t + ∆1) is the size of the
jth population of species x after the catastrophe
(t + ∆1). The term in brackets in equation 4 is constrained to be positive. The inclusion of the densitydependent term in equation 4 allows the presence of an
interspecific competitor to increase the mortality rate
of the other species. This is perhaps unrealistic for
many plant interactions, but it allows us to simulate the
occupancy model’s assumption that the superior competitor can immediately displace the inferior competitor (by setting K = 1, and γy = 1 if species y is the superior
competitor and γx = 0 if species x is the inferior competitor). Further, the occupancy model only considers
mortality due to catastrophes (equation 1), for this reason survivorship (s) is set to 1 when representing the
occupancy model.
Recruitment of seeds into the adult population
determines the population size in the next iteration
Nxj (t + 1):
Nxj (t + 1) = Nxj (t + ∆2) + Gxj,
eqn 5a

N (t + ∆ 2) + α y N yj (t + ∆ 2) 
Gxj = gx SBxj 1 − xj
,
K


eqn 5b
Here Gxj is the number of seeds germinating, (SBxj) is
the seed density of species x at patch j and gx is the germination rate of species x (for all simulations presented
here we assume g = 1); the term in brackets allows
intraspecific (Nx(t + ∆2)) and interspecific (αy Ny(t + ∆2))
densities to influence germination (this term is constrained to be positive). To simulate the occupancy
model’s assumption that the inferior competitor
cannot recruit in patches occupied by the superior competitor we set K = 1, and αy = 1 if species y is the superior
competitor or αx = 0 if species x is the inferior competitor.
Seeds that do not germinate decay at rate ux:
SBxj (t + 1) = SBxj − ux[SBxj − Gxj].
eqn 6
  
Long-distance dispersal and colonization rates in
metapopulations
We first perform a simulation experiment that aims to
determine the factors that define colonization rates. In
the occupancy model the colonization (seed arrival)
rate equals the rate of seed production (that is, all seeds
land in suitable habitat). To facilitate comparison
between models we define colonization rate as the
mean number of seeds arriving per habitat patch per
year. We use a factorial experimental design to quantify the effect of fecundity (r), the proportion of longdistance dispersal ( p2), the local dispersal ability (λ1),
the long-distance dispersal ability (λ2), and the interpatch distance (the linear dimension of the square
landscape, h, was varied to create different interpatch
distances) on the observed colonization rate. A 25
design was used: each of five factors was assigned a
high and a low level (Table 1). The parameter levels
were selected to be consistent with values reported in
the literature. The complete factorial was replicated 10
times. Each run was initiated with a single species (at
an initial density of half K; K = 10 000) occupying all
habitat patches and run for 1000 years; the colonization rate (mean number of seeds arriving in a habitat
patch per year) was recorded from each run.
Competition, local population dynamics, dispersal and
coexistence in metapopulations
The second simulation experiment investigates interactions between superior and inferior competitors in
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Table 1 Factor levels used in the 25 factorial simulation
experiment used to quantify the effect of fecundity (r), the
proportion of long distance dispersal ( p2), the mean local
dispersal distance (λ1), the mean long distance dispersal (λ2),
and the interpatch distance (the linear dimension of the
square landscape, h, was varied to create different interpatch
distances) on the observed colonization rate (mean number of
seeds arriving in a habitat patch per year)
Factor
Low level
High level
r
p2
λ1
λ2
h*
1
0
2m
50 m
5 km
5
0.01
10 m
500 m
20 km
*Landscape size (h, km) is varied to change the interpatch
distance (for h = 5 km interpatch distance is approximately
0.25 km; for h = 20 km interpatch distance is approximately
1 km).
© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
metapopulations by sequentially relaxing two important assumptions of the occupancy model: no local
population dynamics and global dispersal. These
simulations cover a broad range of assumptions one
could make when simulating dispersal, competition,
patch configuration and life histories of plants.
For dispersal we consider either global dispersal,
local dispersal or stratified dispersal. Global dispersal
is simulated by allocating seeds to patches with equal
probability (i.e. the relative position of source and destination patches does not influence the seed rain and
no seeds fail to land in a patch). Global dispersal
mimics the dispersal assumptions of the occupancy
model. Kernel dispersal (both local and stratified
cases) assumes that seed rain declines with distance
from source (equation 3) and that seed may fail to land
in a suitable patch. The local dispersal case assumes
that dispersal is short tailed ( p1 = 1), whereas the stratified dispersal case assumes that the dispersal kernel
has a longer tail ( p2 > 0 and λ2 > λ1). In this simulation
experiment, we parameterize the local and stratified dispersal kernels in such a way that the overall mean dispersal distances of both kernels are equal; this means
that the variance of the stratified dispersal kernel is
greater than the variance of the local dispersal kernel.
The mean dispersal distances used are consistent with
values reported by Willson (1993).
For competition we consider either instantaneous
competition (instant competitive displacement), absolute competition (absolute competitive advantage but
delayed competitive displacement), two-way competition (reduced competitive advantage), or neighbourhood competition (reduced competitive advantage
where competitive effects are restricted to local neighbourhoods). In the instantaneous competition case, a
patch can only be occupied by a single species and competition instantaneously favours the superior competitor, i.e. the inferior competitor cannot persist, produce
seeds or recruit in the presence of the superior competitor and is invisible to the superior competitor. The
instantaneous competition case closely mimics the
demographic assumptions of occupancy models, in
that local population dynamics do not occur. In the
absolute competition case, the inferior competitor
remains invisible to the superior competitor but the
performance of the inferior competitor is dependent on
the density of the superior competitor: when the superior competitor is at high density the inferior competitor cannot persist, produce seeds or recruit. Thus, in
effect, competitive exclusion is time-lagged in the absolute competition case.
In the two-way competition case, competitive effects
are not absolute or instantaneous and the inferior competitor is not invisible to the superior competitor:
although both species experience interspecific competitive effects, the superior competitor has a sufficient
advantage to drive the inferior competitor to extinction
in a closed local population. Neighbourhood competition
is the most realistic competition model we consider:
here competition is simulated as for two-way competition, but competitive interactions are restricted to the
local neighbourhood. That is, the competition neighbourhood (w) is smaller than the patch size (d, Fig. 1).
To explore the broad effects of life history, we consider either annual plants with seedbanks or perennial
plants without seedbanks. These different life histories
are simulated by varying adult longevity (s) from an
annual to a long-lived perennial and rates of seedbank
decay (u) from no seedbank to a moderately persistent
seedbank. Finally, to explore the effects of landscape
configuration we consider either adjacent patches or
disjunct patches by varying the size of the landscape (h,
Fig. 1).
The four types of competition modes, three types of
dispersal, two types of landscape configuration and
two life history types yield 48 potential cases of the simulation model. Some (8) of these cases are identical: for
global dispersal no distinction between adjacent and
disjunct patches can be made. The parameter values in
Table 2 define the scenarios. Fecundity (r) values are
varied for each simulation to create a range of colonization rates. For the superior competitor we draw
fecundity values from a uniform distribution that
spans a range from values that allow persistence in single species cases to values that allow quasi-equilibrium
patch occupancies < 1. For the inferior competitor
fecundity values are also drawn from a uniform distribution that spans a range from values that allow persistence in single species cases to values that allow
quasi-equilibrium patch occupancies of 1. For each
simulated case 1000 simulations, each with a different
pair of fecundity values, were run. Each simulation was
run for 500 years before an outcome (both species persist, both species go extinct, only the superior competitor persists, or only the inferior competitor persists)
was assigned. For the catastrophe rates we consider
500 years is long enough for a true outcome to emerge.
The outcome, the time it takes to reach the outcome and
the colonization rate (mean number of seeds arriving
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Table 2 Parameter levels used for the different cases of the simulation experiment that explores the conditions under which
persistence and coexistence can occur between competing species in metapopulation models. The average interpatch distance for
simulations where patches are disjunct is 80 m (h = 12.15 km). For all simulations we draw values of r from uniform distributions;
the ranges of r are scaled so as to ensure persistence in single species cases
Competition modes
Instantaneous
Absolute
Two-way
Neighbourhood
Parameter
Superior
Inferior
Superior
Inferior
Superior
Inferior
Superior
Inferior
β
α
γ
K
c (m)
d (m)
1
1
1
1
9
9
0
0
0
1
9
9
1
1
1
81
9
9
0
0
0
81
9
9
1.5
1.5
0
81
9
9
0.75
0.75
0
81
9
9
1.5
1.5
0
9
3
9
0.75
0.75
0
9
3
9
Dispersal modes
Global
p1
λ1 (m)
λ2 (m)
Local
Stratified
Superior
Inferior
Superior
Inferior
Superior
Inferior
–
–
–
–
–
–
1
6.9
–
1
6.9
–
0.98
5
100
0.98
5
100
Life history modes
e
s*
u
g
Annual with seedbank
Perennial no seedbank
Superior
Inferior
Superior
Inferior
0.2
0
0.25
1
0.2
0
0.25
1
0.15
0.95
1
1
0.15
0.95
1
1
*To mimic the occupancy model when instant competition is simulated s = 1.
in a habitat patch per year) of each species were recorded for analysis.
All the coexistence simulations were initiated with
33% of the patches occupied by the superior competitor. The superior competitor was allowed to reach
a quasi-equilibrium occupancy by running the model
for 100 years, after which the inferior competitor was
introduced to 15% of sites (the superior competitor was
removed from the sites where the inferior competitor
was introduced). These initial conditions were selected
because they represent a relatively strict coexistence
criterion where an inferior competitor must invade,
from low densities, an established superior competitor.
Results
-  
    
© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
There was no relationship between mean colonization
rate and the overall dispersal distance (mean of both
short- and long-distance dispersal) for a range of
fecundity, dispersal and landscape connectivity parameterizations (Fig. 2a). This implies that the overall
mean dispersal distance, which largely reflects local
dispersal abilities, may often be a poor predictor of colonization rates. A more detailed understanding of what
constitutes colonization rate is provided by the factorial experimental design described in Table 1. An 
of this design showed that colonization is influenced by
fecundity, the proportion of seeds that move long distances, mean distance of long-distance dispersal and
interpatch distance (P < 0.001 for all factors), but not
by the mean distance of local dispersal (P = 0.504). In
addition, the colonization rate is affected by interactions between all factors, except those involving the
mean distance of local dispersal (Fig. 2b). Because
interaction effects were often as strong as main effects,
one cannot predict colonization ability from information on a single factor alone.
,    
,    
 
We examine the simulation model’s behaviour in the
context of the prediction of occupancy models that
the inferior competitor can persist only when it has a
JEC_694.fm Page 622 Saturday, July 27, 2002 10:37 AM
needed for coexistence (Fig. 3b,d), particularly if dispersal is stratified (compare Fig. 3b and 3d).
In general we found that coexistence can occur for a
broader range of conditions than those suggested by
the occupancy model: in the majority of cases considered, coexistence can occur when the inferior competitor does not have a colonization advantage (Fig. 4).
These results suggest that the inferior competitor does
not need a colonization advantage to coexist with the
superior competitor providing competition is not
instantaneous (Fig. 4). An exception is that when dispersal is local and habitat patches are disjunct no colonization advantage is needed for coexistence, even
when competition is instantaneous. This is because, when
dispersal is local and patches disjunct, interpatch colonization is extremely rare and the opportunity for competitive displacement is consequently severely restricted.
Overall, these results suggest that colonization
advantages may only be a prerequisite for coexistence
when competition is highly asymmetric.
622
S. I. Higgins &
M. L. Cain
Discussion
Fig. 2 (a) Mean colonization rate (number of seeds arriving
in habitat patches per year) is not a simple function of the
overall mean dispersal distance. These data were generated from
stratified dispersal simulations in which dispersal, fecundity
and landscape connectivity parameters were varied. (b) Effects
of fecundity (r), the proportion of long-distance dispersal
( p2), the local dispersal ability (λ1), the long-distance dispersal
ability (λ2) and the interpatch distance (the linear dimension
of the square landscape, h, was varied to create different
interpatch distances) and interactions between these factors
on colonization rates. Effects were estimated using the 
of the experimental design described in Table 1.
© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
strong colonization advantage (ci > c 2s /e, where ci and cs
are the colonization rates of the inferior and superior
species and e is the rate at which all occupants are removed
from a patch; Hastings 1980). We devise a statistical
test for whether a colonization advantage is needed by
the inferior competitor for it to coexist with the superior competitor using the empirical cumulative density
function of ci− cs. If more than 0.1 of the simulations
where coexistence is reported have ci − cs < 0, we conclude
that the inferior competitor does not need a colonization advantage to coexist with the superior competitor.
The results show that when we assume global dispersal and instantaneous competition, the predictions
from our simulation model match those from the occupancy model, i.e. a colonization advantage is needed
for coexistence (Fig. 3a); in addition, the coexistence
criterion (ci > c 2s /e) mentioned above holds (results not
shown). This suggests that our discrete analogue adequately mimics the behaviour of the occupancy model.
Relaxing global dispersal assumption to stratified dispersal reduced, but did not eliminate, the need for the
inferior competitor to be a superior colonizer (Fig. 3c).
When we relax the assumption of instant competitive
displacement, a colonization advantage is no longer
We constructed a spatially explicit metapopulation
model that includes the occupancy model of Hastings
(1980) as a special case. We used the model to sequentially relax two important assumptions of the occupancy model, namely no local population dynamics
and global dispersal, with the aim of detecting the
dependency of the conclusions of the occupancy model
on its simplifying assumptions. The occupancy model
assumes that fecundity is equal to colonization rate.
Colonization rates in the spatially explicit metapopulation model were not related to mean dispersal distance or fecundity in the simple manner suggested by
previous models, but rather were a result of a complex
set of interactions also involving other factors. These
results indicate colonization rates cannot be predicted
from information on a single factor, such as fecundity
or mean dispersal distance; data on infrequent longdistance dispersal, fecundity and habitat connectivity
are all essential. The interacting set of factors that
influenced colonization rates suggests that important
information may be lost in applications of models, such
as patch-occupancy models, that use a single parameter to describe colonization rates.
With respect to species coexistence, we showed that
an inferior competitor could coexist with a superior
competitor under a much broader set of conditions
than allowed in patch-occupancy models, such as
Hastings (1980) or Tilman (1994). By varying how life
history, competition, dispersal and the location of
patches were represented in our model, our simulations
covered a broad set of cases. We found that local or
stratified dispersal reduced the need for an inferior
competitor to have a colonization advantage in order
to coexist with a superior competitor and that using
realistic local competition functions totally removed
this need. Overall, our results suggest that results from
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Spatially realistic
metapopulation
models
Fig. 3 The occupancy model predicts that an inferior competitor needs a colonization advantage to coexist with a superior
competitor. The outcome of four cases of the simulation experiment that tests this idea are shown (all 48 cases are summarized
in Figure 4). In each of these four cases, patches are disjunct and plants are perennials without seedbanks: (a) the discrete analogue
of the occupancy model (asymmetrical and instant competition, global dispersal); (b) two-way (asymmetrical but not instant)
competition and global dispersal; (c) asymmetrical and instant competition and stratified (mostly local dispersal, with some longdistance dispersal) dispersal; and (d) two-way competition and stratified dispersal. For each case we ran 1000 simulations using
parameter values as defined in Table 2. If a colonization competition trade-off is necessary for coexistence we should only observe
coexistence when ci > cs, that is, all points should lie above the ci = cs line (left panels). The right panels show empirical cumulative
density functions of the test statistic ci − cs for simulations where coexistence was recorded. The arrows indicate where ci − cs = 0.
P-values are the proportion of simulations where coexistence occurred when ci − cs < 0.
© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
patch-occupancy models concerning species coexistence depended strongly on the assumptions of no local
population dynamics and global dispersal, neither of
which is realistic.
While our results show that there are many situations in which colonization competitor trade-offs are
not needed to explain species coexistence in patchy
environments, it is not immediately clear what mechanisms are preventing competitive exclusion. We believe
that, in our model, coexistence is promoted by refuges
from competition: kernel dispersal creates spatial refuges, whereas local population dynamics create temporal refuges. Under local and stratified dispersal most
(all in the case of local dispersal) dispersal is restricted,
leading to spatial aggregation amongst populations
of a species (i.e. spatial mixing is prevented). The
consequent aggregation of local populations means that
a population is more likely to have intraspecifc than
interspecific neighbours, hence effects of interspecific
competition become less important than the effects of
intraspecific competition. Spatial refuges therefore
reduce the need for the inferior species to have a large
colonization advantage in order to coexist with a superior competitor. Temporal refuges are created by using
competition functions that allow the inferior competitor to persist, albeit sometimes only briefly, in a local
patch before being excluded by the superior competitor. This brief period of persistence may be sufficient
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S. I. Higgins &
M. L. Cain
Fig. 4 Summary of the life-history, dispersal, competition and patch distribution conditions for which an inferior competitor
could coexist with a superior competitor while also being an inferior competitor (how these different model assumptions are
defined is summarized in Table 2). Open blocks are cases where simulations yielded coexistence when the inferior competitor was
also an inferior colonizer (ci < cs for > 0.1 of simulations that yielded coexistence). Shaded blocks are cases where simulations
suggested that coexistence was dependent on the inferior competitor being a superior colonizer (ci < cs for < 0.1 of simulations
that yielded coexistence). The top-left block represents the discrete analogue of the occupancy model.
© 2002 British
Ecological Society,
Journal of Ecology,
90, 616 – 626
time for the inferior competitor to disperse seeds into a
patch where the competitor may be absent.
This interpretation of spatial and temporal refuges
is consistent with Lavorel et al. (1994) who, in a
simulation model of annual plant competition, showed
how temporal and spatial storage mechanisms both
promoted species coexistence. More formally, a recent
analysis of spatial and temporal mechanisms of plant
competition (Bolker & Pacala 1999) has shown that
spatial and temporal refuges increase the likelihood
of coexistence. Bolker & Pacala (1999) used moment
closure techniques to derive an analytically tractable
model that approximates the spatial aggregation and
temporal refuge effects operating in our models. Bolker
and Pacala show how spatial aggregation of the superior competitor can increase the space available for the
inferior competitor to invade. Similarly, any negative
spatial association between the two competitors allows
the inferior competitor to avoid competitive confrontations with the superior competitor. Bolker and Pacala
also analyse the temporal refuge hypothesis; they show
that temporal refuges can be exploited when the inferior competitor has a higher reproductive rate and
when the carrying capacity (K ) is large. As reproductive
and colonization rates are not always correlated (cf.
Figure 2b), this temporal refuge can potentially operate independently of differences in colonization rates.
We found that the mechanisms that Bolker & Pacala
(1999) identified can explain the qualitative behaviour
of our model. This is even true for the cases of our
model that consider situations where the analytical
approximations used by Bolker and Pacala are not
strictly valid (such as disjunct patches).
Bolker & Pacala (1999) emphasized the importance
of short-distance dispersal for promoting spatial aggregation. However, for landscapes comprised of disjunct
patches, local dispersal kernels have the disadvantage
(relative to kernels that incorporate long-distance dispersal) that they need to be combined with unrealistically high fecundity rates to ensure that colonization
rates exceed extinction rates. In this regard, stratified
dispersal kernels hold the advantage in that they allow
colonization rates to exceed extinction rates, while
maintaining biologically reasonable fecundity parameters. In addition, because stratified dispersal kernels
mean that most seeds travel only short distances, stratified dispersal offers the same positive effect that localized dispersal displays in allowing individuals to escape
interspecific competition. It therefore seems that stratified dispersal offers the advantages of both global
dispersal (metapopulation persistence) and local
dispersal (interspecific avoidance).
Few studies of species coexistence provide explicit
links between empirical systems and theoretical models.
Empirical tests of analytical models are rare because of
the abstract nature of the models and their parameters.
In contrast, simulation models hold the advantage that
their parameters can often be estimated directly from
empirical systems (e.g. Pacala et al. 1993), although
their behaviour can be complex and difficult to interpret. However, new analytical tools allow simulation
models to be interpreted with the help of analytical
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Spatially realistic
metapopulation
models
approximations (e.g. Bolker & Pacala 1999; Law &
Dieckmann 2000). These developments suggest that
better links between analytical, simulation and
empirical systems are now possible.
To develop links between our model predictions and
empirical systems we would need to parameterize our
model for a real plant metapopulation and investigate
the empirically defined range of behaviours that the
model exhibits. Such an enterprise highlights the need
to obtain elusive empirical data, such as estimates of
the location of suitable habitat (a major challenge as
occupied patches may be ‘sinks’ and unoccupied habitat may be suitable but not yet colonized; Eriksson &
Kiviniemi 1999), measures of landscape connectivity
(Tischendorf & Fahrig 2000) and estimates of longdistance dispersal ability (Cain et al. 2000). Only with
such data will we be able to estimate parameters for more
realistic metapopulation models, and then use those
models with confidence to address important applied
issues, such as the response of species to habitat loss,
habitat fragmentation and other forms of environmental
change.
Finally, as model predictions change greatly
depending on how competition and colonization are
represented, and as the modes of competition and
colonization assumed in patch-occupancy models
are highly unrealistic, our results urge caution when
interpreting the result of patch-occupancy models in
an applied context (McCarthy et al. 1997). Patchoccupancy models have played and can continue to play
an important heuristic role, as illustrated by the large
impact of Lande (1987) on forest management practices in the United States of America. However, with
respect to the details of how habitat destruction affects
patterns of species extinction and other important
issues, we think it would be premature to base management decisions on results from occupancy models (e.g.
Nee & May 1992; Tilman et al. 1994) until those results
are confirmed by models that consider more realistic
competition and dispersal processes.
Acknowledgements
We thank the anonymous reviewers for suggestions
that substantially improved the manuscript. Thanks
also to Karin Johst, Björn Reineking and Eckart Winkler
for insightful comments on earlier versions and to Lindsay
Haddon for helping to improve the style.
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Received 13 December 2001
revision accepted 11 March 2002