Spatial Dynamics in Model Plant Communities: What Do We Really


Spatial Dynamics in Model Plant Communities: What Do We Really
vol. 162, no. 2
the american naturalist
august 2003
Spatial Dynamics in Model Plant Communities:
What Do We Really Know?
Benjamin M. Bolker,1,* Stephen W. Pacala,2 and Claudia Neuhauser3
1. Zoology Department, University of Florida, Gainesville, Florida
2. Department of Ecology and Evolutionary Biology, Princeton
University, Princeton, New Jersey 08544-1003;
3. Department of Ecology, Evolution, and Behavior, University of
Minnesota, St. Paul, Minnesota 55108
Submitted March 14, 2002; Accepted December 20, 2002;
Electronically published July 21, 2003
abstract: A variety of models have shown that spatial dynamics
and small-scale endogenous heterogeneity (e.g., forest gaps or local
resource depletion zones) can change the rate and outcome of competition in communities of plants or other sessile organisms. However, the theory appears complicated and hard to connect to real
systems. We synthesize results from three different kinds of models:
interacting particle systems, moment equations for spatial point processes, and metapopulation or patch models. Studies using all three
frameworks agree that spatial dynamics need not enhance coexistence
nor slow down dynamics; their effects depend on the underlying
competitive interactions in the community. When similar species
would coexist in a nonspatial habitat, endogenous spatial structure
inhibits coexistence and slows dynamics. When a dominant species
disperses poorly and the weaker species has higher fecundity or better
dispersal, competition-colonization trade-offs enhance coexistence.
Even when species have equal dispersal and per-generation fecundity,
spatial successional niches where the weaker and faster-growing species can rapidly exploit ephemeral local resources can enhance coexistence. When interspecific competition is strong, spatial dynamics
reduce founder control at large scales and short dispersal becomes
advantageous. We describe a series of empirical tests to detect and
distinguish among the suggested scenarios.
Keywords: spatial, competition, competition-colonization, successional niche, phalanx, endogenous.
Spatial heterogeneity at many different scales can change
the rate and outcome of competitive interactions. In the
past decade, attention has focused on the effects of short* Corresponding author; e-mail: [email protected]
Am. Nat. 2003. Vol. 162, pp. 135–148. 䉷 2003 by The University of Chicago.
0003-0147/2003/16202-020110$15.00. All rights reserved.
distance dispersal and small-scale, endogenous resource
heterogeneity such as light gaps in forests (Dalling et al.
1998). Empirical and theoretical studies have demonstrated the effects of small-scale endogenous heterogeneity
on invasion, productivity, succession, and diversity in
communities ranging from bacterial assemblages to forests
(Chao and Levin 1981; Tilman 1993, 1994; Pacala and
Deutschman 1995; Rees et al. 1996). Given that dispersal
and competitive interactions are fundamentally restricted
in space, often to scales only a few times the size of a
single organism, we might expect short dispersal and
small-scale heterogeneity to have nearly universal effects
on community dynamics.
The dynamical richness and technical challenge of spatial
processes have encouraged the proliferation of mathematical
models for endogenous spatial pattern formation (Levins
and Culver 1971; Shigesada et al. 1979; Durrett and Levin
1998; Gandhi et al. 1998). However, most of the models
have focused attention on a single class of phenomena (clustering and competition-colonization trade-offs) at the expense of other possibilities. In addition, studies that attempt
any kind of analysis have made strong simplifying assumptions about the scales and nature of spatial competition and
lead to controversy and confusion when different models
have conflicting assumptions (Grace 1990).
This article tackles three basic questions: How does endogenous spatial structure affect competition and coexistence in simple competition models? Do these effects
differ depending on the way in which space is modeled?
How can these theoretical predictions be tested in the field?
We find, reassuringly, that a variety of different models all
make qualitatively similar predictions, provided that they
incorporate the basic building blocks of discrete individuals, local competition, and local dispersal (Durrett and
Levin 1994).
When interspecific competition is weak relative to intraspecific competition and species would coexist nonspatially (with global dispersal and competition) and when
competing species have similar life histories, endogenous
spatial structure may tip the community from coexistence
to exclusion in the long term (Neuhauser and Pacala 1999;
The American Naturalist
fig. 2). However, spatial structure can also change the rate
of community dynamics and slow competitive exclusion
(Neuhauser and Pacala 1999).
Short dispersal can handicap a competitively dominant
species and allow competitive subordinates to coexist with
a short-dispersing dominant species; this phenomenon is
a form of the classical “competition-colonization tradeoff” (CC; Tilman 1994; Holmes and Wilson 1998).
Short dispersal can benefit a competitively subordinate
species that reproduces faster than the dominant by allowing it to concentrate its reproductive effort in areas
that are temporarily free from interspecific competition;
this strategy has variously been called “competition”
(Grime 1977), “exploitation” (Bolker and Pacala 1999), or
a “successional niche” (Pacala and Rees 1998). Here, we
call it the “spatial successional niche” (SSN), to avoid confusion with temporal successional niches that are maintained by disturbance.
When interspecific competition is strong relative to intraspecific competition, short dispersal and small competitive neighborhoods benefit individuals by allowing local control of the environment. In spatial arenas, local
founder control (dependence of competitive success within
a small region on initial density) can still occur, but global
founder control disappears; in a large enough habitat, one
species will always win as long as both species start with
positive densities everywhere in the habitat (Neuhauser
and Pacala 1999). The combination of spatial localization
and interspecific interactions strongly retards competitive
dynamics (Gandhi et al. 1998).
Next, we briefly review the different model frameworks
that we and others have used to explore spatial community
dynamics. We synthesize recent results from these different
frameworks and show that all of them produce similar results that we can classify into the four basic phenomena
previously described. We then outline a set of field experiments that test for and distinguish among various spatial
phenomena that occur in natural communities in the absence of exogenous heterogeneity. (Although exogenous heterogeneity is obviously important in structuring natural
communities, high-diversity communities do exist where no
obvious exogenous heterogeneity is acting to maintain diversity.) Such experiments would fill a major gap because
the rich theory of community spatial dynamics remains
largely untested by manipulative experiments except in bacterial communities (Chao and Levin 1981; Korona et al.
1994; Rainey and Travisano 1998).
Review and Synthesis of Spatial Phenomena
Modeling Frameworks
The wide range of simplifying assumptions about the
structure of space and the nature of competition used by
different models makes it hard to compare their predictions. We have found, however, that several different model
frameworks that we and others have explored over the
past few years converge independently to the same conclusions. Interacting particle systems (IPS) are stochastic,
continuous-time models with discrete individuals located
in the cells of a square lattice; these models often offer
qualitative insights into spatial dynamics rather than quantitative predictions (but see Silvertown et al. 1992 or Law
et al. 1997 for counterexamples). Space is discrete rather
than continuous in IPS models, but it is contiguous; competitive neighborhoods overlap and are connected in an
explicitly spatial structure.
Stochastic point processes also track discrete individuals
that compete and disperse locally but assume that they
occupy a single point in continuous (and, hence, also contiguous) space (Gandhi et al. 1998; Bolker and Pacala 1999;
Dieckmann and Law 2000; see Diggle 1983 for the theory
of static point processes). Stochastic point processes represent a low-density limit of IPS models where very few
lattice cells are occupied but competition and dispersal
also become long range. We have simulated stochastic
point processes and also analyzed them with approximate
moment equations that track the mean densities and spatial covariances within and between species (Bolker and
Pacala 1999).
Finally, in a limit where competition remains local but
dispersal is global, we obtain noncontiguous patch models.
These models differ from the metapopulation or patchoccupancy models used in many studies (Levins and Culver 1971; Tilman 1993) in that they allow multiple individuals and multiple species per patch. They allow some
offspring to remain in the parental sites, which in turn
allows spatial pattern to develop in the form of a multivariate distribution of the fractions of patches with particular combinations of species densities.
Each of these approaches has its own strengths and
weaknesses; using all three allows a choice of strengths and
lends generality to the results. The IPS framework allows
rigorous proof of the long-term coexistence or exclusion
of species with different competitive and dispersal abilities
in an infinite habitat (Durrett 1992; Durrett and Neuhauser 1994; Neuhauser and Pacala 1999) or approximation
by means of pair approximations (Iwasa 2000). Spatial
point processes provide the closest link to individual-based
field studies and models, with all processes defined in
terms of field-measurable life-history and spatial properties such as per capita fecundity and the density of seeds
or offspring at a given distance from the parent. In addition, spatial moment equations allow analytical calculations of the shape and scale of equilibrium spatial patterns in a community. Finally, patch models provide a
model framework that is much simpler to analyze or sim-
Spatial Dynamics in Model Plant Communities
ulate than either IPS or moment equations, and they give
a point of comparison with popular and even simpler
patch-occupancy or metapopulation models (Keeling
We will discuss results from IPS, point processes, and
patch models in order to demonstrate the strengths and
weaknesses of each kind of model and to show how all
three approaches converge on the same qualitative
Model Results: Classes of Phenomena
Nonspatial Foundations. We start with the classical nonspatial Lotka-Volterra (LV) competition model:
( Kn ⫺ b Kn ) ,
ṅ p r (1 ⫺ ⫺ b
ṅ1 p r1 1 ⫺
where ri (for i p 1, 2) is the intrinsic exponential growth
rate, Ki is the carrying capacity when each species is grown
in monoculture, and bij gives the relative strength of
within- and between-species competition. (Our definition
of bij gives the competitive effect of a density of species j
relative to its own carrying capacity Kj rather than relative
to the target species’ density Ki, which simplifies the notation; using the more common notation would replace
bij with aij p [K i /K j]bij.) The parameters in the model
describe whole-population demography rather than individual life history and competitive abilities and as such
may be misleading in an evolutionary context; for example,
increasing density-independent fecundity or decreasing
density-independent mortality would increase both r and
K rather than just r. However, the model can also be easily
derived from an individual-based competition model with
parameters fi (fecundity), mi (density-independent mortality), and aij (per capita effect of competition; Yodzis
1989; Royama 1992; Bolker and Pacala 1999). The
population-level parameterization is simply more convenient for describing the criteria for species coexistence.
Although the LV model is a caricature of real competition processes, it provides a useful framework for discussing the possible outcomes of competition (fig. 1). It
covers scenarios ranging from simple single-resource competition (when b12 b21 p 1) to strongly asymmetric competition, as is commonly incorporated in patch-occupancy
models (when b12 r ⬁, b21 r 0). The invasion criterion—the conditions on life-history, competitive, and spatial parameters that allow one species to invade an established monoculture of another species—for species i to
invade species j in this model is bij ! 1. With appropriate
definitions, this invasion criterion also applies to a wider
range of models, where per capita competition is a possibly
nonlinear function of linear combinations of species
densities (Chesson 2000b; B. M. Bolker, unpublished
The LV model allows any of the possible outcomes of
two-species competition: dominance by either species, coexistence of both species, or founder control (exclusion of
one species by another depending on initial conditions,
usually exclusion of an initially rare species by an initially
common one). If we derive the LV equations from an
underlying model where individuals compete for a single
limiting (but well-mixed) resource, we obtain the same
equations, but the plane of possible competitive parameters shown in figure 1 shrinks to a hyperbola that allows
only single-species dominance. Although many processes
(e.g., allelopathy or responses to herbivory or pathogens)
can move interactions off of the single-resource hyperbola
and allow coexistence or founder control in a nonspatial
model, we focus on how spatial competition allows coexistence when the underlying nonspatial rules would allow only one dominant species to persist.
The LV parameter plane also helps organize the different
possible effects of spatial pattern formation on competitive
dynamics. The possible phenomena, which we discuss in
the following sections, are spatial segregation and clustering when the parameters lie in the coexistence region of
figure 1 (quadrat I), competition-colonization and spatial
successional niche dynamics in the single-species dominance regions (quadrats II and IV), and phalanx growth
in the founder control region (quadrat III).
Before going any further, we should clarify the meaning
of “spatial coexistence.” Spatial coexistence occurs when
endogenous or exogenous spatial structure allows species
to coexist indefinitely (in the absence of demographic stochasticity) where they could not if all individuals experienced the average conditions in the environment, including spatially averaged population densities. Real
communities can be difficult to homogenize in this way.
Conversely, determining the nonspatial outcome of competition is simple in a model where we can set the effective
density of each species to its spatial mean density (although
finding the nonspatial equivalents of patch-occupancy
models can be surprisingly difficult). Heuristically, one can
think about the nonspatial outcomes as the hypothetical
result of competition if all individuals competed and dispersed globally.
Spatial Segregation and Clustering. The first, simplest competitive scenario is where two species could coexist in the
nonspatial, homogeneously mixed case (fig. 1, quadrat I).
Although such coexistence is not possible under competition for a single essential resource, there are many sit-
The American Naturalist
Figure 1: Invasibility and coexistence regions for the Lotka-Volterra
model and the nonlinear (nonspatial) competition model, with regions
of interest marked.
uations such as competition for multiple resources or attack by host-specific pathogens that lead to coexistence
even in a homogeneously mixed environment. In this case,
we are interested in whether endogenous spatial structure
can prevent (rather than enhance) coexistence and in how
it changes the timescale of competitive dynamics.
When interspecific competition is much weaker than
intraspecific competition (both b12 and b21 are small; fig.
1, region d), space does not change qualitative outcomes
of competition; species that coexist in the absence of space
still coexist in the IPS. We can prove coexistence rigorously
in the high-density limit, where fecundity is much greater
than mortality for both species and there is no empty space
left in the habitat (Neuhauser and Pacala 1999).
In contrast, if interspecific competition is nearly as
strong as intraspecific competition (b ! but ≈ 1) for one
species or the other (fig. 1, regions a1 and a2), the combination of local competition and discrete individuals actually reduces coexistence in the high-density limit. In the
shaded regions, where the stronger competitor would almost outcompete the weaker in the nonspatial model, the
stronger competitor can actually exclude the weaker because of the discreteness of the competitive neighborhood.
In this parameter regime, only a low density of the weaker
species can survive in the nonspatial model. If the competition neighborhood is small enough, fewer than one
individual could survive within a competitive neighborhood; because individuals are discrete in this model, the
weaker species can never establish. Neuhauser and Pacala
(1999) prove this phenomenon rigorously for the highdensity limit.
Similar phenomena occur in patch models and in the
spatial moment equations. In patch models, we restrict
movement by reducing the probability of global interpatch
movement rather than the size of overlapping neighborhoods, but the results are roughly comparable. To explore
this effect, we used an approach that uses forward equations to track the entire probability distribution of patch
occupancies in an infinite ensemble of patches (Renshaw
1991). Within a patch, population dynamics follow a stochastic process that would converge on the LV equations
(1) for large patch sizes; deaths occur at a constant densityindependent rate, and fecundity decreases linearly according to the combination of con- and heterospecific population densities shown in equations (1) down to 0 when
N ≥ K. If births are globally distributed among patches
when they occur (m p 1), the model converges to the LV
model. If we restrict interpatch movement to 50% in a
model with patch size (carrying capacity) K p 10, the coexistence region shrinks by approximately 10% (1/K) in
the same regions where coexistence fails in an IPS (fig. 2).
Reducing interpatch movement further (to 1%) more than
doubles this contraction, and it reduces the size of the
founder control region (see “Phalanx Strategies”).
In spatial point process models, intraspecific competition is always strengthened by a self-competition term
(Bolker and Pacala 1999). An individual-based parameterization of the LV model starts with the statement that
competition with an individual of species j at distance r
decreases individual fecundity rate or increases individual
mortality rate by an amount aijUij(r). (The connection
with the population-level parameterization previously
given is that K i p ri /aii and bij p [aij /aii][K j /K i].) In a
randomly distributed or well-mixed conspecific population of density Ni, the total strength of competition is
aii(Ni ⫹ Uii[0]), where the second term represents selfcompetition that does not occur in the nonspatial model.
Self-competition changes competitive outcomes only if it
changes bij, which in turn changes only if the ratio of
carrying capacities K i/K j p (ri ⫺ aiiUii(0))/(rj ⫺ ajjUjj (0)) is
different from the original ratio K i /K j, which in turn requires Uii(0)/K i ( Ujj (0)/K j. This criterion depends on the
relative carrying capacities and the shape of the competitive neighborhood. In contrast, the more general neighborhood exclusion phenomenon proved for the IPS does
not depend on details of spatial neighborhoods (Neuhauser and Pacala 1999) but may depend on the high-density
A final spatial phenomenon that does not alter the qualitative outcome of competition but does change the apparent strength of interspecific competition (changing the
estimated location of a community in the parameter space
shown in fig. 1) is the effect of spatial segregation among
coexisting species. In nonspatial models with symmetric
Spatial Dynamics in Model Plant Communities
species, increasing interspecific competition (e.g., from increasing niche overlap) continuously increases the observed strength of competitive interactions between species. In the equivalent spatial model, however, interspecific
spatial segregation also increases with increasing interspecific competition and reduces interspecific interaction. Removal experiments—observing the increased performance
of plants in an established community after the experimental removal of conspecific or heterospecific competitors—are a standard assay of competitive strength. As we
have shown (Pacala and Levin 1998), the endogenous spatial segregation generated by competition can lower the
estimated strength of competition. For strongly competing,
symmetric species (fig. 1, region b), spatial segregation is
so strong that interspecific population dynamics are nearly
neutral, as proved in the high-density limit by Neuhauser
and Pacala (1999). This phenomenon might explain observed differences between quadrat-based and individualbased field estimates of interspecific competition based on
removal experiments.
Competition-Colonization Trade-Offs. A more familiar scenario is the case where one species is clearly competitively
dominant in a nonspatial setting (e.g., in small-plot experiments pitting two heterospecific individuals against
each other). Strong competitors may be better at preempting resources or may have some other advantage such
as herbivore or pathogen resistance (Holt et al. 1994). In
the LV model and other models that assume both spatial
and temporal homogeneity (Tilman 1982), a superior
competitor always wins, regardless of handicaps in fecundity or dispersal, so any coexistence must be the result
of spatial structure. Our conclusions come from a variety
of models that all converge to the LV in the appropriate
nonspatial limit of large competition neighborhoods or
patch sizes, so we can interpret nonspatial parameters such
as relative strength of interspecific competition in terms
of the LV parameters (fig. 1). In this case, we are looking
at quadrats II and IV of figure 1, where one species dominates the other.
One particular form of spatial coexistence is the competition-colonization trade-off (CC), where two or more
species coexist in a spatial environment because the competitively dominant species have poorer colonizing ability:
low fecundity, short dispersal, or both. The CC has been
studied with a variety of models (IPS, spatial point processes, patch-occupancy models) and analytical tools (formal proofs [Durrett and Swindle 1991], pair approximation [Harada and Iwasa 1994], numerical studies [Holmes
and Wilson 1998], and second-order moment equations
[Bolker and Pacala 1999]). Competition-colonization
trade-offs in patch-occupancy models allow a large number of species to coexist (Tilman 1994; Kinzig et al. 1999),
but most fully spatial analyses have considered only twospecies interactions. The basic mechanism underlying all
forms of CC is that colonization limitation of the dominant species leads to a lower overall (spatial mean) density
of the dominant, which in turn allows the inferior species
to invade. In contiguous-space models such as IPS and
point processes, the dominant’s density decreases because
of spatial clustering, which enhances intraspecific competition and reduces mean density (Harada and Iwasa
1994; Holmes and Wilson 1998; Bolker and Pacala 1999).
The CC is fundamentally spatial but is strongly enhanced if the dominant species also has low fecundity or
high mortality so that its density is low (or it occupies
only a small portion of the habitat) even in the absence
of competition (Bolker and Pacala 1999). Formally similar
trade-offs can also be constructed using nonspatial tradeoffs between seed size and number, for example (Adler
and Mosquera 2000).
In classical patch-occupancy models (Levins and Culver
1971), fecundity and dispersal ability are confounded in
a single measure of nonlocal colonization (the number of
offspring available to colonize outside the parent’s site is
determined by both the number of offspring produced
Figure 2: Changes in coexistence regions with limited movement in the
metapopulation Lotka-Volterra (LV) model. When all births are dispersed
globally (m p 1), the coexistence regions are equal to those of the standard LV (fig. 1). When a fraction 1 ⫺ m of births are retained locally,
the coexistence and founder control regions (calculated by searching
numerically for the b values where the invasion rate of each species is
0) shrink as shown. The triangles show the equivalent region of contraction for the interacting particle systems in the high-density limit.
Parameters: intrinsic rate of increase r1 p r2 p 3.0 , intrinsic reproductive
number (competition-free expected offspring per generation) R1 p
R2 p 4.0, carrying capacity K1 p K2 p 10.0.
The American Naturalist
and the fraction that disperse beyond the site boundaries).
The distinction between low fecundity and short dispersal
is important because the two forms of limitation have
different implications; fecundity limitation is nonspatial
while dispersal limitation is spatial. Fecundity and dispersal limitation combine to produce colonization, or recruitment limitation, which can be tested by adding recruits (seeds or young juveniles) to a site and observing
whether population density increases. While the influence
of recruitment limitation has been extensively tested in
both marine and terrestrial communities (Clark and Ji
1995; Clark et al. 1999; Hubbell et al. 1999; Warner et al.
2000), the effects of dispersal limitation have begun to be
studied more recently (Dalling et al. 1998; Ehrlen and
Eriksson 2000; Jacquemyn et al. 2001; Verheyen and
Hermy 2001; Webb and Peart 2001).
There are biological as well as technical reasons to expect
difficulty in separating fecundity from dispersal limitation.
Late-successional species that invest heavily in structures
for resource competition may, as a result, have fewer resources to invest in dispersal structures for their offspring.
They are also under less pressure to colonize because their
survival does not depend on escaping competition, although competition both with kin and with other conspecific neighbors will always give some reasons to disperse. They may also choose to produce a small number
of large, short-dispersing seeds that have higher germination probabilities and early growth rates (Ezoe 1998;
Levin and Muller-Landau 2000). Finally, these species are
able to maintain long-term control of their local resource
environment and may be playing a “phalanx” strategy
against other strong competitors.
Although CCs have been intensively studied by theoreticians, their influence is actually limited in continuousspace models. In simple models of homogeneous environments, CCs are limited to a small region of parameter
space where strong competitors have extremely low fecundity and short dispersal (Bolker and Pacala 1999).
When fecundity is realistically high (intrinsic reproductive
number or lifetime reproduction in the absence of competition greater than two), intraspecific spatial clustering
is weak or even negative and leads to evenly spaced populations and rules out a CC (Bolker and Pacala 1997).
Competition-colonization trade-offs are more important
in models such as patch-occupancy models that assume
some spatial structure or heterogeneity, but even these
models require the dominant species to be such a poor
colonizer that its patch occupancy never gets high enough
to exclude the inferior competitor. We suspect that exogenous heterogeneity and disturbance, which both increase the variance in population densities above the level
that can be generated by endogenous processes, will enhance the effects of CC; for example, the apparent CC
identified by Tilman (1988) in Minnesota sand plain communities was driven by disturbance at the scale of agricultural fields. Whether it is safe to extend the same framework we have used for endogenous dynamics to include
disturbance and exogenous variability is an open question,
although Chesson’s (2000a) framework for spatial coexistence, which we discuss later, does lump endogenous and
exogenous processes.
Spatial Successional Niches. Competition-colonization
trade-offs are not the only form of spatial coexistence under asymmetric competition; competitively inferior species
can also gain from short dispersal. In previous work, we
have termed this scenario a “successional niche” (Pacala
and Rees 1998) or an “exploitative strategy” (Bolker and
Pacala 1999); here, we call it a “spatial successional niche”
(SSN). Even when the dominant species does not cluster
(which it must do to make a CC work), the inferior species
can persist by dispersing its offspring nearby to exploit
local, endogenous heterogeneity in the resource environment. The SSN strategists gain the greatest advantage from
short dispersal by reproducing quickly to exploit gaps before competitively superior species arrive and take over.
We have found this result in both moment equations
for point process models, where it appears as part of the
spatial advantage arising from spatial segregation between
species (Bolker and Pacala 1999), and IPS. In IPS, the SSN
requires that the inferior species reproduce on a fast timescale and that the empty spaces unfilled by the dominant
species percolate—form unbroken corridors across the environment. When these conditions hold, the inferior species can always find empty space to occupy; the corridors
do open and close because of stochastic birth and death
in the dominant species, but because of its fast timescale,
the inferior species can never be trapped and become extinct. (While it would seem that long-distance dispersal
benefits the inferior species in this case, it is true only if
dispersal is somehow restricted to corridors and avoids the
majority of the habitat that is occupied by the dominant
As with the CC, the SSN encompasses both spatial
mechanisms (local dispersal) and nonspatial mechanisms
(fast growth; Pacala and Rees 1998). Grime’s (1977) “competitive” species exploit a nonspatial successional niche by
growing rapidly. Local dispersal and rapid growth reflect
different geometries of local growth. New growth of a
modular organism can be distributed horizontally, as in
clonal swards and grasses, or vertically, as in woody trees.
The asymmetric and time-dependent nature of heightstructured competition makes vertical growth qualitatively
different from horizontal growth and harder to model;
there are no analytical models that fully incorporate the
trade-offs among vertical growth, horizontal growth, and
Spatial Dynamics in Model Plant Communities
dispersal (Pacala et al. 1993; Pacala and Tilman 1994). In
contrast to CCs, where nonspatial traits (high fecundity)
and spatial traits (good dispersal) co-vary in existing species, nonspatial aspects of successional niches may go along
with spatial CCs. For example, early-successional canopy
trees often have fast growth, possibly corresponding to a
nonspatial successional niche; long dispersal, corresponding to a spatial CC; and high fecundity, which benefits
both strategies. There are too many other trade-offs in
plant strategies (seed mass and number, germination probability, dispersal ability, etc.) to do a complete accounting
or a complete comparative analysis (Rees 1996), but the
partitioning of spatial strategies into categories that are
robust to differences in model structure is an important
first step.
All three of our modeling approaches (IPS, pointprocess, and patch models) suggest that SSN strategists
must have a short generation time relative to their competitors. We suspect, therefore, that SSNs will be more
important for strongly asymmetric competition between
life forms such as grasses and trees rather than for withinguild competition or competition between more similar
guilds such as early- and late-successional trees. Although
competitive interactions between grasses and trees are often assumed to reflect a trade-off between utilization of
water and light, grasses could also persist through a smallscale SSN strategy (and through an interaction with fire
frequency in many ecosystems). As in the case of CC,
disturbance may also increase the strength of SSN (preliminary explorations of patch models suggest that both
CC and SSN increase with mild amounts of correlated
within-patch disturbance (B. M. Bolker, unpublished manuscript), but adding either disturbance or other forms of
exogenous heterogeneity blurs the distinction between
SSN and other forms of spatial coexistence such as temporal successional niches.
It may seem hard to distinguish between CC and SSN
strategies; among other things, loose semantics allow one
to say that any spatial persistence strategy represents a
“colonization advantage.” However, there is a simple operational test that distinguishes CC from SSN. If a focal
species benefits from CC, increasing its dispersal distance
will give it minor benefits (because it has good colonization
ability already), while increasing a competitor’s dispersal
distance will hurt the focal species by eliminating the colonization advantage. In contrast, increasing the dispersal
distance of a focal species that benefits from SSN will hurt
it by removing its ability to retain seeds in good areas; in
this case, increasing the competitor’s dispersal will have
little effect.
Phalanx Strategies. Finally, we come to the scenario where
interspecific competition is stronger than intraspecific
competition (either above the single-resource hyperbola
in fig. 1 or more strictly in the upper right quadrat, regions
c1 and c2). Here, spatial dynamics affect both the timescales of community dynamics and the outcome of
Strong interspecific competition leads to strong spatial
segregation in this parameter regime, which in turn slows
down competitive dynamics. Gandhi et al. (1998) have
shown that in a spatial point process starting from random
initial conditions, monospecific patches form in the first
phase of competitive dynamics as the locally denser species
in each neighborhood excludes the sparser species. Thereafter, the stronger competitor encroaches on patches of
the weaker competitor but at a drastically reduced pace
that would be equivalent to decades or millennia in most
ecological systems.
The qualitative change in competitive dynamics is that
unlike in the nonspatial model, founder control never occurs; in infinite space and time, the stronger competitor
(species 1 in region c1 or species 2 in region c2 of fig. 1)
always wins eventually. Neuhauser and Pacala (1999) rigorously proved that the founder control region is reduced
in the high-density limit model; in a long-range limit, one
can show that the founder control region disappears.
From an ecological point of view, however, one may
observe either local founder control—dominance of a local
region by a weaker competitor that started at higher density—or, at a slightly larger scale, apparent coexistence—stability of large monospecific patches over ecological timescales. Some authors have concluded on the
basis of simulation studies that different species can coexist
in this regime (Solé and Bascompte 1997; Molofsky et al.
2001), even though competitive exclusion can be proved
rigorously for long enough timescales; moment equations
and pair approximations also give the wrong result in this
regime (strong interspecific competition) because they underestimate the effects of large-scale spatial structure
(Iwasa et al. 1998). While indefinite coexistence is impossible in this regime, apparent coexistence over many
generations is a real possibility (Frelich et al. 1993).
These timescale effects apply more generally and weaken
interspecific competition relative to intraspecific competition whenever strong spatial segregation occurs. Spatial
segregation among similar species could amplify the effects
of recruitment limitation that have been shown to slow
down competitive exclusion in some cases (Hurtt and Pacala 1995). In particular, recent theories of the dynamics
of neutral communities (Hubbell 2001) could be strengthened and reconciled with the apparent importance of competition on an individual scale, through the effects of
small-scale interspecific and interguild spatial segregation.
If species within a guild are symmetric and similar in their
ability to compete for resources, endogenous spatial seg-
The American Naturalist
regation could slow dynamics to effective neutrality on
ecological timescales (although Chesson and Huntly
[1997] argue broadly that mechanisms that simply slow
exclusion are not enough to account for diversity). This
phenomenon could be tested by analyzing the neighborhood densities within a community to see whether interspecific association (within guilds) is much rarer than expected from a random spatial distribution, although the
confounding effects of environmental heterogeneity would
have to be taken into account.
We call the scenario of strong spatial segregation in the
founder-control region “phalanx growth,” by analogy with
the phalanx/guerrilla dichotomy described by observers of
growth forms in clonal plants (Lovett-Doust 1981). In
nutrient-rich habitats, clonal plants tend to form tightly
aggregated patches, whereas in nutrient-poor habitats, they
extend their stolons and disperse new ramets farther from
existing ones. Spatial moment equations show that this
strategy works best when the intrinsic reproductive number (offspring per generation in a noncompetitive environment) is large (Bolker and Pacala 1999), which corresponds well with a nutrient-rich environment. This
empirical observation suggests that a spatial competition
strategy may be an important part of the community dynamics of competing clonal swards. As we will discuss later
in more detail, we could test this prediction by relocating
ramets randomly to see whether they gain a competitive
advantage by forming patches or whether the patches have
more to do with the spatial scale and variability of the
resource environment.
Empirical Tests
No matter how clearly it seems to explain the organization
of competitive communities, any categorization of spatial
strategies stands or falls on its ability to be tested in the
field. While recruitment limitation (which could arise from
any combination of fecundity, microsite, or dispersal limitation) has been both empirically tested and noted in the
field (Clark and Ji 1995; Clark et al. 1999; Hubbell et al.
1999), tests of explicitly spatial phenomena are rarer. These
tests usually consist of carefully calibrating a spatial model
and then using it to contrast the theoretical estimate of
the nonspatial behavior of the community with the dynamics (productivity, species coexistence, etc.) actually observed (Pacala and Silander 1987, 1990; Pacala and
Deutschman 1995; Rees et al. 1996; Clark et al. 1998). A
related approach uses more generic models of competition
to extract competition coefficients from observational data
on an appropriate scale (Law et al. 1997; Freckleton and
Watkinson 2001); these estimates could, in principle, be
compared with nonspatial estimates averaging competition
over the entire habitat to observe the effects of endogenous
spatial structure on the effective strength of intra- and
interspecific competition. We know of only one published
example of a manipulative experiment designed to test
endogenous spatial mechanisms in plant communities
(Stoll and Prati 2001), although there are several such tests
of bacterial communities, as previously cited (Chao and
Levin 1981; Korona et al. 1994; Rainey and Travisano
We propose empirical tests to discriminate between different mechanisms of spatial coexistence. These range from
purely observational tests of field systems to calibration of
data-driven models to manipulative experiments. Clearly,
there is no magic bullet and no substitute for the natural
history and experimental work required to establish the
basic processes that operate in a community. We hope,
however, that this discussion will clarify what observations
are necessary to establish the existence of spatial competition mechanisms and clarify their nature in a particular
plant community.
One obvious shortcoming of the tests we propose is that
they distinguish only among different endogenous spatial
phenomena; they attribute any spatial structure observed
to endogenous processes. In experimental systems, one can
eliminate exogenous heterogeneity by the usual means of
tilling, shade cloth, uniform watering, or fertilization, and
so forth, but one can never be absolutely sure that the
system has really been homogenized. We feel that exploring
endogenous mechanisms is a first step toward understanding the full complexity of plant communities, including
both endogenous and exogenous heterogeneity and their
How can we discriminate among the different spatial
scenarios: symmetric species, competition-colonization,
SSN, and phalanx growth? We might already know something about the relative strengths of inter- and intraspecific
competition and the relative competitive abilities of the
two species; then, we could tentatively locate the system
on the plot shown in figure 1, which would narrow the
possibilities. However, figure 1 is qualitative rather than
quantitative, and it would be difficult in practice to reduce
the complexity of a real community to a point in the
parameter plane. In addition, finding competitive asymmetry between two species would not resolve whether CCs,
SSNs, or both were operating. Instead, we suggest a series
of qualitative experiments that will discriminate among
the different alternatives.
The most straightforward way to test the effects of local
dispersal and spatial structure is to relocate individuals so
that their local competitors become a random sample of
the population. (Note that these experiments are very different from classical recruitment limitation experiments,
which supply additional seeds or recruits to an area rather
than randomizing the spatial patterns of individuals.) All
Spatial Dynamics in Model Plant Communities
Table 1: Experiments for discriminating different spatial scenarios
Randomize species 1
Randomize species 2
1 0/F
No change
2 0/F
No change
Species 2 maintained by CC
Species 1 maintained by CC
Species 1 maintained by SSN
Species 2 maintained by SSN
Phalanx/spatial segregation
Note: Up and down arrows denote increases and decreases, respectively, in fecundity, growth,
or survivorship relative to control treatments; 0 denotes no change for one species. CC p
competition-colonization trade-off; SN p spatial successional niche.
of the experiments we suggest are, unfortunately, practically limited to communities of small- or medium-size
plants, both by the logistics of transplanting trees and by
the timescales required to see an effect, which are on the
order of the generation time; forest ecologists will have to
continue to calibrate models to determine spatial effects.
For logistical reasons, for the ability to distinguish ecologically distinct individuals (those unconnected by a network of underground structure), and in order to avoid
transplant shock, it will be easiest to do these experiments
with plants that reproduce primarily from seed. Annual
plants would be best of all; everything that we describe in
terms of transplants could be done by much easier seed
manipulation. Randomizing the positions of all individuals
is a test of spatial coexistence itself and reduces the system
to its nonspatial equivalent by forcing all individuals to
compete equally (or at least randomly) with all other individuals. The control treatment is to swap the positions
of conspecifics randomly, which will mimic the effects of
the randomization treatment but preserve the hetero- and
conspecific neighborhood densities of all individuals.
If randomization leads to a significant change in densities or population persistence, we would then want to
see which spatial mechanism is operating. We modify the
previous experiment by randomizing the position of only
one species at a time (always controlling for transplant
effects by digging up and replanting the “sedentary” species in the same positions or randomly interchanging the
positions of sedentary individuals; table 1).
If the focal species benefits from randomization and
increases in relative abundance or drives the other species
to extinction, we conclude that it is naturally colonization
limited. If the focal species is hurt by randomization and
decreases in relative abundance or becomes extinct, we
conclude that it exploits endogenous spatial pattern in the
environment to survive. If one species clearly dominates
in the control treatment and the other exploits spatial
pattern, then we conclude that it occupies a SSN. If neither
species dominates and both are hurt by randomization,
we can infer a phalanx scenario.
If neither species benefits consistently from the exper-
imental treatment but the timescale of competitive dynamics accelerates significantly, we infer that the system
is symmetric and experiences either spatial segregation of
coexisting species or phalanx competition. In the spatial
segregation case, randomization will not change the coexistence or diversity of the community, but it will increase
resilience or the speed at which the community recovers
from a change in relative population densities. In the phalanx case, randomization will increase the speed of extinction and possibly induce founder control (as in Chao
and Levin’s [1981] experiments), although the timescales
of plant communities may be too slow to see this effect.
In this case, if seeds or seedlings (preferably grown in
a common garden) are available, we can assemble communities from scratch by planting or transplanting individuals of different species in predetermined random or
structured configurations. These assembled communities
distinguish between clustering in the coexistence region
and clustering in the “founder control” or phalanx region
of figure 1. These cases are hard to separate by nonmanipulative methods because spatial segregation slows the
rate of population dynamics severely. In the coexistence
region, single individuals or small clusters of an invading
species planted within a monoculture of a resident species
should increase their local density and/or expand over the
course of a few seasons; in the phalanx region, they should
disappear as they are overwhelmed by the greater local
density of the resident. If interspecific dynamics are too
close to neutrality, it could still be impossible to tell coexistence from exclusion, but this protocol at least offers
a way to separate individual-level neutral dynamics from
the reduced interspecific interactions caused by spatial
Stoll and Prati (2001) sowed herbaceous plants in random and “intraspecifically aggregated” mixtures and measured biomass and reproductive output; they found that
stronger competitors performed worse and weaker competitors performed better in the aggregated treatment. In
our terminology, this experiment shows the results of both
CC (reduced performance of stronger competitors) and
SSN (increased performance of weaker competitors; if the
The American Naturalist
overall neighborhood density is maintained, intraspecific
aggregation implies interspecific segregation).
A completely different way to test and measure the
strength of different spatial scenarios is to calibrate a phenomenological or mechanistic individual-based model of
the plant community by measuring local dispersal and
neighborhood competition strengths. Ideally, such a model
should also be tested, either against independent observational data or (even better) by using it to predict the
outcome of controlled experiments (see van den Bosch et
al. 1988 for an example from plant epidemiology). Once
the model is calibrated and tested, one can run any experiment in the model, including making competition or
dispersal global for one or both species, and can measure
the effect of endogenous spatial structure on the persistence or invasibility of different species. Such a model can
test and discriminate among any of the scenarios listed.
Completely field-calibrated models have strong advantages
but require a great deal of effort to parameterize and test;
only a few models of this sort have actually been constructed and used to test spatial dynamics (Cain et al. 1991;
Pacala et al. 1993; Rees et al. 1996).
These experiments will require considerable investment
and, as we previously discussed, are practically limited to
small plants that reproduce from seed and mature and
reproduce quickly. These specifications obviously rule out
many important natural communities (forests, old-field
communities dominated by long-lived perennials such as
Solidago, etc.), but they are a starting point. We can always
fall back on carefully calibrated models for those communities that resist experimental manipulation, but we
should start with systems where we can do qualitative,
manipulative experiments that are independent of specific
models, and, if we do use models, we can check them
against independent experimental (not just observational)
evidence. Given the potential importance of spatial dynamics and given the near complete absence of experimental tests, experiments such as the ones we have suggested should be well worth the effort.
We believe that our work and the work of many other
mathematicians and mathematical ecologists over the past
few years have finally made it possible to understand the
basic spatial phenomena that underlie both simple strategic models and more complicated models used to understand applied ecological issues such as the generation
and conservation of biodiversity. The results presented
here suggest that we can understand many of the basic
phenomena that underlie spatial competition equally well
in patch, lattice, or point-process models. This is good
news; in particular, it strengthens the case that the con-
clusions of the voluminous literature on patch dynamics
could carry over, qualitatively, to landscapes that are not
inherently divided into discrete patches. Models with explicit neighborhood structure such as lattices or point processes will still be needed to make quantitative or explicitly
spatial predictions (such as exploring spatial scales and
patterns of dispersal) and measure the relative strengths
of different phenomena, but we can hope that patch models will suffice for elucidating basic spatial processes.
While we have intentionally titled this article “Spatial
Dynamics in Model Plant Communities,” it is reasonable
to ask how important we think the various mechanisms
outlined and endogenous spatial structure in general will
be in real communities. We argued in the introduction to
this article that short scales of dispersal and competition
are ubiquitous; conversely, so are exogenous disturbance
and heterogeneity, which might be expected to swamp
endogenous effects. Some important natural and experimental systems such as the Cedar Creek Long Term Ecological Research site or the Rothamstead Park Grass experiment apparently support high diversity with minimal
exogenous heterogeneity. However, these represent only a
few (perhaps special) systems, and one can never be sure
that exogenous heterogeneity is really absent; color polymorphisms in Linanthus parryae, long cited as an example
of endogenous pattern in population genetics but now
believed to be maintained by subtle exogenous heterogeneity, are but one cautionary example (Schemske and Bierzychudek 2001).
More evidence for the importance of some kind of spatiotemporal mechanism, either endogenous or exogenous,
comes from reviews of plant competition experiments and
observations (Goldberg and Barton 1992; Gurevitch et al.
1992; Freckleton and Watkinson 2001). These reviews suggest that, in general, intraspecific competition is not much
weaker than interspecific competition (as would be expected from models of resource partitioning), which rules
out region Id in figure 1 but does not necessarily rule out
regions Ia and Ib. (Then again, Rees et al. [1996] did find
that interspecific competition was relatively weak.) These
reviews did not generally try to separate systems on the
basis of competitive symmetry (quadrats II and IV vs.
quadrats I and III in fig. 1); however, asymmetry is at least
anecdotally common in natural communities (two examples are the communities studied by Law et al. [1997]
and Tilman [1988]). We do not know, in general, the
likelihood of competitive near equivalence (b12 or b21 ≈
1), which would put species pairs near the boundary of
quadrat I, where strong segregation can occur and where
small spatial effects could have important qualitative differences. (Hubbell [2001] argues for equivalence or near
equivalence and bases his argument on macroscopic patterns of diversity rather than on competition experiments.)
Spatial Dynamics in Model Plant Communities
As for founder control (quadrat III in fig. 1), we know of
no direct evidence for this scenario in plant communities
(even in clonal plant communities where one might expect
little to no intraspecific competition between ramets), although it has been found in bacterial communities (Chao
and Levin 1981). Despite all the effort that has been put
into field studies of competition, we still cannot make
strong a priori predictions about what kinds of endogenous effects we expect.
The models previously discussed are simple “strategic” or
“toy” models (Nisbet and Gurney 1982). As such, they
neglect many important complexities such as exogenous
spatial and temporal heterogeneity (including disturbance
and habitat fragmentation) and nonlinear per capita effects
of competition (Chesson 2000a, 2000b), which are certain
to shape natural competitive communities. How can we
be sure that these phenomena do not somehow reverse
our conclusions or that we have not missed some important strategic spatial axis?
We have explored nonlinear patch models (like the
patch models previously discussed but with nonlinear per
capita competitive effects), with and without correlated
disturbance that drives an entire patch population extinct
simultaneously. In the absence of disturbance, SSN strategies still dominate CCs over much of parameter space.
Alternatively, disturbance, which makes the model more
similar to patch-occupancy models by emptying entire
patches simultaneously, strengthens the effect of CCs (B.
M. Bolker, unpublished manuscript). The only novel phenomenon seen in these nonlinear disturbance models is
the effect of different curvatures in the effect of competition, which allows one species to do better in rare “gaps”
while the other does better in common, medium-density
patches (relative nonlinearity; Chesson 2000b).
Peter Chesson (1984, 1985, 1990, 1994, 2000a, 2000b)
has extensively explored and categorized the (largely exogenous) effects of spatial and temporal heterogeneity, and
their interactions, in patch models. While Chesson’s results
may not carry over to contiguous-space models unchanged, the framework is sufficiently general that we can
expect that many of the phenomena will be the same.
Chesson’s scheme divides spatiotemporal mechanisms into
three general categories: relative nonlinearity, which allows
some species to capitalize on rare, high-quality sites while
others exploit average sites; temporal or spatial storage
effects, where carryover from good sites and time periods
allows an inferior competitor to persist; and densitygrowth correlation, where a species manages to inhabit
good sites and time periods preferentially, either actively
(through directed dispersal) or passively (by retaining
propagules in good habitats). In terms of this framework,
the SSN (and interspecific spatial segregation generally)
represents a positive density-growth covariance for the inferior species; by reproducing on a fast timescale and retaining its propagules locally, a plant species induces a
positive association between population density and good
(enemy-free) habitat. In contrast, the CC represents negative density-growth covariance; in intraspecific competition, more individuals are necessarily found in bad (highdensity) neighborhoods. We have found that in the
presence of exogenous heterogeneity, short dispersal adds
an additional component to the density-growth correlation
(B. M. Bolker, unpublished manuscript). Chesson’s results
suggest that nonlinearity, disturbance, and heterogeneity
will add new spatiotemporal mechanisms but not qualitatively change the ones discussed here.
In contrast to disturbances and heterogeneity, which can
be understood at the level of a patch, the effects of fragmentation—a particular form of exogenous heterogeneity
that partially or completely cuts patches off from colonization—are explicitly spatial and, thus, harder to understand. They are captured in some ways by our models
when movement rates drop very low in patch models—for
example—but a full exploration of fragmentation probably
requires explicitly spatial models (Bascompte and Solé
1997; Fahrig 1998).
We are strongly encouraged by the convergence of three
structurally different types of models—IPS, spatial point
processes, and patch models—on the same qualitative conclusions. This convergence strengthens our conclusions,
and it suggests, as one would hope, that the fine details
of mathematical models do not affect qualitative predictions. However, there are two strong caveats to this statement. First, quantitative predictions of competitive outcomes in natural communities definitely will depend on
choosing an appropriate structure and spatial scale with
which to model the system. Second, there are important
details as well as unimportant ones (“Everything should
be made as simple as possible but not simpler”; Albert
Einstein); as we stated at the outset, the properties of individual discreteness and local competition and dispersal
are vital. In addition, models that describe only site occupancy or assume that sites are always filled at some stage
in the annual cycle may miss important aspects of spatial
dynamics, although increasing the number of different
types of patches to allow temporary within-patch coexistence (Pacala and Rees 1998) or multiple resource levels
(Wilson et al. 1999) is one way around this problem.
We have shown that despite the potential richness of
spatial dynamics, it is possible to generalize about the ef-
The American Naturalist
fects of endogenous spatial structure on competitive outcomes. Spatial effects are context dependent; whether
space enhances or undermines coexistence and whether it
speeds up or slows down the rate at which communities
approach equilibrium varies according to the degree of
asymmetry (in fecundity, mortality, size, etc.) and the position in the competitive plane (fig. 1). More to the point,
this variability and context dependence is measurable; we
have described a series of experiments that could allow
empiricists to distinguish among different spatial scenarios. We believe that experiments along the lines we have
suggested may help in the ongoing effort to establish which
mechanisms are actually most important in maintaining
diversity in natural communities.
C.N. was partially supported by National Science Foundation (NSF) grant DMS-97-03694, and S.W.P. and B.M.B.
were partially supported by NSF applied math grant
9807755. We thank P. Chesson, R. Snyder, and four anonymous reviewers for helpful comments.
Literature Cited
Adler, F. R., and J. Mosquera. 2000. Is space necessary?
interference competition and limits to biodiversity.
Ecology 81:3226–3232.
Bascompte, J., and R. V. Solé. 1997. Models of habitat
fragmentation. Pages 127–150 in J. Bascompte and R.
V. Solé, eds. Modeling spatiotemporal dynamics in ecology. Springer, New York.
Bolker, B. M., and S. W. Pacala. 1997. Using moment
equations to understand stochastically driven spatial
pattern formation in ecological systems. Theoretical
Population Biology 52:179–197.
———. 1999. Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. American Naturalist 153:
Cain, M., S. W. Pacala, and J. Silander, Jr. 1991. Stochastic
simulation of clonal growth in the tall goldenrod, Solidago altissima. Oecologia (Berlin) 88:477–485.
Chao, L., and B. R. Levin. 1981. Structured habitats and
the evolution of anticompetitor toxins in bacteria. Proceedings of the National Academy of Sciences of the
USA 78:6324–6328.
Chesson, P., and N. Huntly. 1997. The roles of harsh and
fluctuating conditions in the dynamics of ecological
communities. American Naturalist 150:519–553.
Chesson, P. L. 1984. The storage effect in stochastic population models. Lecture Notes in Biomathematics 54:
———. 1985. Coexistence of competitors in spatially and
temporally varying environments: a look at the combined effects of different sorts of variability. Theoretical
Population Biology 28:263–287.
———. 1990. Geometry, heterogeneity and competition
in variable environments. Philosophical Transactions of
the Royal Society of London B, Biological Sciences 330:
———. 1994. Multispecies competition in variable environments. Theoretical Population Biology 45:
———. 2000a. General theory of competitive coexistence
in spatially varying environments. Theoretical Population Biology 58:211–237.
———. 2000b. Mechanisms of maintenance of species
diversity. Annual Review of Ecology and Systematics 31:
Clark, J. S., and Y. Ji. 1995. Fecundity and dispersal in
plant populations: implications for structure and diversity. American Naturalist 146:72–111.
Clark, J. S., E. Macklin, and L. Wood. 1998. Stages and
spatial scales of recruitment limitation in southern Appalachian forests. Ecological Monographs 68:213–235.
Clark, J. S., B. Beckage, P. Camill, B. Cleveland, J.
HilleRisLambers, J. Lighter, J. McLachlan, J. Mohan, and
P. Wyckoff. 1999. Interpreting recruitment limitation in
forests. American Journal of Botany 86:1–16.
Dalling, J. W., S. P. Hubbell, and K. Silvera. 1998. Seed
dispersal, seedling establishment and gap partitioning
among tropical pioneer trees. Journal of Ecology 86:
Dieckmann, U., and R. Law. 2000. Relaxation projections
and the method of moments. Pages 412–455 in U.
Dieckmann, R. Law, and J. A. J. Metz, eds. The geometry
of ecological interactions: simplifying spatial complexity.
Cambridge Studies in Adaptive Dynamics. Cambridge
University Press, Cambridge.
Diggle, P. 1983. Statistical analysis of spatial point patterns.
Academic Press, New York.
Durrett, R. 1992. Multicolor particle systems with large
threshold and range. Journal of Theoretical Probability
Durrett, R., and S. A. Levin. 1994. The importance of being
discrete (and spatial). Theoretical Population Biology
———. 1998. Spatial aspects of interspecific competition.
Theoretical Population Biology 53:30–43.
Durrett, R., and C. Neuhauser. 1994. Particle systems and
reaction-diffusion equations. Annals of Probability 22:
Durrett, R., and G. Swindle. 1991. Are there bushes in a
forest? Stochastic Processes and Their Applications 37:
Spatial Dynamics in Model Plant Communities
Ehrlen, J., and O. Eriksson. 2000. Dispersal limitation and
patch occupancy in forest herbs. Ecology 81:1667–1674.
Ezoe, H. 1998. Optimal dispersal range and seed size in a
stable environment. Journal of Theoretical Biology 190:
Fahrig, L. 1998. When does fragmentation of breeding
habitat affect population survival? Ecological Modelling
Freckleton, R. P., and A. R. Watkinson. 2001. Nonmanipulative determination of plant community dynamics.
Trends in Ecology & Evolution 16:301–307.
Frelich, L. E., R. R. Calcote, and M. B. Davis. 1993. Patch
formation and maintenance in an old-growth hemlockhardwood forest. Ecology 74:513–527.
Gandhi, A., S. Levin, and S. Orszag. 1998. “Critical slowing
down” in time-to-extinction: an example of critical phenomena in ecology. Journal of Theoretical Biology 192:
Goldberg, D. E., and A. M. Barton. 1992. Patterns and
consequences of interspecific competition in natural
communities: a review of field experiments with plants.
American Naturalist 139:771–801.
Grace, J. B. 1990. On the relationship between plant traits
and competitive ability. Pages 51–65 in J. B. Grace and
D. Tilman, eds. Perspectives on plant competition. Academic Press, San Diego, Calif.
Grime, J. P. 1977. Evidence for the existence of three primary strategies in plants and its relevance to ecological
and evolutionary theory. American Naturalist 111:
Gurevitch, J., L. L. Morrow, A. Wallace, and J. S. Walsh.
1992. A meta-analysis of competition in field experiments. American Naturalist 140:539–572.
Harada, Y., and Y. Iwasa. 1994. Lattice population dynamics for plants with dispersing seeds and vegetative propagation. Researches on Population Ecology (Kyoto) 36:
Holmes, E., and H. Wilson. 1998. Running from trouble:
long-distance dispersal and the competitive coexistence
of inferior species. American Naturalist 151:578–586.
Holt, R., J. Grover, and D. Tilman. 1994. Simple rules for
interspecific dominance in systems with exploitative and
apparent competition. American Naturalist 144:
Hubbell, S. P. 2001. The unified neutral theory of biodiversity and biogeography. Monographs in Population
Biology. Princeton University Press, Princeton, N.J.
Hubbell, S. P., R. B. Foster, S. T. O’Brien, K. E. Harms,
R. Condit, B. Wechsler, S.-J. Wright, and S.-L. De Lao.
1999. Light-gap disturbances, recruitment limitation,
and tree diversity in a Neotropical forest. Science (Washington, D.C.) 283:554–557.
Hurtt, G. C., and S. W. Pacala. 1995. The consequences
of recruitment limitation: reconciling chance, history
and competitive differences between plants. Journal of
Theoretical Biology 176:1–12.
Iwasa, Y. 2000. Lattice models and pair approximation in
ecology. Pages 227–251 in U. Dieckmann, R. Law, and
J. A. J. Metz, eds. The geometry of ecological interactions: simplifying spatial complexity. Cambridge Studies
in Adaptive Dynamics. Cambridge University Press,
Iwasa, Y., M. Nakamaru, and S. A. Levin. 1998. Allelopathy
of bacteria in a lattice population: competition between
colicin-sensitive and colicin-producing strains. Evolutionary Ecology 12:785–802.
Jacquemyn, H., J. Butaye, and M. Hermy. 2001. Forest
plant species richness in small, fragmented mixed deciduous forest patches: the role of area, time and dispersal limitation. Journal of Biogeography 28:801–812.
Keeling, M. J. 2000. Metapopulation moments: coupling,
stochasticity and persistence. Journal of Animal Ecology
Kinzig, A. P., S. A. Levin, J. Dushoff, and S. W. Pacala.
1999. Limiting similarity, species packing, and system
stability for hierarchical competition-colonization models. American Naturalist 153:371–383.
Korona, R., C. H. Nakatsu, L. J. Forney, and R. E. Lenski.
1994. Evidence for multiple adaptive peaks from populations of bacteria evolving in a structured habitat.
Proceedings of the National Academy of Sciences of the
USA 91:9037–9041.
Law, R., T. Herben, and U. Dieckmann. 1997. Nonmanipulative estimates of competition coefficients in a
montane grassland community. Journal of Ecology 85:
Levin, S. A., and H. C. Muller-Landau. 2000. The evolution
of dispersal and seed size in plant communities. Evolutionary Ecology Research 2:409–435.
Levins, R., and D. Culver. 1971. Regional coexistence of
species and competition between rare species. Proceedings of the National Academy of Sciences of the USA
Lovett-Doust, L. 1981. Population dynamics and local specialization in a clonal perennial (Ranunculus repens). I.
The dynamics of ramets in contrasting habitats. Journal
of Ecology 69:743–755.
Molofsky, J., J. D. Bever, and J. Antonovics. 2001. Coexistence under positive frequency dependence. Proceedings of the Royal Society of London B, Biological Sciences 268:273–277.
Neuhauser, C., and S. W. Pacala. 1999. An explicitly spatial
version of the Lotka-Volterra model with interspecific competition. Annals of Applied Probability 9:
The American Naturalist
Nisbet, R., and W. Gurney. 1982. Modelling fluctuating
populations. Wiley, New York.
Pacala, S. W., and D. H. Deutschman. 1995. Details that
matter: the spatial distribution of individual trees maintains forest ecosystem function. Oikos 74:357–365.
Pacala, S. W., and S. A. Levin. 1998. Biologically generated
spatial pattern and the coexistence of competing species.
Pages 204–232 in D. Tilman and P. Kareiva, eds. Spatial
ecology: the role of space in population dynamics and
interspecific interactions. Princeton University Press,
Princeton, N.J.
Pacala, S. W., and M. Rees. 1998. Field experiments that
test alternative hypotheses explaining successional diversity. American Naturalist 152:729–737.
Pacala, S. W., and J. A. Silander, Jr. 1987. Neighborhood
interference among velvet leaf, Abutilon theophrasti, and
pigweed, Amaranthus retroflexus. Oikos 48:217–224.
———. 1990. Field tests of neighborhood population dynamic models of two annual weed species. Ecological
Monographs 60:113–134.
Pacala, S. W., and D. Tilman. 1994. Limiting similarity in
mechanistic and spatial models of plant competition in
heterogeneous environments. American Naturalist 143:
Pacala, S. W., C. D. Canham, and J. A. Silander, Jr. 1993.
Forest models defined by field measurements. I. The
design of a northeastern forest simulator. Canadian
Journal of Forest Research 23:1980–1988.
Rainey, P. B., and M. Travisano. 1998. Adaptive radiation
in a heterogeneous environment. Nature 394:69–72.
Rees, M. 1996. Evolutionary ecology of seed dormancy
and seed size. Philosophical Transactions of the Royal
Society of London B, Biological Sciences 351:1299–1308.
Rees, M., P. J. Grubb, and D. Kelly. 1996. Quantifying the
impact of competition and spatial heterogeneity on the
structure and dynamics of a four-species guild of winter
annuals. American Naturalist 147:1–32.
Renshaw, E. 1991. Modelling biological populations in
space and time. Cambridge Studies in Mathematical Biology. Cambridge University Press, Cambridge.
Royama, T. 1992. Analytical population dynamics. Chapman & Hall, New York.
Schemske, D. W., and P. Bierzychudek. 2001. Evolution of
flower color in the desert annual Linanthus parryae:
Wright revisited. Evolution 55:1269–1282.
Shigesada, N., K. Kawasaki, and E. Teramoto. 1979. Spatial
segregation of interacting species. Journal of Theoretical
Biology 79:83–99.
Silvertown, J., S. Holtier, J. Johnson, and P. Dale. 1992.
Cellular automaton models of interspecific competition
for space: the effect of pattern on process. Journal of
Ecology 80:527–534.
Solé, R. V., and J. Bascompte. 1997. Emergent phenomena
in spatially extended model ecosystems. Pages 1–26 in
J. Bascompte and R. V. Solé, eds. Modeling spatiotemporal dynamics in ecology. Springer, New York.
Stoll, P., and D. Prati. 2001. Intraspecific aggregation alters
competitive interactions in experimental plant communities. Ecology 82:319–327.
Tilman, D. 1982. Resource competition and community
structure. Princeton University Press, Princeton, N.J.
———. 1988. Plant strategies and the dynamics and structure of plant communities. Princeton University Press,
Princeton, N.J.
———. 1993. Species richness of experimental productivity gradients: how important is colonization limitation? Ecology 74:2179–2191.
———. 1994. Competition and biodiversity in spatially
structured habitats. Ecology 75:2–16.
van den Bosch, F., H. D. Frinking, J. A. J. Metz, and J. C.
Zadoks. 1988. Focus expansion in plant disease. III. Two
experimental examples. Phytopathology 78:919–925.
Verheyen, K., and M. Hermy. 2001. An integrated analysis
of the spatio-temporal colonization patterns of forest
plant species. Journal of Vegetation Science 12:567–578.
Warner, R. R., S. E. Swearer, and J. E. Caselle. 2000. Larval
accumulation and retention: implications for the design
of marine reserves and essential fish habitat. Bulletin of
Marine Science 66:821–830.
Webb, C. O., and D. R. Peart. 2001. High seed dispersal
rates in faunally intact tropical rain forest: theoretical
and conservation implications. Ecology Letters 4:
Wilson, W. G., C. W. Osenberg, R. J. Schmitt, and R.
Nisbet. 1999. Complementary foraging behaviors allow
coexistence of two consumers. Ecology 80:2358–2372.
Yodzis, P. 1989. Introduction to theoretical ecology. Harper
& Row, New York.
Associate Editor: Per Lundberg

Similar documents