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Functional
Ecology 2001
15, 615 – 623
Asymmetric competition between plant species
Blackwell Science, Ltd
R. P. FRECKLETON *† and A. R. WATKINSON ‡
*Department of Zoology, University of Oxford, Oxford OX1 3PS, UK, and ‡Schools of Environmental and Biological
Sciences, University of East Anglia, Norwich NR4 7TJ, UK
Summary
1. Asymmetric competition is an unequal division of resources amongst competing
plants. Thus, competition may be asymmetric in the sense that some individuals remove
a disproportionately large amount of resource. Alternatively, competition may be
asymmetric in that one species removes a disproportionately large amount of resource.
The mechanisms determining the two forms of asymmetry may be similar, for example
through initial size advantage or over-topping.
2. We explore the consequences of these two forms of asymmetry for competition
models that predict mean performance as a function of the density of interacting
species. We do so using neighbourhood models that explicitly consider the allocation
of resources to individuals within an interacting mixture.
3. Asymmetric individual competition is modelled by assuming that individuals are
formed into a competitive hierarchy such that individuals at the top of the hierarchy
are able to remove more resources than those at the bottom. Mean performance declines
exponentially, moving from top to bottom of the hierarchy. Asymmetric species-level
competition is modelled by assuming that one species occupies all of the upper positions
in the competitive hierarchy and hence dominates the resource.
4. When competition is asymmetric at the species level, yield–density responses follow
an exponential decline. Otherwise, arithmetic mean performance follows a classic
hyperbolic response.
5. Using this approach, we explore the asymmetry of competition between wheat and
three species of weeds.
Key-words: Contest competition, non-linear model, maximum likelihood, resource competition, scramble
competition
Functional Ecology (2001) 15, 615 – 623
Introduction
The outcome of competition in mixtures of plant
species within a community will be determined by a
variety of processes, including the spatial distribution
of individuals; the resources being competed for; and
the ability of the species to compete for these resources.
In single-species stands, a variety of studies have
emphasized the importance of variation in competitive ability and resource acquisition at the level of the
individual (Hara 1984a; Hara 1984b; Firbank &
Watkinson 1985a; Weiner 1986; Firbank & Watkinson
1987; Weiner & Thomas 1986; Pacala & Weiner 1991;
Hara & Wyszomirski 1994; Nagashima et al. 1995).
In particular, these studies have concentrated on how
variability in individual growth rates affects size hierarchy formation and the response of mean performance to changing density.
© 2001 British
Ecological Society
†Author to whom correspondence should be addressed.
E-mail: [email protected]
In studying competition within monocultures, it has
been found useful to distinguish between two forms of
competition (Weiner 1988): symmetric competition is
regarded as a sharing of resources amongst individuals,
whilst asymmetric competition is an unequal sharing
of resources as a consequence of larger individuals
having a competitive advantage over smaller ones.
Asymmetric competition may arise, for example, as a
consequence of variation in emergence times within a
population, with those plants emerging first gaining an
advantage over later-emerging ones (Ross & Harper
1972). The degree to which the outcome of competition
is either symmetric or asymmetric plays a fundamental
role in determining the strength of the effects of increasing population density and shape of the response
curve (Watkinson 1980; Firbank & Watkinson 1985a).
This form of asymmetric competition may be
viewed as a competitive hierarchy. Individuals at the
top of the hierarchy (for example, those plants that
emerge first) obtain the most resources, are affected
little by competition from individuals lower in the
615
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R. P. Freckleton &
A. R. Watkinson
© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
hierarchy, and hence grow largest. Individuals lower in
the hierarchy grow smaller as they have access to fewer
resources than those at the top of the hierarchy. In the
extreme case of asymmetric hierarchy formation, individuals are affected only by competition from those at
higher positions in the hierarchy, and are unaffected
by those lower down. This form of competition has
been explored in a number of models for competition
in single-species populations. (Firbank & Watkinson
1985a; Pacala & Weiner 1991; Hara & Wyszomirski
1994).
In the same way that plants within monocultures
form competitive hierarchies (Ross & Harper 1972;
Weiner & Solbrig 1984), it would be expected that in
mixtures of species, competition is not equal for all
members of the interacting populations. Furthermore,
species differ in their ability to capture resources.
Watkinson (1985), for example, refers to data from
Butcher (1983) on competition between varieties of
peas and Avena fatua. In that study, it was found that
the form of the frequency distributions of individual
biomass between and within species depended on
which variety of pea competed with the Avena: in one
case, the frequency distributions of the two species
were very similar, whilst in another case the largest
plants were all Avena, indicating that the two species
were organized in a competitive hierarchy. Similarly,
Weiner (1985) found that competition in mixtures
simultaneously determined the distribution of sizes of
individuals within species, as well as the distribution of
biomass across species. Work that has manipulated the
relative emergence times of competing species (Kropff
& Spitters 1991; Kropff & Spitters 1992; Connolly &
Wayne 1996) shows that the degree to which one species may be able to overtop another, and thus dominate
light resources, may be influenced by relative emergence time, and that this affects the relative amounts
of resource captured by each species. Furthermore,
asymmetric competition may interact with the spatial
distribution of the interacting species in determining
mean performance (Weiner et al. 2001).
In defining asymmetric competition between species, it will be useful to distinguish two components of
asymmetry of resource capture. In addition to the division of resource amongst individuals, in species mixtures it is also necessary to consider the division of
resources between the species, and the degree to which
one species or the other is able to pre-empt resources.
To date, these processes have not been separated in
studying asymmetric competition between plant species. Most studies that have considered asymmetric
competition between species have considered just onesided competition, where one species is completely
dominant over another (Crawley & May 1987; Rees &
Long 1992). Alternatively, models have simply considered differences between resource capture at the
species level, but ignore individual-level asymmetric
competition (Kropff & Spitters 1991; Kropff &
Spitters 1992; Reynolds & Pacala 1993; Benjamin &
Aikman 1995; Rees & Bergelson 1997). The aim of this
paper is to present and test a simple model that incorporates and contrasts these two components of asymmetry, and to consider the implications of interspecific
competitive asymmetry for yield–density responses in
two-species mixtures.
Materials and methods
   

The model we analyse is a simple model of resource
competition based on the simulation of Firbank &
Watkinson (1985a).The model is formulated in the
following way:
1. The model considers two species, x and y, the
densities of which are denoted by Nx and Ny individuals of each species, respectively.
2. Individuals are located at discrete points in space
and remove resources from spatially restricted
neighbourhoods. The neighbourhoods of individual plants of species x and y are of area qx and qy,
respectively. For simplicity we assume that the
neighbourhoods are circular, although we could
assume that the neighbourhoods are of any shape;
the key assumption is that the neighbourhoods
are spatially restricted.
3. Individuals remove resources from their neighbourhood. The amount of resource removed determines
the size of the plant. Adjacent plants compete
for resources when their neighbourhoods overlap.
We incorporate three rules for determining how
resources are allocated between individuals that
imply different degrees of asymmetry of resource
capture. These are illustrated schematically in Fig. 1.
4. In the first case, resources are shared evenly
between individuals (Fig. 1a). Hence if N individuals overlap an area of habitat, a fraction 1/N of
the resource contained within this area is allocated
to each individual. We term this symmetric competition, as this corresponds to the mechanism of
symmetric competition in single-species stands.
5. The second case assumes that competition between
species is asymmetric, but that neither species is
able to pre-empt the resource (Fig. 1b). (i) The
individuals of the two species are organized into a
linear hierarchy. There are 1 to N positions in the
hierarchy, where N is the total number of individuals
of the two species. Individuals are assigned to positions in the hierarchy with individuals removing
resources in the order in which they are assigned to
the hierarchy. (ii) The hierarchy is randomly assembled such that if there are Nx and Ny individuals
of species x and y, respectively, the probability of
a given position being occupied by species x is
Nx(Nx + Ny)–1. (iii) Each individual of x or y successively removes a proportion, dx or dy, respectively,
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Asymmetric
competition
between plant
species
(a)
(b)
case of light extinction within a canopy. We term
this asymmetric hierarchy formation.
The model is solved analytically to predict the
expected mean weight of an individual of species x
interacting with Nx and Ny, other individuals of species
x and y, respectively.
 
(c)
Fig. 1. Schematic diagram illustrating the three modes of
competition employed in the modelling. For the sake of
illustration it is assumed that the two species (differentiated by
shading) are competing for light, such that taller individuals
are able remove more resources and are unaffected by the
presence of smaller plants. (a) Symmetric competition: all
individuals are able to remove the same amount of resource
and no individual achieves competitive dominance. The effects
of competition are thus more-or-less equal for all individuals, and resources are simply shared amongst competing
individuals. (b) Asymmetric individual-level competition:
individuals vary in competitive ability, with the result that
some individuals are able to remove more resources than
others. No one species is on average competitively superior
to the other. (c) Asymmetric competition between species: one
species is able to dominate the resource, and all individuals of
this species are able to pre-empt resources and make them
unavailable to the second species.
© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
of the resource from its neighbourhood. Competition is asymmetric as the performance of an individual
at a given position in the hierarchy is affected only
by those individuals above in the hierarchy, and not
by those below. There is an exponential decline in
expected performance moving from the top of the
hierarchy to the bottom, and the relative variance in
performance increases with increasing density. We
term this asymmetric individual-level competition.
6. The third case assumes that one of the species is
able to pre-empt the resource (Fig. 1c). In particular,
species y is competitively dominant and, if there are
Ny individuals of species y, then the top Ny positions in the hierarchy are occupied by species y, and
lower positions in the interspecific hierarchy are
occupied by species x. At the level of the individual,
competition is again asymmetric, with each individual
of x or y successively removing a proportion, dx or
dy, respectively, of the resource from its neighbourhood. This value is set at a constant for each species irrespective of position in the hierarchy. This
implies that although the proportion of resources
removed from an individual’s neighbourhood does
not vary with rank, the net amount of resources
removed will decline as an exponential function of
the position of an individual, for example, as in the
We compared the model predictions with field data on
competition between winter wheat and three species of
arable weed: Galium aparine, Anisantha (= Bromus)
sterilis and Papaver rhoeas. A detailed description of
the experiment is given elsewhere (Lintell Smith et al.
1999). The experiment consisted of 48 3 × 3 m plots
marked out in an area of field (36 × 48 m) that had
been ploughed and rolled prior to the start of the
experiment. Plots were separated by a 3 m discard
area. The field was drilled with wheat at a depth of
4 cm at a density of 370 seeds m–2 on 23 October 1990,
following roterra cultivation to 6 cm depth. Three replicates of each of eight weed treatments (all three species, all pairwise combinations, each species alone and
weed-free) were sown at two nitrogen levels (240 and
120 kg ha–1) and laid out in a fully randomized design.
Each species was sown at a density of 50 seeds m–2.
Weeds were allowed to set seed at the end of each season. These germinated to form the weed population in
the next season. The experiment was repeated using
the same protocol in 1990, 1991 and 1992.
The data analysed are the yields of wheat recorded
from within small (20 × 20 cm) quadrats taken within
the main experimental plots. All above-ground biomass of plants was removed from these areas, and the
number of weeds recorded. The wheat plants were
dried at 70 °C for 48 h, and the total dry weight of
these was recorded. As these data are taken from small
neighbourhoods (an average of 10·8 wheat plants per
quadrat), they are ideal for comparison with the prediction of the model, which similarly considers competition between plants within small neighbourhoods.
We used non-linear regression analysis to fit models
that predict the yield of wheat as a function of the combined density of surviving weeds. We fitted a model of
the form ¥ = ym f (N ).
The yield of wheat is related to ym, the yield of wheat
in the absence of competition, and f (N ), a function
that predicts the reduction in yield owing to competition from the weeds. The particular forms of f used
were generated from the solutions to the competition model (below). We used a maximum-likelihood
approach to fit the models. Exploratory analysis
indicated that model fits were extremely sensitive to a
small proportion of residual values. We therefore fit
the model using a maximum-likelihood approach
assuming a Cauchy distribution of error, which is less
sensitive to outliers and changes in residual variance
than other distributions (Hilborn & Mangel 1997).
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R. P. Freckleton &
A. R. Watkinson
Data were logarithmically transformed prior to analysis
and the likelihood was numerically maximized using
a Rosenbrock pattern search (Rosenbrock 1960). As
wheat was grown under weed-free conditions in each
year of the experiment, we were able to estimate ym
independently through calculating the average yield of
the weed free plots. These estimates were used as ‘hard’
parameter estimates, and the non-linear fitting procedure was used to estimate the parameters of the competition function.
Results
 
In general terms, the model may be solved by expressing mean performance in terms of the probability of an
individual occupying a given position of the combined
hierarchy and its expected performance at that position. The model may be expressed in the following
form, where E [wx] is the expected mean weight of a
target individual of species x interacting with Nx and
Ny neighbours of species x and y, respectively:
E[wx] = wm
Nx+Ny
∑ p(k) f (A N ,A N | k),
x
x
y
y
eqn 1
k=0
© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
where p(k) is the probability of occupying position k of
the 0 to Nx + Ny positions in the combined hierarchy of
the two species. wm is the expected weight of an isolated
individual in the absence of competition; f (AxNx, AyNy | k)
is the reduction in mean performance experienced by
an individual at position k in the hierarchy, where Ax
and Ay are the average proportions of the neighbourhood of the target plant overlapped by any neighbour
of x and y, respectively. The form of f will depend on
the degree to which neighbours of both species remove
resources and hence make them available to other individuals, as well the organization of the competitive
hierarchy, both intra- and interspecifically, as shown
schematically in Fig. 1. In terms of the implementation
of the model as a series of overlapping circular neighbourhoods, the competition function f is a composite
of the removal function (whether individuals share
resources, or a proportion dx or dy of resources is
removed by each individual from its neighbourhood)
and the formation of the hierarchy (the number of
dominant individuals overlapping the neighbourhood
of a target individual and hence removing resources).
Equation 1 predicts the expected performance of a
single target individual with given numbers of intraand interspecific competitors. To develop a full neighbourhood model to predict the mean weight of an
individual within a stand, it would be necessary to
model the probability distribution of the numbers of
intra- and interspecific (Pacala & Silander 1985). In
general, however, the behaviour-of-neighbourhood
models are essentially direct functions of the model
assumed for mean individual-level effects (Pacala &
Silander 1985; Pacala & Weiner 1991). Therefore the
form of relationship between performance and density
predicted at the individual level by equation 1 will be
identical to the form of this relationship at the population level.
  

When competition between individuals is symmetric
(Fig. 1a), expected performance is the same irrespective of which position in the combined hierarchy
the individual occupies, and each individual has the
same probability of occupying each position in the
hierarchy. Thus, px(k) = py(k) = (1 + Nx + Ny)−1 and
f (k) = (1 + AxNx + Ay Ny)−1 for all positions in the hierarchy, assuming that individuals are approximately
randomly distributed at the scale of local neighbourhoods (Firbank & Watkinson 1985a). Hence the
competition function is:
E [wx] = wm
Nx+Ny
1
k=0
x
1
- -------------------------------------------∑ -------------------------------(1 + N + N ) (1 + A N + A N )
y
x
x
y
eqn 2
y
wm
= ----------------------------------------- .
1 + Ax Nx + Ay Ny
eqn 3
Mean weight thus declines as a hyperbolic function of
the density of both species. This is the familiar form of
response of mean size to density (Firbank & Watkinson
1985b).
  

When competition between individuals is asymmetric,
the competition function f() has to consider both the
probability that a neighbour occupying a given position is of one species or the other, and the amount of
resources removed by neighbours at each position.
Thus at position i in the hierarchy, the amount of
resource removed by species x is the probability that
the position is occupied by species x (Nx px, where px is
the probability that the position is occupied by a given
individual of species x, as above) multiplied by the
resource removed by an individual, (1 − Ax dx). Hence
f (k) has the following form, where px and py are, as
above, the probabilities that a given individual of x or y,
respectively, will occupy a given position in the hierarchy:
k
f (k) = ∏ [Nx px(i )(1 – Axdx) + Ny py(i)(1 – Aydy)]. eqn 4
i=0
In equation 4, the competitive effect is calculated across
the k dominant competitors in the hierarchy above the
target individual. The values of px and py determine whether resource pre-emption by one or other of the species
occurs through determining whether one species is more
likely to occupy the higher positions in the hierarchy.
FEC558.fm Page 619 Tuesday, September 4, 2001 5:19 PM
  
;  
 
Hyperbolic model
When interspecific hierarchies are formed randomly,
that is, there is no asymmetry for resource access
between species, a target plant has an equal probability
of occupying any position in the hierarchy. Hence
px(k) = py(k) = (1 + Nx + Ny)−1 for all k positions in the
combined hierarchy. Equation 4 therefore becomes:
k
Nx(1 – Axdx) + Ny(1 – Aydy)
f (k) = ∏ ------------------------------------------------------------------1 + Nx + Ny
i=0
Mean weight (log scale)
619
Asymmetric
competition
between plant
species
Exponential model
Density (log scale)
eqn 5
(Nx(1 – Axdx) + Ny(1 – Aydy))
= --------------------------------------------------------------------------.
k
(1 + Nx + Ny)
k
Fig. 2. Contrasting competition–density responses: the
exponential function (equations 10 and 11) and hyperbolic
model (equations 3 and 7) predicting the mean performance
of species x as a function of species y, plotted on a doublelogarithmic scale.
Substituting this into equation 1 then yields:
Nx+Ny
(Nx(1 – Axdx) + Ny(1 – Aydy))
wm
E[wx] = ------------------------- ∑ ---------------------------------------------------------------------k
(1 + Nx + Ny)
1 + Nx + Ny k=0
k
wm
= --------------------------- ×
1 + Nx + Ny
[1 – (Nx (1 – Ax dx) + Ny (1 – Ay dy))
( 1 – Ay dy )
Ny
k – Ny
∏ (1 – A d )
x x
i=0
f (k) = --------------------------------------------------------------.
1 + Nx
eqn 8
Substituting these expressions into equation 1 then
yields:
1+Nx+Ny
(1 + Nx + Ny)
– (1+Nx+Ny)
]
--------------------------------------------------------------------------------------------------------------------------------- .
–1
1 – (Nx (1 – Ax dx) + Ny (1 – Ay dy) ) (1 + Nx + Ny)
eqn 6
Ny
1+Nx
wm(1 – Aydy) (1 – Axdx)(1 – (1 – Axdx) )
E[wx] = ------------------------------------ -------------------------------------------------------------------------.
Axdx
1 + Nx
eqn 9
By evaluating this equation in the limits that Nx → ∞
and Ny → ∞, it is possible to approximate equation 6
by:
wm
E[w] = ------------------------------------------------ ,
(1 + αxxNx + αxyNy)
eqn 7
where αxx = Axdx and αxy = Aydy. In this case, therefore,
the form of competition is of the same form as predicted by equation 3. The difference between the
model for symmetric competition and this model for
asymmetric competition, however, is that there is considerably more variance in performance in the model
for asymmetric competition, as equation 5 predicts
an exponential decline in performance moving down
from the top of the competitive hierarchy. This variance forms the basis for diagnosing the form of interactions at the individual level from field data (below).
  

© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
When species y is able to pre-empt resources through
domination of the interspecific hierarchy, then for the
first k = 1 ... Ny positions in the combined hierarchy,
py(k) = Ny–1 and px(k) = 0. Then for the k = Ny + 1 ...
1 + Nx + Ny, py(k) = 0 and px(k) = (1 + Nx)–1. Hence, for
the first k = 1 ... Ny positions in the combined hierarchy,
f(k) = (1 −Ay dy)k, and for the k = Ny + 1 ... 1 + Nx + Ny
other positions,
This then may be approximated by the model:
wm exp (βNy)
E[wx] = ------------------------------- ,
1 + αxxNx
eqn 10
where β = ln(1 – Aydy). The important difference
between this equation and the forms of yield–density
responses predicted by equations 3 and 7 is that in
equation 10 the mean weight of species x declines
exponentially as the density of species y increases. The
yield–density response predicted by equation 10 is
therefore very different from that predicted under the
other forms of competition (Fig. 2). Under the hyperbolic model, log mean weight declines linearly with
increasing log density at high densities. In contrast,
under the exponential model, the rate of decline in
log weight with increasing density is proportional to
density. Since the sensitivity of the model (the rate of
change in log weight with log density) to changing the
density of species y is very much higher than that for
changing the density of species x, mean performance
according to equation 10 is much more sensitive to
changing the fraction of species y in the community.
Note that when hierarchies are formed asymmetrically such that the combined hierarchy is always
dominated by species y, the yield–density relationship
is always of the form of equation 10, irrespective of the
form of symmetry of competition between individuals
of species x.
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R. P. Freckleton &
A. R. Watkinson

The aim is to test data in order to distinguish between
competition that is asymmetric at the individual level,
that is, in terms of the removal of resources by individual plants, and asymmetry in terms of resource
pre-emption by one species or another. The distinction
between asymmetric competition in terms of hierarchy
domination by one of the species, and symmetric
competition where interspecific hierarchies are randomly formed, is straightforward as the yield–density
responses predicted by the models for these two
forms of competition are very different. Specifically,
as shown in Fig. 2, a plot of log mean plant weight
versus log density should reveal clear differences in
response.
Distinguishing between symmetric and asymmetric
competition at the level of individual plants, assuming
that hierarchies are randomly formed, requires further
analysis of the model. Specifically, when competition
between individuals is asymmetric, the relative variance
1000
(a)
in mean performance should increase dramatically as
density increases. This is characteristic of asymmetric
competition in single-species populations. A simple
way to analyse this behaviour is to look at geometric
mean performance. The geometric mean is always
smaller than the arithmetic mean, the difference
between the two being a function of the relative variability of the data. Specifically, if the variance in size
changes systematically with density, then the geometric
mean will respond differently from the arithmetic mean
to changing density.
Under symmetric competition, the difference between
the predictions of the two means will be minimal,
as there is no assumed mechanism for generating
variance as a function of competitive intensity.
By contrast, for the asymmetric model without
resource pre-emption by one of the species, the geometric mean should differ considerably from arithmetic
mean performance. Specifically, geometric mean performance may be predicted by modifying the general
form of model (equation 1) to consider log performance. In the case of asymmetric competition without
resource pre-emption, the particular form (equation 6)
is modified to:
N +N
x y
wm
E[log wx] = --------------------------- ∑ k log [Nx(1 – Axdx)
1 + Nx + Ny k=0
+ Ny(1 – Aydy)] – k log [1 + Nx + Ny]
100
wm
=  ------------------------  (log [Nx(1 – Ax d x) + Ny(1 – Ay dy)]
1 + Nx + Ny
Nx+Ny
10
– log [1 + Nx + Ny] ) ∑ k
Yield of wheat (g m–2)
k=0
wm
= ------ (Nx + Ny)(log [Nx(1 – Ax dx) + Ny(1 – Ay dy)]
2
1
1
10
100
1000
10000
1000
– log [1 + Nx + Ny] ).
Hence geometric mean performance is given by:
(b)
1
Nx(1 – Axdx) + Ny(1 – Aydy) --2- (Nx+Ny)
.
GM(wx) = wm -------------------------------------------------------------------
1 + Nx + Ny
100
eqn 11
10
1
0·1
1
10
100
1000
10000
Density of weeds (m–2)
© 2001
British
Fig.
3. Yield–density
responses in mixtures of winter wheat and three species of weeds
Ecological
(see
text forSociety,
details). The curves show the best-fit exponential and hyperbolic models
Functional Ecology,
(parameters
in Table 1). (a) Raw data; (b) smoothed response, calculated from a
running
mean of the ordered data.
15, 615 –geometric
623
This again is an exponential yield–density relationship that contrasts with the hyperbolic relationship
between arithmetic mean performance and density.
The difference between the response of log AM and
log GM performance to increasing density should
therefore measure the degree of asymmetry of performance at the individual level.
   
Figure 3a shows the relationship between yield and
total weed density. There were no clear differences
between the yield–density responses with the different
FEC558.fm Page 621 Tuesday, September 4, 2001 5:19 PM
Table 1. Fits of competition models to the data presented in Fig. 3a. The functions
621
fitted
were either the exponential or hyperbolic models, n = 112 in both cases. In both
Asymmetric
cases the estimate of ym was obtained from data on plants grown in the absence of
competition
weeds (n = 20). The model-fitting procedure is described in the text
between plant
species
Model
Exponential
Hyperbolic
y = ym exp(–aN )
y = ym /(1 + aN )
ym
Parameter estimate (± SE)
Log likelihood
0·002885
0·021050
697·1105
–205·61
–205·95
–
0·000335
0·006401
42·41
(a)
100
10
Biomass of species x
1
0·1
1
10
1000
100
(b)
100
10
1
0·1
1
10
100
Density of species y
Fig. 4. Simulated yield–density responses for comparison with Fig. 2. The simulation
results were derived from a spatially explicit realization of the model described in the
text. Plants were allocated to random positions within a habitat area of 200 × 200 units.
Neighbourhoods of both species were 300 square units in area. Species x was sown at
a constant density of 10 plants; the density of species y varied from 1 to 100 plants, with
10 replicates at each density. The competitive hierarchy was formed randomly such that
individuals could occupy any position within the combined hierarchy of the two
species. The arithmetic mean yield–density response for this model is predicted to
follow a hyperbolic yield–density model (equation 7), whereas, the geometric mean or
smoothed responses are predicted to follow an exponential response (equation 7).
(a) Raw results; (b) smoothed response based on a running mean of the ordered data.
© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
weed combinations or clear effects of nitrogen (Lintell
Smith et al. 1999). Hence we do not consider these
differences further, but analyse the data as a function of
total weed density. Wheat yield is not a simple function
of weed density (Fig. 3a), and visually does not appear
to correspond well to either of the forms shown in
Fig. 2. Whilst at low densities there is little response of
wheat yield to weed density, at high weed densities the
response is highly variable. Notably, however, the
variance in performance increases as density increases.
One consequence of this is that at intermediate weed
densities some plots yield mean plant sizes as high as
those at lower densities, whereas other plots yield
plants less than 1% of the size of those at lower densities.
Table 1 summarizes the analysis of the raw data. The
exponential model fit the data better, as indicated by
the value of the log-likelihood function. This improvement of fit is minimal, however, and it is clear from
Fig. 3a that neither model describes the data entirely
satisfactorily, and there are clear systematic deviations
in both cases.
In order to look at trends in geometric mean yield–
density responses we generated smoothed responses.
Smoothing is often used, for example, in time-series
analysis in order to damp local stochastic variation
with the aim of discerning long-term trends. By analogy,
we employed smoothing in order to remove some of
the variation about the mean response for the data in
Fig. 3a, and hence to determine which function underlies the yield–density response. We calculated running
means for successive observations of the ordered data
(Fig. 3b). Presented in this way, it is clear that the yield–
density response is essentially intermediate between
the two forms of model. Except at the highest densities,
the correspondence between the exponential model
and the smoothed data is considerably better than that
of the hyperbolic model.
Our interpretation of the yield–density response is
therefore that competition is asymmetric at the level of
individual plants, but that neither one species nor the
other completely dominates the combined competitive
hierarchy of the two species. This interpretation is reinforced by the results of an explicit simulation of the
model where competition is asymmetric, but without
pre-emption of resources by either of the species
(Fig. 4). Although the arithmetic mean response follows the hyperbolic model (equation 7), the geometric
mean response follows the exponential model (equation 11). Both the raw simulation results (Fig. 4a) and
the smoothed response (Fig. 4b) show remarkable
qualitative similarity to the original data (Fig. 3). At a
high densities in both the data and simulation, mean
weight may vary by two orders of magnitude or more.
The main difference is that there appear to be fewer
points in the very top right of the response observed in
the data (Fig. 3a) than the simulation (Fig. 4a). This
difference probably relates to the incomplete dominance
postulated above. Presumably in reality the hierarchy
is not completely dominated by the weed species, but
contains an intermediate area of overlap where either
species may occur. This could be modelled, for example,
by using a continuous switching function to model
asymmetric species competition (Freckleton 1997).
FEC558.fm Page 622 Tuesday, September 4, 2001 5:19 PM
622
R. P. Freckleton &
A. R. Watkinson
© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
Discussion
The notion of competitive asymmetry in plants has
generally been applied to competition within singlespecies stands, and has only rarely been extended
to explore competition between species (Weiner 1985;
Schwinning & Fox 1995; Connolly & Wayne 1996;
Weiner et al. 2001). In contrast, animal ecologists have
tended to use the term asymmetric competition specifically in relation to competition between pairs of
species, without reference to the individuals that are
competing (Lawton & Hassell 1981; Calow 1998).
Here we have highlighted how asymmetric competition between species should be considered as having two components: the asymmetry of competition
between individual plants; and the asymmetry of competition at the level of the species. The models and
data we present demonstrate that these effects can
have dramatic impacts on yield–density responses,
and are readily detected under field conditions.
The asymmetry of competition between individuals
in mixed-species stands is basically the same in nature
as asymmetric competition between individuals in
single-species stands. This asymmetry is modelled phenomenologically by generating an asymmetric division
of resources amongst individuals. This is because those
individuals at the top of a competitive hierarchy, such
as those that emerge first, are able to remove a disproportionately large amount of resource, for example
through size advantage (Weiner 1988). More generally,
this asymmetry results from individual-level variations
in resource capture resulting from, for example, variations in initial emergence.
Asymmetric competition between species results
from the differential ability of the species to be able to
occupy higher positions in the competitive hierarchy.
This may result, for example, from height differences
between species with one species being able to completely over-top another and hence pre-empt access to
light. The determinants of this competitive asymmetry
may be similar to those that determine competitive
asymmetry at the level of individuals. The difference
is that whereas competitive asymmetry at the level of
individuals results from variance among individuals,
asymmetry in competition between species results
from mean differences between species.
The approach we have taken to explore the consequences of asymmetric competition is set within the
framework of model yield–density responses. The
advantage of this approach is that such models allow
predictions of yields ( Weiner et al. 2001), as well as
allowing the population- and community-level impacts
of asymmetric competition on performance to be
modelled (Schwinning & Fox 1995). Several studies
have explored how the parameters of single-species
yield–density models are affected by varying the symmetry of competition (Firbank & Watkinson 1985a;
Pacala & Weiner 1991; Hara & Wyszomirski 1994;
Freckleton 1997). In single-species stands the degree of
asymmetry determines how a fixed amount of resource
is allocated amongst competing individuals. The impact
of changing the degree of asymmetry on model parameters basically depends on the nature of resource use
at the individual level (Firbank & Watkinson 1985a).
Under asymmetric competition, the yield–density
response always follows the hyperbolic form defined
above. When competition is symmetric, however, underor over-compensating yield–density responses may be
predicted to occur, depending on the efficiency with
which individuals convert resources into biomass
(Firbank & Watkinson 1985a; Freckleton 1997). The
impacts of symmetric competition on yield–density
responses in plants are thus fundamentally different
from those predicted for animal populations (Royama
1992) where ‘collapsing’ competition–density responses
result from the inability of individuals to survive below
some threshold level of resource acquisition. Plants
generally do not show such thresholds for survival or
reproduction (Rees & Crawley 1989), and hence the
consequences of symmetric competition for yield–density
responses in plant monocultures are quite different.
In contrast to the predictions for single-species
stands, our models suggest that in mixtures of species,
the form of yield–density responses may be quite profoundly changed by altering the degree of competitive
asymmetry (Fig. 2). When competition was asymmetric at the level of individual plants, but the combined
hierarchy was formed at random such that neither
species asymmetrically dominated, mean performance was predicted to follow the hyperbolic model
(equation 7), whereas geometric mean performance
followed the exponential model (equation 11). This
distinction is important if we are interested in predicting the effects of competition on long-term dynamics,
as the model predictions are isotropic only on the logarithmic scale. An isotropic distribution implies that
the weight of a randomly chosen individual is likely
to be bigger or smaller than the average with equal
probability (Lande 1998). The predictions of the geometric mean model are isotropic as the distribution of
plant sizes (which decline exponentially moving from
the top of the hierarchy to the bottom) is linear on the
logarithmic scale. Hence, on the logarithmic scale 50%
of individuals will be smaller than average, and 50%
will be larger. On the arithmetic scale most individuals
will be smaller than the average. The predictions of
the geometric mean model (equation 11) may thus be
more relevant to understanding the impacts of asymmetric competition on interspecific interactions for
either species. One consequence of this may be to generate founder effects in community dynamics, resulting
from the disproportionately intense impacts of competition at high densities under the exponential model.
Although founder effects have been postulated to arise
through asymmetric competition between species
(Reynolds & Pacala 1993; Rees & Bergelson 1997), the
mechanism postulated here is very different, resulting
from variance in performance at the individual level.
FEC558.fm Page 623 Tuesday, September 4, 2001 5:19 PM
623
Asymmetric
competition
between plant
species
In conclusion, we present models and data that
demonstrate important impacts of the asymmetry of
competition between individuals and species on the
form of competition density response. Particularly if
we look at geometric mean performance, the consequences of asymmetric competition at the level of individuals may be important, irrespective of whether
one species or the other tends to dominate access to
resources. As a wide body of evidence has shown
asymmetric competition to be characteristic of competitive hierarchies in single species, asymmetric competition between species may be an important – but
largely overlooked – phenomenon.
Acknowledgements
We would like to thank Richard Law and Jake Weiner
for extensive discussion of this work, and two anonymous
referees for helpful comments. Also many thanks to
Graham Hopkins for providing the impetus for this work.
References
© 2001 British
Ecological Society,
Functional Ecology,
15, 615 – 623
Benjamin, L.R. & Aikman, D.P. (1995) Predicting growth in
stands of mixed species from that in individual species.
Annals of Botany 76, 31– 42.
Butcher, R.E. (1983) Studies on interference between weeds
and peas. PhD Thesis, University of East Anglia, UK.
Calow, P. (ed.) (1998) The Encyclopaedia of Ecology and
Environmental Management. Blackwell Science, Oxford.
Connolly, J. & Wayne, P. (1996) Asymmetric competition
between plant species. Oecologia 108, 311–320.
Crawley, M.J. & May, R.M. (1987) Population dynamics and
plant community structure: competition between annuals and
perennials. Journal of Theoretical Biology 125, 475 – 489.
Firbank, L.G. & Watkinson, A.R. (1985a) A model of interference within plant populations. Journal of Theoretical
Biology 116, 291–311.
Firbank, L.G. & Watkinson, A.R. (1985b) On the analysis of
competition within two-species mixtures of plants. Journal
of Applied Ecology 22, 503–517.
Firbank, L.G. & Watkinson, A.R. (1987) On the analysis of
competition at the level of the individual plant. Oecologia
71, 308–317.
Freckleton, R.P. (1997) Studies on variability in plant populations. PhD Thesis, University of East Anglia, UK.
Hara, T. (1984a) Dynamics of stand structure in plant monocultures. Journal of Theoretical Biology 110, 223–239.
Hara, T. (1984b) A stochastic model and the moment dynamics of the growth and size distribution in plant populations.
Journal of Theoretical Biology 109, 173 –190.
Hara, T. & Wyszomirski, T. (1994) Competitive asymmetry
reduces spatial effects on size-structure dynamics in plant
populations. Annals of Botany 73, 185–197.
Hilborn, R. & Mangel, M. (1997) The Ecological Detective:
Confronting Models with Data. Princeton University Press,
Princeton. NJ.
Kropff, M.J. & Spitters, C.J.T. (1991) A simple model of crop
loss by weed competition from early observations of on relative leaf area of the weeds. Weed Research 31, 465 – 471.
Kropff, M.J. & Spitters, C.J.T. (1992) An eco-physiological
model for interspecific competition, applied to the influence
of Chenopodium album L. on sugar beet. I. Model description and parameterization. Weed Research 32, 437– 450.
Lande, R. (1998) Demographic stochasticity and Allee effect
on a scale with isotropic noise. Oikos 83, 353–358.
Lawton, J.H. & Hassell, M.P. (1981) Asymmetrical competition in insects. Nature 289, 793–795.
Lintell Smith, G., Freckleton, R.P., Firbank, L.G. & Watkinson, A.R. (1999) The population dynamics of Anisantha
sterilis in winter wheat: comparative demography, and the
role of management. Journal of Applied Ecology 36, 455–
471.
Nagashima, H., Terashima, I. & Katoh, S. (1995) Effects of
plant density on frequency distributions of plant height in
Chenopodium album stands: analysis based on continuous
monitoring of the height-growth of individual plants.
Annals of Botany 75, 173–180.
Pacala, S.W. & Silander, J.A.J. (1985) Neighbourhood
models of plant population dynamics I. Single-species
models of annuals. American Naturalist 125, 385–411.
Pacala, S.W. & Weiner, J. (1991) Effects of competitive asymmetry on a local density model of plant competition.
Journal of Theoretical Biology 149, 165–179.
Rees, M. & Bergelson, J. (1997) Asymmetric light competition and founder control in plant communities. Journal of
Theoretical Biology 184, 353–358.
Rees, M. & Crawley, M.J. (1989) Growth, reproduction and
population dynamics. Functional Ecology 3, 645–653.
Rees, M. & Long, M.J. (1992) Germination biology and the
ecology of annual plants. American Naturalist 139, 484–
508.
Reynolds, H.L. & Pacala, S.W. (1993) An analytical treatment of root-to-shoot ratio and plant competition for soil
nutrient and light. American Naturalist 141, 51–70.
Rosenbrock, H.H. (1960) An automatic method for finding
the greatest or least value of a function. Computing Journal
3, 175 –184.
Ross, M.A. & Harper, J.L. (1972) Occupation of biological
space during seedling establishment. Journal of Ecology 60,
77– 88.
Royama, T. (1992) Analytical Population Dynamics. Chapman
& Hall, London.
Schwinning, S. & Fox, G.A. (1995) Population dynamics consequences of competitive symmetry in annual plants. Oikos
72, 422– 432.
Watkinson, A.R. (1980) Density-dependence in single-species
populations of plants. Journal of Theoretical Biology 83,
345–357.
Watkinson, A.R. (1985) Plant responses to crowding. Studies
on Plant Demography: A Festschrift for John L. Harper (ed.
J. White), pp. 275–289. Academic Press, London.
Weiner, J. (1985) Size hierarchies in experimental populations
of annual plants. Ecology 66, 743–752.
Weiner, J. (1986) How competition for light and nutrients
affects size variability in Ipomoea tricolor populations.
Ecology 67, 1425 –1427.
Weiner, J. (1988) Variation in the performance of individuals
in plant populations. Plant Population Ecology (eds A.J.
Davy, M.J. Hutchings & A.R. Watkinson), pp. 59–81.
Blackwell Scientific Publications, Oxford.
Weiner, J. & Solbrig, O.T. (1984) The meaning and measurement of size hierarchies in plant populations. Oecologia 61,
334 –336.
Weiner, J. & Thomas, S.C. (1986) Size variability and competition in plant monocultures. Oikos 47, 211–222.
Weiner, J., Griepentrog, H.-W. & Kristensen, L. (2001)
Increasing the suppression of weeds by cereal crops.
Journal of Applied Ecology 38, 784–790.
Received 12 January 2001; revised 28 April 2001; accepted
30 April 2001