Download Interspecific Competition in Plants: How Well Do Current Methods

vol. 157, no. 2
the american naturalist
february 2001
Interspecific Competition in Plants: How Well Do Current
Methods Answer Fundamental Questions?
John Connolly,* Peter Wayne,† and Fakhri A. Bazzaz‡
Department of Organismic and Evolutionary Biology, Harvard
University, Biological Laboratories, Cambridge, Massachusetts
02138
Submitted May 25, 1999; Accepted September 14, 2000
abstract: Accurately quantifying and interpreting the processes
and outcomes of competition among plants is essential for evaluating
theories of plant community organization and evolution. We argue
that many current experimental approaches to quantifying competitive interactions introduce size bias, which may significantly impact
the quantitative and qualitative conclusions drawn from studies. Size
bias generally arises when estimates of competitive ability are erroneously influenced by the initial size of competing individuals. We
employ a series of quantitative thought experiments to demonstrate
the potential for size bias in analysis of four traditional experimental
designs (pairwise, replacement series, additive series, and response
surfaces) either when only final measurements are available or when
both initial and final measurements are collected. We distinguish
three questions relevant to describing competitive interactions:
Which species dominates? Which species gains? and How do species
affect each other? The choice of experimental design and measurements greatly influences the scope of inference permitted. Conditions
under which the latter two questions can give biased information
are tabulated. We outline a new approach to characterizing competition that avoids size bias and that improves the concordance
between research question and experimental design. The implications
of the choice of size metrics used to quantify both the initial state
and the responses of elements in interspecific mixtures are discussed.
The relevance of size bias in competition studies with organisms
other than plants is also discussed.
Keywords: competition, thought experiments, design of competition
experiments, size bias, replacement series, additive series.
* Present address: Department of Statistics, National University of Ireland,
Dublin, Belfield, Dublin 4, Ireland; e-mail: [email protected].
†
Present address: Department of Research, New England School of Acupuncture, 40 Belmont Street, Watertown, Massachusetts 02472; e-mail:
[email protected].
‡
E-mail: [email protected].
Am. Nat. 2001. Vol. 157, pp. 107–125. q 2001 by The University of Chicago.
0003-0147/2001/15702-0001. All rights reserved.
Much research has been devoted toward understanding how
individuals of co-occurring plant species both affect and
respond to one another and how these interactions influence
structure, dynamics, and evolution within plant communities (Harper 1977; Grime 1979; Schoener 1983; Keddy
1989; Grace and Tilman 1990; Bazzaz 1996). Yet, despite its
importance, there is still confusion and considerable debate
regarding how to assess interspecific competitive phenomena among plant species (e.g., Inouye and Schaffer 1981;
Connolly 1986, 1987b, 1988, 1997; Law and Watkinson
1987; Keddy and Shipley 1989; Herben and Krahulec 1990;
Silvertown and Dale 1991; Snaydon 1991, 1994; Grace et
al. 1992; Cousens and O’Neill 1993; Sackville-Hamilton
1994; Shipley and Keddy 1994). Lack of progress in our
understanding of competition has been attributed to a number of factors. These include improper experimental designs
and statistical analyses (Connolly 1986, 1987b; Cousens
1988; Goldberg and Scheiner 1993; Sackville-Hamilton
1994; Gibson et al. 1999), too much emphasis on controlled
environment versus field studies (Goldberg and Barton
1992), the limited duration of many experiments (Keddy
1989), and the lack of a mechanistic understanding of plant
competition (Tilman 1987; Schwinning and Weiner 1998;
Berntson and Wayne 2000). While these issues certainly
have impeded the development of a coherent theory of plant
competition, we believe that progress has also been hindered
by an even more fundamental problem: namely, a widespread lack of concordance between the intuitive questions
ecologists ask about interspecific competition and the experimental procedures that have been employed to address
these questions.
We see at least two major causes for this lack of concordance. First, the questions that ecologists and agronomists have asked about interspecific plant competition have
not always been well articulated. Questions about the
eventuaI outcome of competition have not been sufficiently
differentiated from questions regarding how much neighboring species affect each other and the mechanisms
through which this occurs. In extreme cases, the specific
questions driving interspecific competition studies are not
108 The American Naturalist
even apparent. Cousens (1996, p. 7) noted that it is not
uncommon for researchers to employ popular experimental
designs (e.g., replacement series) in their research “whenever
they wanted to look at competition, without stating clearly
what it was they were trying to achieve.” Moreover, superimposed upon the problem of framing clear questions are
long-standing semantic debates. Controversy surrounding
the meaning and usage of terms such as “competition,”
“interference,” “neighbor effects,” and “plant-plant interactions” have only hindered the ability of ecologists to agree
on the most logical questions to ask and the required methodological tools to address them (Birch 1957; Milne 1961;
Milthorpe 1961; Trenbath and Harper 1973; Keddy 1989;
Connell 1990).
The second difficulty relates to the basis for inferences
in competition experiments. The inferences that can be
drawn from any experiment are limited by the combination of three elements: the experimental design used,
the response and explanatory variables (biotic and abiotic)
measured, and the statistical analyses employed. We subsequently refer to this triad of elements as the “experimental structure.” The relationships between biological
questions and experimental structure in the study of interspecific plant competition have not been properly appreciated, with methods pushed far beyond what the experimental structure can validly sustain. It is the goal of
this article to clarify these relationships.
The core of this article is divided into three sections. In
the first section, we briefly review some of the more common questions pursued by researchers studying plant interactions in mixed-species communities. Second, we review the most common experimental structures used to
quantify interspecific interactions. In the third section, we
employ a series of quantitative thought experiments to
illustrate the potentials and limitations of these experimental structures for addressing particular questions. We
emphasize the establishment of appropriate null hypotheses as assisting in clear thinking (Underwood 1991). The
results of our analysis are summarized in the form of a
table as a guide to experimenters. While our thought experiments emphasize two-species plant mixtures, the conclusions are relevant for multispecies experiments and to
some studies with other organisms. Finally, we introduce
the essential elements for an alternative approach to analyzing interspecific interactions (J. Connolly and P. M.
Wayne, unpublished manuscript).
Questions about Interspecific Competition
Because the nature and consequences of interspecific plant
competition are so broad and touch on so many subdisciplines in ecology, the number of questions that have been
formulated, from community- and ecosystem-level con-
sequences of competition to its physiological and genetic
basis, is very large. It is not the intention of this article to
review these in detail. For a more comprehensive analysis
of the classes of questions asked in interspecific plant competition research, see other recent reviews by Connell
(1983), Schoener (1983), Keddy (1989), Goldberg and Barton (1992), Goldberg and Scheiner (1993), and Cousens
(1996). We limit our scope of inquiry to what we consider
to be three broad classes of questions related to interspecific plant interactions: Which species dominates at a particular point in time? Which species gains (relative to others) over a period of time? and What is the effect of one
species’ presence on a neighboring species’ performance?
How these questions differ from one another and why we
focus on them is developed below.
Which Species Dominates a Mixture/Community
at One Point in Time?
By dominance, we simply mean that at any given time, in
a defined mixture, one species (or any mixture component
such as a genotype or functional group) is more abundant
(e.g., biomass, seed number, leaf area, or population size)
than another. As such, it is clear right from the start that
this question has more to do with issues of community
composition than with the process of competition, per se.
Nevertheless, we include it for two reasons. First, in most
interspecific plant competition studies, the primary response variable measured (and used to derive indices) is
a single, end of experiment measure of species’ abundance—that is, dominance. Second, the dominance question serves as an important contrast with question 2
(Which species gains over an interval of time?), with which
it is commonly confounded.
The dominance question frequently arises in studies that
aim to assess the effects of environmental factors on the
composition of mixed-species stands. Controlled environment studies of this nature are typically designed with
mixtures of identical composition in terms of species’ densities repeated across a range of abiotic (e.g., soil fertility,
light, CO2 level) or biotic (e.g., herbivores, mycorrhizae,
pathogens) factors. The response variable is some measure
of species’ relative size in the stand at harvest time.
While studies with an experimental structure designed
to address the dominance question are useful in assessing
how abiotic or biotic factors influence a given species’
relative abundance at one point in time, the answer to this
question tells us very little about the demographic or physiological processes underlying compositional shifts, that is,
how the present composition arose and in which direction
it is likely to develop. Single, static measures of species’
abundance cannot be used to assess whether, in a given
mixture or environmental regime, one species is gaining
Competition: Theory and Questions 109
(or performing “better”) relative to another (i.e., question
2). For example, if two species ended up the same size in
a mixture, our interpretation of their relative performances
would be quite different if we were told that in one case
both started at the same seed or seedling size or that, in
another case, one species started 20 times smaller than the
other. Moreover, single static measures of dominance offer
very little insight into whether the presence of one species
is suppressing, or possibly even enhancing, the performance of another (question 3). For example, biomass estimates of the components of forest communities will always
reveal that canopy trees are much larger than understory
herbs, but this does not necessarily mean that the trees
“suppress” or “outcompete” the herbs. In many cases, understory herbs benefit from being subordinate to dominant
canopy species. Yet despite the intuitive logic of these examples, many studies rely on one static, single harvest to
infer how species are performing relative to one another
and how one species impacts the performance of another.
These misconceptions, and their implications, are articulated more fully in the thought experiments we present
below.
Which Species Gains (“Wins”) in a Mixture?
Most interspecific competition studies aim to characterize
more than a static snapshot of species’ relative dominance.
One very fundamental question is, Over an interval of
time, is one species gaining (e.g., in biomass, population
size, seed number, or access to resources) over the other?
In its simplest sense, the “which species gains” question
can be assessed by comparing the proportionality between
species’ sizes or abundances at both the beginning and the
end of a defined period of time of a given stand’s development. If a species contributes proportionally more at
the end than at the beginning, it has gained more, and it
is fair to say that over that time interval, it outperformed
the other species in some sense. Performance defined in
this way is very close to the traditional concept of growth
efficiency, that is, output per unit input (e.g., g g21), as
defined by Blackman (1919), and is analogous to average
relative growth rate (Evans 1972).
Asking which species gains may provide more insight
into the longer-term dynamics of a given mixture. For
example, if, over an interval of time, the smaller of two
species triples its size while the larger species only doubles
its size, one might infer that, given enough time and assuming no changes in growth trajectories, the smaller species might catch up to or exceed the size of its neighbors.
Examining the outcome of this question over a range of
different conditions may help suggest the outcome of competition in the long term (Connell and Sousa 1983). All
that is required to address the “which species gains” ques-
tion for a given mixture during a particular stage of stand
development is two sequential measures of abundance.
Surprisingly, the importance of making measurements at
several successive times for assessing interspecific competitive phenomena is commonly neglected (Milthorpe
1961; Connolly et al. 1990; e.g., Underwood 1992 for species other than plants).
How Does One Species in a Mixture Affect the
Performance of Another?
While the answer to the “which species gains” question
provides information about species’ relative performances
in a given mixture, it does not necessarily provide information about the causes underlying species’ performances.
More specifically, it tells us very little about the extent to
which the relative success of one plant species was caused
by any direct (e.g., abrasion) or indirect (e.g., resource
competition, pathogen spread, facilitation) interactions
with another species. While it seems logical that the “which
species gains” question should be inextricably linked to
the extent to which co-occurring species affect one another, this may not always be the case. Returning to the
hypothetical forest example used above, it is quite conceivable that, even if understory herbs are outperforming
or gaining on canopy trees by growing 10 times faster per
unit size, they may not be influencing the growth of trees.
Canopy trees may be getting their soil resources (e.g., water
and nutrients) from different strata of the rhizosphere than
herbs, and their shoots would be largely uninfluenced by
the herbs.
To assess how plant species affect one another, we generally need to measure more than relative performance in
one mixture; we need to compare the performance of a
target species across a range of mixtures. A very simple
additive design, for example, in which a set number of
individuals of a target species is grown with and without
individuals of a second species, would tell us whether the
presence of the second species influenced the performance
of the target species. The analysis of this design might be
as simple as a one-way ANOVA; however, its conclusion
regarding the effects of neighbors on targets would also
be simple and limited to a binary answer—yes or no. More
commonly, ecologists want answers to more complex questions, such as, How much of a target species’ performance
is attributable to a neighboring species’ presence? What is
the effect of a heterospecific neighbor relative to a conspecific neighbor? Is the effect of a heterospecific neighbor
contingent on its relative abundance (density/frequency)
and/or on other environmental factors? Can we (hierarchically) rank species within a community with respect to
their effects on a target species performance? While fundamental and intuitive, these questions are far more com-
110 The American Naturalist
plex than the simple yes/no question, Are species influencing one another?
These questions have spurred the development of a
number of experimental designs and associated indices,
including substitutive, or replacement series, designs (e.g.,
deWit 1960; Harper 1977), various additive designs (e.g.,
Harper 1977; Miller and Werner 1987; Snaydon 1991),
and response surface approaches (e.g., Connolly and Nolan 1976; Suehiro and Ogawa 1980; Firbank and Watkinson 1990). To date, each one of these approaches has received criticism, yet few attempts have been made to
compare comprehensively their advantages and limitations
with respect to their ability to answer fundamental questions. Below we attempt to carry out such an analysis
through the use of thought experiments.
Overview of Experimental Structures Employed in
Plant Competition Experiments
A key element of any experimental structure is the design.
The simplest design is the pairwise (PW) experiment (fig.
1A), which consists of a single mixture repeated across a
range of levels of a treatment factor (e.g., fertility). However, historically, many laboratory and field experiments
were designed either as additive series (AS; e.g., Harper
1977; Miller and Werner 1987; Snaydon 1991), replacement series (RS; e.g., Harper 1977; Trenbath 1978; Keddy
and Shipley 1989), or response surface (RE) designs (e.g.,
Suehiro and Ogawa 1980; Firbank and Watkinson 1985).
A review of laboratory plant competition experiments
(Gibson et al. 1999) showed that, of 107 competition experiments reported in 10 leading journals in ecology and
weed science over the period 1984–1993, the frequencies
of PW, AS, and RS experimental designs were 24, 28, and
42, respectively. There were only a few RE designs used.
In the field, manipulation experiments, in which one or
more species is added to, or eliminated from, a community, are also common (e.g., Silander and Antonovics 1982;
Aarsen and Epp 1990; Goldberg and Barton 1992) but will
not be discussed here.
An AS (fig. 1B) design consists of a number of mixtures
in which the density of one species, the target species, is
the same in all mixtures and that of the other species, the
associate species, varies. The AS design frequently includes
the target species in monoculture, often at several densities.
Apart from their use in agriculture for determining the
reduction in crop yield (i.e., target species) by weed species
(i.e., associate species), the AS method is frequently used
to establish competitive hierarchies among plant species
in natural communites (e.g., Harper 1977; Miller and Werner 1987; Goldberg and Landa 1991; Keddy et al. 1994).
The RS design, first introduced by de Wit (1960), consists
of a number of mixtures and monocultures in each of which
the total density of individuals is constant, but species’ relative frequencies vary (fig. IC). A large number of indices
have been developed around the RS method and employed
in plant competition studies (e.g., de Wit and van den Bergh
1965; McGilchrist and Trenbath 1971; Mead 1979; Willey
and Rao 1980; Snaydon and Satorre 1989; Sackville-Hamilton 1994), purporting to measure various facets of species
interactions, such as species’ aggressivity, enhancement and
suppression, competitiveness, and resource use. Replacement series experiments frequently seem to be set up to
address the “which species gains” and the “how does one
species affect the other’s performance” questions, although
the two questions are rarely distinguished. The use of RS
designs became less popular following sustained criticism
in the 1980s (Inouye and Schaffer 1981; Connolly 1986;
Law and Watkinson 1987) but seem to be achieving a resurgence in recent years (Keddy and Shipley 1989; Cousens
and O’Neill 1993; Sackville-Hamilton 1994; Shipley and
Keddy 1994; Gibson et al. 1999).
Partly in response to dissatisfaction with RS and AS
designs, RE designs (fig. 1D) have been advocated (e.g.,
deWit 1960; Connolly and Nolan 1976; Suehiro and Ogawa
1980; Wright 1981; Spitters 1983; Joliffe et al. 1984; Firbank
and Watkinson 1985; Connolly 1987a; Hakansson 1988;
Connolly et al. 1990; Menchaca and Connolly 1990; Connolly and Wayne 1996). In typical two-species plant mixture experiments, the relationship between a species’ response and the explanatory variables (usually the densities
of the two species and sometimes an abiotic factor) is
estimated for each species. A primary index derived from
response surface approaches is the substitution rate
(Wright 1981; Spitters 1983; Connolly 1987a; Connolly et
al. 1990; Menchaca and Connolly 1990), also called “competition coefficients” (Firbank and Watkinson 1985),
which measures the effect on a species of a unit change
in the density of the associate species relative to a unit
change in its own density.
For all methods, responses and biotic explanatory variables (e.g., the yields of associate species), other than density, generally appear to have been measured solely at time
of harvest. As will be seen below, this imposes major limitations on the inferences that can be drawn from any
experimental structure.
Thought Experiments
Thought Experiments as a Tool
The use of thought experiments has a rich history in many
scientific traditions (Brown 1991) but has been surprisingly
underutilized in ecology. Traditionally, they have been employed to clarify concepts and to test the claims of various
proposals without the need to carry out experiments. At
Competition: Theory and Questions 111
Figure 1: Four experimental designs commonly used in competition experiments. The densities of the two species are denoted by d1 and d2. The
designs are (A) pairwise (PW), (B) additive series (AS), (C) replacement series (RS), and (D) response model (RE). For the response model any
selection of mixtures/monocultures that will allow the estimation of the response functions is a valid design.
the heart of thought experiments are the creation of simple,
unambiguous, compelling examples that serve as standards
against which proposed theories must hold true. Thought
experiments can be very useful in demonstrating that certain
approaches will not work because of internal inconsistency
or logical flaws, and they can also obviate the need for
additional research. We believe that thought experiments
can contribute significantly to resolving some of the confusion surrounding the area of interspecific plant competition. The thought experiments below have been designed
to elucidate the inferential range and limitations of a number of currently used experimental structures in answering
fundamental questions regarding interspecific competitive
phenomena among plant species.
The Scope and Limitations of Our
Thought Experiments
For practical purposes, we have limited the scope of our
three thought experiments in a number of ways. First, we
emphasize shorter-term studies on plants. With modification, however, many of the issues and ideas presented
below are equally relevant to longer timescales, that is,
competitive processes that span multiple seasons or generations. Second, our thought experiments address phenomena at the stand level (i.e., considering the average
individual performance of a species). Finally, our thought
experiments focus on the phenomenological level of competition. While mechanistic aspects of interspecific plant
interactions, such as relative foraging efficiencies (Hutchings 1988; Bazzaz 1991), the physiology underlying resource uptake and usage (Caldwell et al. 1987; Schwinning
1996; Wayne and Bazzaz 1997), and nature of indirect
plant-plant interaction mediated through microbes (Sanders et al. 1995), are receiving increasing attention, we feel
that sorting out confusion on the phenomenological level
first will create a clearer arena within which to explore
how these and other mechanisms relate to net species interactions.
112 The American Naturalist
Thought Experiment 1: Without Sequential Measures
over Time, the “Which Species Gains” Question
Cannot Be Answered
In introducing the “which species gains” question above,
we suggested that our interpretation of species’ relative
success at any given point in time in a mixture should
take into account information about the relative sizes of
individuals of the two species at the beginning of their
growth phase. In this thought experiment, we develop this
intuitive, yet generally overlooked, idea more formally.
Competitive outcomes for two hypothetical species, Sp1
and Sp2, are illustrated in figure 2 for three cases in which
the density of each species is the same, but the initial
biomass per individual of Sp2 is twice that of Sp1. In figure
2A, the final biomass ratio (biomass of a species relative
to the other species) is the same (2 : 1) as the initial biomass ratio between the two species. This indicates that,
over this particular growth interval, the species are competitively balanced in some sense; neither has gained or
lost. In figure 2B, the final biomass ratio for Sp2 exceeds
its input biomass ratio, and in figure 2C, the output ratio
for Sp2 is less than its input biomass ratio.
If final biomass data were the only data available, as is
the case in the majority of competition experiments, it is
likely that Sp2 would be judged as more competitive than
Sp1 for all three cases—it is the dominant component in
all mixtures. However, when initial size data are included,
it becomes clear that Sp2 outperforms Sp1 in only one
case (fig. 2B); in both other cases its relative abundance
in mixture either remains the same or declines (fig. 2A,
2C). From the perspective of understanding competitive
interactions, ecologists should be interested in explaining
changes in species’ proportions over time, instead of, or
perhaps as well as, ratios at one point in time. Stated more
formally, the null hypothesis for the “which species gains”
question should be that the final biomass ratio for species
in a given mixture equals the initial biomass ratio, and
not simply that the final biomass ratio equals unity. Thus,
Figure 2: We intuitively discount for initial size differences in answering the “which species gains” question. This figure compares the interpretation
of three possible outcomes of a pairwise experiment when the different initial sizes of individuals of the two species are known with the interpretation
when they are not known. The three outcomes are (A) harvest output size ratio equals initial input size ratio, (B) harvest output ratio shows that
the species with initially larger individuals has gained proportionately more, and (C) harvest output ratio shows that the species with initially smaller
individuals has gained proportionately more.
Competition: Theory and Questions 113
to evaluate competition in a pairwise experiment, it is
important, at a minimum, to know the composition of
the mixture at both the start and end of a given growth
interval. Because the nature of species’ interactions can
vary considerably throughout a given mixture’s development, numerous sequential measurements of species’ relative proportion might be desirable. A feature of this example is that the naive interpretation of final yield leads
to the conclusion that the initially larger species is more
competitive, that is, a size bias in favor of the species with
larger individuals.
Thought Experiment 2: Estimates of Species’ Effects on
One Another Will Exhibit Size Biases When
Competition Indices Rely Solely on Plant
Response at a Single Harvest
In thought experiment 1, we emphasized the role of initial
size and time in assessing the “which species gains” question. Below, we focus on how ignoring initial size and
relying solely on static measures of final yield can lead to
consistent size biases in assessing how one species affects
the performance of another. We do this by analyzing the
results of a simple thought experiment employing commonly used indices associated with three experimental designs: replacement series (RS), additive series (AS), and
response surfaces (RE).
This thought experiment begins by assuming that in a
monoculture of equal-sized individuals of a hypothetical
species (Sp1), we can arbitrarily merge or link two individuals to form a larger individual and label it as a distinct
individual of a second hypothetical species (Sp2). Merged
(Sp2) and unmerged (Sp1) individuals can then be imagined to grow together and can be used to simulate a mixture of two hypothetical species (fig. 3). With more formal
notation, what was once considered a monoculture (at
density d) could now be considered a two-species mixture
of pseudospecies Sp1 and Sp2 (at densities d1 and d2, respectively, where d p d 1 1 2d 2). This partitioning of populations into pseudospecies of different sizes can be envisaged for a range of original monoculture densities, and
different partitions can be selected to simulate different
relative frequencies of pseudospecies across any range of
densities.
For the purpose of this thought experiment we note that
the arbitrary merging and relabeling of individuals (into
pseudospecies) within monocultures in no way changes the
biology of the plants. Thus, the larger pseudospecies have
identical growth, allocation, and physiological characteristics per unit size as do the original, smaller individuals from
which they were aggregated. An appropriate analysis should
lead to the conclusion that these pseudospecies only differ
Figure 3: Illustration of idea behind thought experiments 2 and 3. Total
number of individuals is d; d2 pairs are arbitrarily linked to form individuals of pseudospecies Sp2, and the remaining d1 individuals are nominated as being of pseudospecies Sp1. In this case d p d1 1 2d2.
in size, and that neither is competitively enhanced or suppressed by the other.
To generate predicted final sizes for various mixtures of
pseudospecies so that parameters associated with RS, AS,
and RE designs can be assessed, we assume an inverse
linear model for the relationship between size and density
(e.g., Shinozaki and Kira 1956; Holliday 1960). The relationship between mean yield per individual at harvest
(w) for the true species and its initial density (d) is described by
wp
1
,
a 1 bd
(1)
where the coefficient b (assumed positive) partly determines
the rate at which yield per individual declines with increasing density (the rate also depends on a). Suppose, without
affecting the generality of the argument, that all individuals
at a particular density are the same size, defined by equation
(1). Now suppose, as defined by the assumptions of our
thought experiment, that in a particular plot at density d,
d2 pairs of individuals are selected and each such pair of
individuals is considered as an individual of a pseudospecies
called Sp2 (fig. 3). Let the remaining d1 individuals be labeled as pseudospecies Sp1. It then follows that
d p d 1 1 2d 2 .
(2)
While the density in terms of total numbers of individuals
of the pseudospecies is now not d but d 1 1 d 2 (with d1
114 The American Naturalist
Table 1: Values of a range of indices for RS designs at a range of total densities
RYT
components
Total density
a p .1; b p .02:
2
10
20
40
100
Index value indicating equal competitiveness
a p 0; b p .02:
2
10
20
40
100
Index value indicating equal competitiveness
Relative
crowding
coefficient
RY1
RY2
Sp1
Sp2
Coefficient
of
aggressivity
.44
.38
.36
.35
.34
.5
.56
.63
.64
.65
.66
.5
.78
.60
.56
.53
.51
1
1.29
1.67
1.80
1.89
1.95
1
2.125
2.250
2.286
2.308
2.323
0
.778
.600
.556
.529
.512
1
.33
.33
.33
.33
.33
.5
.67
.67
.67
.67
.67
.5
.50
.50
.50
.50
.50
1
2.00
2.00
2.00
2.00
2.00
1
2.333
2.333
2.333
2.333
2.333
0
.500
.500
.500
.500
.500
1
Competitive
ratio
Note: Indices are calculated assuming, first, that a p 0.1 and b p 0.02 and, second, that a p 0 and b p 0.02 in equation
(3). Included are relative yield components (RYT), the relative crowding coefficients, the coefficient of aggressivity, and the
competitive ratio. The index values that indicate “equal competitiveness” are also shown. Index values are calculated from the
performance of individuals of the two species in the 50 : 50 density mixtures (predicted using eq. [3]) compared with predicted
performance at the appropriate monoculture densities. The indices are computed as follows. The overall density is d, and the
binary mixtures are at densities d/2 for each species. The yields per individual in mixture are w1 and w2 for Sp1 and Sp2,
respectively, and yields per individual in monoculture are w10 and w20 for Sp1 and Sp2, respectively. Then, formulae for index
values are RYT p RY1 1 RY2, where RY1 p dw1/2dw10, RY2 p dw2/2dw20 . The RYT is 1 for all cases considered here, but this
is not so in general. RCCs for Sp1 and Sp2, respectively, are 1/(2w10 /w1 2 1) and 1/(2w20/w2 2 1). Coefficient of aggressivity is
(w1/w10) 2 (w2/w20) and competitive ratio is (w1/w10)/(w2/w20).
standard sized and d2 “double-sized” individuals in the
stand), the yields per individual of Sp1 and Sp2 (w1 p
w and w2 p 2w, respectively) can be written, by substitution of equation (2) in equation (1), as
w1 p w p
1
1
p
,
a 1 bd
a 1 bd 1 1 2bd 2
w2 p 2w p
2
1
p a
.
b
a 1 bd
1 2 d 1 1 bd 2
2
(3)
These simulate response equations for a two-species
mixture where individuals of the species differ only in size:
individuals of Sp2 are twice the size of Sp1, capture twice
as much resource as Sp1, and give twice the yield per
individual, and take up twice the space but have the same
asymptotic yield per unit area in monoculture (1/b). These
equations, and in particular the characteristic that the coefficient of d 2 is twice as large as that of d 1, bring out the
issue that the responses of both species, while still being
density dependent, also reflect the size difference between
individuals of the two species. This is what would naturally
be expected given the construction of the two pseudospecies, since a unit increase in the density of Sp2 is equiv-
alent to an increase of 2 in the density of Sp1 and so
should have a greater effect. This suggests that, in real
stands, if species differ in size at the start of an experiment,
the assessment of a species impact on associates should
allow for both the initial densities and sizes of species.
What conclusions would be reached if an experiment
was carried out on this pseudomixture system using three
experimental designs, an RS, an AS, and a response surface
design? A numerical example using the coefficients a p
0.1 and b p 0.02 in the inverse linear model (eq. [3]) is
used to illustrate the argument. For RS a second case with
a p 0 is used as this simulates constant yield per unit area
for each species, where it has been claimed (Taylor and
Aarssen 1989; Cousens and O’Neill 1993) that RS works
well.
Replacement Series. Five RS increasing in total density from
two to 100 are considered. Each RS comprises the two
monocultures and the mixture consisting of half the
monoculture density of each species. The values of a range
of commonly used indices associated with RS designs
(Mead 1979; Connolly 1986) are calculated (table 1). The
relative yield components (RY1 and RY2; de Wit and Van
den Bergh 1965) are, for a species, the ratio of its total
Competition: Theory and Questions 115
yield in mixture to its total yield in monoculture. The
relative crowding coefficient for each species is the ratio
of the yield per individual of a species in mixture to its
yield per individual in monoculture (de Wit 1960). The
coefficient of aggressivity (McGilchrist and Trenbath 1971)
between two species is half the difference between these
two ratios, and the competitive ratio (Willey and Rao 1980)
is their ratio. In the RS methodology, when competition
is equal between species, these indices take values of 0.5,
1, 0, and 1 for the four indices, respectively, and these
values would naturally form the basis of a null hypothesis
for whatever the index purports to measure.
In the classical interpretation of the results in table 1,
none of the indices returns a value indicating equal competitiveness. Rather, they all take values suggesting that
Sp2 (i.e., merged individuals) outcompetes Sp1. Thus, all
the indices are size biased, and so it is impossible to frame
the appropriate null hypothesis unambiguously since it
would confound bias with the feature of interspecific interaction being investigated. Size bias is even evident in
case B where the RS results are independent of density
and its magnitude varies with density when a ( 0.
Additive Series. Two additive series are considered with a
density of 5 of Sp1 (the target or crop pseudospecies) and
increasing densities of associate (or weed) pseudospecies
Sp2 or Sp3 (Sp3 being formed in the same way as Sp2
except merging three individuals to form an individual of
Sp3). Equations (3), and their analog for mixtures of Sp1
with Sp3, are used to predict the crop yield per individual
at a range of weed densities (fig. 4).
Figure 4 shows that the yield of Sp1 is decreased more
by an individual of Sp3 than of Sp2 and thus Sp3 would
be considered as more competitive than Sp2—but the diagram shows no simple 3 : 2 ratio as might be imagined.
The “aggressiveness” (sensu Harper 1977; the reduction
of the target species by a given density of the associate)
displayed by Sp3 is greater than that of Sp2, but as can
be seen from the artificial nature of the construction of
the example, this aggressiveness should not be overinterpreted; it simply reflects an initial size difference. In the
AS, Sp3 would reduce the yield of Sp1 by more (at an
equivalent density) than would Sp2 because its individuals
were initially larger and grew at the same rate per unit
initial biomass as individuals of Sp2 and hence captured
more resources. However, to use this result as the basis of
a competitive hierarchy in which Sp3 would be judged as
more competitive than Sp2, with any connotation of Sp3
gaining in the mixture, would be quite spurious. This is
clearly seen if one considers a mixture consisting of one
individual of each of the three pseudospecies. In biomass
terms, both the input and output ratios between species
would be 1 : 2 : 3, and so no species has gained or lost
Figure 4: Results from two AS showing the yield per individual of Sp1
at a density of 5 as affected by increasing density of Sp2 (continuous line)
or Sp3 (hatched line). Sp2 and Sp3 are pseudospecies whose individuals
are aggregates of two and three individuals of Sp1, respectively. Response
models for species 1 are derived from equation (3) as w1 p 1/(a 1
2bd2) and w1 p 1/(a 1 3bd3) for Sp1 in mixture with Sp2 or Sp3, respectively. The response of Sp1 individuals in monoculture at density
five is 1/a. Values are calculated assuming a p 0.1 and b p 0.02.
over the course of the experiment. One must be careful
in concluding from an experiment that a species is “competitively better” than another if, perhaps, all that is being
said is that its individuals were initially larger. Without
information on initial sizes, it is impossible to test null
hypotheses as to which species gains. Not knowing initial
size differences is also likely to result in confounding estimates of suppressive/enhancing effects.
Response Surface. Equation (3) gives the response functions
for the relationship between final yield per individual and
initial density for this mixture system. When the competition coefficients (sensu Firbank and Watkinson 1985)
or substitution rates (Wright 1981; Spitters 1983; Connolly
1987a; being the ratio of inter- to intraspecific coefficients
of density) are calculated for this example, they give values
of 2 and 0.5 for Sp1 and Sp2, respectively. This would
traditionally be interpreted as Sp2 being more competitive
than species 1, but again, these values merely reflect initial
size differences. Including information about species relative sizes in estimates of substitution rates is critical for
testing hypotheses about species’ effects on one another
without bias.
In all three methods, there is a danger of size-biased
interpretation when only final yield is measured, favoring
116 The American Naturalist
species with initially larger individuals. While size and sizerelated traits have been shown to modify competitive ability (Thomas and Weiner 1989; Connolly and Wayne 1996),
this example shows that the contribution of initial size
differences to the indices and methods must be discounted
in assessing competitive phenomena. How is this discounting to be done? The third thought experiment suggests avenues for resolving some of these difficulties.
Thought Experiment 3: The Reliance on Species’ Densities
as Explanatory Variables Is at the Heart of the
Size-Biased Estimates of Interspecific
Plant Competition
Thought experiment 2 demonstrated that for two hypothetical pseudospecies in a mixture, identical in all aspects
of biology (e.g., growth rate, physiology, shape) except for
initial size, the initially larger species will be consistently
misjudged as competitively superior by most competition
analyses. This final thought experiment suggests that the
traditional use of initial density in AS, RS, and RE, that
is, the number of individuals of species and not their initial
biomass or size, is at the heart of the difficulties.
We begin this experiment by creating pseudospecies as
we did in thought experiment 2, that is, arbitrarily selecting
d 2 pairs of individuals in a monoculture at density d and
calling them pseudospecies Sp2, leaving d 1 unpaired individuals and labeling them as members of a pseudospecies
Sp1, with d p d 1 1 2d 2. In this experiment, we convert
the models in equation (3) to models that use initial stand
biomass of each species, rather than initial density, as explanatory variables. To do so we let the initial biomass of
individuals (unpaired) of Sp1 be w0, making the initial size
of pseudospecies Sp2 individuals 2w0. A density of d1 individuals of size w0 leads to initial stand biomass for Sp1
of y1 p d 1w0 (from which d 1 p y1/w0). Initial stand biomass for Sp2 is y2 p 2d 2w0, which gives d 2 p y2 /2w0. In
a mixture of d1 and d2 plants of Sp1 and Sp2, respectively,
substituting for densities in equation (3) gives the following equations relating yield at harvest to the initial stand
biomass of the two species:
w1 p
1
1
p
,
a 1 b(y1/w0 ) 1 2b(y2 /2w0 )
a 1 fy1 1 fy2
w2 p
1
(a/2) 1 (b/2)(y1/w0 ) 1 b(y2 /2w0 )
p
(4)
1
,
(a/2) 1 (f/2)y1 1 (f/2)y2
where we use the symbol f to represent b/w0. Changing
variables to the initial biomass scale thus gives response
models in which the coefficient of initial biomass (y1 or
y2) of either Sp1 or Sp2 is f for Sp1 responses and f /2 for
Sp2 responses. As in thought experiment 2, an appropriate
analysis should lead to the conclusion that these pseudospecies only differ in size and that neither is competitively
enhanced or suppressed by the other. Appropriate null
hypotheses are that neither species gains for any mixture
(i.e., that the models of output per unit input, wi /wi0,
readily derived from eq. [4] are identical for both species)
and that neither species suppresses or enhances the other
(i.e., the substitution rates are both 1, or the RS indices
in table 1 take the values 0.5, 1, 0, and 1, respectively). As
before, this is examined for RS, AS, and RE methods.
Replacement Series. An initial biomass replacement series
(IBRS) is a set of mixtures and monocultures all of which
have the same initial total biomass (rather than the same
initial total density as in the more traditional RS). Since
the explanatory variables in model (4) are expressed in
biomass terms, it is easy to use it to simulate the results
from a replacement series design where replacement between the two pseudospecies is on an IBRS basis. Suppose
total initial biomass of y p 0.8 for an IBRS consisting of
three stands, a monoculture of initial stand biomass of 0.8
for each pseudospecies, and a mixture with initial stand
biomass of 0.4 for each. Suppose values of 0.01, 0.1, and
0.02 for w0, a, and b, respectively. This gives a value of
0.02/0.01 p 2 for f. For this (and for any such IBRS), the
replacement series indices suggest (table 2) that neither
pseudospecies is enhanced/suppressed and thus reflects
what is defined to be the truth for this experiment. This
result is very different from the size-biased results obtained
when using a RS based on density (thought experiment
2). When the indices (tables 1, 2) are unbiased, the index
values indicating equal competitiveness (table 1) form the
basis of null hypotheses for the particular aspect of competitiveness captured by the respective index. The traditional RS indices do not address the “which species gains”
question, but clearly in this example, for any mixture along
any IBRS, neither species gains.
Additive Series. An initial biomass additive series (IBAS)
for two species consists of a monoculture of one species
at a given level of initial stand biomass and a set of
mixtures in which the initial stand biomass of that species
is held constant at its monoculture level but the initial
stand biomass of the other species changes over mixtures.
The traditional AS can be readily converted into an IBAS
if the initial sizes of individuals of the two species are
known. Consider the three pseudospecies Sp1, Sp2, and
Sp3, with individuals of Sp2 and Sp3 formed by arbitrarily
grouping two or three individuals of Sp1, respectively, as
described in thought experiment 2. Suppose an additive
Competition: Theory and Questions 117
Table 2: Results of simulated competition experiment based on equations (4) along
an initial biomass replacement series (IBRS)
Monoculture
Sp1
Yield per individual
Yield mixed/mono
.588
Mixture
Sp1
Sp2
.588
1
1.176
1
Monoculture
Sp2
1.176
Values of competition coefficients
Relative yield components
Relative crowding coefficients
Coefficient of aggressivity
Competitive ratio
Relative yield total
.5
1
0
1
1
Note: An initial stand biomass of 0.8 for each species in monoculture and 0.4 for each in mixture
was used. Parameter values of 0.01, 0.1, and 2 for w0, a, and f were used for calculations.
series, with Sp1 always at a density of 5 and Sp2 at densities
0, 2, 4, 6, and 8 and a second additive series for Sp1 and
Sp3 with the same design. We have seen that traditional
AS analyses of these examples gives size-biased results. By
using equation (4), we can simulate the results of such an
experiment conducted on an initial biomass basis. We use
the same parameter values (w0, a, and b equal to 0.01, 0.1,
and 0.02, respectively, and hence f p 2) as for the IBRS
example above. This leads to response models with parameter values a p 0.1 and f p 2 for Sp1 and parameters
values for Sp2 and Sp3, which are one-half and one-third
the size of these, respectively. Harvest biomass per individual is used as the response. Response of Sp1 per individual is plotted against the initial biomass of the associate pseudospecies (fig. 5A) and against the density of
the associate pseudospecies (fig. 5B).
On a density basis (fig. 5B), Sp3 appears more aggressive
than Sp2, but on a per unit initial biomass basis (fig. 5A),
the effect on Sp1 of either pseudospecies is the same (as
would be the effect of Sp1 on itself). This identical effect
of both species per unit initial biomass is what the thought
experiment indicates should occur. This example suggests
that the appropriate null hypothesis to be tested is that
the per unit initial biomass effect on the target species is
the same for both associate species.
Response Surface. Since pseudospecies 1 and 2 are formed
by arbitrarily labeling some pairs of identical individuals,
it follows that the response of a pseudospecies must be
identical for a unit change in initial stand biomass of itself
or its associate, which is just a unit change in total initial
stand biomass. This is what is indicated in model (4),
where the coefficients of initial biomass are the same
within either equation. In any mixture, a unit change in
the initial biomass of either pseudospecies has an identical
effect on its target. In model (3), based on density, the
substitution rates are size biased, but in model (4) they
are unity (f/f and [1/2]f/[1/2]f ), indicating that any size
bias has been eliminated. The appropriate null hypothesis
in this model is that the substitution rates equal 1.
The results here have been derived for output per individual as the response. They also apply where output
per unit input or RGR is used; correcting for differences
in initial biomass would remove size bias. This thought
experiment can be simply extended to show that, even
where output per unit input is identical for both species
in each mixture (as it would be in this thought experiment), estimates of competitive performance are size biased when produced by RS, AS, and RE for equations based
on density but not when based on initial biomass of
species.
A variant of this thought experiment in which initial
(Ni0) and final (Ni1) densities are the measured variables
for the ith species, with the same arbitrary grouping of
individuals in pairs, will lead to a model of the form
N11
N
1
p 21 p
.
N10
N20
g 1 hN10 1 2hN20
(5)
Thus, with respect to which species gains, neither does.
However, with respect to the effects of species on each
other, the relative sizes of individuals appears in the substitution rate between coefficients. This creates difficulties
in establishing a null hypothesis in a real experiment for
the substitution rates or for enhancing/suppressive effects,
unless initial size differences are discounted in some way,
since the observed substitution rates will confound an effect as a result of different initial sizes with suppressive or
enhancing effects. If initial sizes differ, it is also clear that
RS and AS will have similar interpretative difficulties to
those outlined in thought experiment 2 above, even for
the simple model in equation (5). Equation (5) is a simple
118 The American Naturalist
Figure 5: Results of simulated AS competition experiment based on equations (4). Three pseudospecies Sp1, Sp2, and Sp3 are compared where
individuals of Sp2 and Sp3 are composed of two and three individuals of Sp1, respectively. Two additive series are constructed, Sp1 as target with
Sp2 and Sp1 as target with Sp3. For parameter values for equation (4) given in the text, the diagrams show (A) yield per individual of Sp1 versus
initial stand biomass of Sp2 or Sp3 and (B) yield per individual of Sp1 versus density of Sp2 or Sp3.
case of the discrete analog to the Lotka-Volterra equations
(Leslie 1958), and the substitution rates are the competition coefficients for that model. Thus, these difficulties
also attend the interpretation of the competition coefficients of the Lotka-Volterra equations.
In summary, the major difficulties we have identified
relate to two sources of size bias. The first arises in addressing the “which species gains” question on the basis
of final harvest yield and not adjusting for initial size differences between species (thought experiment 1). The second arises in answering the “effects of neighbors” question,
using approaches based on initial density rather than some
measure of initial species contribution to the stand such
as initial biomass (thought experiments 2 and 3). Insofar
as any experimental structure fails to address these two
issues, the estimates of competitive ability and hence tests
of null hypotheses, whether for the “which species gains”
question or the “effects of neighbors” question, will be
potentially size biased in favor of the initially larger species.
Discussion
To synthesize the findings of our analyses and our thought
experiments, we summarize (table 3) the utility of various
experimental structures with respect to their ability to address specific questions. We then discuss a few key issues
Competition: Theory and Questions 119
raised by our analyses and conclude by introducing the
essential elements of an alternative approach for quantifying the effects of interspecific interactions.
Implications of the Analysis for Experiments
Table 3 provides a framework that researchers investigating
plant competition can use for assessing the level of confidence they have in the capacity of a given experimental
structure to address a particular question. The table evaluates eight experimental structures, based on PW, AS, RS,
and RE designs with final harvest data only (H) or also
including appropriate initial size (I) measures. We organize our discussion of this table following the order of the
three questions we have addressed.
Which Species Dominates a Mixture? Of the three questions
we address in this article, Which species dominates a mixture at any one point in time? is the simplest conceptually
and most readily measurable without bias. Since “by dominant” is meant the most abundant species at a particular
time, all that is necessary is a static measure of species’
abundance (e.g., biomass, seed set, ground cover) at that
time. Time trends, size bias, and correction for initial differences do not enter the question. So long as the mixtures
of interest are well replicated and response variables are
accurate estimates of “performance,” addressing this question can be straightforward, whether one is interested in
how species’ dominance in one specific mixture varies with
an environmental factor (e.g., a two-species 50 : 50 mixture repeated across a nutrient gradient) or how a given
species’ dominance changes across a range of mixtures
varying in the initial densities of the two species.
Pairwise experiments give valid but limited information
on dominance, the limitation being that the information
is available for only one mixture (table 3). The AS gives
valid information on dominance but is limited to a single
density of the target species unless it is repeated at several
densities, in which case RE methods can be used. The RS
gives very little information because, in the typical RS, the
monocultures will be of no use with respect to the dominance question and so are a waste of resources if that is
the primary concern. The RE methods will best describe
dominance patterns across a range of densities.
While the results of such one-harvest designs can be
informative regarding how abiotic and biotic factors impact stand-level characteristics such as species composition
and diversity, we hope we have made it clear that, regardless of how sophisticated or complex the sowing design, single static measures of species’ relative abundance
can say very little about which species are gaining in a
mixture or how one species influences the performance
of another. The interpretation of dominance patterns cannot have any connotation of eventual competitive exclusion or current superior ability to preempt resources if
stand history is unknown.
Which Species Gains in a Mixture? The “which species
gains” question in its simplest form (but see “Measurement Metrics”) attempts to characterize how a species’
relative proportion varies over time. Addressing this question thus requires sequential size measures over time: at
least one at the beginning and one at the end of the experiment. For many experiments, it may be desirable to
have initial and final sizes over multiple growth intervals,
as the dynamics of “which species gains” is likely to vary
greatly during the different developmental phases of
mixtures (e.g., early establishment, vegetative growth, reproduction, and senescence). However, whether interest is
focused on one or multiple growth intervals, thought experiment 1 clearly illustrates that only by incorporating
information on species’ sizes at the beginning of a growth
interval (i.e., proportional or relative growth) can measures of performance avoid size biases.
The “which species gains” question is the one closest to
the aim of determining the long-term competitive out-
Table 3: Summary of inferences that can be made from a range of experimental structures in
respect of three common questions
AS
PW
RS
RE
Question
H
I1H
H
I1H
H
I1H
H
I1H
1. Dominate
2. Gain (win)
3. Effects of/on other species
YL
N
N
YL
YL
N
YL
N
YVL
YL
YL
YL
YVL
N
YVL
YVL
YVL
YVL
Y
N
YVL
Y
Y
Y
Note: The experimental structures are defined as the eight combinations of the four designs pairwise (PW), replacement series (RS), additive series (AS), and response equation models (RE) by two measurement cases: (H) only
final harvest data are available or (I1H) appropriate initial and final harvest data are available. The symbols Y, N,
YL, and YVL are used to denote the level of inference that is possible. Y means that valid inference is possible on the
question, N that no information is available, YL that valid but limited information is available, and YVL that valid
but very limited information is available.
120 The American Naturalist
come. However, even the best-executed short-term experiment may not be reliable for longer-term predictions.
There are frequently many other life-history phases (seed
setting, seed and seedling survival, establishment, etc.) and
processes (senescence, nutrient storage, overwintering,
etc.) that may be of equal or greater importance as that
of the direct competitive phase in determining the outcome.
For the “which species gains” question, experimental
structures without information on initial sizes give no valid
inferences unless species begin a growth interval at exactly
the same size (table 3). When initial size information is
available, PW and AS designs can give valid but limited
information, limited in the PW case because it applies to
a single mixture and, in the case of the AS, to a single
density of the target species. For the RS, the only component of relevance for answering the “which species
gains” question is the mixture(s). Thus, the simplest RS
with a single 50 : 50 mixture is only as useful as a simple
PW experiment, and two-thirds of the experimental units
(the monocultures) are of no use. Hence, with respect to
addressing the “which species gains” question, we describe
the RS as very limited. When appropriate initial size information is available, RE methods using RGR or output
per unit input as the response variable will allow modeling
of the “which species gains” question across a range of
species’ densities and relative frequencies or across some
alternative range of initial conditions (e.g., mixtures varying in species’ initial biomass contribution).
How Does One Species in a Mixture Affect the Performance
of Another? Of the three broad questions addressed in this
article, characterizing how one species’ presence influences
the performance of another is by far the most complicated.
Thought experiments 2 and 3 demonstrate that experimental structures that do not adjust for initial size differences generally lead to size-biased estimates of species’
effects, suggesting that species with larger initial size are
more competitive. This occurs even when the response
variable is calculated on a per unit size input basis, for
example, RGR (an extension to thought experiment 3).
Thought experiment 3 suggests that a move from species
densities as explanatory variables to some other measures
of initial conditions (e.g., initial species biomass, height,
leaf area) may be necessary to provide unbiased estimates
of species’ effects on each other.
Pairwise experiments generally contain no information
on the “species’ effects on each other” question (table 3).
The AS without initial size information is very limited,
and comparisons of species with respect to their effects on
a target are potentially size biased. When initial size information is used to adjust for initial interspecific differences, then valid comparisons between associate species
can be made, but these are still limited in that the target
species in the series is at only one density. The RS generally
gives potentially size-biased estimates of competitive phenomena, whether initial information is available or not,
unless initial sizes of species are the same. Response surface
designs without initial information give potentially sizebiased information on the effects of species on each other,
but when appropriate initial information is available, the
question can be validly addressed.
The “Which Species Gains” and the “Species’ Effects on
Each Other” Questions Are Distinct and Not
Necessarily Related
We have tried to convey in our analyses and thought experiments that “which species gains” and “species’ effects
on each other” in a mixture are two very distinct questions
requiring very different experimental structures to answer.
Although there has been some recent effort toward decomposing the process of plant competition into subcomponents (e.g., competitive effects vs. responses; Goldberg
1990), these questions have been generally confounded
and mismeasured. Distinguishing between them is necessary for progress in both practical and theoretical aspects
of competition studies.
While we might conceptually expect there to be a close
relationship between these two aspects of competitive interactions, this is not always the case. For a variety of
reasons, including differences in interacting species’ architecture or allometry (e.g., Connolly et al. 1990; Tremmel
and Bazzaz 1993; Schmid and Bazzaz 1994) or phenology
and timing of resource use (e.g., Elberse and de Kruyf
1979), species that substantially gain over an interval of
time in a mixture may not necessarily have large negative
effects on neighboring species. It is not inconceivable, for
example, for a relatively smaller species in a mixture to
grow more quickly than dominants but to have little effect
on a dominant species’ performance (e.g., Newbery and
Newman 1978). Conversely, a very large dominant species
undergoing a shift from vegetative growth to reproduction
may be accumulating biomass very slowly (i.e., little gain)
but, because of its architecture, may have a very significant
negative shading effect on another species’ performance
per unit initial biomass.
Demographic versus Functional Density
Thought experiment 3 revealed that central to the problem
of answering the “species’ effects on each other” question
without size bias is the general reliance on species’ densities
as explanatory variables in experimental structures (unless
species begin at exactly equal size). To resolve this issue,
we propose a distinction between what we call “demo-
Competition: Theory and Questions 121
graphic” versus “functional” density. We define “demographic density” as the total number of all individuals in
a stand. Defined as such, demographic density says nothing
about the mass, shape, spatial distribution, or any other
characteristic of its units. From this perspective, and using
an extreme example suggested by Snaydon (1991), a density of 100 mature oaks is demographically equivalent both
to a mixture of 50 oaks with 50 daisy plants and to 100
daisy plants. These three stands (two monoculture and a
50 : 50 mixture) are in fact a simple replacement series.
While the demographic density across this RS is constant,
it is obvious that most measures of functional density (e.g.,
the density of biomass, leaf area, root tips) vary a great
deal. We argue that what ecologists generally want to
achieve when addressing the “how one species affects another” question is to assess species’ effects at comparable
functional and not necessarily comparable demographic
densities. Stated another way, they wish to test the null
hypothesis that species react to each other identically on
a unit functional density basis (e.g., initial biomass or leaf
area basis) rather than on a per individual basis.
Problems arising from the use of demographic versus
functional density, although not articulated as such, have
been previously recognized. For example, in critiquing the
RS design, Keddy (1989, p. 116) recognized that the
method’s “concern with constant density is only reasonable if all species are of similar size, so that equivalence
of density translates into equivalence of biomass.” Even
the originator of the RS methodology (de Wit 1960) was
aware of this problem when he chose a 4.5 : 1 ratio of
sowing densities for the replacement series of mixtures of
barley (the smaller) and peas (the larger), essentially forgoing equivalent demographic densities to better approximate equivalent functional densities.
Appreciation of measures more closely related to functional versus demographic density are not limited to the
four designs examined here. For example, some neighborhood-type analyses used to investigate intraspecific
competition have used aggregated neighborhood biomass, as well as neighbor density, to predict target performance (Weiner 1984; Thomas and Weiner 1989). Interspecific process-oriented models (e.g., Shugart 1984;
Pacala et al. 1996) often rely on measures such as the
total leaf area, height, or total biomass of neighboring
trees to model target plant performance. Goldberg and
colleagues (Goldberg 1990; Goldberg and Landa 1991)
have suggested that what they define as competitive effects and responses can be assessed using regression techniques based on total stand biomass (at the end of a
growth interval) and not stand density. Kropf and Spitters
(1991) modeled the impact of weeds on crops using the
leaf area of weeds and not density. Recently, Connolly
and Wayne (1996) demonstrated the use of a new RE
methodology based on initial biomass as the dependent
variable and showed how this experimental structure
could be used to address both the “which species gains”
and the “species’ effect on each other” questions without
bias. Functional density provides a natural framework
for competitive studies on both nonclonal and clonal
plants. Because of difficulties in determining the identity
of ramets, the use of year-to-year changes in biomass for
evaluation of clonal species dynamics seems more appropriate and less problematic than a density-based approach. However, despite this awareness of the value of
using functional density as an explanatory variable, the
vast majority of research in interspecific competition has
relied on demographic density, thus leading to the problems in size bias we have discussed above.
Clarifying the Use of Size as an Explanatory and
Response Variable
Throughout this article, we have made many references
to plant size, both as an explanatory variable and as a
response variable. A few points of clarification are important here. First, while we have regularly used biomass
to characterize size, by no means do we think this is always
the best or the only measure of size or functional density
to use in experimental structures. In contrast, we think
that most measures of size or functional density (e.g.,
height, leaf area, root length) are of some value, each one
giving different insights into how species might be interacting for resources and responding to one another (Weiner and Fishman 1994; Connolly and Wayne 1996; Berntson and Wayne 2000). Second, while we have advocated
species’ size measured at the start of a growth interval as
a logical explanatory variable, it is also possible to use size
at intermediate phases of growth to account for subsequent
species differences. Finally, we suggest that the incorporation of some initial or intermediate measure of plant
size into measures of competitiveness not only results in
unbiased estimates of species performance, but also makes
the link to the mechanisms underlying competition much
simpler. The experimental structures we are moving toward would allow the modeling of many basic integrated
physiological measures, such as RGR and NAR. This shift
would move us a big step closer to meeting the concerns
of researchers who have advocated the need for a more
mechanistic basis for competition theory (Weiner 1990;
Wayne and Bazzaz 1997; Schwinning and Weiner 1998;
Berntson and Wayne 2000).
Measurement Metrics
Characterizing which species gains is relatively straightforward when the experimental response variable (output)
122 The American Naturalist
is measured using the same metric as the input variables
(e.g., biomass or height). But when the metrics of output
and input differ, a broader definition of which species gains
may be required. For example, if leaf area or plant height
were measured at the start of a growth period and biomass
or weight of seeds at the end, then one could frame measures of gain as biomass per unit leaf area, seed weight
per unit height, and so forth. These would capture different facets of the species performance. The appropriate
null hypothesis would then be that the output of the response variable per unit of the initial input variable was
the same for both species. If either the initial or response
variable is measured as counts rather than on a continuous
quantitative scale (e.g., mass, length, area) special care
needs to be taken that the things being counted are comparable for both species. This has already been amply illustrated for density as the starting variable but difficulties
also exist when a count is taken as the response. For example, if species have very different seed sizes a null hypothesis that the number of seeds per unit initial leaf area
is the same for both species may be intrinsically uninteresting. On the other hand, if both metrics are counts of
the same thing (e.g., seeds or numbers of individuals of
each species) a null hypothesis that output of seeds per
seed input (a standard measure in population dynamics)
are the same for the two species may be very relevant.
When noncount variables are used, considerations of scale
may enter into the determination of appropriate measures
of gain. For example, yield of seeds per unit leaf area may
be considered more biologically appropriate than per unit
leaf length for broad-leaved herbs versus narrow-leaved
grass species.
For characterizing species’ effects on each other when
input and output metrics are different (and where the
input variable is not a count), one testable null hypothesis
is that the response of a species to a unit of the initial
variable is the same, irrespective of whether the unit is of
the responding species or its associate. If models with a
linear component (e.g., inverse linear) are fitted to experimental responses of a species to the initial value of
the input variable for both competing species, this null
hypothesis reduces to the substitution rate being 1. Interpretative difficulties arise in models using counts as the
input variable in respect of this hypothesis as has been
shown above, irrespective of whether the output metric is
a count or otherwise. Also, unlike the which species gains
case, if both input and output variables in the response
model are counts of the same type, there are still problems
in framing a suitable null hypothesis since the substitution
rates may confound differences in initial relative sizes and
competitive ability. This is a difficulty with the interpretation of the competition coefficients of the Lotka-Volterra
model.
Experiments may be concerned with multiseason
changes. Where the measurement of change over seasons
is based on change in some quantitative continuous measure such as biomass, leaf area, and so forth, then yearto-year changes per species will form the basis for answering which species gains. In some designs, there is
also the possibility of modeling these changes over a
range of initial starting levels of both species (from any
one year to the next) to allow assessment of species’
effects on each other, essentially the same broad approach
as would be taken to multiple measurement periods
within a year. For counts of species numbers, the difficulties of addressing the “effects of species on each other”
discussed above apply equally across seasons. Competition studies are often concerned with the growth of populations over several generations where immigration, emigration, birth, and mortality all may interact in complex
fashion. Many of the issues that may arise in such multigenerational studies go far beyond the concerns of this
article. However, the distinction between gaining and the
extent of the effect on other species is still valid; the
difficulties with the use of counts as initial input variables
and the concern to measure development through time,
perhaps over several phases, will be particularly relevant.
Size Bias in Competition Studies with Nonplant Species
While this article has concentrated solely on competition
between plant species, some of the points raised have relevance for other organisms. For example, in animal (e.g.,
livestock) competition studies where densities are used in
the experimental design and biomass is used as a response
variable, size bias is likely to occur in assessing the effects
of species on each other unless the starting sizes of the
competing species are identical (e.g., ConnolIy and Nolan
1976). Size biases can also emerge when metrics other than
biomass are employed. For example, many approaches to
quantify competition between animals or microorganisms
are based on the Lotka-Voltera models as a starting point.
In these models, density is generally used as an initial
variable, and output is often a discrete variable other than
biomass such as (surviving) density or numbers of progeny. These experimental structures are analogous to the
plant example discussed above at equation (5), where designs begin with density and output is also density. The
biases can influence traditional interpretation of LotkaVolterra. Where the initial variable is density and the response is a variable such as amount of resources (food,
territorial area, etc.) preempted by each species, the
thought experiments with plants apply directly. Where the
initial sizes of individuals are not comparable, important
size bias may occur. Thus, the general problems with using
density as an initial explanatory variable apply to questions
Competition: Theory and Questions 123
such as the examination of whether intraspecific competition is less than interspecific (Connell 1983) or the establishment of hierarchies based on asymmetric competition (sensu Connell 1983; see Connolly 1997). As
discussed earlier, biases can also occur in assessing which
species gains in a mixture if output units are not comparable (e.g., one fly vs. one elephant). As with plants, the
issue of size bias is related to the selection of the correct
metrics and a major concern in designing experiments is
selection of appropriate testable null hypotheses.
Toward an Alternative Approach to
Competition Experiments
The results of our thought experiments and analyses point
toward an alternative approach to the design and analysis
of interspecific competition experiments with plants. Some
elements of this approach have already been presented by
Connolly and Wayne (1996) and Gibson et al. (1999), and
a more comprehensive treatment of the methods and their
application is presented by J. Connolly and P. M. Wayne
(unpublished manuscript). Our alternative approach incorporates three refinements to the study of interspecific
competition: the use of functional instead of demographic
density as the independent variable in experimental designs; the incorporation of successive (at least two) measurements over time; and recognition of the distinction
between the “which species dominates,” “which species
gains,” and “species’ effects on each other” questions.
Acknowledgments
We wish to thank the very many people with whom we
have discussed these ideas. In particular, we thank S.
Gaines, D. Gibson, J. Grace, R. Keane, A. Luescher, E.
Macklin, L. Tzu, J. Weiner, B. Zaitchik, and two anonymous referees for their very constructive comments. This
work has been supported by Forbairt International Collaboration Programme grants in 1997 and 1998 and by
the European Union Concerted Action Unification of Indicator Quality for Assessment of Impact of Multidisciplinary Systems (UNIQUAIMS) grant ERBIC18CT970156.
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Associate Editor: Steven D. Gaines