Download Using constraint lines to characterize plant

OIKOS 83: 237-245. Copenhagen 1998
Using constraint lines to characterize plant performance
Qinfeng Guo, James H. Brown and Brian J. Enquist
Guo, Q., Brown, J. H. and Enquist, B. J. 1998. Using contraint lines to characterize
plant performance. - Oikos 83: 237-245.
We used the self-thinning law to model constraints on the performance of desert
annual plants in competitive situations which were also influenced by many other
biotic and abiotic factors. We then applied the model to field data for two desert
annual species measured over 384 permanent quadrats. Based on the self-thinning
law, competition for limiting resources from conspecifics and heterospecifics places an
upper boundary on the survivorship of desert annuals. Simple trade-off models are
also developed to define constraint lines for another measure of performance, density
under interspecific competition. Similar constraint lines can be potentially fitted to
field data where still other factors also influence plant performance. Compared with
most conventional methods, such as linear regression and multivariate techniques,
models of constraint lines are better able to characterize the effects of the major
limiting processes in complex systems in which many other factors contribute to the
observed variation. This approach offers a powerful tool in dealing with complex
ecological patterns; it should be generally applicable to many other systems.
Q. Guo, J . H. B r o w and B. J . Enquist, Dept of' Biology, Uniu. o f New Mexico,
Albuquerque, N M 87131, U S A (present address of' QG: School of AVatural Resources,
Unic. of',Vebraska, Lincoln, N E 68583-0814, USA [[email protected]]).
Ecologists and evolutionary biologists are searching for
ways to understand the organization of complex systems. Individual organisms, populations, communities,
and ecosystems all contain many components that differ from each other. Each of these components interact
with one another and with their extrinsic abiotic and
biotic environment to produce complex structures and
dynamics. When studying such systems, it is difficult to
sort through the complex patterns of variation to identify the effects of fundamental processes. Application of
traditional univariate statistics is often unsatisfactory,
because no single hypothesis or variable is able to
account for a large proportion of the variation. Attempts to test multiple hypotheses are complicated by
the fact that they are rarely mutually exclusive. Results
of multivariate statistical analyses are difficult to interpret because they fail to elucidate the functional interdependencies among the multiple processes that have
produced the complex patterns of variation.
Recently, Brown and Maurer (1987, 1989, Brown
1995; see also Maller et al. 1983, Maller 1990, Enquist
Accepted 27 March 1998
Copyright Q OIKOS 1998
ISSN 0030-1299
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OIKOS 83:2 (1998)
et al. 1995, Hubert et al. 1996, Thompson et al. 1996)
have suggested that the effects of fundamental processes may often be revealed as constraints on the
observed patterns of statistical variation. The exercise
of fitting constraint lines to enclose the observed distribution of values within constraint envelopes can be a
useful aid in developing and evaluating mechanistic
hypotheses. Like most approaches, this one is not entirely new. There is a long history of recognizing, at
least implicitly and qualitatively, the limits on variation.
For example, there have been many discussions of
"threshold effects" and "boundary conditions" (see
Goldberg and Scheiner 1993, Thompson et al. 1996)
but these have rarely been quantified, or predicted by
mathematical models.
There are both practical and conceptual difficulties in
describing any complex pattern of variation. Consider
the approaches that have been applied to even a relatively simple case: the distribution of variables in bivariate space shown in Fig. 1. Such distributions are often
obtained, for example, when some measure of plant
variation and it violates parametric assumptions about
homogeneity of variance.
Alternatively, one of several multivariate techniques
(e.g. multiple or partial regression, principal components analysis, analysis of covariance, or multivariate
analysis of variance) can be applied to the data, provided that the investigator has measured other variables
that could potentially account for some of the variation
(e.g., Rees et al. 1996, Thompson et al. 1996). Often,
however. as in the case of many ecological experiments,
additional variables have not been measured. Even
when they have. the results of the multivariate analyses
provide assumption-laden descriptions of the relationships among the variables, and these can be difficult to
interpret in terms of the underlying mechanisms that
have produced the pattern.
Still another approach to the kind of patterns shown
in Fig. 1 is to fit an upper boundary that encloses all or
nearly all of the data points, creating a constraint
envelope. Most frequently, this is done qualitatively,
either by discussing verbally the apparent existence of a
boundary (e.g. Goldberg 1987, Goldberg and Scheiner
1993) or by fitting a line or curve by eye (e.g. KodricBrown and Brown 1978, Brown and Maurer 1987,
Fowler 1990). Although such methods obviously lack
Traditional approaches
quantitative rigor. they may lead to conceptual adPatterns of variation, such as the one shown in Fig. 1, vances. While the hypothetical constraint line may be
typically have been treated in several ways. Some inves- qualitative, it nevertheless represents a conceptually
tigators simply have fitted a linear or curvilinear regres- new way of thinking about the pattern of variation.
sion (e.g. Pacala and Silander 1990, Lindquist et al. Proposing the existence of a constraint line, either
1994; see also Goldberg and Scheiner 1993). If there are verbally or by a line fitted by eye. is often the first stage
large numbers of data points, such a procedure will in developing and evaluating a mechanistic hypothesis.
often yield a highly significant relationship. even For example, Brown and Maurer (1987, 1989; see also
though it accounts for only a small proportion of the Damuth 1991. Brown 1995) fitted lines by eye to the
upper boundaries of log-log plots of population density
as a function of body mass for a variety of animal
species, and the similarity of the slopes to the exponents
of well-established allometric equations suggested interesting hypotheses about mechanisms of resource
Enogonurn aabdianurn
Haplopappus gmcilis
allocation.
A more quantitative alternative is to use some standard statistical convention to isolate the data points
near the boundary (e.g. some fraction of the highest
points within equal increments along the abscissa) and
then use parametric regression to fit a line or curve
through these extreme points (e.g. Blackburn et al.
1992). This approach has the practical advantage of
quantifying the extremes of variation. It has one serious
conceptual shortcoming, however; it cannot characterInitial Density of Conspecifics
ize very precisely the limits set by the constraining
Fig. 1. Examples of field data that suggest a complex, but mechanism. There will be approximately equal numbers
constrained pattern of variation. Here we have plotted the
survivorship of two species of desert annuals, E. ubertiarzutn of data points on either side of any fitted regression
and H. gracilis, as a function of the initial density of conspe- line: if the upper boundary is hypothesized to reflect
cific individuals. The distribution of data points suggests that some absolute constraint, however. then it should be
maximal survivorship decreases with increasing density, implying that intraspecific competition sets an upper limit on the logically impossible for data points to fall beyond the
line.
proportion of ~ndividualsthat survive.
performance, such as survivorshlp, growth rate, reproductive effort, or biomass, is plotted as a function of
some measure of intra- or interspecific competition,
such as the density, size, and/or distance of neighboring
plants (e.g. Pacala and Silander 1985, 1990. Firbank
and Watkinson 1987, Goldberg 1987, Goldberg and
Fleetwood 1987, Goldberg and Scheiner 1993,
Lindquist et al. 1994, Rees et al. 1996. Thompson et al.
1996). Here we discuss approaches that have been
applied to this and more complex patterns reflecting the
influence of competition and other environmental conditions on plant performance. Then we show how explicit mechanistic mathematical models can be used to
define constraint lines that bound the observed distribution of data points. One value of such a modeling
approach is that it develops an explicit hypothesis for
each constraint line, and each hypothesis is potentially
subject to an empirical test. To the extent that the
hypotheses are supported by empirical evidence, they
suggest fundamental mechanisms that powerfully constrain the performance of plants, despite the complexities and the magnitude of unexplained variation.
Mathematical models of constraint lines
Probably the most rigorous and productive way to
draw constraint lines and to begin evaluating hypothesized mechanisms is to use mathematical models. One
example in plant ecology is the application of Yoda's
thinning equation to show the effects of competition in
constraining local density (Yoda et al. 1963, Osawa and
Siguta 1989, Zeide, 1991, Osawa and Allen 1993). We
will use this relationship to model survivorship as a
function of density. Charnov and Skinner (1984)
provide another and elegant example of this modeling
approach. They used Lack's theory of clutch size to
derive a predicted maximal brood size for the parasitic
wasp Nasonia, and then showed that the vast majority
of data points fell below this theoretical constraint line.
Here we use a similar approach to define hypothesized
constraints of competition on the performance of
plants.
Source of data
We use field data on survivorship of annual plants on
our Chihuahuan Desert study site as the basis for this
modeling exercise. A close relationship between modeling and data is appropriate, because the effects of
fundamental constraining processes will vary with the
environmental conditions, and will most readily be
interpreted in the context of a specific sampling regime
or experimental design.
At our long-term study site near Portal, Arizona,
where the annual precipitation is distinctly bimodal,
most annual plant species are present as vegetative
individuals during either the winter or summer rainy
seasons. During the last decade, densities of annuals
and perennials on 384 0.25-m2 permanent quadrats
were censused at the end of both the winter (AprilMay) and the summer (August-September) seasons, so
as to coincide with the period of cessation of growth,
highest biomass, and maximal flowering and fruiting
(for additional details, see Guo and Brown 1997).
For the purpose of this study, we analyzed data for
two species, Eriogonum abertianum Torr. (Polygonaceae) and Haplopappus gracilis (Nutt.) A. Gray (Asteraceae) in the winter and summer of 1992. Among the
annual plants at the study site, these are two common
species that are unusual in having a biseasonal life
history (Inouye et al. 1980, Fox 1989). They germinate
after the winter rains. and grow and occasionally flower
during the spring. However, the critical phase of their
life history is survival through the spring drought; such
survival is influenced by plant size and root development attained before the drought while competing with
other winter-active annual and perennial species. These
biseasonal annuals usually suffer high mortality during
the May-June drought, but surviving plants grow
rapidly, flower, and produce seeds following the onset
of summer rain, and then die during the next drought
(Fox 1989). Our models are based on two measures of
population-level performance: (1) survivorship during
the spring drought, measured as the number of individuals present in each quadrat at the end of the summer
season divided by the number present on that quadrat
at the end of the winter season (i.e., final densitylinitial
density); and (2) final density, measured as the number
of individuals in each quadrat at the end of the summer
season.
Models of survivorship
We first consider the effect of the population density of
neighbors on the survival of a target species. Assume
the following experimental or observational design. We
have a large number of replicated quadrats, similar in
soil, water, nutrients, and other abiotic conditions, but
varying in the initial number and/or spacing of
seedlings. Further assume that some fixed quantity of
limiting resource, constant across all quadrats, is available to support the growth and survival of the
seedlings.
This design fits our field data set well. The limiting
resource is the water supplied by winter rains, and
depletion of soil moisture during the spring drought is
a major cause of mortality. Plots of survivorship of E.
abertianunz and H. gracilis as a function of initial
density of conspecifics in all 384 sample quadrats are
shown in Fig. 1. In each graph the points all fall within
a space that is roughly triangular but has a distinct
curvilinear upper boundary, which appears to reflect
some maximum limit on survival as a function of initial
density. We will now apply the well-known allometric
thinning law in plant ecology to characterize this constraint of interspecific competition on survival.
Changes in plant size and survivorship under competition for a limiting resource can be modeled by the
allometric thinning law described by Yoda et al. (1963).
This is usually written as
where M is the average mass of an individual plant, N
is the density of plants in the neighborhood, b is the
slope of the thinning relationship and c, is a fitted
constant that varies with species and local environmental conditions. Yoda et al. (1963) and some subsequent
authors have suggested on theoretical grounds, based
on the geometry of plant growth and competition, that
the value of b should be - 3'2. Empirical studies,
however, have typically obtained values of b closer to
- 413 (Lonsdale 1990), and B. J. Enquist et al. (unpub].) have derived a theoretical model, based on plant
energetics and resource use, that predicts this value.
Nevertheless, as will be shown below, the values of b and
c , d o not affect the generality of our results. While the
thinning law has most often been applied to intraspecific
competition in forestry and other monocultures, it is
very general and can often be used to characterize the
combined effects of intra- and interspecific competition
in forests and other polycultures (White 1980, Westoby
1984, Guo and Rundel 1998, Enquist et al. in press).
Although the thinning law has traditionally been used
to assess the effect of density and competition on
population biomass, because of interest in timber yield
in forestry and similar applications, it can also be used
to model survivorship. As mentioned above, traditional
studies have usually focused on the trajectory and slope
of the thinning relationship. Instead, the emphasis in this
study is on the final density attained. The simple model
presented here assumes that for any initial density, N,,
plants will grow in size until resources become limiting
(see Fig. 2). At this point the thinning line is reached and
competition for resources begins. Further increase in size
Log (density m-2)
is only possible if some plants die. Others will then Fig. 2. A graphical representation of the allometric thinning
continue to grow at their expense. If populations are relationship over time in which the average mass, M, of an
individual plant is plotted as a function of the neighborhood
grown at various densities, high-density plots will reach population
density, N, on logarithmically scaled axes. The
the thinning line sooner than low-density plots.
solid line indicates the upper boundary set by the thinning line,
In this model, we have further assumed that since we where M = cNh. The solid line with arrows indicates the
have samples (stands) of plants growing under identical trajectory described by three population over three hypothetical time steps as plants increase in size and decrease in density
conditions of resource limitation, except for their initial during the thinning process. Individuals growing in non-comdensities, they are subject to the same thinning line. petitive environments will continue to grow at the same maxiNow, as populations experience greater time under mal rate ( R , ) until resources are limited due to competition. In
this model, competition for resources is intense when populathinning conditions, mortality will continue to occur. If tions reach the thinning boundary. Once populations reach the
the thinning process is allowed to continue, eventually all thinning boundary, growth is only allowed by the death of
populations will reach a similar uniform density (Nf) other individuals. Movement along the thinning line (R,) is
(i.e. see Kays and Harper 1974, Harper 1977, Chapin dictated by the rate at which individuals can gain mass at the
expense of others. The location of the final density attained by
1993). This density appears to be set by species-specific all populations ( N f )was designated in this graph as (*). In this
requirements for space and resources, and the rate at example, population 3 has undergone the most mortality due
which resources are renewed (not a factor in our appli- to competitively induced thinning, while population 1 has not
undergone thinning. If all populations were allowed to concation of the model, but see Chapin 1993, Enquist et al. tinue to thin throughout the growing season, then each popuin press). It is at this point that no further increase in lation would reach the same final density (*).
size is possible because resources are exhausted.
If the final density of all populations is roughly
similar, then it is possible to predict the maximal survivorship at a given initial population density. Given a
fixed amount of time (i.e. the growing season), high- Maximum values of population survivorship depend
density plots will experience larger changes in density upon the initial population density, N,, and the species(lower survivorship), due to longer time experiencing and environment-specific constant. ,c The exponent - 1
thinning conditions, than low-density plots (higher sur- simply reflects the fact that all populations attain the
vivorship). Thus, in our example (Fig. 2), high-density same final density.
plots experience proportionally more thinning (mortalIn the case of desert annuals where all of the limiting
ity) than low-density ones. The maximum survivorship resource is used and the initial densities are sufficiently
value for a given population can then be shown as
high that all (or nearly all) populations experience
thinning mortality, we expect the exponent of this
relationship to approximate - 1. Together, thinning
mortality and the attainment of a similar final density
If all populations varying in initial density N, thin to a for each population set the conditions under which
final density (Nf) that is species and environment spe- maximal survivorship can be achieved. Equation 3 precific, characterized here by the constant c? then maxi- dicts that for any value of cf all values of S will lie along
mum survivorship values can be rearranged as
a concave line as a function of N, (Fig. 3).
240
OIKOS 83.2 (1998)
Our survivorship function, based on a simple model
of growth and thinning predicts a maximum survivorship value for a given population size. The maximum
survivorship value is that which would occur if all
mortality were associated with thinning. In order to
apply the thinning law to define a constraint line for
our data on desert annuals, we assumed that for the
existing conditions of resource limitation, intraspecific
competition sets only an upper boundary on survivorship. Many other factors, such as physical stress, disease, disturbance, herbivory, and interspecific
competition, could cause mortality and result in values
of survival below the upper boundary. Therefore, we
fitted the position of the constraint line by varying the
constant cf, so that the line just encloses the most
extreme values (Fig. 3).
The model provides a robust mathematical description and offers a mechanistic hypothesis for the constraint of resource-based competition on the observed
pattern of variation in survivorship. In particular, the
allometric thinning process appears to accurately predict maximal survivorship over several orders of magnitude of variation in initial density. Further, the model
leads logically to several additional testable predictions.
If we had data on growth rates of individual plants which we do not - we could make and test predictions
about ~ l a n size
t
as a function of both initial and final
density. We d o have data on the densities of other plant
species, however, so we can extend the approach to
assess interspecific competition
If plants of all species are competing for a common
resource - a reasonable assumption in this water-limited arid ecosystem (Went 1948, Beatly 1967, Kemp
1983), then we should see the outcome reflected in
similar constraint lines defining similar patterns of variation in survival as a function of initial density of
heterospecific, as well as conspecific, individuals. That
is just what we observe for both E. abertianum and H.
gracilis (Fig. 4). Even though both species differ in
many evolutionary and ecological aspects the slope
constraint line appears identical. It is only necessary to
0
100m3004M)m
-.-
0
m ~ 6 0 0 8 0 0 l m
Initial Density of Conspecifics
Fig. 3. The same data on survivorship of E. abertianum and
H. gracilis as a function of initial density of conspecific
neighbors as in Fig. 1, but with the upper bound fitted using
the thinning function, and selecting a value of the constant cf
OIKOS 8 3 : 2 (1998)
8
.-,
Initial Density of Heterospecific Annuals
Initial Density of Perennials
Fig. 4. Field data for survivorship of E. abertianum and H .
gracilis as a function of initial density of different kinds of
interspecific neighbors, either all other species of annuals
(above) or perennials (below). The patterns of variation are
consistent with an upper limit on survival being set by the
density of competitors in the neighborhood.
fit the constant cf to obtain a function describing the
upper maximum survivorship boundary that fits the
data quite well. This indicates that all populations
experienced a similar thinning process, and thinned to a
unique species-specific final density. Since the sizes and
competitive effects of different species vary, different
values of cf have to be fitted to define the constraint of
density of neighbors on the maximum survival of the
target species.
The differing cf values observed for each target species are due in part to their unique requirements for
space and resources. Variation among species in their
final densities provides insight into their relative requirements for resources and their tolerance for competition. Some of the variation in cf will also be due to the
competitive environment, the quality of resources available, and especially the resource requirements of the
competing species. Note the differences in c, in the
scaling of the abscissa in Fig. 3 is due to the varying per
capita effects on the target species of different kinds of
competitors: intraspecific vs interspecific, annuals vs
perennials.
Further, interspecific competition can be hypothesized to account for some of the variation in survivorship within the constraint envelope set by intraspecific
competition. This leads to the testable prediction that a
significant component of the as yet unexplained survivorship in Fig. 3 should be inversely correlated with
the initial density of heterospecific individuals. We
tested this prediction by fitting a regression to the
log-transformed data on survivorship as a function of
density of conspecifics, and then regressing the residuals
on the densities of all other annual plants. The regressions were highly significant (E. abertianum - survivorship vs density of conspecifics: r = - 0.67, P < 0.0001;
residuals vs density of all other annuals: r = - 0.36,
P < 0.0001; H. gracilis: survivorship vs density of conspecifics: r = - 0.61, P < 0.0001; residuals vs density of
all other annuals: I. = - 0.20, P < 0.0006). Thus, much
of the variation in survival not explained by intraspecific competition can be attributed to interspecific competition. Together, these analyses show that between
40'% and 60'% of the variation in survival can be
explained by competition. Hypotheses about effects of
other factors, such as herbivores, physical stress, and
disturbance, could be tested in a similar way if one had
independent estimates of these sources of mortality.
to fall within a triangular constraint envelope. This
suggests that interspecific competition causes a linear
trade-off in maximal densities of the two species.
In terms of our study, this trade-off can be modeled
very simply. It is only necessary to assume that within
each replicate quadrat some quantity of limiting resource, W, has been divided among individuals of two
species, A and B, such that
where the N's are the final (after competition) densities
of each species; and the c's are species-specific constants that reflect the influence of such characteristics as
initial seedling size, growth rate. and root system on the
rates at which resources can be taken up from the
environment and converted into numbers of plants. For
a design such as ours, it is convenient to reparameterize
eq. 3, as
Models of density and biomass
When other variables are used to measure plant performance, different patterns of distributions of data points
can be obtained and different models can be developed
to define the constraint lines. For example, in plots of
the density of one species of annual as a function of the
density of the other species (Fig. 5), the points appear
WINTER
SUMMER
DENSITY OF ALL BISEASONAL SPECIES
DENSITY OF E ABEfiTiAlVUM
Fig. 5. Examples of apparently linear constraint lines reflecting the effect of a common limiting resource on the densities of
competing plants. A and B: densities of other annuals as a
function of the joint densities of the two biseasonal annuals
(E. abertianurn and H. gracilis) in winter and summer of 1992.
C and D: Competitive relationships between the two dominant
biseasonal species in both winter and summer in 1992. Note
that the linear trade-off model (eq. 5 ) described in the text has
been used to fit the line defining the upper boundary of the
triangular constraint space.
where the c, and c, give the maximal number or density
of species A and B, respectively, when each is growing
in monoculture (Fig. 5A). Differences in these constants
among species reflect variation in per capita resource
demand and resulting asymmetries in competitive effects. Thus, interspecific competition for these resources
sets the upper boundary on the joint maximal densities
of each species. Other sources of mortality (abiotic
stresses, interspecific interactions, and intraspecific thinning) will cause the observed densities to be less than
the maximal possible values.
The same kind of model can be applied to any kind
of linear resource division between competing plants.
For example, when plotted against each other. the
densities of annual and perennial plants on our study
quadrats fall within triangular-shaped constraint envelopes (Fig. 5B), suggesting that they reflect the outcome of a similar linear competitive process of resource
division. The observed species-specific constants convey
the relative interspecific competitive impact per individual. In this case, widely different values of c, and c,
presumably reflect the asymmetrical constraint of competition from established perennials on the densities of
annuals.
Very similar linear trade-off models can be used to
characterize the constraints on biomass of competing
species. For example, Goldberg (1987) conducted field
experiments in which she assessed the effect of competition from intra- and interspecific neighbors on growth
rates and biomass of single target plant individuals.
When she plotted biomass of the target individual at
the end of the growing season as a function of the
combined biomass of its neighbors, she obtained triangular distributions of data points similar to those we
obtained for population density in Fig. 5. This is con-
sistent with a simple linear model of competition for
some limited resource that is used to produce plant
biomass (see also Firbank and Watkinson 1987).
Discussion
This paper is about the use of models of constraint lines
and constraint envelopes as a conceptual approach to
investigate complex patterns of variation and the mechanistic processes that influence them. We have illustrated this approach by presenting examples of how
mechanistic models of intra- and interspecific competition can be applied to characterize limits on the performance of annual plants. Many applications of this
approach to both perennials and annual plants in differing habitats. and to other kinds of systems are
possible.
The advantage of fitting constraint lines is that, on
the one hand. the lines quantify hypothesized effects of
important processes on the structure andlor dynamics
of the system, but, on the other hand they do not
attempt to account for all of the variation in those
structures or dynamics. This point is well illustrated by
studies of the effects of intra- and interspecific competition on plant performance. In the laboratory (and
sometimes in manipulative field experiments) it is possible to analyze the effects of competition by designing
experiments to vary the identity, density, size, or distance of neighbors and to hold constant effects of other
potentially confounding variables such as other kinds
of competitors. herbivores, and abiotic characteristics
of the microsite. The effects of the variable of interest
can be assessed by appropriate statistical techniques,
such as analysis of variance, analysis of covariance or
regression analysis (linear or non-linear) (see Goldberg
and Scheiner 1993). The experimental design has effectively simplified the system by removing or holding
constant all of the variables that might potentially
affect plant performance except for competition. Thus,
it should not be surprising that in well-designed and
carefully conducted laboratory experiments it is often
possible to find by regression analysis that a measure of
competition, such as neighbor density or biomass, accounts for a large proportion of the variation in a
measure of plant performance, such as growth or survival (e.g. Waller 1981, Firbank and Watkinson 1987.
Goldberg 1987).
Field systems. including many experimentally manipulated ones. are much more complex. Many factors
other than competition can affect the performance of
plants (Harper 1977). These include variation in: (1)
limiting resources (e.g. water, nutrients, light); (2) physical disturbance (e.g. animal activities, fire, flood); and
(3) non-competitive interactions with other organisms
(e.g., herbivores. granivores, seed dispersers, pollinaOIKOS
x i z (199x1
tors, pathogens, mycorrhizal fungi). Thus it is not
surprising that some measure of the competitive neighborhood often fails to account for more than a small
proportion of the variation in plant performance (e.g.
Turkington and Aarssen 1984, Goldberg 1987. Stadt et
al. 1994. Guo and Rundel 1998). In part for this reason,
there has been a long-standing debate over the importance and general role of competition within and across
plant communities (see Fowler 1986. Schoener 1986,
Tilman 1987. Wilson and Tilman 1991, Campbell and
Grime 1992, Goldberg and Barton 1992). Oftentimes.
the highly variable results of such experiments are often
interpreted as indicating that competition "is not very
important" in the field. This may not be true, especially
when there is good evidence for the existence of a
constraint line, such as in Fig. 1. In some of these
systems. competition on resources does play an important role in the system by setting the ultimate deterministic limits on plant performance and ecological
variation.
Given evidence for the existence of a constraint line
and some limiting process, there are several advantages
in trying to characterize them using mathematical models. Compared to a verbal or other qualitative description of the pattern of variation, the mathematical
model has the obvious advantage of rigorously and
quantitatively specifying the relationships among the
variables. A statistical characterization of the variation
(e.g. univariate regression or multivariate analysis)
would also provide such quantification. Such a purely
statistical model, of the sort advocated by Peters (1991),
however. provides only a description of the system. A
statistical description of a complex ecological system is
of limited utility. For one thing, it is only valid within
the range of variables used to create it: it cannot be
used to make predictions beyond this range. Further,
such statistical models are sensitive to assumptions
about the distribution and independence of the data.
For example. fitting regressions to the data in Fig. 1 is
problematical for two reasons: (1) variance in the dependent variable, S. is not distributed homogeneously
as a function of the independent variable, Y,, and (2)
the dependent variable includes the independent variable in its denominator. because S = N, N,so that a
negative relationship between S and N, would be expected by chance alone. Note. that this is true only
when N, is the initial density of conspecifics, as in Fig.
1, but not when it is the density of other species as in
Fig. 4.
If constraint lines are to be used to describe quantitatively how critical processes limit variation, then it will
be necessary to develop statistical procedures for fitting
the lines to data. This raises a number of issues that
have not been thoroughly addressed above or in the
literature of statistics or biometry (E. Bedrick pers.
comm.: see also Thompson et al. 1996). There are both
conceptual and practical difficulties. For example. if the
243
line describes an absolute constraint, as is hypothesized
by our models, then it should be theoretically impossible to have any data points outside the boundary. This
implies that the line should be fitted so that the line
passes through the most extreme values, as we have
done in Fig. 3. Doing so also assumes, however, as does
our model, that the replicate plots were all initially
identical, so there was no effect of variation in initial
conditions on the maximum observed values of survivorship. While this may be a reasonable assumption
for a mathematical theory of thinning process, it is
unlikely to be upheld in the field. Further, a reasonable
sample of data points is usually required to fit a function statistically. but the very concept of an absolute
constraint boundary implies that there will often be few
points on or near the line.
Given these practical problems, one might ask why
attempt to define constraint lines and envelopes at all,
especially when standard regression techniques can "explain" 40-60'Y0 of the variation in the empirical measures of plant performance. There are two answers. The
first is statistical. While standard parametric and nonparametric statistics are relatively good at describing
central tendencies in data. they are not intended to
characterize boundary conditions. When the limits of
variation rather than the central tendency become the
phenomena of interest, different statistical procedures
are called for. Fortunately, statistics is an active and
evolving discipline, so new techniques can be expected
to be developed to meet the need (see also Blackburn et
al. 1992, Thompson et al. 1996). The second reason for
modeling constraint lines is conceptual. Statistical fits
to data do not imply any mechanisms, they just describe quantitative relationships among the variables. In
contrast, even our very simple mathematical models
imply a particular mechanism and specify its functional
form. When a model is fitted to data, the mathematical
function and the fitted constants provide a precise
quantitative description of the process. While the constraint lines that we have derived above must be regarded as hypotheses, they have the advantage of
specifying a mathematical form and the quantitative
limits of intra- and interspecific competition on annual
plant performance.
The great advantage of an analytical mathematical
model is that usually it implies a specific quantitative
mechanism. This serves as an hypothesis that can be
evaluated by some combination of analysis of existing
data and collection of new experimental or observational data. The testing of such hypotheses about constraining processes and boundary conditions can form
the basis of a productive research program designed to
elucidate the fundamental processes that influence the
structure and dynamics of complex systems. Elsewhere,
Brown (1995) has suggested that the statistical distributions of large numbers of data points with respect to
potentially important ecological variables may hold
some of the best clues to the general, lawlike processes
that underlie much ecological complexity.
Acknowledgements
We thank P. Dixon and N. Huntly for
helpful comments on the early version of this manuscript. This
work was supported by NSF grant DEB-9221238 to J. H.
Brown. Q.G. was also partially supported by National Geographic Society grant (5523-95) and a biology graduate research fund at the Univ. of New Mexico. B.J.E. was supported
by a Fulbright Fellowship and NSF grant GER-9553623.
-
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