1 Experiments on competition between plant species are frequently designed without considering the analysis stage of the study. We argue that this omission may lead to over-complication of the issue of designing experiments.

2 An overwhelming number of studies have shown that the effects on performance of competition in plant mixtures may be described by simple (hyperbolic) regression models. The most natural view of the problem of measuring plant competition is therefore as a problem in regression.

3 Only with experiments designed explicitly to apply regression analyses can phenomena such as frequency dependence and the size dependence of measures of competition be identified. In contrast to previous assertions, this means that designs that are based around, or allow, regression analysis are the most robust as such effects may be tested for using appropriate statistics.

4 Experiments are probably most easily designed to measure competition as a function of the density of interacting species, rather than biomass. This is because the per unit biomass effect of competition on performance is a function of density. Competition measures based on biomass will hence be dependent on the density at which the experiment is performed. Furthermore, the most effective way to manipulate biomass is through changing species’ densities.

5 In terms of economy of design, we would recommend simple additive series. Whilst this does not allow the role of frequency dependence to be analysed, this phenomenon appears to be rare in any case.

The design and analysis of experiments for competition studies has occupied considerable research effort in plant population biology. Opinions differ, however, as to what constitutes the appropriate approach to designing such experiments (Gibson et al. 1999) or how analyses should be conducted (Freckleton & Watkinson 1997a, 1999). A recent review by Gibson et al. (1999) explored the design of competition experiments and critically examined the merits of commonly employed designs. This review did not, however, consider the analysis of experiments as a factor guiding the suitability of different designs. In this paper we argue that this lack of consideration of the analytical component of competition studies leads to a number of difficulties with experimental designs that are more apparent than real and hence lead to misleading recommendations. Furthermore, as the ultimate point of performing greenhouse experiments is to relate experimental results to hypotheses (either framed qualitatively or quantitatively) that can be tested under field conditions, we call for an approach to the design and analysis of greenhouse experiments that considers the analysis of competition under field conditions.

Common designs for competition experiments

There are three categories of experimental designs that are commonly employed for analysing competition under greenhouse conditions. These are: (i) the substitutive replacement series, whereby the proportion of species within mixtures is varied at a constant density; (ii) the simple additive design, where competition is treated in a factorial manner, and plants are grown either with or without competitors; and (iii) the full additive design where competition is analysed by means of a trend analysis with the density of each species being varied through a range of densities.

The replacement series has been roundly criticised and many authorities no longer recommend its use (Connolly 1987, 1987; Law & Watkinson 1987; Snaydon 1991). Nevertheless, this design is commonly employed because of its simplicity. It may yield more information than a simple additive approach in certain circumstances since the components of the mixture are varied through a continuous range, but it confounds the effects of intra- and interspecific competition and there are problems with the analysis and interpretation of the coefficients it generates (Law & Watkinson 1987).

Of the two forms of additive design, the simple additive design where plants are grown ‘with’ and ‘without’ competition is by far the most commonly employed (Gibson et al. 1999). This very simple design is commonly employed when large numbers of species and/or other factors (such as nutrient levels, symbionts and parasites) are manipulated as it allows the size of the experiment to be minimized. The main criticisms that have been levelled at this kind of approach have concerned the analytical stage of such experiments. Typically, the effects of intra- and interspecific competition are not dissociated, and the experimental results are affected by the density at which the experiment is performed (Freckleton & Watkinson 1997a,b, 1999).

Fully additive designs are the most labour-intensive to perform. The most complete design, the response surface methodology (Law & Watkinson 1987) allows complete quantification of the effects of competition within the range of densities explored in the experiment, but at the expense of very large designs. Less intensive additive designs include, for example, the methodology of Goldberg & Landa (1991), which enabled all 49 competition coefficients for 7 × 7 species to be estimated. This was achieved by considering only pairwise interactions and ignoring those of higher order. The additive approach is often criticised on the grounds that it confounds density and proportion and because the results may be biased by initial or final size (Connolly 1986, 1987; Snaydon 1991; Gibson et al. 1999).

The problems associated with the replacement series design have been extensively debated (Connolly 1986, 1987; Law & Watkinson 1987; Snaydon 1991). We therefore do not consider this design further, but explore the analytical properties of data derived from the simple and full additive designs and look at how these properties might affect the design of competition experiments.

Competition: what can we expect?

If plants are competing, then as we increase the density of plants, mean plant performance (e.g. individual plant size, mean plant fecundity, per plant probability of survivorship) will decline. Furthermore, most studies that have measured competition in detail have shown that the effects of competition on mean plant performance can be predicted by an equation of the form (Firbank & Watkinson 1990):

In this case, for a two-species mixture, the mean performance per individual of species 1, f¯_{1}, is a function of F_{1}, its mean performance per individual in the absence of competition as well as two coefficients (α_{11} and α_{12}, respectively) that measure the decline in performance with increasing densities of the two species (N_{1} and N_{2}). If this particular function does not describe the data well then an alternative may be employed (e.g. Law & Watkinson 1987), but we will restrict our discussion to eqn 1, without loss of generality.

Equation 1 makes it clear that the problem of measuring competition is a problem in regression, and more specifically multiple non-linear regression. Performance, according to eqn 1, varies continuously with the densities of the interacting species and the competition coefficients (α_{11} and α_{12}) are density-independent. There is no sense in which plants are either ‘subject to competition’ or ‘not subject to competition’. Hence, a simple additive approach where plants are grown either singly or in the presence of competitors does not reflect well the continuous non-linear nature of competition-density responses. The outcome of such experiments would be inadequate to measure the coefficients in eqn 1 (but see the suggestion below) and sensitive to the densities at which the experiment is performed. The most that a simple additive approach can achieve is to demonstrate that competition is taking place. The inadequacies of the simple additive approach are further reinforced when we consider the interpretation of the results of such experiments if used to measure competition under different environmental conditions, for example, under two nutrient levels. While there may be considerable changes in the ratio of the inter- to intraspecific coefficient in eqn 1, and hence in the importance and intensity of interspecific competition, the simple additive design would be inadequate to detect this (Freckleton & Watkinson 1997a, 1999).

We note that if (i) intraspecific competition is introduced as a factor in a simple additive experiment (although this is commonly not done), (ii) very high densities of neighbours are used, and (iii) the weight of isolated plants were measured, it may be possible to estimate the α variables from eqn 1 through re-arrangement of the equation, albeit not with great certainty (Firbank & Watkinson 1985; Baylis & Watkinson 1991). It would, however, be difficult to specify in advance what constitutes a high enough density and it has to be assumed that the exponent in eqn 1 is equal to unity; this technique may be worth further exploration.

By considering eqn 1 and approaching the design of competition experiments with the analytical stage in mind, we are also able to deal with a recurring concern with competition experiments. The problem is that experiments that vary densities are said to confound densities with species proportions or total density (Gibson et al. 1999). However, if competition follows the form given by eqn 1, this is not a problem: in principle, eqn 1 contains no frequency-dependent elements and the proportion of a species in the mixture should not impact on model parameters. Furthermore, if we have conducted an experiment that allows a fitting of the full response surface represented by eqn 1, we could then test for frequency dependence by fitting the model in a form such as:

or in some other form in which frequency dependence or total density is included. A significance test on the parameter β in eqn 2 allows us to determine whether an interaction effect is occurring. Studies that have looked for such interactions typically have not found them (Turkington & Jolliffe 1996; McCloskey et al. 1998; P. Jolliffe, personal communication), and frequency dependence can probably be discounted even if a full response surface design was not employed (as in the experiment of Goldberg & Landa 1991).

Biases vs. determinants

A second major cause for concern is the likely introduction of biases in the measured outcome of competition as a consequence of the way an experiment is designed. In particular, Snaydon (1991) and Gibson et al. (1999) expressed their concernes about the impacts of initial size differences (e.g. seed and seedling size) or final size on measured competition coefficients. The problem here, it is argued, is that measures of competition may be simply a function of the relative size of competing species. Gibson et al. (1999) recommend that experiments should be designed to ‘control’ for the effects of initial or final size on competition. The problem we have with this notion is that it is not clear why a correlation of measured competition coefficients with simple ecological traits, such as emergence time or seed size, should be such an undesirable property. At an extreme level, of course, this could lead to trivial experiments, such as competing oak trees and grasses (cf. Snaydon 1991). On the other hand, one can argue that if most variation in the per individual strength of competition were explained by size (which could be interpreted as a measure of resource capture) we should have a simple predictor or determinant of competitive ability.

Many competition experiments have studied the effects of some discrete external factor (e.g. mycorrhizas, pathogen or nutrients) on competition (e.g. Tables 3 in Gibson et al. 1999). In this case, initial size (i.e. seed size and seedling size) will generally not differ across treatments, and will therefore not present a problem in interpreting the effects of variation in the factor on the outcome of competition. In an analysis of the effects of mycorrhiza on competition between two grasses, for example, we found that the strength of intra- and interspecific competition increased at nearly the same rate as total plant size and, as a consequence, the net effects of mycorrhizas on competition tended to be minimal at high densities (Watkinson & Freckleton 1997). Initial size was not an issue, since this was the same for both colonized and non-colonized plants. Some forms of manipulation, such as varying the timing of emergence will, of course, affect the initial size of plants. In such cases both per capita, density-based coefficients (e.g. Cousens et al. 1987) as well as models based on per unit biomass or per unit leaf area have been shown to provide good descriptions of competition between species (e.g. Kropff & Spitters 1991; Connolly & Wayne 1996). We would not wish to factor out size in such situations as it has an impact on resource capture and hence on competition.

The one major consequence of considering size biases at the design stage is that it leads to the recommendation that experiments are designed such that competition is explored as a function of some variable other than density. Whilst this appears, superficially, to be a reasonable way around the problem of size bias, it is a surprisingly complicated one. The reason for this is that the biomass of the interacting species is nonlinearly related to their density. If Y_{1} and Y_{2} are the total biomasses per unit area of species 1 and 2 (given by multiplying the mean weight per plant by density), and W_{1} and W_{2} are the mean yields per isolated individual (substituting for F in eqn 1), then we may use eqn 1 to predict the total yield of each species as a function of the two densities. We can then re-arrange these equations in order to express plant densities as a function of the biomass of the two species. These quantities may then be substituted into eqn 1, yielding for species 1:

(eqn 3)

where w¯_{1} is the mean weight per plant of species 1. The main feature of eqn 3 is its complexity, showing that if we use species’ biomass as our independent variable then the interpretation of competition through measures of per unit biomass becomes complex. The per gram effects of one or other of the species on the mean weight per plant of species 1 are given by the terms within square brackets in eqn 3 and depend on the biomasses of both species. Only at very high densities of one or other of the species (but not both), when either Y_{1} or Y_{2} approach their asymptotic maximum, are the per gram competition coefficients approximately constant. In these cases the per gram competition coefficients are given, approximately, by a_{1}/W_{1} for intraspecific competition and a_{12}/W_{2} for interspecific competition, respectively. We therefore would derive no more information from a study that measured competition as a function of biomass than from one designed to measure competition as a function of density, as the coefficients measured are simply functions of each other. If maximum mean plant sizes (W_{1} and W_{2}) were not measured then we would in fact derive less information.

Again, by considering the analysis at the design stage, we can predict that competition coefficients based on biomass rather than on densities will be both density- and frequency-dependent: meaningful independent measures of competition will not be easy to generate if the data require the use of eqn 3. It is also difficult to see how one can effectively manipulate neighbour biomass without altering density, as manipulating, for example, germination or emergence times would confound other effects such as canopy height development.

From greenhouse to field

The greenhouse environment is such that extrinsic variability can be minimized and the effects on competition of single factors can be explored with other variables held constant. Although often not explicitly stated, the ultimate aim of greenhouse experiments must be to generate some prediction (at whatever scale or level) that, in principle, relates to or can be tested under field conditions. Of course it would be unrealistic to expect parameters measured in the greenhouse to be exactly the same as those measured in the field. What the greenhouse allows us to do, however, is to remove the variability of the real world, thus allowing the impact of some factor to be measured in isolation, and the use the results to generate a hypothesis that can be tested under field conditions. For example, when we analysed the effects of arbuscular mycorrhizal fungi on plant competition, the rationale behind the experiment was to find whether the presence of these fungi affected the outcome of competition between two species (West 1996). We found that changes in plant size were accompanied by essentially compensatory changes in the intensity of inter- and intraspecific plant competition. Although the actual parameter values under field conditions will differ from those measured in the greenhouse, we have generated a qualitative prediction based on the greenhouse study that can be tested in the field.

The easiest and most obvious way to link greenhouse and field studies more closely is to employ the same or comparable measures of competition in both the greenhouse and in the field. Broadly speaking, the same kinds of designs (substitutive, simple additive and full additive) have been used to analyse competition in both situations. Studies on community dynamics, as opposed to weed–crop interactions, have mainly employed a removal approach whereby interspecific neighbours are removed from some plots, but left intact in others, and differences between the two treatments are measured in some aspect of performance (Goldberg & Barton 1992). This design is closely allied to the simple addition design used in greenhouse experiments (Gurevitch et al. 1992) and is commonly analysed in the same way using simple competition indices (Goldberg & Scheiner 1993).

The commonly employed removal method for quantifying competition (Gurevitch et al. 1992) has the fatal limitation that it is unable to detect competition in spatially structured natural assemblages. Pacala & Levin (1997) have shown that in mixtures of species that are competing strongly, extreme spatial segregation develops between the interacting species and that under such conditions, competition coefficients based on simple removal experiments indicate that no competition is occurring. While this phenomenon may be important in generating coexistence, it is disastrous from the point of view of experimenters wishing to measure competition because a low removal coefficient is compatible with both strong and weak underlying interactions. It is important to note therefore that the measures of competition derived from the simple additive approach in the greenhouse are not directly comparable with those derived from the removal approach in the field, despite their apparent similarity.

This need for generating comparable measures of competitive ability from both field and greenhouse studies further exposes the limitations of measures of the intensity of competition based on per unit biomass. It is difficult and time-consuming to measure biomass under field conditions, and clearly requires destructive sampling. For measuring the per individual and per capita effects of competition in a range of unmanipulated multispecies communities under field conditions it is, however, possible to use a regression approach that generates competition coefficients that are analogous to those in eqn 1 (Rees et al. 1996; Law et al. 1997; R. P. Freckleton & A. R. Watkinson, unpublished data). With this technique, the performance of individuals of each species within small quadrats is related to the density of neighbours. This method appears particularly promising for quantifying the competitive interactions between all the component species of a community and has the advantage that the same approach can be used to analyse fully additive experiments in the greenhouse. The analysis of Rees et al. (1996) has, for example, generated competition coefficients that are directly comparable with those of eqn 1, whilst the analysis of competition between tillering grasses by Law et al. (1997) was based on a similar rationale to the greenhouse-derived measures of competition between tillers used by Thorhallsdottir (1990) and Silvertown et al. (1992). More work is required to examine the degree to which the technique can be applied to longer-lived, size-structured species, but we see no reason why the technique should not be successful.

Concluding remarks

The main point we wish to emphasize here is that the design of competition experiments should centre on their analysis and likely outcome. Consideration of the general form of competition in plant mixtures allows us to predict that some designs will be inadequate to address directly the issues of bias and frequency dependence and to provide unequivocal measures of the strength of competition. Moreover, the power of regression-based methods to characterize interactions accurately under field conditions offer a powerful link between the artificial environment of the greenhouse and the real world. Of course, per individual and per capita measures of the effect of competition on plant performance are only one component of per capita effects of competition on net population growth in the field (Tilman 1988; Goldberg & Barton 1992; Chesson & Huntly 1997). Consequently, whilst competitive hierarchies derived from measuring competition-density responses may translate directly into community dynamics, this need not necessarily be the case (Moloney & Chiariello 1998) and unequivocal measures of competition are required to assess its contribution to population and community dynamics.

There is no ‘optimum’ design for competition experiments since the aims, objectives and practicalities vary from study to study and species to species. Whatever their design, greenhouse experiments have always been criticised on the grounds that the greenhouse environment is too simplistic and unrepresentative of the real world. Theory has shown that, of the available methods, the regression approach is generally the most robust for analysing competition under field conditions. By designing experiments to generate comparable measures in the greenhouse, it may be possible to form closer links between the two systems. This must have important consequences for research on weeds, symbiosis, pathogens, climate change and community theory.

Acknowledgements

We should like to thank Peter Jolliffe for his comments on an earlier draft of this paper. This work was supported by NERC grant GR3/11458 to A.R.W.

Received 1 April 1999revisionaccepted 16 December 1999