Distinction between ‘outcome of competition’ and ‘effects of species on each other’

We deal with two main aspects of interspecific interaction. ‘Outcome of competition’ refers to the relative long-term success of species, i.e. the end point for the community in terms of its composition and we are concerned with what indications short-term experiments can give about this. ‘Effects of species on each other’ refers to the impact of each species on the other (Goldberg & Werner 1983; Goldberg 1990) and may be an important part of the process that determines the end point, but is distinct from it. So while these two aspects of interaction may often be related, they are not equivalent, and their study may require different techniques. We believe that they are regularly confounded in current practice.

Since competitive exclusion rarely occurs in short-term experiments, the primary indicator available, however inadequate it may be, of long-term prospects is increased dominance of a species in a mixture, i.e. greater gain in terms of greater output per unit input. Although an estimate of this may be made from a single mixture, assessment of the effects of species on each other generally requires the inclusion of a range of mixtures and/or monocultures in the design.

Many experiments use methods and indices (e.g. relative crowding coefficients, coefficients of aggressivity, relative yield total) that purport to reflect the outcome of competition (Keddy & Shipley 1989) but actually address the questions of effects of species on each other or an amalgam of both. Furthermore, the indices and analytical methods used are generally susceptible to bias because they ignore initial differences between components, and therefore tend to favour larger individuals (Connolly 1986; Grace et al. 1993).

The role of time, initial conditions and plant size/life history

Analysis on the basis of final yield alone may be misleading, as although final yield represents a summation of the effects of plant interaction over the course of the experiment, it may also partly reflect initial differences. Initial size differences must be discounted to assess plant interactions adequately over the experimental period and make the comparison fair to both species. The final per unit size of a species will depend on both initial size and interspecific interactions (e.g. by an asymmetric effect such as increasing its shading impact more than pro rata to its size). These effects can be included as part of the explanation of subsequent performance (e.g. Connolly & Wayne 1996). This double role and use of initial size allows the experimenter to deal with situations where species ontogeny, other life-history traits, or direct experimental manipulation (e.g. of sowing date) lead to considerable differences in size between species at the commencement of the experiment. The sole use of final yield will also miss dynamic changes in species interaction (e.g. Connolly et al. 1990; Turkington & Jolliffe 1996) and is possibly the single most neglected and important issue in current practice.

The difficulty with density

Many experimental designs (e.g. including most RS designs or some additive designs) equate species on the basis of their numbers. However, simple equivalence on the basis of density can introduce size bias (Connolly 1986, 1997; Silvertown & Dale 1991; Grace et al. 1993) and thus distort an assessment of interspecific relations (e.g. Connolly 1986; Snaydon 1991). Snaydon (1991) gives the extreme example of the nonsense of equating densities of oak trees and daisy plants. Size differences may of course reflect life-history traits or natural conditions, and, where present, must therefore be allowed for, for example in the double way suggested in the section above.

Competition and single mixtures

Most assessments of interspecific interactions have used a single mixture (usually 50:50) in addition to the relevant monoculture(s). Data from additional mixtures or monocultures may be useful if they allow generalizations (since interaction may depend on the proportions and densities of the components) and will increase the precision of estimation of what is observable in the single mixture. However, a problem arises when such extra data contradict the findings obtained for a particular mixture (e.g. Benner & Bazzaz 1987; an example in Connolly 1997). In other words, because of the issues of size and density equivalence raised in the second and third sections above, a monoculture may not always be the appropriate reference point for assessing the interactions in a particular mixture.

Limitations on inferences (logical limitations vs. misuse)

There are very few useless experiments (Cousens 1996). However, the inferences that can be validly drawn from a particular experiment depend on the design used, the measurements taken and the analysis of the data. If these logical limitations to inference are not fully appreciated, e.g. the second and third sections above, then biased assessments will result. We distinguish these logical limitations, which lead to pushing the inferences beyond what the design and measurements would support, from inappropriate interpretation resulting from faults with the design per se (i.e. misuse) (Cousens 1996).

Predictive power

Most studies of interspecific interactions are short-term, frequently lasting less than a year, often measuring only vegetative growth rather than reproductive success, and based on one phase of the life-history of species, whilst largely ignoring the rest. We must not expect too much predictive power from such experiments, unless we are convinced that the phase being tested is critically important (e.g. vegetative biomass can be used as a measure of fitness in many annual plants; Goldberg & Fleetwood 1987). They will usually supply no more than indicators to the answers required by ecologists, although they may be of more direct use to the interests of agronomists (e.g. Shrefler et al. 1994).

Our discussion develops analyses of the relationships between (i) questions that appear to be asked in studies of interspecific interaction, (ii) variables that are measured, and (iii) the designs used, with a view to identifying the range and limits of questions that can be validly addressed using particular combinations of design and variables measured (J. Connolly, P. Wayne & F. A. Bazzaz, unpublished data). J. Connolly et al. concluded that the omission of initial information severely limits the inferences that can be drawn using several of the most common designs. In the case of RS, even with the provision of this initial information the range of inference is still quite limited, and in other designs the comparison of species may remain problematic. J. Connolly et al. also draw attention to the distinction between the outcome of competition and the effects of species on each other (see the first section above) as an issue that has led to confusion in interpretation of studies on species interaction.

Sp designs

In SP experiments (also called additive, equal proportions; Austin et al. 1988), mixtures consisting of a fixed, usually 1:1, ratio of the two species are maintained (Fig. 1a). SP designs have been used to examine the role of numerous factors in plant interactions, frequently using a range of treatments applied to a particular mixture of two species (see the WWW archive for examples). Additions of monocultures at appropriate densities can convert SP designs to AD (e.g. Gurevitch et al. 1990) or RS (e.g. Berendse et al. 1992) experiments. Some studies are difficult to classify as strictly SP, diallel or AD studies (e.g. Allen & Allen 1984, where the design is a partial diallel, with pair-wise comparisons of Salsola kali with two other species, but not between the other two species).

SP designs at a single relative frequency and density can be used, in a limited way, to address questions about the outcome of competition between two species. Measurements over time should be included to allow assessment of changes in relative abundance. However, SP designs do not allow assessment of the effects of species on each other, unless one or other species completely disappears. If final yield is the only parameter available then all that one can safely say is whether both species survived and which contributed most to final biomass. If an experiment includes pair-wise mixtures between more than two species, then comparisons of interspecific interactions for different mixtures may be problematic (J. Connolly et al. unpublished data).

SP designs therefore provide a useful, if limited, tool for screening the effects of a treatment gradient on the outcome of competition; they are efficient in that no resources are allocated to monocultures, which may not provide useful information on the question addressed. In addition, they are amenable to fairly straightforward statistical treatment, and the difficulty raised by the probable correlation between responses in a mixture can often be avoided by combining the responses to give a single per-pot measure. Perhaps SP designs are used less frequently than they should be.

Rs designs

In an RS, the planting density of the two constituent species may vary but the total density is held constant (Fig. 1b). The effect of other factors (e.g. a soil nutrient) on interaction between the components is tested by using a replicate RS for each of several levels of the factor. Ratio and replacement diagrams (Harper 1977) offer graphical presentation of results. Of the several indices that have been proposed to present the results of RS experiments (Trenbath 1978; Connolly 1986), relative yield total (RYT; de Wit & Van den Bergh 1965) is generally the most popular. The objective of some of these indices (e.g. relative crowding coefficients, de Wit 1960; competitive ratio, Willey & Rao 1980; coefficient of aggressivity, McGilchrist & Trenbath 1971) generally appears (although this is not often clearly stated) to be an attempt to assess the outcome of competition, whereas the RYT, a single value for the stand, relates to the joint capture and use of resources by the competing species (i.e. it describes a niche relationship). Although niche relationships contribute to understanding why a particular outcome occurs, they rarely predicate any particular outcome; thus the value of an index like RYT does not determine one way or another what impact niche separation will have on the outcome of competition.

Replacement designs have been widely used since they were introduced by de Wit (1960). Applications include aspects of inter- and intraspecific interactions between wild plants (e.g. Solbrig et al. 1988; Fone 1989), between wild plants (weeds) and crops (e.g. Ogg et al. 1993; Wall 1993), and between commercial cultivars of forage grasses (e.g. Frankow-Lindberg 1985). In the majority of studies, yield is the only character assessed, although other measures such as shoot/root ratios may perhaps help to illustrate the physiological basis of species’ interactions (Bi & Turvey 1994).

We identified five problems with the RS design that seriously undermine its usefulness as an experimental tool (see also Cousens 1996). (i) It is generally used with final yields only, which can lead to size bias in interpretation if species differ in initial size (Connolly 1986, 1997; Grace et al. 1992; but disputed by Shipley & Keddy 1994). (ii) The validity of the RS method rests on the assumption that individuals of the competing species are exactly equivalent at the start of the experiment (Keddy 1989). If seedlings are of quite different sizes then it is both difficult to see how they can be regarded as equivalent (Connolly 1986; Snaydon 1991) and impossible to eliminate size bias (J. Connolly et al. unpublished data). (iii) The outcome of competition is frequently confused with the effects of neighbours when interpreting results of RS. Including information from monocultures in the analysis can introduce bias in the assessment of species’ effects both on each other in a mixture and on the long-term outcome (Connolly 1986; J. Connolly et al. unpublished data). (iv) RS are carried out at a fixed, and often arbitrarily chosen, density (Inouye & Schaffer 1981; Taylor & Aarssen 1989; Snaydon 1991; Silvertown & Lovett Doust 1993) and results at that density may not generalize. Some of the density problem can be overcome by using replicate RS at different total densities derived from an additive series over a range of densities (Fig. 1d) (Firbank & Watkinson 1985; Cousens & O’Neill 1993). (v) Logistically, RS experiments necessitate tying up large numbers of experimental units (66% if only a singe mixture is used) in monocultures that may not contribute significantly to the analysis.

These problems lead to difficulty in correctly interpreting both RS diagrams and competition indices (Connolly 1986, 1988, 1997; Snaydon 1994). We are led to agree with the critics of this method (e.g. Inouye & Schaffer 1981; Jolliffe et al. 1984; Connolly 1986, 1988, 1997; Law & Watkinson 1987; Snaydon 1991, 1994): while RS may yield some useful information (Cousens 1996) it will be on a very limited range of questions. The tendency to misuse the method is so pervasive that its continued use should be discouraged.

Ad and target–neighbour designs

In the simplest form of AD designs (i.e. the partial additive) the density of the focal species is maintained across all mixtures and the density of the associate species is varied, usually with the goal of assessing the response of the focal species to increasing levels of the associate (Fig. 1c). More complex designs involve simultaneously varying the proportions of focal and associate species (i.e. addition series; Fig. 1d). This approach has useful applications, such as studying the impact of varying densities and distributions of weed populations on a crop sown at fixed density (Zimdahl 1980; Radosevich 1987; see the WWW archive). Additive designs have also been used to assess the role of various factors (e.g. relatedness, genotype, emergence time, initial plant size, maternal effects, herbivory) on a focal species’ response to its associate in situations where comparing intra- vs. interspecific interactions and distinguishing effects of species’ proportions from those of total density were less important objectives (see the WWW archive). They have been used for distinguishing allelopathic effects from resource exploitation due to density-dependent phytotoxic effects (Weidenhamer 1996; Weidenhamer et al. 1989).

A problem with this design is that the overall density and the proportions of focal and associate species can vary simultaneously and this confounding of variables makes the interpretation of results difficult (although not necessarily unrealistic compared with field situations) (Harper 1977; Silvertown & Lovett Doust 1993). Some of the problems of confounding the effects of species’ proportion and density can be overcome by independently manipulating densities of both species and analysing performance of the focal species via response surface methods (Firbank & Watkinson 1985; Law & Watkinson 1987; Fredshavn 1994).

Target–neighbour designs involve growing an individual of a ‘target’ species with varying abundances of ‘neighbours’, which could be either an associate species or itself. This is essentially an AD design in which the density of the focal or target species is reduced to a single individual or to a density low enough to preclude significant intraspecific interactions. This design has been used to address a variety of mechanistic questions about plant interactions (see the WWW archive for examples). Goldberg & Landa (1991) used it to determine which plant traits are responsible for differences in the effects and responses between species, and whether these two measures of interaction are related.

The per unit (per capita or per unit biomass) effect of neighbours on individuals of a target species is measured as the slope of a regression of target plant performance against the number (or biomass) of immediate neighbours. The target–neighbour design focuses on individual plant responses rather than the mean population response and estimates the importance of interspecific interactions relative to other factors in determining the fate and performance of individuals. While these measures on the target do give information at the individual plant level, they do not allow direct assessment of the outcome of competition for the target since the factors affecting the target may also affect the neighbours to the same or greater degree. For example, increasing the density of an associate may greatly reduce the performance of the target but it may also reduce the performance of the associate. Comparison of the impact on both species is essential in assessing the outcome of competition.

This approach claims numerous additional advantages. By measuring interference on a per unit basis it incorporates asymmetries in individual plant size at harvest among competing species. The relationships can be useful in interpreting features of interspecific interactions. Comparison of the slopes of the target performance–neighbour abundance regressions can be used as a quantitative measure of the effect (sensuGoldberg & Fleetwood 1987) of different neighbour species (Goldberg & Landa 1991). Statistical comparison of these slopes under different conditions may be made using ancova (e.g. Hartnett et al. 1993). However, as in all ancova, care must be taken in interpretation if the covariate is estimated after the commencement of the experiment as it may also contain effects of treatments that are discounted in the comparison of slopes. For example, the method compares the effects of two associate species on the target as if they had the same final yield and, if this is not the case, may lead to an unfair comparison. The competitive ‘response’ can be estimated from the slopes of regression coefficients when different target species are grown with the same neighbour species (Goldberg & Landa 1991; Hartnett et al. 1993). This general approach has been described in some detail by Goldberg & Werner (1983) for use in field-based studies. Discussion of some statistical considerations for these types of additive experiments is found in Goldberg & Scheiner (1993).

An advantage claimed for target–neighbour experiments is their economy in terms of both space and plants (Thijs et al. 1994; compare Hartnett et al. 1993 with Hetrick et al. 1994). However, caution is necessary in claiming greater efficiency for one design over another. The statistical criterion used to compare the efficiency of different designs should in each case be the experimental resource required to achieve a particular precision in the estimation of a particular parameter(s). However, considerations other than statistical efficiency may influence the selection of design and measurement: a design that is somewhat less efficient for one particular purpose may provide a far wider basis for inference and may thus be usable to address a wider range of questions.

A variation of the target–neighbour approach incorporates measurements of the distance, as well as biomass or numbers, of neighbours, and so allows the decreasing effects of ‘non-nearest neighbours’ to be incorporated.

In practice, AD and target–neighbour designs often consider only final yield (but see Gibson & Skeel 1996) and so suffer from ignoring the time–course of interactions and initial differences in species’ size. As well as confounding species density and relative frequency, they sometimes equate species simply on the basis of density (e.g. in comparing regression coefficients for different neighbour species where density is the independent variable in the regression). Thus conclusions may well be affected by size bias in a manner similar to RS, leading to certain species being judged more competitive simply because they were initially larger. Even if all information on initial sizes is available, a partial additive or additive series will not allow the same range of questions to be addressed as a response surface approach would (e.g. Connolly & Wayne 1996).

Despite the biases that can occur with these methods, comparisons among a range of treatments applied to the same additive series and species will give a basis for ranking treatments relative to each other, even if the absolute level of the effects may be biased. However, comparisons of treatments across species potentially suffer from difficulties unless initial size differences are measured and accounted for.

Response surface methods

An experimental design that includes a range of densities and relative frequencies of the species under study (not necessarily including any monocultures, e.g. Connolly & Wayne 1996; Ramseier et al. 1996) may be used to generate response models for each species. Such a design allows the fitting of regression-style response models relating some measure of per capita performance for each species to the density of each species (e.g. Suehiro & Ogawa 1980; Spitters 1983; Connolly 1987; Law & Watkinson 1987), the initial biomass of each species (Connolly & Wayne 1996) or some other initial measure of biological potential, such as early leaf area index of each species (e.g. Kropf & Spitters 1991). The response models and their parameters are used to assess species interaction. These methods avoid some of the problems inherent in the analysis of replacement and additive designs, and in diallel analysis (e.g. Law & Watkinson 1987; Bullock et al. 1995; Connolly & Wayne 1996). As with additive designs, the inclusion of initial and intermediate measurements allows the study of species’ interactions over time. Connolly et al. (1990) and Menchaca & Connolly (1990) report changes in species’ interactions over time that would have been totally overlooked in an analysis of final yield only. Indeed, the conclusions drawn from a response surface analysis incorporating the time–course of plant–plant interactions can be qualitatively different from, and more effectively predict the outcome of competition than, those derived from an RS (Connolly et al. 1990; Grace et al. 1993). The inclusion, for example non-destructive leaf demographic measurements, can provide a tool for time series/growth dynamics to be made.

Several ways of designing experiments for response surface models have been described. These include establishing an RS at several total densities, called an addition series (e.g. Spitters 1983; Connolly 1987; Radosevich 1987; Rejmánek et al. 1989; Rodriguez 1997) or establishing additive series (Fig. 1d, which can be regarded as either an additive design or a number of RS at different densities), similar to the bivariate factorial defined by Snaydon (1991). Any set of mixtures that allows the fitting of bivariate response models will suffice. In the absence of a statistical assessment the choice of optimal method is a moot point and may vary with the question being addressed.

Despite their definite superiority to RS and AD designs and methods, the response surface methods may also suffer from similar size bias in the estimations of species’ effects and responses and the outcome of competition, unless initial differences are allowed for and the appropriate response measurements are analysed (J. Connolly et al. unpublished data). An example that corrects for initial differences is given in Connolly & Wayne (1996). Furthermore, there are several statistical issues in the fitting of some of these models (and those for AD), e.g. there is often a decrease in variance with decreasing plant size (Connolly et al. 1990) that should be allowed for. In estimating hyperbolic yield–density relationships it is preferable to use weighted regression, non-linear methods or the generalized linear model approach (Nelder & Wedderburn 1972) available in many statistical packages.

Diallel designs

Diallel designs use ‘all possible combinations of n species’, i.e. ‘n(n − 1)/2 separate RS of two species, each represented by two pure stands and one equiproportioned mixture’ (Harper 1977, p. 268; Trenbath 1978; but see also Gleeson & McGilchrist 1980 for unequal proportion extensions). Interspecific interactions are assessed using RS methods (Gurevitch et al. 1990) or by analysing a matrix of species’ performance using anova and/or covariance analysis (Trenbath 1978). A matrix of ‘competition coefficients’ (sensuFirbank & Watkinson 1985) calculated as slopes of regressions in a series of pair-wise target–neighbour experiments can also be analysed by diallel methods. Data from diallel designs carried out at more than one density may be analysed by response surface methods (Connolly 1987).

Diallel designs are derived from genetic analysis (Durrant 1965) and have been used extensively in the greenhouse and field by plant breeders and agronomists to assess interspecific interactions between cereal varieties and among forage grasses (e.g. Norrington-Davies & Hutto 1972; Rousvoal & Gallais 1973). Applications to better understand natural systems include Aarssen's (1988) study of four pasture species, Taylor & Aarssen's (1990) study of the interactions among 10 genotypes of three perennial grasses, and Aplet & Laven's (1993) study of the competitive hierarchy of four Hawaiian shrubs. The debate on competitive hierarchies (see below) relies heavily on results from experiments using diallel designs.

The diallel design at a single density is subject to the same difficulties in interpretation as RS.

Hexagonal fan designs

Most experiments on interspecific interaction focus on the mean population responses of species (e.g. species yield) under varying densities and species’ proportions. However, an important feature of plants and other sessile organisms is that they do not sense or respond to overall population density or frequency, but only interact with their immediate neighbours (Harper 1977). This principle argues strongly for designs, such as the target–neighbour and fan designs, that focus on the interaction between a plant and its immediate neighbours. Mead (1979) lists five spatial factors that may affect interspecific interactions between two species and hence can be included as factors in design, namely the density and the intraspecific spatial arrangement of each species and the intimacy of their interspecific arrangement. There are many approaches to the study of intraspecific interactions between individuals (Firbank & Watkinson 1987), and some of the statistical issues were reviewed by Mead (1979). Fan designs were the only ones found in our literature survey.

Hexagonal fan designs utilize a particular plant spacing pattern involving a honeycomb of overlapping hexagons such that each individual is surrounded by zero to six intraspecific neighbours and six to zero interspecific neighbours. This array of hexagons is arranged in a plant spacing gradient (fan design) with plants positioned in a particular pattern, such as a polar coordinate grid or a parallel row design (Nelder 1962; Bleasdale 1967). Thus fan designs vary density and frequency and select a particular form for intraspecific spatial arrangement and interspecific intimacy (see illustrations in the studies listed in the WWW archive).

Hexagonal fan experiments developed as a combination of the fan designs used in agronomic trials to examine the effects of plant density (Nelder 1962), and hexagonal planting designs were developed by Boffey & Veevers (1977) to study the effects of neighbour species’ frequencies. Thus they have the advantages of incorporating variation in species’ proportions and densities and the local spatial distribution of neighbours in assessing the response of individual plants to neighbours. In addition, they can be significantly more efficient in use of greenhouse space relative to other designs (Antonovics & Fowler 1985). Schmid & Harper (1985) used a fan design to show that interspecific interactions change in varying ways with changing density, sometimes with complete reversals of competitive outcomes between two species at different total densities. In addition, hexagonal fan experiments help in the assessment of optimal planting arrangements in mixed-cropping systems and facilitate the analysis of frequency and density-dependent selection in genotype mixtures (Antonovics & Fowler 1985).

The primary advantages of hexagonal fan designs are their focus on neighbourhood interactions, their efficiency in use of space and plants, and their ability to allow assessment of interspecific interactions across a range of densities or plant spacing patterns. However, there are statistical problems associated with the analyses of such designs (Mead 1979; Antonovics & Fowler 1985): they are unrandomized and so may be biased due to underlying trends in fans, the correlated responses in neighbouring plants may require a more complex analysis, and they may have limitations in situations in which second or third nearest neighbour effects and more diffuse interactions are significant. Often the analysis of these designs assumes that ‘non-nearest neighbour’ effects are insignificant. In addition to these statistical difficulties, size bias may arise if initial size differences are not discounted. Like all studies of individual rather than mean response, they require a greater input of time and labour. These designs can be extended to study multi-species interactions (see below).

Designs to assess multi-species interactions

Despite attempts to provide the greatest degree of realism to interaction experiments, greenhouse experiments involving interspecific interactions among mixtures of three or more species, i.e. diffuse or multi-species interactions (MacArthur 1972), have been infrequent. This is perhaps not surprising given the logistical and statistical problems inherent in the effective design and interpretation of just the multiple pair-wise interaction experiments of the diallel design (Mitchley 1987). The growth of multi-species mixtures under various treatments can be used simply to assess the outcome of competition (e.g. Grime et al. 1987), but we have identified five further main approaches to assessing multi-species interactions in the greenhouse.

(i) Fowler (1982) showed that, in a three-species RS design (de Wit 1960), predicted yield per plant was statistically related to the observed yield per plant. Interpretations agreed with results obtained from pair-wise RS experiments. This method is subject to the same criticisms as the RS with two species. (ii) The performance of each species in a multi-species mixture compared with its performance in monoculture is a form of multi-species AD (see the WWW archive and Ellenberg 1954; Mueller-Dombois & Sims 1966; Pickett & Bazzaz 1978; Austin 1982). (iii) Rejmánek et al. (1989) applied reciprocal yield regression models to a three-species complete additive experiment but the results did not support the interpretations drawn from previous two-species investigations of the species. (iv) Plants can be grown in hexagonal arrays (see above) in which a species has each of the different species under investigation as a neighbour, but never itself. Thórhallsdóttir (1990) and Turkington (1994) used this approach to investigate the role of interspecific interactions on the spatial dynamics of grasses. While elegant, problems with this design include those mentioned above for hexagonal designs and colleagues discussed in Thórhallsdóttir (1990). (v) Ramseier et al. (1996) proposed a simplex design (Cornell 1990) for multi-species experiments in which all species appear in each of a number of mixtures (the minimum number of mixtures is the number of species + 1) but in different relative frequencies, each species in turn being the largest component of a sown mixture with the other species being equally represented, with an additional mixture having all species equally represented. Repeated at a number of densities and with initial sizes of species measured, this design allows a response surface analysis in which questions of outcome and effects of species on each other may be assessed. Additional design points may be added and the order of interaction terms that can be assessed in the model depends on the structure and number of design points. Advantages claimed for the particular simplex design used are that each mixture is an experimental community with all species represented, that the spread in community type allows the examination of interspecific interaction over a wide range of systems, that resource use is efficient in that there are no resources devoted to monocultures, and that it can be readily extended to larger numbers of species in a coherent manner without a major increase in experimental size. Disadvantages are the possible complexity of a full statistical treatment. The problems raised earlier with other designs must be borne in mind when using any of these multi-species approaches.

Background species

Interspecific interactions are sometimes examined by establishing a spacing gradient grid of one species and overseeding the entire grid with a second species. This attempts to assess the effects of varying intensities of intraspecific interaction under the constant influence of a ‘background’ (Radosevich 1987), although the idea of a constant influence on all species may be illusory. Such an approach ignores the reciprocal nature of many interspecific interactions, such that the introduced individuals generally influence the background species as well as being influenced by it. Over time the background will tend to respond differentially to different species and so what started as a common influence may rapidly cease to be so. This will occur at a localized level in the vicinity of the introduced individuals. Further, an individual or a unit of initial biomass will tend to have less effect at high compared with low density. In addition, a given density of a background species will have a smaller per unit effect on a fixed density of large rather than small introduced individuals of other species, since the overall effective density with large introduced individuals is greater than with small introduced individuals and so like is not being compared with like.

Competitive hierarchies

Results from using several types of design (i.e. AD, target–neighbour, RS and diallel designs) have been prominent in the search for competitive hierarchies in which a species will out-compete (in the sense of outcome of competition) all species ranked below it in the hierarchy and be out-competed by those above it (Keddy & Shipley 1989; Shipley 1993). The occurrence of reversals of rank order, indicating either a network of competitive performance (intransitivity) (Herben & Krahulec 1990; Silvertown & Dale 1991; Shipley 1993) or competitive combining ability, is controversial (Taylor & Aarssen 1990). However, these designs as usually analysed are prone to the misinterpretations and dangers of size bias (Silvertown & Dale 1991; Grace et al. 1993; Connolly 1997; J. Connolly et al. unpublished data), with factors such as differences in seed size or initial seedling mass providing a mechanism for that bias (although see Shipley & Keddy 1994).

Received 6 October 1997