Introduction

Competition is widely accepted as a major process in shaping plant communities. ‘One of the underlying assumptions for both theoretical and empirical community ecology is that the processes determining community composition and abundance of species are interactions specific to particular pairs of species’ (Goldberg & Werner 1983, p. 1098). Pairwise (1 : 1) competition experiments have therefore become a major tool for the study of plant competition and are still widely used to assess different functional aspects of competitive interactions. This approach has been successfully used to test the effect of varying resource availability (Van Mierlo et al. 2000), resource distribution (Fransen et al. 2001), herbivory (Siemens et al. 2002), species density (Howard 2001) and facilitation (Tewksbury & Lloyd 2001), as well as below-ground aspects such as root competition (Cahill 2002; Rajaniemi 2003), mycorrhization (Marler et al. 1999) and root herbivores or pathogens (Ridenour & Callaway 2003). Pairwise competition experiments have also been used to analyse competitive hierarchies and predict the outcome of multi-species competition in natural communities (Keddy & Shipley 1989; Keddy et al. 1994, 1998, 2000; Roxburgh & Wilson 2000a,b). However, an individual plant in the field will rarely encounter a single-species neighbourhood, but rather a mixture of different neighbouring species. The effects of such multi-species environments might be separated into: (i) direct effects of a neighbour species on a target individual; and (ii) indirect effects produced when one neighbour species changes the direct effect of one or more different neighbour species on the target individual (Miller 1994; Wootton 1994a,b). The latter is also referred to as ‘higher order interactions’ in animal ecological studies (Vandermeer 1969; Abrams 1983).

To date, it is not clear whether a plant's multi-species environment can simply be described as the sum of its pairwise (direct and indirect) interactions. An extrapolation of results from pairwise experiments to whole communities was successful in some animal (Vandermeer 1969; Pomerantz 1981) and plant studies (Fowler 1982), but others revealed partly non-additive effects leading to significant differences between the results of multi-species mixtures and what would have been expected from the species pairs (for animals: Wilbur 1972; Neill 1974; Case & Bender 1981; Wilbur & Fauth 1990; Wootton 1993; but see Pomerantz 1981 for criticism; for plants: Miller 1994; Dormann & Roxburgh 2005). In this paper, non-additive effects will be referred to as deviations of target plant growth in multi-species mixtures from the expected performance calculated as a simple sum of effects of individual neighbour species. This definition of non-additive effects includes indirect interactions based on per gram effects of neighbours, which are more generally considered additive (Miller 1994; Dormann & Roxburgh 2005).

Many experimental ecologists applied removal techniques in natural communities as a more realistic approach, often including multi-species competitive interactions. Single species (Miller 1994; Herben et al. 1997; Dormann et al. 2000; Gerdol et al. 2000; Bertness & Ewanchuk 2002; Fowler 2002; Gebauer et al. 2002; Brewer 2003) or functional groups of species (Köchy & Wilson 2000; Bret-Harte et al. 2004) have been removed to study competitive interactions in plant communities. In this investigation, we quantify the effect of a ‘natural’ community on a target species by measuring target performance with and without the competing matrix, but we gather no information about the fraction each single species contributes to this effect or if there is any competitive release or amplification due to interactions between neighbouring plants.

A different methodological approach yielding similar results is the study of natural or experimental biodiversity gradients, where the species pool is increased by seeding (or planting) instead of removing species. The ongoing debate regarding the effect of biodiversity on ecosystem functioning often involves the relationship between competition intensity and biodiversity of the system. These changes, however, are yet to be proven. Several examples of transgressive overyielding (i.e. higher biomass production in mixtures than in the most productive monocultures) have been published, which might indicate that species differences in resource demand or acquisition can result in competitive release in multi-species mixtures (Schmid et al. 2002; Tilman et al. 2002; Roscher et al. 2005; but see Hooper & Dukes 2004). The effect of changing biodiversity on single species or even individual plants, rather than on plant communities, is mainly addressed in studies of plant invasion within biodiversity experiments (Tilman 1997; Knops et al. 1999; Levine 2000; Naeem et al. 2000; Symstad 2000; Dukes 2001; Hector et al. 2001; Pfisterer et al. 2004; Turnbull et al. 2005). Several studies have indicated that competition of multi-species mixtures against invaders depends on the specific species combination, which might hint towards indirect and partly non-additive interactions (Fargione et al. 2003; Van Ruijven et al. 2003; Fargione & Tilman 2005). Here again, inferences concerning the effect of specific multi-species mixtures relative to their monocultures are missing and experimental approaches to unravel the mechanistic pathways of multi-species competition are rare (Milbau et al. 2005). Although the current study was not designed to investigate functional aspects of biodiversity, the results may provide some insight for this field of research.

Finally, in theoretical community ecology, many models simulating competitive interactions in plant communities are based on two-species interactions (see Klausmeier & Tilman 2002 for review; Chesson 1994; Ohsawa et al. 2003). However, recent models increasingly focus on the simultaneous growth of multiple plant species with different competitive or physiological traits (Huisman & Weissing 2001; Zhai et al. 2004). Most interestingly, one of the main conclusions of Huisman & Weissing's modelling investigation was that ‘the winners of multi-species competition can be as predictable as a throw of the dice’ and that several outcomes of their model may exhibit transient chaos. From this result it seems reasonable to assume the non-additivity of competitive interactions between plants. A recent study by Dormann & Roxburgh (2005) indeed found mainly non-additive effects of multi-species competition in a combined experimental and modelling approach. Their model comprises indirect interactions between neighbouring species and defines all additional effects as non-additive, while neighbour density is not considered.

In this study, we investigated whether the competitive effect of multi-species mixtures is an additive function of the effects of single species. We used four common species (Corynephorus canescens, Jasione montana, Festuca ovina ssp. psammophila and Hieracium pilosella) of dry acidic grasslands in a field-like competition experiment on sand and studied the effects of both pairwise and multi-species mixtures on the target species Hieracium pilosella. Our data were analysed using two different approaches: (i) we calculated the relative yield and the relative yield total to compare expected and observed target plant performance; and (ii) we fitted a series of hyperbolic models based on neighbour density to test whether target growth in multi-species mixtures can be predicted from pairwise mixtures and what might be the mechanisms behind possible non-additivity.

Materials and methods

competition experiment

Multi-species competition was tested in a target-neighbour design with H. pilosella as the target and H. pilosella, C. canescens, F. ovina and J. montana as neighbouring plants. We planted individual plants (single growing controls) and individuals with six conspecific neighbours of H. pilosella (monoculture) or with the other three species (one-species neighbour mixtures), as well as all combinations of two- and three-species neighbour mixtures with H. pilosella as the target species (Fig. 1), resulting in 15 different treatments (one control, one monoculture, three one-species neighbour mixtures, six two-species and four three-species neighbour mixtures with Hp target). Each combination had six replicates. Plants were grown under field-like conditions in an experimental ‘sand pit’, divided into four isolated chambers (length, 6 m; width, 5 m; depth, 1.2 m) filled with pure sand in a common garden area next to the University of Bielefeld, Germany.

Plants of all species were grown from seed and sown for germination 2 months prior to the start of the experiment. Seed material was collected from former sand-pit plants for C. canescens and H. pilosella (which were grown from seeds collected from approximately 20 maternal plants in a sand dune area near Bielefeld 08°40′ E, 51°57′ N), and bought for F. ovina and J. montana: (Bauerreiß-Company, Bad Windsheim, Germany). Mean total dry weight (± SE) per plant at the start of the experiment was greater for the herbs (H. pilosella, 0.074 ± 0.007 g; J. montana, 0.060 ± 0.004 g) than for the grasses (C. canescens, 0.052 ± 0.004 g; F. ovina, 0.038 ± 0.003 g). Treatments and controls were planted on 23 and 24 April 2001 in a complete randomized block design in hexagonal plots as shown in Fig. 1. Following 5¹/₂ months of growth, all plots were completely harvested (above and below ground) from 9–11 October 2001. The root systems were carefully washed out and separated in the laboratory. The above- and below-ground plant material was dried for at least 72 hours at 70 °C.

data analysis

To compare the expected and measured effect of competition by specific multi-species mixtures we used an altered version of the relative yield (RY) index (Fowler 1982). It is a rather coarse measure calculating the expected performance of the target plant in multi-species mixtures as the mean performance from the respective pairwise plots. The observed RY is calculated as RY_observed = F_t/F_m, where F_m is total dry weight of the H. pilosella target plant in monocultures and F_t is the total dry weight of the target plants grown in one-, two- or three-species neighbour mixtures. From the observed RY of one-species neighbour mixtures (RY_i,observed), the expected RY of multi-species neighbour mixtures is than calculated as follows:

(eqn 1)

where the summation is over all neighbouring species. The difference between both values (ΔRY = RY_observed −RY_expected) is zero if the measured target biomass equals the expected biomass. ΔRY > 0 indicates a better performance of the target plant than would have been expected from monoculture effects.

The idea is similar to the relative yield total (RYT), which is often used to quantify the overyielding of diverse mixtures relative to monocultures in studies of biodiversity effects on ecosystem function (see Fridley 2001 and references therein). The only difference is that this index is calculated on the total yield per plot basis while the above RY was calculated using target plant biomass alone.

(eqn 2)

where s is the total number of species in a mixture. RY_i = F_T/F_M, where F_T is the observed mixture yield per plot of species i and F_M is the monoculture yield per plot of species i. RYT > 1 indicates overyielding.

Target and neighbour biomass were tested for differences between species and treatments using factorial anova and post-hoc Tukey (HSD) tests. All dry weight data were log transformed to meet assumptions of normality and homogeneity of variances. All statistical analyses were performed with Statistica for Windows (Version 6.1, Statsoft Tulsa).

regression model analysis

The comparison of target plant biomass as well as the calculation of a competition index (ΔRY and RYT) represents a straightforward method analysing the differences in treatment effects and also the one most often used in studies on plant competition. For multi-species mixtures, however, they only provide an incomplete reflection of the competitive interactions because they ignore differences in neighbour densities and biomass. For a more detailed analysis of possible non-additive interactions and their mechanisms in multi-species mixtures we used a regression model approach. A variety of studies (Watkinson 1981; Firbank & Watkinson 1985; Law & Watkinson 1987; Silander & Pacala 1990; Freckleton & Watkinson 1997, 1999, 2000a,b, 2001a,b) have shown that plant competition is well described by the hyperbolic function:

F_t = F_m(1 + α₁N₁ + α₂N₂ + α₃N₃ + α₄N₄)⁻¹(eqn 3)

where F_m is control plant biomass and F_t is biomass of Hieracium pilosella target plants in intra- or interspecific mixtures. N₁–N₄ are the densities of the four species, with 1 = Hieracium pilosella, 2 = Corynephorus canescens, 3 = Festuca ovina and 4 = Jasione montana. The parameters α₁–α₄ quantify the species-specific competitive effects on the target. The model is fitted by the maximum likelihood method assuming a gamma distribution of F_t with shape parameter k (Pacala & Silander 1990), which was confirmed to be superior to the normal distribution when describing the random deviations from the model, for our data (data not shown). Equation 3 describes model number 1 in Table 1 and will henceforth be referred to as ‘model 1’. The hyperbolic model was originally derived for systems with annual plants where the number of individuals per pot is related to the individual's biomass in a hyperbolic way in equilibrium. However, even though we worked with perennial and clonal plants for a period of 6 months, all competition treatments showed pronounced species interaction well before the end of the experiment and the vegetation cover resembled natural early successional stages on sand dunes. The good fit of the hyperbolic models also suggests that it is suitable for our investigation.

Table 1.  Regression models to describe target plant biomass production and the respective log-likelihood (loglike) and AIC values for model fit (significant differences from model 1 are indicated in bold). The basic model (density only) fits one parameter for the density of each competing species separately. β is the non-additivity parameter added to the model given different preconditions, with species abbreviations as follows: Hp = Hieracium pilosella, Cc = Corynephorus canescens, Fo = Festuca ovina and Jm = Jasione montana. R² is the coefficient of determination for the regression between estimated and observed mean values per competition treatment. P is the number of parameters in the model. For detailed model descriptions see Appendix S1

Nr	Model description	β	R2	P	loglike	AIC
1	Density only		0.60	5	1.924	6.152
2	Full model		0.82	12	7.893	8.214
3	Density +β if Hp > 0	0.315	0.50	6	2.851	6.298
4	Density +β if Cc > 0	0.397	0.67	6	3.555	4.890
5	Density +β if Fo > 0	−0.272	0.62	6	2.692	6.616
6	Density +β if Jm > 0	0.085	0.59	6	2.026	7.948
7	Density +β if Hp > 0 and Cc > 0	0.692	0.77	6	4.597	2.806
8	Density +β if Hp > 0 and Fo > 0	−0.132	0.59	6	2.053	7.894
9	Density +β if Hp > 0 and Jm > 0	0.403	0.43	6	3.055	5.890
10	Density +β if Cc > 0 and Fo > 0	0.039	0.60	6	1.934	8.132
11	Density +β if Cc > 0 and Jm > 0	0.502	0.66	6	3.979	4.042
12	Density +β if Fo > 0 and Jm > 0	−0.287	0.64	6	2.678	6.644
13	Density +β × total neighbour biomass	−0.012	0.5	5	2.213	7.575
14	Density +β × root neighbour biomass	−0.040	0.60	5	2.245	5.510
15	Density + total biomass per species		0.83	9	12.150	−6.300
16	Density + root biomass per species		0.78	9	9.014	−0.028

If the results from pairwise competition experiments were directly transferable to multi-species communities, model 1 should provide a comparably good fit for both the one-species neighbour mixtures and the multi-species communities. To investigate non-additivity, we predicted target plant performances in multi-species treatments based on species-specific effects, estimated from all treatments containing H. pilosella (Hp) and at most, one additional species (neighbour species: Hp, Cc, Fo, Jm, Hp Cc, Hp Fo, Hp Jm using species abbreviations given in Table 1). For each multi-species treatment, the degree of deviation from additivity was quantified by an additional parameter β_i (i = 1, … ,7). The full model in multi-species competition (model 2 in Table 1) based on neighbour species density could therefore be written as:

(eqn 4)

The seven different multi-species treatments (neighbour species: Cc Fo, Fo Jm, Hp Cc Fo, Hp Cc Jm, Hp Fo Jm, Cc Fo Jm) comprised half of the plots in our experiment. A similar approach was used in the models 3–12 (Table 1 and Appendix S1 in Supplementary Material) where the competition coefficients (α₁–α₄) were all estimated based on the pairwise treatments and those multi-species treatments that are considered to be adequately described using additive effects (e.g. all multi-species treatments not containing neighbouring Hp individuals for model 3). It is important to note that our models are restricted to interactions between target and neighbour plants. Hence, all interactions between neighbours are defined as non-additive, including indirect interactions in a stricter sense as defined by Dormann & Roxburgh (2005). In their approach Dormann & Roxburgh (2005) included a neighbour interaction term, while neighbour density was not considered. These differences originate from the different experimental designs used in both studies. The overall fit of all models was compared using Akaike's Information Criterion (AIC). Smaller values of AIC indicate a higher predictive power of the respective statistical model. A better fit of model 2 (including the non-additivity parameters) compared with model 1 (excluding non-additivity parameters) could therefore be interpreted as an important missing factor to describe target plant growth in multi-species mixtures (see Dormann & Roxburgh 2005). As non-additivity of competitive effects may be caused by particular species or specific species combinations, instead of being a general mechanism, we also fitted a set of regression models where the non-additivity is described in a species-specific, more parsimonious way. Here again, we compared these models with model 1, where additivity is assumed.

To get further information about possible mechanisms of non-additivity, we fitted the models 13–16 (Table 1) with additional parameters. As neighbour size or biomass may be important parameters for the outcome of competitive interactions we used total neighbour biomass (F_total) or species-specific neighbour biomass (F₁–F₄) as additional variables to predict performance of target plants under multi-species competition. For a detailed description of the models see Appendix S1. The most comprehensive of our models uses densities and the biomass of all neighbouring species as predictors of target plant performance:

F_t = F_m/(1 + α₁N₁ + α₂N₂ + α₃N₃ + α₄N₄ + β₁F₁ +β₂F₂ + β₃F₃ + β₄F₄)⁻¹(eqn 5)

This model is a generalization of model 1, where the parameters β_i can be interpreted as adjustments of the competition coefficients α_i according to differences in realized per-individual biomass:

F_t = F_m/(1 + (α₁ + β₁F₁/N₁)N₁ + (α₂ + β₂F₂/N₂)N₂+ (α₃ + β₃F₃/N₃)N₃ + (α₄ + β₄F₄/N₄)N₄)⁻¹(eqn 5a)

The competitive effect of a particular species i on the target plant is therefore described by a combination of parameters α_i and β_i and not by the individual parameters. Again, a comparison of the model's predictive power with the original model 1 would provide information on the importance of these extra parameters.

All model parameters were estimated by maximum likelihood using the optim-function of the statistical software package R (Version 2.2.0). Approximate standard errors for the parameter estimates were derived from the Hessian matrix provided by the optim-function.

Results

target plant biomass andΔry

The mean total dry weight of H. pilosella target species revealed substantial competitive interaction between target and neighbouring plants with a mean biomass reduction of 57 ± 3.5% (mean ± SE) compared with individual control plants. There was a significant overall effect of competition treatments on target plant biomass (anova, F_14,6 = 3.84, P < 0.001), but only a few significant differences between specific combinations of neighbour species (Fig. 2a). The results reveal that the target biomass of the control plants and of those competing with a mixture of H. pilosella and J. montana (Hp Jm) was significantly higher than the biomass of plants in competition with F. ovina (Fo), with a mixture of H. pilosella and C. canescens (Hp Cc), with the three-species neighbour mixtures of C. canescens, F. ovina and J. montana or H. pilosella, C. canescens and J. montana (Cc Fo Jm and Hp Cc Jm, Tukey HSD P < 0.05, n = 6). Targets in this last mixture also remained significantly smaller than targets grown in competition with J. montana (Jm). The target biomass of H. pilosella indicates a trend of a competitive hierarchy with F. ovina > C. canescens = H. pilosella > J. montana. The mean target biomass revealed no overall significant influence of the number of neighbouring species on competition intensity (anova, F_2,23 = 1.32, P = 0.28). This competitive response (sensu Goldberg & Werner 1983) of H. pilosella targets was equal for monocultures (or one-species neighbour mixtures) and two-species neighbour mixtures, and not significantly lower for three-species neighbour mixtures (mean target biomass 0.61 ± 0.07, 0.58 ± 0.05 and 0.48 ± 0.06, respectively).

The ΔRY is significantly different from zero in two out of eight cases (Hp Fo and Hp Cc Jm) and marginally different (P = 0.08) for one more neighbour mixture (Hp Cc) (Fig. 2b). In all three cases the difference is negative, indicating that the target grows worse in mixtures than would have been expected from monocultures.

target plant regression models

The simple hyperbolic model reveals that H. pilosella target plant biomass production can be described using species-specific neighbour plant density alone as a descriptive variable (AIC = 6.15, model 1, Table 1). A linear regression between the observed and estimated target plant mean biomass production per competition treatment yielded an R² of 0.60 (Table 1, Fig. 4a). In contrast to this simple model 1, which assumes additivity, models 2–16 (Table 1) all include additional parameters to describe a possible non-additive effect of multi-species mixtures. According to our definition of non-additivity, this effect would include indirect effects in general. From this set of models, model 2 (Table 1) is the most general and complex one that includes a separate parameter for each specific multi-species mixture. The fit of model 2 is good (AIC = 8.21), but the predictive power is significantly less compared with model 1, even though the linear regression has a much higher R² (0.82). In the search for a simpler model that describes our data set in a more parsimonious way, we performed a more specific analysis adding β only in the presence of single species or species pairs (models 3–12, Table 1). These show that in eight out of 10 cases there is no model improvement compared with the density model 1. In combination with the presence of C. canescens and one of the herbs, however, the non-additivity parameter β significantly increases the predictive power of the model (AIC = 2.81 and 4.04, models 7 and 11, Table 1, respectively). Hence, in these cases, models that take an additional parameter into account result in more accurate predictions of target plant biomass than do models that assume additive pairwise effects.

To test for possible mechanisms of the additional effect we also analysed a set of four models including neighbour biomass. We separately added the summed total neighbour biomass as well as the summed root neighbour biomass to the density model. Neither resulted in a significant improvement in model fit (AIC = 7.58 and 5.51, models 13 and 14, Table 1, respectively). The inclusion of species-specific biomass, however, has a large and significant effect on the predictive power of the model (AIC = −6.3, model 15, Table 1). The regression between observed and estimated target plant biomass of the latter model yields an R² of 0.83 (Fig. 4b). Adding species-specific root biomass alone also results in a significant increase in the predictive power of the model (AIC = −0.03, model 17, Table 1), with a slightly lower R² of 0.78 for the comparison between observed and estimated target plant biomass.

neighbouring plants and ryt

Generally, we would expect target performance to be strongly affected by neighbour biomass. There was a significant overall treatment effect on neighbour biomass (anova, F_14,6 = 17.48, P < 0.001). The one-species neighbour mixtures with H. pilosella and J. montana and the combination of both herbs, had significantly less biomass than all other neighbour treatments (Fig. 2c) and target biomass for these three treatments indicates a lower competitive effect of the herbs compared with neighbour mixtures where grasses are included (Fig. 2a).

Apart from the effect on the target, however, neighbour species have an effect on each other, which in turn can either release or magnify their combined effect on the target (indirect effects). We therefore looked at the biomass of individual neighbour plants of the four study species. For C. canescens and J. montana, the dry weight per plant did not significantly change depending on whether these plants grew as one-species neighbour mixtures or different two- or three-species neighbour mixtures (Fig. 3a,b). The growth of neighbour plants of H. pilosella was significantly greater in combination with J. montana (Hp Jm), while individuals of all other treatments showed similar growth (Fig. 3c). In addition, F. ovina plants grew better in combination with J. montana (Fo Jm), but this effect was only significant when compared with two three-species neighbour mixtures (Fo Hp Cc and Fo Cc Jm, Fig. 3d). Moreover, the comparison between biomass per plant in intraspecific competition compared with interspecific competition (indicated by dashed lines in Fig. 3) indicates that both herbs tend to grow less well in neighbour mixtures (Fig. 3b,c) while the grasses tend to grow better or equally (Fig. 3a,d). Significant differences, however, were only found in the aforementioned cases.

These differences between species growth in mixtures compared with their respective monocultures (or one-species neighbour mixture) are summed up in the RYT. The RYT per plot shows little indication of overyielding in the multi-species mixtures. It is only marginally different from unity in the two-species neighbour mixtures of F. ovina with either C. canescens or J. montana (Cc Fo and Fo Jm). It is important to note that these are not the mixtures indicating non-additivity from target plant growth (Fig. 2b). Overyielding does not seem to be related to non-additivity.

Discussion

Overall, we find additive competitive effects for the majority of multi-species combinations in our study, which would imply that multi-species competition can be predicted from additive functions of single species effects. However, we also find significant deviations from this result depending on specific species combinations. Overall, target plant biomass production is therefore best predicted using a non-additive model including density as well as species-specific total biomass of neighbouring species, where root biomass is one crucial factor.

The target biomass of H. pilosella for the monoculture and the one-species neighbour mixtures shows a trend towards a competitive hierarchy with F. ovina > C. canescens = H. pilosella > J. montana. From the measures included here, we can say that neighbour biomass is one important reason for this ranking in competitive effect, but additionally some species-specific traits must play a role as well. Previous studies on competitive interactions between dominant plants of inland sand dunes revealed similar results. However, in those cases, strict competitive hierarchies were mainly due to below-ground competition (Weigelt et al. 2002, 2005), where neighbour biomass had an important effect but still left a considerable part of the variation unexplained (Weigelt et al. 2002). The comparison between pairwise and multi-species competition, revealed that the competitive response of target individuals of H. pilosella was species-specific and did not depend on the species richness of the neighbours. There was little indication of a general competitive release or amplification in multi-species environments. This changes if we include the specific comparisons of all species combinations (Fig. 2a).

From the competitive hierarchy, it could be expected that neighbour mixtures with both C. canescens and F. ovina should have the highest competitive effect on the target, but instead the trend points towards a mixture with C. canescens, H. pilosella and J. montana as being most competitive. However, the combined effect of H. pilosella and J. montana in a two-species neighbour mixture does have the lowest effect on the target as expected. The ΔRY, as a coarse estimate of non-additivity, reveals that in two out of 10 cases, the difference between the observed and expected target biomass significantly differs from zero. The most significant of these mixtures is the above-mentioned with C. canescens, H. pilosella and J. montana. In all cases, targets remain smaller than would have been expected from the pairwise treatments, thereby indicating an increase in competition intensity in multi-species mixtures of specific species combinations. So far, experiments studying the predictability of multi-species competition from single species effects in plants are rare and contradicting. Fowler (1982) predicted relative yield of mixtures based on monocultures of increasing densities and found strictly additive effects for six grassland species. In contrast, a similar approach by Miller (1994) found partly non-additive effects for five species in an old-field community.

To unravel the complex coherences in these multi-species mixtures, the analysis of yield-density models proved to be a helpful tool. According to our assumption that multi-species competition is not a simple additive function of single species effects, a model integrating one or several parameters of non-additivity should provide a better fit to our data than a simple model without any additive parameters. Our simplest model (model 1) is the standard hyperbolic model of Freckleton & Watkinson (1997), which uses neighbour species density alone to describe target plant performance. Modelling target plant biomass using this simple model resulted in a good fit even though the total density of neighbouring species was kept constant in the present design. This once more demonstrates the applicability of the hyperbolic model to answer a wide variety of questions concerning plant competition, as has been shown previously for quantifying competition for resources (Firbank & Watkinson 1985; Connolly 1987; Freckleton & Watkinson 1997, 1999, 2001a), competitive asymmetry (Connolly & Wayne 1996; Freckleton & Watkinson 2001b), spatial competition (Rees et al. 1996; Law et al. 1997; Freckleton & Watkinson 2000b), the impact of mycorrhiza (Watkinson & Freckleton 1997) and weed-crop competition (Rejmanek et al. 1989; Kropff & Spitters 1991). Model 1 does not include a non-additivity parameter and therefore predicts target performance in mixtures as a fully additive function of pairwise competition.

To test our assumption of non-additivity, we analysed a set of models comprising additional parameters to describe target performance and compared them with model 1. From the series of non-additive models, the ‘full model’ (model 2) is the most complex one, adding a separate parameter for each specific multi-species combination. Model 2 does not significantly improve the overall predictive power of the model. This finding leaves two ways of interpretation: (i) the effects are additive and hence competition intensity measured with pairwise mixtures could predict multi-species competition; or (ii) non-additive effects do occur but they are specific to single species or species combinations, and the full model is therefore overparameterized. In the latter case the inferior performance of model 2 would be caused by the need to estimate a large number of parameters that are essentially zero. We therefore checked the effect of a non-additivity parameter in the presence of single species and species combinations. In eight of 10 possible species combinations there is still no significant improvement of model prediction, which strongly indicates that the competitive effect of multi-species mixtures is indeed well predictable from pairwise interactions. For two of the species combinations, however, the additional parameter significantly enhances the predictive power of the model. Neighbour mixtures including C. canescens and one of the two herbs show a better fit if one additional predictive parameter is taken into account. Applying yield-density models, Moloney & Chiariello (1998) found non-additive effects on seed production for three out of four annual species and recently Dormann & Roxburgh (2005) used Lotka–Volterra models and also found mainly non-additive competitive effects for seven species in a lawn community.

Our approach is similar to that of Dormann & Roxburgh (2005), but there are two differences: their model comprises competitive interactions between neighbouring individuals while neighbour density is not included. As a result the authors are able to separate indirect but additive effects between neighbours (effects strictly predictable from pairwise designs) from indirect and non-additive effects leading to a stricter definition of non-additivity. Our experimental design did not include all pairwise treatments (only H. pilosella was used as target species) and therefore doesn't allow for this separation. In this paper, non-additive effects were defined as all deviations of target plant growth in multi-species mixtures from the expected performance based on the sum of single species effects. This definition includes indirect interactions based on the per gram effects of neighbours and therefore the non-additive effects found for C. canescens in combination with the herbs could also be additive indirect effects. Nevertheless, we think that our definition of non-additivity reflects a relevant and possibly more pragmatic approach to the problem of predicting competition effects in multi-species communities if there is a particular target species. In this special case, the question of whether the combined effect of a mixture of neighbouring species can be predicted as a simple additive function of single species effects on the target (without considering all possible indirect effects, where no information may be available) appears natural. Overall, there is a growing body of evidence clearly demonstrating that non-additive interaction in one or the other definition, might in fact be more common than is widely accepted and the results presented here further support this finding.

In our experiment, multi-species mixtures including C. canescens and either H. pilosella or J. montana clearly show interactions that are not predictable from our pairwise design. What might be the major factors influencing these interactions? One possible reason could be a disproportional change of neighbour plant biomass relative to their density. It has long been known that biomass- or size-dependent effects are important for plant competition (Goldberg & Werner 1983; Goldberg 1987; Miller 1994). In the resource-poor inland dune habitat, competition is predominantly below ground and hence root biomass might be a more decisive parameter (Weigelt et al. 2002, 2005; Bartelheimer et al. 2006).

This assumption was supported for species-specific neighbour biomass. Including species-specific total biomass together with neighbour density in the model yields an enormous increase in its predictive power. The same is true if species-specific root biomass is added, although the increase in predictive power is significantly less than for total neighbour biomass. Hence, specific density and species-specific total biomass of neighbours together do best predict target plant biomass production, and root biomass is one crucial determinant in this process. Disproportional changes in neighbour biomass relative to their density are therefore important in our study system. This could be due to differences between intra- and interspecific competition for strong and weak competitors. Intraspecific aggregation has been shown to be advantageous for competitively inferior species due to overall lower competition intensity with conspecifics, while dominant species benefit from segregation leading to interspecific interaction (Stoll & Prati 2001; Monzeglio & Stoll 2005). Our results indicate that both H. pilosella and J. montana are poor competitors, thus C. canescens benefits and grows larger in these specific neighbour mixtures, thereby increasing its relative biomass while total biomass is constant. A corresponding trend might be seen for C. canescens neighbour biomass per species (Fig. 3a) but this is not significant. However, F. ovina is an equally strong competitor, showing no non-predictive effects even though this species reveals significant changes in neighbour biomass per plant. Obviously, there is no predictable pattern of relationship between competitively strong and weak species but rather a species-specific network of interactions.

Looking at the relevance of these results for biodiversity studies, we are well aware of the fact that four species are not sufficient when discussing species richness. Nevertheless, some of our findings might harbour interesting relationships. (i) We found no significant differences in competitive response with changing number of neighbour species, and at given densities, changes in total neighbour biomass did not have a significant effect on target growth. Many experimental studies found a significant increase of community biomass with increasing biodiversity (reviewed by Schmid et al. 2002), which is more pronounced at lower but stable densities (He et al. 2005). Thus, this might indicate a constant intensity of competition with increasing species richness. (ii) The RYT revealed only marginal differences from unity and these were not related to our findings of non-additivity. This means our findings of non-additive interactions in multi-species mixtures do not result in overyielding. For biodiversity experiments, the importance of competition is mainly attributed to the complementarity effect, which explains part of the overyielding (see Hooper et al. 2005 and references therein) and therefore one might expect that overyielding mixtures would be particularly prone to non-additivity. However, overyielding concentrates on positive effects while non-additive interactions can also be negative, leading to increased competition, as in our case. Future studies with a more suitable design to test this relationship will hopefully shed more light on these patterns. (iii) The present results stress the importance of unpredictable and species-specific effects on competition intensity in certain species combinations. With increasing biodiversity the probability for particularly influential or ‘key’-species should increase by a pure sampling effect and cause an increasing importance of species identity at higher biodiversity levels. As a consequence the relative frequency of non-additive effects should increase with increasing species richness (also see Dormann & Roxburgh 2005), a correlation that still remains to be proven.

In theoretical ecology, additive competitive effects are an integral part of many models that increasingly contribute to our understanding of the effect of competition on plant communities and stable coexistence as one important mechanism of maintaining biodiversity. In experimental ecology, inferences from simple pairwise competition experiments towards strict competitive hierarchies of species in natural multi-species communities are common (Keddy & Shipley 1989; Keddy et al. 1994, 1998, 2000; Roxburgh & Wilson 2000a,b) and apart from a few exceptions (Fowler 1982; Turkington & Klein 1993; Miller 1994; Moloney & Chiariello 1998; Stoll & Prati 2001; Dormann & Roxburgh 2005), the focus is still on pairwise approaches to study plant competition. Such studies are important to understand the underlying processes of competing species pairs. In view of the increasing evidence of at least species-specific non-additivity of competitive interactions, there is also a strong need to include multi-species competitive approaches, studied either in removal experiments or in artificial communities to describe natural communities.

Acknowledgements

The skilful assistance of Elke Furlkröger and Christine Schlüter during the experiments and the technical support of Ulrich Richhardt and Gerd Drexler are gratefully acknowledged. We thank two anonymous reviewers for comments on an earlier version.