Measuring the importance of competition: a new formulation of the problem

Authors

  • Christian Damgaard,

    Corresponding author
    1. Department of Terrestrial Ecology, NERI, University of Aarhus, Vejlsøvej 25, 8600 Silkeborg, Denmark
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  • Adeline Fayolle

    1. Cirad, Environments and Societies Department, Natural Forest Dynamics Research Unit, Campus International de Baillarguet, TA 10C, BP 5035, 34035 Montpellier, France
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*Correspondence author. E-mail: cfd@dmu.dk

Summary

1. Currently, there is a debate among plant ecologists on the concepts of the intensity of competition and the importance of competition, which is central to many issues of modern plant population ecology and plant community ecology.

2. It is problematic that the current measures of intensity and importance of competition, typically, are reported as dimensionless indices because they hide the fact that both indices are functions of plant density and the level of the environmental gradient.

3. Here, a new formulation of the concepts is suggested, which explicitly highlights the functional dependencies on plant density and the level of the environmental gradient. The new measures are a generalization of the previous indices and correspond to the previous indices in the case of a simple experimental design.

4. The suggested measures of the intensity and importance of competition are exemplified using data from a response surface competition experiment between Agrostis capillaris and Festuca ovina along a herbicide gradient, where the expected clear effect of plant density was demonstrated.

5.Synthesis. As the suggested measures of the intensity and importance of competition explicitly highlight the functional dependencies on plant density and the level of the environmental gradient, we think that they will help to ensure a closer connection between experimental plant ecology and the attempts to model plant populations and communities.

Introduction

The concept of competition has a strong explanatory power in plant ecology and is central to most hypotheses on the structure and dynamics of plant communities (e.g. Grime 1979; Tilman 1988; Huston 1994). In the light of the huge variability of life histories and ecosystems, this explanatory power and the apparent success of generating fairly general hypotheses is to some degree a result of a rather loose definition of the term ‘competition’. However, a loose definition is also dangerous and may lead to unfruitful dialogues due to differences in semantics, especially if the use of mathematical models in the scientific tradition has been downplayed. A systematic experimental approach has led to an amazing amount of evidence that competition exists in natural communities [for example, Goldberg & Barton (1992) reviewed 101 experiments]. However, the role of competition in the structure and dynamics of plant communities is still poorly understood (Suding, Goldberg & Hartman 2003; Agrawal et al. 2007; Lamb & Cahill 2008), even though the key questions have already been addressed, as exemplified by the quotations below.

Documenting that competition occurs does not necessarily imply that this competition has important consequences for ecological communities (Goldberg & Barton 1992).

The first part of any analysis of competition is to determine whether it occurs and then what its form is. Having done this, we may ask, does it matter? (Rees, Grubb & Kelly 1996).

The distinction between importance and intensity of competition

A central refinement in the theory of plant competition is the distinction between the intensity and the importance of competition. This distinction was first introduced by Welden & Slauson (1986) and later explored by several authors (e.g. Grace 1991; Brooker et al. 2005; Brooker & Kikvidze 2008; but criticized by Freckleton, Watkinson & Rees 2009). The intensity of competition refers to the degree to which a plant population is reduced by the presence of neighbours, whereas the importance of competition refers to the relative reduction of a plant population by competition compared to the reduction due to other forces, such as herbivory or unfavourable abiotic conditions (both definitions due to Grace 1991). To illustrate the distinction between intensity and importance of competition, imagine a plant community where neighbouring individuals have strong negative effects on each other (intense competition), but where the interspecific competitive interactions do not play an important role in the structuring of the plant community compared to the filtering effect of an abiotic factor on species composition. Both the intensity and the importance of competition have been investigated experimentally; but where the intensity of competition has been examined in numerous studies, only relatively few studies have made empirical investigations of the importance of competition (e.g. Greiner La Peyre et al. 2001; Gaucherand, Liancourt & Lavorel 2006).

The significance of the distinction between intensity and importance of competition cannot be downplayed. Indeed, it is at the heart of the prolonged and painful ‘Grime-Tilman-debate’ in plant ecology on the role of competition in structuring plant communities along productivity gradients (Grace 1991). It is now recognized that the two authors employed different definitions of competition and competitive success or ability (Goldberg, Grace & Tilman 1990; Grace 1991; Brooker et al. 2005; Brooker & Kikvidze 2008): Grime’s theory postulates that the importance of competition in structuring plant communities increases along a productivity gradient. In his view, competition corresponds to the mechanism whereby a plant suppresses the fitness of a neighbour, and competitive ability is linked to resource acquisition. Tilman’s theory, on the other hand, postulates that the intensity of competition is constant along a productivity gradient. In his view, competition corresponds to the mechanism whereby a plant tolerates a low level of resource, i.e. if a species has a high competitive ability at low levels of a specific limiting resource; this is because the plant is able to grow at low levels of that resource.

The need for an integrated approach

To date, plant ecological literature has mainly focused on how to measure the intensity and importance of competition (Freckleton, Watkinson & Rees 2009). Indices are calculated from transplant or replacement experiments, comparing a set of interacting species and/or community position along an environmental gradient. The indices originally suggested by Welden & Slauson (1986) measure intensity of competition as OC, where O is the state of a plant in the absence of competition and under optimal growing conditions, and C is the state of the plant when competition is taking place again under optimal growing conditions. The importance of competition is measured by (OC)/(OA), where A is the state of a plant when both competition is taking place and the plant is growing in a stressful environment. Except in the case of facilitation (Bruno, Stachowicz & Bertness 2003; Brooker et al. 2008), we expect that > A, and, consequently, that the index of importance of competition takes values between 0 and 1 and may be interpreted as the decrease in the state of a plant due to competition relative to the decrease in the state of a plant due to both competition and a stressful growing environment.

Some of the other indices used in the literature generalize the above index by Welden & Slauson (1986) to include the effect of facilitation, but all measures of the intensity and importance of competition have in common that they are not functions of either density or the level of the environmental gradient (see Weigelt & Jolliffe 2003 for a review of indices measuring competition intensity). This is unfortunate, as we know that competition, generally, depends on plant density and the level of the environmental gradient (e.g. Goldberg et al. 2001; van Andel & Aronson 2005; Turkington et al. 2005; Freckleton, Watkinson & Rees 2009).

Here, we present a generalization of the indices of importance and intensity of competition that causes them to be functions of density and the level of an environmental gradient in such a way that (i) the competition intensity and importance indices can be calculated on any measure of ecological success at the population or individual level; (ii) the functions contain the same ecological information as the indices, but for general densities and levels of an environmental gradient and (iii) in the case of a simple experimental design, e.g. if only one density is used in a competition experiment (a replacement design), the functions will degenerate to the indices used presently.

Materials and methods

Proposed measures of intensity and importance

If a measure of the ecological success of a plant species i (e.g. size, biomass, fecundity, increases in density or increases in cover) is measured at variable densities (or cover) of other plant species, d, and levels of one or more environmental gradients, x, then the expected ecological success of an individual plant of species i may satisfactorily (in a statistical sense) be expressed by an empirical function, Fi(d, x). For example, if biomass is measured at variable densities in a response surface competition design along an environmental gradient, then the effect of the densities and the gradient on mean plant biomass may be expressed by a generalized hyperbolic function with d and x as variables (see Appendix S1 in Supporting Information).

The intensity of competition of species i at specified densities of the other species and at specified levels of the environmental gradient may be defined as the negative change in the ecological success of an individual plant of species i by changing the densities of the studied species, that is,

image( eqn 1)

which is the partial derivative of the measure of ecological success with respect to the density of one of the plant species (d). If n competing plant species are investigated, then n2 different indices may be computed.

The above index (eqn 1) will be positive if the effect of competition is negative, and negative in the case of facilitation. This index (eqn 1) measures the change at the level of the individual plants to ensure correspondence with individual fitness measures and allows a direct connection with plant population dynamics (Rees, Grubb & Kelly 1996; Freville & Silvertown 2005). However, if, for some reason, it is preferable to obtain the change at the stand level, then the index should be multiplied by the intraspecific density, xi. Alternatively, it is has been argued (e.g. Grace 1995) that, in some cases, it would be preferable to study the relative intensity of competition, which may be defined as

image( eqn 2)

The importance of competition of species i may be defined as,

image( eqn 3)

where inline image is the absolute change in the ecological success of species i by changing the levels of one environmental gradient (x). The use of the absolute functions assures that the importance of competition takes values between 0 and 1, independent of whether changes in the environmental gradient lead to increasing or decreasing ecological success of the plant species and in case of facilitation. This means that the measure of the importance of competition, as was the intention with the old index by Welden & Slauson (1986), may be interpreted as the proportion of the overall change in ecological success that is caused by plant–plant interactions.

In the case of a simple experimental design, the proposed measures of the intensity and importance of competition degenerate to the old indices by Welden & Slauson (1986). For example, if only a single species is investigated either with or without neighbours along an environmental gradient, then eqn 1 degenerates to OC, except for a scaling factor due to different units. If the above species additionally is examined in only two environments, then eqn 3 degenerates to |OC|/(|OC| + |CA|), except for a scaling factor due to different units, and if A, then eqn 3 degenerates to (OC)/(OA), which is equal to the index of the importance of competition suggested by Welden & Slauson (1986).

The competition experiment

To illustrate the use of the proposed measures of intensity and importance of competition, we considered the competition between two plant species, Agrostis capillaris L. and Festuca ovina (agg.), along a herbicide gradient. The competitive interactions were determined in a response surface experimental design, where both the density and the proportion of both species were varied (Inouye 2001; Damgaard 2008). The two grasses were transplanted into 40 × 40 cm boxes in 26 different combinations of density and proportions with three replicates (Fig. 1). This experimental design was repeated at three levels of the herbicide glyphosate (Roundup Bio; 0, 20 and 60 g active ingredient per ha2) to create an environmental gradient that a priori was known to affect the species composition in long-time experiments with the two species; A. capillaris is relatively sensitive to glyphosate compared to F. ovina, and relatively less abundant in plots sprayed with glyphosate (Holst et al. 2008). Five weeks after spraying, plants were harvested and the dry above-ground biomass was weighed (for more experimental details, see Holst et al. 2008).

Figure 1.

 The densities (individuals per m2) used in the competition experiment between Agrostis capillaris and Festuca ovina (Holst et al. 2008).

Data analysis and indices calculation

The observed dry weight of the plants was fitted to a generalized hyperbolic model (Damgaard 2003, 2004; Damgaard, Mathiassen & Kudsk 2008). The hyperbolic model and its underlying assumptions, as well as the fitting procedure, are explained in the Appendix S1. To be able to calculate the uncertainties of the indices and to make statistical inferences, the joint Bayesian posterior distribution of the model parameters were calculated using Markov Chain Monte Carlo (MCMC) methods (see Appendix S1). The two measures of the intensity and importance of competition were calculated by numerically differentiating the generalized hyperbolic model after inserting one set of joint parameter values from the MCMC chain with respect to either the densities or the concentration level along the glyphosate gradient. This numerical differentiation was carried out for each set of joint parameter values from the MCMC chain in order to obtain the posterior distribution of the intensity and importance of competition.

The posterior distributions of the intensity and importance of competition have the same dimensions as Fi(dx), and in the case of two plant species competing along a herbicide gradient, the posterior distributions are functions of the plant densities of the two plant species and the concentration level along the environmental gradient. It is not possible to plot a four-dimensional function, and here we have chosen to show only two of the possible graphical representations of the posterior distributions: the relative intensity of competition for the two species will be shown by the 2.5%, 50% and 97.5% percentiles of the posterior distribution as a function of the intraspecific density (at a fixed interspecific density and level of the herbicide), thus showing the statistical uncertainty of the calculated indices. The importance of competition of the two species will be shown by the mean of the posterior distribution as a function of intraspecific density and the level of the herbicide, thus keeping the interspecific density fixed. However, depending on the research questions, other representations of the distributions of the indices may be more relevant. For example, it may be relevant to investigate the indices at different interspecific densities.

Results

The 95% credibility interval of the relative intensity of competition on the biomass of F. ovina and A. capillaris as a function of the intraspecific densities are shown in Fig. 2. It is apparent that the relative intensity of intraspecific competition depends on the intraspecific density and that the relationship is unimodal. Furthermore, in this particular case, the statistical uncertainty of the relative intensity of competition was sizeable, and judged by the overlapping credibility intervals, the two species do not differ in their relative competition intensity.

Figure 2.

 Relative intensity of competition on the biomass of Festuca ovina and Agrostis capillaris as a function of the intraspecific densities (individuals per m2) shown by the 2.5%, 50% and 97.5% percentiles of the posterior distributions. The distributions of the relative intensity of competition were calculated using eqn 2 in the text, and the measure should be interpreted as the expected relative decrease in individual plant biomass if the intraspecific density is augmented by one plant per m2. For both species, the interspecific density and the level of the herbicide were fixed at zero (see text).

The mean of the importance of competition also depends on the intraspecific density, as shown in Fig. 3, and there is an apparent difference between the two plant species; whereas the importance of competition is relatively unaffected by the level of environmental gradient in the case of A. capillaries, the importance of competition seems relatively inferior at low intraspecific density and low herbicide levels in the case of F. ovina. However, the statistical uncertainty in this area of the domain (low intraspecific density and low herbicide levels) is substantial in the case of F. ovina (results not shown).

Figure 3.

 Importance of competition on the biomass of Festuca ovina and Agrostis capillaris as a function of the intraspecific density (individuals per m2) and the concentration of the herbicide glyphosate (g active ingredient per ha2) shown by the mean of the posterior distribution. The distribution of the importance of competition is calculated using eqn 3 in the text, and the measure should be interpreted as the proportion of the change in individual plant biomass that is due to the change in densities (i.e. competition) if the intraspecific density is augmented by one plant per m2 and the concentration of glyphosate is augmented by one g active ingredient per ha2.

Discussion

The new measures of intensity and importance

One advantage of the proposed measures of the intensity and importance of competition is that the measures may be calculated for many different types of competition experiments, including designs where the densities and proportions of more plant species are varied along different environmental gradients. The novelty of the proposed measures of the intensity and importance of competition is that they are explicit functions of the densities and the levels of the environmental gradient, which allows a quantification of the relative roles of the two factors.

In the studied case, where A. capillaris competed against F. ovina at different levels of the herbicide glyphosate, there was a clear effect of density on both the intensity and importance of competition. This result was expected and is a consequence of the approximately hyperbolic relationship between plant size and density, which has been observed in a large number of competition studies (e.g. Shinozaki & Kira 1956; Law & Watkinson 1987; Cousens 1991). We expect that most studies of the intensity and importance of competition will show a similar effect of density if the studies are performed at more than one density (see also Damgaard 2008). Generally, we expect that the importance of competition decreases with density, as the augmentation of the density with one plant per m2 has a larger effect on the density at low densities compared to high densities. For example, if a density of 10 plants per m2 is augmented with one plant per m2, then the density increases by 10%, whereas, if a density of 100 plants per m2 is augmented with one plant per m2, then the density only increases by 1%. Furthermore, in the literature, the indices of intensity and importance of competition are mostly reported as non-dimensional values, even though it is apparent from the proposed measures of the intensity and importance of competition that the measures, generally, depend on the units used in the study, and in the studied example some care was taken to report the units of both the densities and the environmental gradient.

In the reported case study, the relative intensity of competition experienced by the two grasses was comparable at the same levels of herbicide (Fig. 2). Competitive response, i.e. the intensity of competition measured on a target species, has previously been shown to be a property of a particular taxon (Goldberg 1996), and the similarity between the two species may be due to the fact that both of them are perennial grasses with similar functional traits. The functional dependency of the importance of competition on the intraspecific density and the level of the herbicide appeared to be species specific (Fig. 3). However, as the statistical uncertainty is relatively large at low intraspecific density and low herbicide levels, further investigations are needed to interpret this result. Notice that the reported competition experiment was made under constraints on space and time; plant biomass was measured after only 5 weeks in a greenhouse environment (Holst et al. 2008). Nevertheless, the reported case study gave important ecological information on the relationship between the effects of competition and the level of the herbicide that was needed for regulating herbicide use at agricultural field boundaries (Holst et al. 2008).

The way forward

Recently, Freckleton, Watkinson & Rees (2009) questioned the need of a single concept or measure of the importance of competition. They argued that a single measure would be an inappropriate simplification of the ecological processes occurring at the different life stages compared to an explicitly developed population model, and we agree. Nevertheless, often the experimental designs used in ecological studies fall short of measuring all the relevant features to construct a population model, and we argue that it is relevant to develop general indices that can be used in a broad experimental context. However, to link plant interactions to community structure, intensity and importance of competition must be investigated and calculated based on meaningful measures of ecological success and not only in terms of biomass depletion (Goldberg & Barton 1992; Aarssen & Keogh 2002; Freckleton, Watkinson & Rees 2009).

Numerous studies have reported that the measured competition intensity depended on the fitness component (survival, biomass or reproduction), and on the life stage considered (seedlings or adult plants) (e.g. Eckstein 2005; Schiffers & Tielbörger 2006; Violle, Richarte & Navas 2006; Fayolle, Violle & Navas 2009). Sometimes one fitness component or life stage may suggest facilitative interactions whereas another fitness component or life stage may suggest competitive interactions. Thus, it is natural to ask which fitness component or life stage is most relevant to measure in order to report on competition intensity and importance. A possible and promising way to solve these problems is to adopt a demographic approach in competition experiments that integrates the whole life cycle of the competing plant species (e.g. Freville & Silvertown 2005; Fayolle, Violle & Navas 2009). This approach is relatively easy to follow when it comes to annual plants (e.g. Rees, Grubb & Kelly 1996; Damgaard 2004; Freckleton, Watkinson & Rees 2009) where the population growth rate may be calculated by measuring reproduction, mortality and seed germination. However, few integrated life cycle studies exist for perennial and clonally propagated species, but, in many plant communities, it may be feasible to express the ecological success of different perennial and clonally propagated plant species by their relative change in cover and compactness as a function of the other species (Damgaard, Riis-Nielsen & Schmidt 2009).

Another possible route is to adopt a community-based approach instead of a species-centred approach. A good example is given in the study of Lamb & Cahill (2008). These authors pooled the competition intensity recorded for several target species and then compared the pooled competition intensity to other factors in order to asses the importance of competition on community diversity (species richness and evenness) using structural equation modelling. Another trait-based approach was proposed by McGill et al. (2006), who suggested characterizing communities as an interaction milieu described by size related trait distribution, since these traits are involved in the competitive effect (Goldberg, Grace & Tilman 1990), with large or tall plants competitively depressing smaller ones (e.g. Gaudet & Keddy 1988; Keddy & Shipley 1989; Violle et al. 2009). A combination of the two previously described approaches would be to record competition on the fitness components of several transplanted target species in communities that create a gradient of size-related traits. Such experiments could be used to disentangle the variation of competition intensity and importance along gradients at the community scale by pooling the fitness components of several target species.

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