## Introduction

One of the key aims of plant population and community ecology is to measure the effects of biotic interactions, especially competition, and then understand and predict how these translate into consequences at the level of the whole community (Grime 1979; Tilman 1988; Hubbell 2000). In a recent critique of other studies on competition in plant communities, Brooker & Kikvidze (2008) argue that researchers are confusing the issue by conflating two key measures of competitive interactions. Following Welden & Slauson (1986), they argue for the distinction between competitive intensity (a measure of absolute, proximate strength) and importance (broadly, a long-term, relative outcome). Citing two studies, one of which did not use either term in this framework, they argue that ‘widespread confusion produces continued debate’. Here we argue that Brooker and Kikvidze do not provide a resolution because they do not make the distinction between studies that focus on the outcome of competition on, say population growth rate, and those that focus on components of competition, and by accepting the arguments presented by Welden & Slauson (1986) too uncritically. Moreover, they only consider ‘importance’ in one context, that is, competition along environmental gradients, whereas the fundamental distinction between proximate effects of competition and fitness consequences can be applied more generally.

In this article we argue that the inconsistencies in the work by Welden & Slauson (1986) preclude a meaningful interpretation of either the importance or intensity of competition, and caution against reading their definitions and concepts too uncritically. We emphasize that studies that do not take a population or community dynamic perspective, can, necessarily, say little about the importance of competition to population or community ecology. Finally we question whether a single concept or measure termed ‘importance’ is really useful, especially in the light of the developments in the theoretical and statistical literature over the past 20 or so years.

## Importance and intensity

### simple concepts

The ideas of importance and intensity are put deceptively simply by Welden & Slauson (1986) in the abstract to their article:

‘The intensity of competition is a physiological concept, related directly to the well-being of individual organisms but only indirectly and conditionally to their fitness ...’

and

‘The importance of competition is primarily an ecological and evolutionary concept, related directly to the ecology and fitness of individuals but only indirectly to their ecological states.’

The subsequent discussion in that article elaborates on this idea, but unfortunately does not keep to these key concepts. The basis of the idea is, however, that the importance of competition should be measured relative to other processes and should summarize the impact of competitive interactions on, say, fitness (i.e. per capita rate of increase) relative to these other processes, whereas intensity is a measure of the proximate effects of competition on individuals.

### example with annuals

This initial distinction may be illustrated simply using annual plants as an example, and here we outline a hypothetical example of how we might use the basic notions of intensity and importance in an application. We use this example not to suggest or promote a new or particular index of competition. Rather, we use the model to explore how simple measures of competitive intensity and importance with respect to population growth behave in a simple model, and to contrast that with the generic index proposed by Brooker & Kikvidze (2008).

Annual plants frequently compete for resources (e.g. water, light and nutrients). In the absence of competition a plant will grow to a size* w*_{m}, but in the presence of competitors at density *N* (it does not matter whether *N* is the density of con- or heterospecifics) will grow to a smaller size *w*(*N*). The ratio of *w*_{m} and *w*(*N*) or the difference between them – the distinction is immaterial, we do not wish to relive here the protracted debates on relative vs. absolute measures of competition – is a measure of competitive intensity according to the definition of Welden & Slauson (1986) ‘the amount by which the competition-induced component of the sub-optimal state differs from the optimal’:

(eqn 1) This ratio ranges from 0 (relative performance is zero and intensity is maximal) to 1 (relative performance is 1 and no effect of competition).

In this example, we can measure the importance of competition for population dynamics by relating *w*(*N*) to population growth rate: this is because the ultimate aim is to measure the effects of competition with other processes in the life cycle, and to summarize an effect on net fitness. In our example of annuals this is easily done as there is usually a linear relationship between biomass and seed production (Watkinson 1980; Rees & Crawley 1989; Thomson *et al.* 1991). If *S* is the production of seeds per unit biomass and *g* is the proportion of seeds that survive to become plants the following year (i.e. includes all mortality in the life cycle), then population dynamics are given by:

(eqn 2) Eqn 2 integrates all the effects of competition and other processes (reproduction, mortality and seed germination) on the life cycle. At a given density, the importance of competition for population growth rate is calculated from the ratio between this quantity and the maximal per capita population growth rate, *N*_{t}_{+1}/*N*_{t} = *gSw*_{m}. Most usefully, this is calculated at the equilibrium density:

(eqn 3) Similarly this ratio varies from 0 (no population growth and maximal effect of competition) to 1 (maximal growth and no effect). From eqn 2 we note that at equilibrium, *w*(*N*_{eq}) equals 1/*gS* so that:

(eqn 4) It may appear that quantities (1) and (3 or 4) are the same; however, two important points need to be made. First, eqn 1 is derived solely from measures on performance *within* a growing season and with no reference to population growth; specifically, the density *N* is arbitrary. Thus, if the density *N* is held constant, the intensity of competition will also remain constant, and if density is varied, then the intensity will vary (Freckleton & Watkinson 1997*a*). However, in eqn 4 the importance of competition will vary with factors that affect *g* or *S*, because they influence *N*_{eq}, but do not affect the intensity of competition at a given density. Thus, in this example, increasing *g*, *S *or *w*_{m} will decrease the importance of competition for population growth rate.

Competition need not only affect biomass; for example, if *g* could be affected by density (e.g. through density-dependent emergence or predation of seed). If this were the case, a study on the effects of competition that estimated biomass effects via eqn 1 would not be relevant to understanding the net effects of competition through its combined effects on germination and biomass and consequent impact on population growth rate. For example, Lintell Smith *et al.* (1999) showed that in the annual weed *Anisantha sterilis *the intensity of competition measured on biomass alone was approximately 10 times lower than the combined effects of germination and biomass. In this example, information on competition between individual plants would yield no information on how populations are regulated by competition in the wider sense, and this can only be achieved by considering the whole life cycle and population growth rate.

### intensity and importance over gradients

So far we have considered measures of the intensity and importance of competition within a site. One frequently explored issue is how competition varies along experimental environmental gradients. For comparisons along a gradient, Brooker & Kikvidze (2008) suggest using an index of the form:

(eqn 5) where *w*_{MAX} is the maximum weight mass of an isolated plant along the gradient, and *w*_{m} and *w*(*N*) are as defined above. This index varies between 0 (no competition) and 1 (maximal effect of competition). Simplifying and assuming *w*_{m} >> *w*(*N*) we find:

(eqn 6) The assumption that *w*_{m} >> *w*(*N*) can always be justified by increasing the sowing density (*N)*, as this is usually fixed by the experimenter (e.g. see examples analysed in Brooker & Kikvidze 2008). From eqn 6, we conclude that the importance of competition increases as productivity (measured by maximum rates of individual growth) increases. However, this is a consequence of the way the index is constructed rather than of any deep biological significance, because the values estimated are not generated with respect to the whole life cycle or population growth.

In our example of annuals, what would the measure of importance that we used for population growth tell us about the importance of competition for population growth rates in contrasting environments? Consider the case when we measure competition in two environments, denoted 1 and 2. Hence we can estimate the importance of competition for population growth rate in both:

We can calculate how important competition is in environment 1 relative to environment 2, and ask how the importance of competition for population growth changes along the gradient by looking at the ratio of these:

(eqn 7) The key point about eqn 7 is that any of the parameters might vary between the two environments. For instance, if environment 1 were a drier environment, individual productivity (*w*_{m}) would decline as might seed germination and subsequent survival (decreasing *g*), or facilitation in such an environment could increase *g*. An alternative scenario is that in more productive environments, predation of seeds or seedlings might increase because higher-productivity environments harbour more natural enemies, and consequently *g* or *S* could decline with increasing *w*_{m}. Many outcomes are possible and the relative measure in eqn 7 can encompass a range of behaviour that the restricted index (eqn 5) cannot.

We note that if we assume that *g* and *S* are constant between environments, then eqn 7 reduces to:

(eqn 8) At first glance, this might seem to be the same as eqn 6. However, eqn 8 is arrived at by considering population growth rates and has a population dynamic interpretation: if productivity were twice as high in environment 2 compared with 1, then eqn (8) says that competition will reduce population growth rates by twice as much in environment 2 compared to 1. Furthermore, eqn (8) is a null prediction under the simple expectation that *g* and *S* are constant, and this may or may not be the case in reality. In contrast, the result in eqn (6) simply follows from the way the index is constructed.

### conclusions

What do we learn from this example? First, to measure the effects of competition it is necessary to include the proximate effects of competition on individuals, as well as the effects on the population growth rate from other sources of flux in the life cycle. Second, an index that is based solely on measurements of plant performance is not adequate to characterize competition as this fails to integrate other sources of mortality, variability or competition. This can only be achieved by measuring population growth rate. Third, the question of how competition scales along gradients or within habitats requires that competition is characterized fully in this way within each habitat. Finally, the difference between eqns 6 and 7 is that the latter allows all factors operating on population growth within each habitat to be accounted for, whereas the former only focuses on the effects of competition on vegetative growth. Although we have framed this example in terms of annual plants for simplicity of presentation, the same conclusions apply to perennials: all processes in the life cycle need to be measured and integrated into a single relevant measure of fitness or population growth.

## Critique

### what is meant by ‘population-level effects’?

Brooker & Kikvidze (2008) claim that there is ‘widespread confusion’ in the literature. In sole support of this contention they argue that Freckleton & Watkinson (2001) in particular employed a confusing definition of importance. They argue that Freckleton & Watkinson (2001) define importance at the ‘population level’ and intensity at the ‘individual level’. And indeed, Weldon and Slauson also do this, or at least in some parts of their article.

More specifically, Brooker & Kikvidze (2008) say in critique of Freckleton & Watkinson (2001): ‘even if you can demonstrate a population-level effect of competition, it remains a measure of competition intensity if you fail to place that effect within the context of population-level effects from other sources’. It is clear from both of our original definition (explicitly given in terms of population growth rate), from the equations given by Freckleton & Watkinson (2001), and the foregoing expanded rationale that this criticism is quite unjustified. If examining the dynamics of a species within a community, the population growth rate is the only measure that allows this integration to be achieved, and we specifically defined importance with reference to population growth rate.

Although not defined explicitly, it appears that what Brooker & Kikvidze (2008) mean by ‘population-level effects from other sources’ are differences in population growth rate or performance with respect to other environments, basically with respect to *W*_{MAX} in eqn 5. Thus, if a population in an ‘optimal’ habitat can achieve a maximum mean mass per individual of 10 g per plant, and in a suboptimal one achieves 5 g per plant, this twofold difference in growth is a ‘population-level’ effect of ‘stress’ on the population in the sub-optimal environment. In the sub-optimal environment it is argued that plants experience this stress in addition to the effects of competition and the importance of competition has to account for this.

Brooker & Kikvidze (2008) are therefore asking about importance in only one respect, that is, how the effects of competition vary along environmental gradients. This is only one context in which the effects of competition might be explored, and can by no means be used to ascribe a general index of ‘importance’ to competition. The issue is that, although performance with respect to *W*_{MAX} could be used to analyse relative performance in a series of populations, it cannot be used to predict performance within a focal population or community, is irrelevant to population persistence, and, within multispecies communities, is irrelevant to the outcome of competition. The question of how competition scales with the effects of competition along gradients is of course potentially interesting, but is not directly relevant to understanding the long-term effects of competition within communities, what determines whether species are able to coexist or not, or whether communities can be invaded by new species.

### the importance of competition is 42

What does this mean? Symptomatic of much of the literature on the importance of competition is the failure to state to what competition is important and what this is measured relative to. Without stating this explicitly, any statement, such as ‘evidence for the decline in importance of competition for resources in unproductive vegetation’ (Grime (1979, p. 20) becomes open to misinterpretation. Brooker & Kikvidze (2008) discuss Grime's theories and present an analysis of two field experiments quantifying the within-season effects of competition on plant performance, and a pot study looking at the effects of competition on relative growth rate. They then use their measure of importance measured on gradients to interpret these.

However, Grime (2001), when discussing the importance of competition, seems to be primarily interested in controls on community membership. He writes:

‘However, it has never been the contention in CSR theory that competition disappears completely under conditions of low productivity or intense disturbance. It does seem logical, however, to use terminology which reminds us of the circumstances where competition for resources is the dominant (but not exclusive) influence on the **membership** of plant communities’

and later on page 37 he says

‘As Welden and Slauson (1986) recognised, there is an obvious and continuing conceptual gulf dividing those ecologists who are measuring the role of competition in modulating the relative abundance of species co-existing in plant communities and those pursuing the larger predictive framework that identifies the circumstances where competition determines the kind of plant species **admitted to** communities.’

Particular confusion can arise here as competitive intensity or importance (i.e. effects on performance, population growth or abundance) is not a direct predictor of invasibility (i.e. whether a species can enter a community from which it is absent; see Lonsdale 1999). Measuring invasibility (i.e. whether a population can exhibit positive population growth under given conditions) is a different matter from measuring competitive intensity or importance and there is no simple way to predict one from the other. In summary, without clearly spelling out what competition is important to and relative to what, statements about the importance of competition are largely devoid of meaning.

## The way forward?

### the quantitative basis

An uncritical reading of Welden & Slauson (1986) could result in a focus on an inappropriate measure with which to estimate ‘importance’. For example, if competition experiments focus only on the growth phase, failure to measure survival or reproduction would yield incomplete and potentially faulty estimates of the net effects of competition. If there is confusion in the literature regarding the ideas of importance and intensity, this in part arises because the examples used by Weldon and Slauson are unhelpful and even contradict the broad aims of the distinction between importance and intensity that they propose at the outset of their article. For instance, in their fig. 1, Weldon and Slauson attempt to distinguish the importance and intensity of competition by reference to survival of individuals. However, survival is only one component of fitness and the fitness of an individual (or equivalently the population growth rate) may be sensitive to other factors (e.g. reproduction, growth). Thus, despite proposing an apparently appealing idea, the examples given are unclear and even contradictory and we would caution against reading this article uncritically.

This contradiction is pointed out by Brooker & Kikvidze (2008), but they do not attempt a resolution. To us the only logical way to resolve this contradiction is to regard within-generation competitive effects as being incapable of providing information about the importance of competition in broader terms. For instance, the information in fig. 1 of Welden & Slauson (1986) (or fig. 1 of Brooker *et al.* 2005) can only be used to measure the intensity of competition as it does not integrate processes operating across the whole life cycle via estimating population growth rate or fitness.

A general issue is that the article by Weldon and Slauson was written over 20 years ago before many of the considerable developments in theoretical population ecology and the recent closer integration of theoretical and statistical ecology. Brooker & Kikvidze (2008) do not (by necessity) incorporate the ideas that have been developed in this time, such as quantitative measures of invasibility (Geritz *et al.* 1997, 1998), population persistence in the face of stochasticity (Lande 1998; Lande *et al.* 2003), community dynamics and invasibility (Chesson 2000) or measurements of evolutionary fitness (Metz *et al.* 1992; Metz & Gyllenberg 2001). Moreover, an increasing number of studies have dissected population and community dynamics from first principles, especially in the zoological and epidemiological literatures (e.g. Rees *et al.* 1996; Pacala & Rees 1998; Fromentin *et al.* 2000; Lande *et al.* 2003; Clark & Bjornstad 2004; Grenfell *et al.* 2004; Morris *et al.* 2004; Cornell *et al.* 2008). In plant community ecology there has been a shift towards looking at the fundamental factors determining life-histories and the trade-offs between traits to understand how plants evolve to fit into their environment (Rees *et al.* 2001). The analytical tools exist with which to analyse population and community dynamics from a purely quantitative basis and it seems that far more progress could be made by doing so (e.g. Sears & Chesson 2007).

### measuring competition in the short-term

By focussing on gradients and not examining long-term outcomes of competition within communities it is possible to be misled into believing that the underlying determinants of competition have been isolated. In particular, the long-term impacts of competition relative to stochastic factors such as climate or herbivory could be missed. The examples by Brooker & Kikvidze (2008), together with those given in an earlier article (Brooker *et al. *2005) would seem to contradict the call by Brooker and Kikvidze to evaluate the effects of competition relative to other factors: Brooker *et al.* (2005) attempt to calculate the ‘importance’ of competition based on a 10–12 week field trial with the perennial grass *Poa pratensis* and using data from a 1-month glasshouse experiment with seedlings of a perennial shrub.

To calculate ‘importance’ using such data is clearly inconsistent with a broad reading of the recommendations by Brooker & Kikvidze (2008). Our simple example with annuals makes it clear why this is not correct. Even if asking a simple question such as ‘what is the importance of competition for population growth in an annual plant?’, a short-term experiment would only be able to measure the effects of competition on part of the whole generation. Thus, it would only be possible to estimate eqn 1, and not to estimate the quantity in eqn 4. Even if this were done across a gradient, such that the importance index of Brooker & Kikvidze (2008) could be measured, it would be an incomplete measure, particularly if the gradient is an experimental one at fixed density.

It is, almost by definition, not possible to estimate the importance of competition relative to other processes operating on a population unless all components of the whole population growth cycle have been estimated. In a long-lived perennial, a short-term experiment involving only one life stage, for example, will tell us nothing about the effects of competition or other process operating at other stages of the life cycle. At the least, all stages of the life cycle need to be measured, preferably via multi-generational studies.

Of course that does not mean that short-term studies have no place in research on competition. These are of considerable value in testing hypotheses and in elucidating mechanisms. There are many types of experiment and manipulation that are impossible to carry out in the field or under uncontrolled conditions and that can only be performed in the short-term under controlled conditions. However, at the same time it needs to be realized that there are limitations in using data from highly controlled conditions to infer the strength of competition under field conditions.

### defining importance

We would question whether the concept of ‘importance’ as a single concept or measure is of any value in plant population or community ecology. The idea behind calculating ‘importance’ is to estimate the effect of competition relative to other processes as a single index. In many circumstances this seems to be of questionable value for several reasons. First, as an example, consider the case of single-species dynamics (such as the general model in eqn 2), where the product of *g*, *S* and *w*_{m} is (when appropriately averaged) greater than one, then the population will grow exponentially. Density-dependence via competition will serve to reduce population growth so that an equilibrium is achieved. In one sense it is immaterial whether the density-independent population growth rate is 1.1, 10 or 100 in three different environments, that is whether the required net reduction in population growth rate via competition to achieve equilibrium is 1%, 90% or 99%, because in all three cases competition is ‘important’ in preventing unlimited population growth. That is, as we emphasize above, specifying the ‘important to’ is critical.

A second reason why general measures of the importance of competition are of questionable value is that such indices are often not related to the fundamental parameters generating population dynamics. This is true whether one is considering one, two or many species. For instance, in eqn 1 or 4 point estimates of either the intensity or importance values are not sufficient to fully characterize population dynamics. This is because the function *w*(*N*) is a nonlinear function of density and measuring this requires data on performance that was measured at a range of densities. In two species mixtures the problem of trying to infer the nature of competition from indices becomes worse (see Freckleton & Watkinson 1997a,b, 1999) and, despite claims to the contrary, the situation becomes hopeless if there are more than two species. This is because in an *n*-species mixture there are at least *n*^{2} pairwise interaction coefficients (ignoring higher order effects) as well as variance in *n* densities, all of which combine in a nonlinear way to determine proximate effects of competition and their net effects on populations. A single estimate of ‘importance’ is averaging across at least *n*^{2}+*n* unknown variables, and is therefore unlikely to yield any useful information on the underlying community dynamics.

## Concluding remarks

In short, we would advocate less reliance on *ad hoc* and questionable index-based measures of competition from artificial conditions as they have to date not thrown much light on how natural population and communities function and are unlikely to do so in the future. We urge plant ecologists to study real populations and communities using model-based approaches. At present, semantic debates about terminology, indices and measuring competition are hindering plant ecology. What is needed is a mechanistic, model-based approach to measuring competition and the other factors determining population dynamics.

## Acknowledgements

RPF is funded by a Royal Society University Research fellowship.