The lack of underlying ecological theory

A central argument proposed for the rejection of the concept of competition importance, also imposed on the index C_imp (point 3 of Rees et al.'s list of ideal properties for ecological indices), is that it has no underlying theoretical basis.

To us, the theoretical basis is clear. Grime's C-S-R theory (Grime 1974, 1979) classifies plant strategies according to their level of adaptation to three main drivers: stress, disturbance and competition. It predicts that in either highly disturbed or stressed environments, the relative role of competition as a community structuring or selective force is reduced: competition becomes less important. Grime (2007) specifically states that it is ‘evident that we must recognize that competition declines in importance under the impacts of reduced productivity and/or severe disturbance’. This provides the basic rationale for examining the shape of the relationship between the importance of competition for regulating plant success or community structure and gradients of stress and/or disturbance.

Rees et al.'s argument that the concept of competition importance has no theoretical basis might stem from one of two sources. First, they may not think that the work of Grime and colleagues constitutes an ecological theory. Alternatively, they may be referring to a lack of an underlying mathematical as opposed to conceptual model. Interestingly, Rees (2012) examines the application of short-term manipulation experiments to the questions raised by the Grime–Tilman debate. Rees states that ‘there is currently no theory that predicts when particular patterns will be observed or guides the interpretation of the experimental studies’. Perhaps, it is with respect to within-experiment processes and specific types of severity gradient (as discussed below) that Rees believes there is a lack of theoretical underpinning.

We prefer to leave consideration of what constitutes theory (in a broad sense) to those more philosophically inclined (for example see Colyvan 2011; Gorelick 2011), but the work of Grime and colleagues is widely regarded as one of the major theories within plant ecology, and we take it as the theoretical basis for our work. Furthermore, other theories are relevant to the concept of competition importance. The original work by Welden & Slauson (1986) is theoretical in nature, whilst the exploitation ecosystems hypothesis (EEH) theory of Oksanen et al. (1981) also explores changes in the relative roles of community-structuring processes along environmental gradients. In the EEH, it is the relative role of trophic levels in regulating one another's productivity that is examined, generating largely convergent predictions to the work of Grime and colleagues but through different mechanisms.

The importance of what

A second argument put forward for the rejection of the concept of competition importance is ‘the importance of what’ argument, that is, that we must define the factor and response variable (the importance of what to what) when discussing the concept or its measurement. We argue that this is not necessary. The definition of competition importance, as given above, explicitly states that the factor is the impact of competition (relative to the impact of other environmental factors), and the response variable is some measure of community structure or plant success.

As also noted above, the definition and measurement of plant success are a problem that is not specific to the topic of competition importance. Success in an evolutionary sense is the contribution of an individual to the next generation, and competitive ability is then ‘the relative ability of the individual to leave descendants when resources are contested’ (Aarssen & Keogh 2002). However, measuring the lifetime contribution of an individual plant to the next generation of reproductive adults (and consequently, the extent to which competition in an environment suppresses this contribution and therefore acts as an evolutionary force) is extremely difficult: it necessitates following all offspring of an individual to reproductive maturity. To handle this problem, plant ecologists generally measure what are assumed to be proxies for this ultimate measure of success: biomass, levels of flowering, seed production and seed viability have all been used (Goldberg et al. 1999; Malkinson & Tielbörger 2010). However, under different circumstances, these may or may not be good proxies, and it is widely recognized that the use of different measurements can result in different conclusions being drawn with respect to the impact of regulatory factors such as competition (Goldberg et al. 1999; Aarssen & Keogh 2002; Trinder, Brooker & Robinson 2013). What is critical to remember here is that the measurement of plant success, or the forces structuring community composition (which can be assessed as the average role of competition in regulating success for the suite of species within a community), and our ability or otherwise to assess this through the use of proxies, is not a problem specific to the issue of competition importance. Defining and developing measurements of genuine success – and understanding how they relate to commonly used proxies – are challenges for plant ecology in general.

To return to our main subject, the answer to the question ‘the importance of what’ is then ‘the importance of competition’ with ‘importance’ having already been defined; the answer to the associated question of ‘to what’ is ‘plant success or community structure’, noting the general challenge that this represents in terms of measurement. Analytical approaches, particularly with respect to assessing the theories of Grime, must then allow us to assess changes in the relative role of interactions – that is, changes in their importance – along environmental gradients, and in particular, the underlying shape of this relationship (as discussed in more detail below under ‘variance decomposition and other analytical alternatives to competition indices’).

The use of short-hand terminology, avoiding the need to repeat detailed definitions, is not unusual in ecology (for example, use of the term ‘evolution’). The term ‘competition importance’ comes from the work of Welden & Slauson (1986). We would be interested in workable proposals for an alternative and less cumbersome terminology, but also wonder whether it is genuinely needed. As pointed out by Rees et al., the concept of competition importance is being increasingly discussed throughout the ecological literature and with, we suggest, generally a good level of accuracy irrespective of any problems associated with its measurement.

Rees et al.'s analytical approach

Rees et al. conclude that the relationship between the indices and severity is very simple and can be estimated as the ratio P_NC/P_max (where P_NC is plant success in the absence of competition, and P_max is the maximum value of P_NC observed along the gradient). This conclusion is supported by analyses of randomly generated data and data from the study by Kadmon (1995). There are at least three areas where these conclusions can be challenged:

1. The use of P_NC as a measure of ‘productivity’

   Because P_NC is one of the variables used to calculate C_imp, the use of P_NC as the metric of severity (i.e. the use of P_NC as the explanatory variable, as well a component of the response variable) leads to the conclusion that severity – or ‘productivity’ – and C_imp are closely related. The use of P_NC in this way is unusual, although some of us have previously adopted Rees et al.'s approach (Brooker et al. (2005), where we used P_NC to analyse the data of Pugnaire & Luque (2001)) and we must recognize that this might have influenced the reported results.

   Rees et al. utilize P_NC as a measure of productivity on the grounds that this ‘simplifies presentation, helps clarify the underlying relationships between variables of interest and is the approach used in several studies’. If we examine the studies listed by Rees et al. in support of this approach, none of them use P_NC as an indicator of habitat productivity in their analyses (Table S1 in Supporting Information). The nearest equivalent is the use by Gross et al. (2010) of P_NC/P_optimum as a measure of ‘strain’ (sensu Welden & Slauson 1986). Gross et al. do not claim that this represents habitat productivity or environmental severity.

   So what should we use as an explanatory variable? The fundamental question being addressed is how the relative impact of competition (or interactions) on plant success or community structure changes along gradients of environmental severity, that is, stress or disturbance or some combination of the two. The measurement of severity would then be the key stress or disturbance variable(s) that compose the severity gradient and these would be measured independently from the success of the target species. However, environmental severity is often assessed using biomass proxies, commonly standing crop or NPP, and the use of such proxies leads to ‘productivity’ being equated to ‘severity’. A common solution is to use stand-level metrics of productivity, for example, biomass produced per unit area per unit time (NPP) as a more independent measure of severity than P_NC? But depending on the proportion of standing crop or NPP contributed by our target species, they may correlate closely with P_NC, and hence, our explanatory variable will not be independent of our response variable. Alternatively, severity can be assessed using abiotic environmental proxies, such as nutrient or water availability, but as has been pointed out recently with respect to debate concerning the stress gradient hypothesis, severity gradients can involve multiple abiotic environmental drivers, necessitating refinement of theoretical predictions with respect to particular types of abiotic stress (Brooker et al. 2008; Maestre et al. 2009).

   Ultimately, and as for the measurement of success, the measurement of severity is a generic problem for plant ecology and is not specific to the issue of competition importance. The fundamental point is that if a measure of system severity that is independent of the biomass of the target species is used, there is no inevitability in the relationship between C_imp and severity. We believe that this is a key point of differentiation between ourselves and Rees et al.: the way in which ecologists perceive variation in the abiotic environment and its relationship to other factors, such as the impact of competition on plant success, is completely different depending on whether we estimate it using individual plant success (e.g. P_NC), community-level measures such as NPP, or abiotic variables, such as temperature. Depending on which of these is used as the explanatory variable, there will be a more or less ‘inevitability’ in the relationship between our measure of environmental severity and response variables such as C_imp that are derived from measurements of plant growth.

   This possibility for confusion is why we have used the terminology ‘severity gradient’ throughout this study, in an attempt to distinguish these larger-scale gradients of environmental severity – which impact upon both stand-level and individual productivity – from finer scale variation in the productivity of the target species itself. In addition, if the aim of a study is to test community-structuring forces – for example, the importance of competition for regulating the composition of a community in order to test some of the predictions of models such as the CSR theory of Grime (as well as the EEH theory of Oksanen et al. (1981)) – then a community-level metric of severity is necessary. We return to this fundamental difference in approach in our simulation analyses, below.

2. The use of C as an index of competition

   Rees et al. use C (P_NC/P_C) as an index of competition, where P_C is plant success in the presence of competition. We see two particular problems with their use of C. The first concerns the ease with which C can be interpreted. Apparently, simple relationships between P_NC and C used by Rees et al. (Rees et al.'s Fig. 1a) hide underlying complexity in the raw data, particularly P_C. We know both P_NC and C, and so can calculate P_C, and from this realize that its relationships with P_NC (Fig. S1 in Supporting Information) are not readily assessed from the relationships between P_NC and C. These more complex relationships of P_NC and P_C then in part drive the relatively complex pattern of relationship between P_NC and C_imp shown in Rees et al.'s Fig. 1b (see also The complexity of the response of C_imp to ‘productivity’, Appendix S1).

   Secondly, Rees et al. use randomly generated values of C to assess the relationship of C_imp to P_NC. The problem with this approach is that C is a ratio, more often expressed as RR (the response ratio). lnRR is commonly used to account for the inherent statistical issues associated with ratios (see, for example, Kikvidze & Armas 2010). General problems associated with the analysis of ratios have been discussed widely, and we return later to this topic; here, we are concerned with the use specifically of C in this particular set of analyses. In theory C can vary between 0 and ∞, with values >1 indicating situations of competition. Generating a uniform distribution of C between 1 and 100, as used by Rees et al. to provide synthetic data to test the C_imp–P_NC relationship, results in a non-uniform distribution of P_C such that in general P_C << P_NC (as shown in Fig. S2b). This strongly biases the simulated data towards very high competition, such that a very tight relationship between C_imp and P_NC is found. It is also worth noting that real world values for C are often relatively small. For example in the studies by Carlyle, Fraser & Turkington (2010) and le Roux & McGeoch (2010), values of C range from 1.15 to 2.8 and from 0.02 to 1.34 (based on Figs 1 and 2c of these studies, respectively). Assessing the relationship between P_NC and C_imp using values of C evenly spread between 1 and 100 may under-sample that area where many experimental studies sit, but the use of C to illustrate data obscures this bias in P_C. Generation of random values of P_C allows better coverage of parameter space (Fig. S3) and provides a better assessment of the potential variability in the C_imp–P_NC relationship (see also Selection of random data, Appendix S1).

3. The use of only the data from Kadmon (1995) to explore the C_imp–P_NC relationship in field studies

   As for the synthetic data of Rees et al., the empirical data from the work by Kadmon (1995) show also a strong relationship between C_imp and P_NC. Again, Rees et al. use P_NC as the measurement of severity (productivity), and as discussed above, this has predictable consequences for detected relationships. But in addition to this, Kadmon's study is of a very specific situation, focussing on annual plants in an extreme desert habitat with small scale variation in soil surface conditions leading to highly localized changes in severity over small spatial scales (metres). Because of the extreme environmental conditions, it is not surprising that a strong positive correlation between interaction importance and severity is found in this study. In such harsh conditions, the availability of water is the main factor governing plant abundance and thus also the nature of biotic interactions within the area. In addition, because this study was conducted on a very limited geographical scale, it is perhaps unrepresentative of the responses found in gradient studies in general. Consequently, it is unwise to assume that data from this study, interesting though it is, are broadly representative of patterns found in all field data. It is notable that data from other field studies do not necessarily show the same tight relationship between P_NC and C_imp (Fig. S4; see also ‘An exploration of relationships using field data’, Appendix S1).

A better approach to assessing the severity–C_imp relationship

We take a different approach to assessing the properties of C_imp and I_imp. Our simulation (Fig. 1) explores the response of three hypothetical plant species to variation in environmental severity. We assume that – as in the ideal case described above – the gradient of severity is assessed using a metric such as water or nutrient availability which represents the key severity drivers, but which is also independent of the success variables used to calculate C_imp. C_imp is therefore free to hold any value at a given point along the severity gradient, and we can assess independently the responses of P_C, P_NC and C_imp to variation in severity.

We use three alternative assumptions about the way in which plant success without competition (P_NC) varies in relation to environmental severity, relating to three different models of variation in success along severity gradients.

1. As in Rees et al., success is directly and linearly related to the severity gradient, declining with increasing severity.
2. The success optimum of each species is in a different location along the severity gradient, and success varies in a Gaussian distribution around this optimum. In addition, absolute maximum success is higher for species with an optimum towards the less severe end of the severity gradient. This distribution pattern is based on the concept of differences in species' fundamental niches.
3. Species' success increases up to an asymptote, and species reaching this asymptote under more severe environmental conditions have a lower maximum success. This pattern of distribution is based on the concept of shifting competitive hierarchies (Keddy 1989).

In all simulations, species with optima in the more severe environments might be considered typical stress tolerants (sensu Grime) and are capable of achieving a lower level of maximum success relative to more competitive species from the less severe end of the gradient. Further details of the assumptions behind our simulation are given in Fig. 1, and the R script used for these analyses is available in Appendix S2.

Overall, the analyses (Fig. 1) show that for any type of variation in success along the gradient, variation in C_imp is related to variation in P_NC. This is no surprise: as we have pointed out (above and in Appendix S1), P_NC is one of the components of C_imp, and so their distributions will inevitably be coupled. However, there is substantial potential for C_imp to deviate away from P_NC. But perhaps, more importantly, it is not inevitable that C_imp declines with increasing environmental severity. Although this can occur (Fig. 1d), it depends on the related response of plant success to the severity gradient (Fig. 1a). If success has a different response to the severity gradient, this can result in a different response pattern for C_imp, including bell-shaped distributions (Fig. 1e) and asymptotic relationships (Fig. 1f). What is critical is that C_imp can in theory have any kind of relationships with severity depending on variation in plant success.

Critically, relaxing the assumption that P_NC varies linearly along the severity gradient demonstrates that C_imp is not designed to show a decline with increasing severity to support a particular conceptual framework (such as that of Grime). Instead, C_imp depends on the response of plant success. The question of how plant success (P_NC) varies along the severity gradient therefore is crucial in the analysis of the importance of competition, linking it to the very considerable body of literature focusing on the processes that regulate species distributions in relation to environmental drivers including, for example, the works by Whittaker (1975), Keddy (1989), and Bigelow & Canham (2002).

Variance decomposition and its application to gradient analyses

One of the alternative approaches suggested by Kikvidze, Suzuki & Brooker (2011b) is the use of variance partitioning. Rees et al. also suggest the use of variance partitioning – or as they describe it variance decomposition – illustrating its application by re-analysing the data of Kadmon (1995).

As discussed, key ecological theories, such as those of Grime et al., the EEH (Oksanen et al. 1981), or the Stress Gradient Hypothesis (Bertness & Callaway 1994), predict a particular pattern of change in the relative role of interactions across environmental gradients. Any analytical approach relevant to these theories must therefore allow us to visualize how the relative role of plant–plant interactions in regulating plant success or community structure changes along the environmental gradient in question. Rees et al.'s approach does not allow us to answer this question, it assesses instead how much of the variation in plant success across the entire environmental gradient is due to severity and how much is due to competition, giving a single value (the ratio of var(C) to var(S)) for the relative roles of competition and stress. Consequently, this approach cannot be considered an alternative to the types of analyses we have performed using interaction indices. Its inability to explore the shape of the relationship between plant interactions and the severity gradient can be illustrated readily. We simulate a number of different relationships between P_NC (success in the absence of competition) and a hypothetical gradient of environmental severity. As we can see from these simulations (Fig. 2), there are quite distinct differences in the underlying relationships between P_NC and the gradient of environmental severity. However, the final analysis of these relationships, using the approach applied by Rees et al., is insensitive to the substantial differences in their form and appears no more responsive than the calculation of an average value of C_imp (Fig. 2). Interestingly, Rees et al.'s proposal for the use of variance decomposition appears implicitly to acknowledge this. They state:

If we are examining the effects of competition and the quality of the environment along an environmental gradient, we are likely to be interested in three aspects of the data: (i) how do changes in the quality of environment in the absence of competition affect plant performance? (ii) How do changes in competition affect performance? And (iii) how do the effects of competition and the environment interact with each other? Of these, the interaction is likely to be of greatest interest, that is, how does the environment affect the reduction in performance due to competition?

Only the third question here is related to the relative role of competition in regulating plant success, and it does not consider the shape of this relationship.

Sears & Chesson (2007) use variance decomposition to test for the spatial storage effect, a proposed mechanism by which dominant and subdominant plants can coexist in a spatially heterogeneous environment. Both Sears & Chesson (2007) and Rees et al. define the severity gradient on a species-by-species basis. Sears & Chesson (2007) state ‘the storage effect theory defines habitat quality separately for each species based on its response to spatially covarying conditions, rather than by discrete characteristics of the physical environment (such as nitrogen levels). In practice… gradients are defined by individual plant responses to the sum of environmental conditions at each growing location’. This again highlights the fundamental difference between approaches that define environmental gradients through individual species’ success (or some proxy of it such as biomass of plants in the absence of neighbours, i.e. P_NC), and those that use a metric of stand-level severity such as nutrient or water availability. Both are valid in different circumstances, but there are potentially confusing overlaps in terminology. Consequently, we need to be careful that we do not simply assume that a common parameter name such as ‘productivity’ or ‘severity’ is referring to the same type of variable and thus promote the use of a particular analytical technique for addressing a question to which it is not relevant.

Other possible alternatives

Our aim is not to provide a comprehensive assessment of alternative analytical approaches to these problems, but it seems sensible to discuss a few of the possible alternatives, remembering that our focal question is now more clearly defined as: what is the shape of the relationship between the relative role of competition and environmental severity?

First, it may be possible to adapt the variance partitioning approach. The key change would be to describe the severity gradient with a variable independent of P_NC, for example nutrient or water availability, or temperature. This would also require modelling of the response of plant success in the absence of neighbours (P_NC) to this gradient. The underlying assumption of Rees et al. that a linear relationship occurs between the gradient and P_NC is the simplest, but many different assumptions for the shape of this response curve can be proposed. The approach proposed by Rees et al. is therefore very promising, but needs to be developed such that it tests whether variation in P_NC is driven by the severity gradient or not.

Modifying the application of indices such as C_imp or I_imp might provide another alternative approach. Specifically, given that there is constraint of these indices in relation to P_NC (with upper boundaries for values of C_imp determined by P_NC/P_max), it may be possible to disentangle the relative contribution to C_imp of two underlying elements: (i) variation in plant success in the absence of neighbours (P_NC) along the gradient and (ii) variation in the intensity of competition along the gradient. For instance, it would be possible to simulate C_imp under a null model of random competition (as used in our critique of Rees et al.'s analysis, above) to show the implications of P_NC variation alone. The second step would be to analyse how variation in competition intensity contributes to the remaining variation in C_imp.

Finally, Jasienski & Bazzaz (1999) suggest the use of approaches that allow partitioning of variation in an explicit causal model, for example structural equation modelling (SEM) or path analysis. This approach is increasingly common in plant ecology and can be applied to understanding the relative role of different drivers in systems with potentially complex networks of interactions (see for example, the studies by Matias et al. (2011) and Lamb & Cahill (2008)). An exciting opportunity is that the Bayesian approach proposed by Rees et al. could be combined with the SEM approach in a hierarchical model (see Lee (2007) for a discussion of Bayesian SEM).

We conclude that there are some possible alternatives to the simple use of indices, but that the use of these approaches has to be appropriate to the question. The work of Rees et al. has been helpful in forcing us to define more clearly the question which is the focus for many studies, and the pitfalls associated with different measurements and proxies of success and severity. These more explicit definitions can now be used to ensure the relevance of the approach.