REPORT Predicting competition coef cients for plant mixtures: reciprocity, transitivity and correlations with life-history traits

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1 Ecology Letters, (2001) 4: 348±357 REPORT Predicting competition coef cients for plant mixtures: reciprocity, transitivity and correlations with life-history traits R.P. Freckleton 1 and A.R. Watkinson 2 1 Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, U.K. 2 Centre for Ecology Evolution and Conservation, Schools of Environmental and Biological Sciences, University of East Anglia, Norwich NR4 7TJ, U.K. Abstract There are few empirical or theoretical predictions of how per capita or per individual competition coef cients for pairs of plant species should relate to each other. In contrast, there are a considerable number of general hypotheses that predict competitive ability as a function of a range of ecological traits, together with a suite of increasingly sophisticated models for competitive interactions between plant species. We re-analyse a data set on competition between all pairwise combinations of seven species and show that competition coef cients relate strongly to differences between the maximum sizes, root allocation, emergence time and seed size of species. Regressions suggest that the best predictor of competition coef cients is the difference in the maximum size of species and that correlations of the other traits with the competition coef cients occur through effects on the maximum size. We also explore the patterns of association between coef cients across the competition matrix. We nd signi cant evidence for coef cient reciprocity (inverse relationships between the interspeci c coef cients for species pairs) and transitivity (numerically predictable hierarchies of competition between species) across competition matrices. These results therefore suggest simple null models for plant community structure when there is competition for resources. Keywords Community dynamics, resource competition, yield-density equation. Ecology Letters (2001) 4: 348±357 INTRODUCTION Quantitative models of plant competition are of two types that are commonly argued to be mutually exclusive (Grover 1997). Resource-based approaches (Tilman 1982, 1988; Grover 1997) analyse competition by explicitly considering the partitioning of resources between and within competing plants and species. When combined with information on births and deaths, these models allow longterm projections of the consequences of the short-term process of competition between individuals for resources. In contrast, empirical approaches (Law & Watkinson 1987; Pacala & Silander 1990), that trace their formulation back to the classical Lotka±Volterra models (Hassell & Comins 1976), measure competition coef cients over the short term, but do not explicitly consider resource competition (Tilman 1988; Grover 1997). Consequently, it is not possible to predict how competition will be affected by changed conditions or resource levels. The advantage of the former approach is that it is possible to derive a complete understanding of the mechanics and outcome of competitive interactions, as well as to derive general rules concerning the outcome of competition (reviewed by Tilman 1988 and Grover 1997). The disadvantage, however, is that the experimental data requirements are large compared with the simple empirical approach (e.g. Pacala et al. 1996). Nevertheless, the common rationale behind both approaches is the prediction of the outcome of competition, and it is consequently important to link them. To this end, in lieu of a detailed understanding of the mechanisms of resource competition, a number of authors have suggested that competitive ability may be related to simple ecological traits. These include seed size (MaranÄoÂn & Grubb 1993; Rees 1995; Rees & Westoby 1997), seedling size (Schwinning & Fox 1995), emergence time

2 Predicting competition coef cients for plant mixtures 349 (Cousens et al. 1987; Kropff & Spitters 1991), plant size (Goldberg & Landa 1991) and population productivity (Gaudet & Keddy 1988). The advantage of identifying single factor correlates of this nature is that patterns of competition may be predicted using easily measured ecological variables. The disadvantages, however, are that most ecological and life-history traits are correlated through either trade-offs, constraints or allometry so that the underlying determinants of competition may be poorly or misunderstood. Also, the identi cation of particular correlates of competitive ability will almost certainly depend on the way in which competition is measured. A series of attempts has also been made to de ne the competitive relationships between plants within communities. In particular, studies have concentrated on crossspecies comparisons of competitive effect and response (e.g. Jacquard 1968; Goldberg & Landa 1991), as well as studying competitive hierarchies within communities (e.g. Keddy & Shipley 1989; Goldberg & Landa 1991; Shipley 1993; Keddy et al. 1994). The identi cation and interpretation of competitive hierarchies may also depend on the measure of competition employed (Silvertown & Dale 1991; Goldberg 1996). While some general patterns emerge when studies are compared (Goldberg 1996), this kind of analysis is typically qualitative, i.e. a hierarchy of species of the form A > B > C >¼ is generated. In this sense, transitivity is a simple topological property of competition matrices, and there has been no attempt to explore whether such relationships have a quantitative basis, for example using series of pairwise competition coef cients. There are few predictions of either the coef cients of simple ecological models or how coef cients for interactions should vary across competitive hierarchies. This is despite the extensive use of coef cient-based competition models for dozens of statistical, theoretical and applied problems (e.g. Watkinson 1980, 1981; Firbank & Watkinson 1985, 1986; Law & Watkinson 1987; Kropff 1988; Cousens & Mortimer 1995; Schwinning & Fox 1995; Rees et al. 1996; Rees & Westoby 1997; Freckleton et al. 2000). In an attempt to make such predictions, Keddy & Shipley (1989) argued that pairwise interaction coef cients should relate closely to the height differences of the interacting species and, moreover, that in a multispecies community species should be organized into a transitive hierarchy. In this paper, we use previously published data on competitive interactions under glasshouse conditions to explore these issues, as well as the potential for linking empirical models to basic life-history parameters. Although the system we study is considerably simpler than eld communities, the analysis serves to demonstrate the potential for exploring multispecies community structure in terms of competition coef cients. In particular, the dominant form of interaction should be competition for limiting resources, and the effects of, for example, litter dynamics, pathogens and herbivores, that exist in natural communities are factored out. Speci cally, we explore: (i) how coef cients from simple competition models may be related to basic plant traits; and (ii) how competition between plants under controlled conditions may generate simple patterns of association between competition coef cients. METHODS Data for analysis The data we analyse are taken from Goldberg & Landa (1991). They performed an experiment in which seven species were grown according to an additive design in all pairwise combinations. Trays (25.4 cm 25.4 cm 6.3 cm), lled with a nutrient-rich soil mix, were sown with nine target individuals and 0, 8, 16, 32, 64, 128 or 256 neighbours. The density of target individuals was low enough that it could be assumed that little or no interference occurred between them. The mean whole plant biomass of the target plants was recorded from each tray after 5 weeks of growth. The seed mass for each species was determined from weighing ve lots (20±50 seeds per lot) of seeds. The emergence time was measured as the median number of days to emergence of lots of at least 50 seeds of each species. The other life-history traits (allocation to roots, maximum plant size and relative growth rate (RGR)) were measured at harvest on plants grown in isolation. Goldberg & Landa (1991) measured the effects of varying density through tting of the linearized model: w ij ˆ 1 a ii N i a ij N j 1 w m;i Performance was measured as the reciprocal of the ratio of mean performance in a mixture (w ij ) to mean performance in isolation w m;i. The effects of increasing intraspeci c densities (N i ) and interspeci c densities (N j ) are measured through two coef cients, a ii and a ij, respectively. This model is a linearized form of the classic hyperbolic yield-density model (see below). As such the coef cients measured are per capita competition coef cients and this experiment is therefore exceptional in that these coef cients have been measured in all 49 possible pairwise combinations of the species. The values estimated by Goldberg & Landa (1991) are given in Table 1a. Goldberg & Landa (1991) did not present standard errors for these coef cients. We therefore re-analysed the data from their paper (Goldberg & Landa 1991, g. 2) in order to generate these, which are given in parentheses in Table 1a. The mean values for the life-history variables reported by Goldberg & Landa (1991) are summarized in Table 1b.

3 350 R.P. Freckleton and A.R. Watkinson Table 1 Details of competition coef cients estimated by Goldberg & Landa (1991) for seven plant species. The species were Lolium perenne L. [1], Trifolium pratense L. [2], Trifolium repens L. [3], Rumex crispus L. [4], Chenopodium album L. [5], Amaranthus retro exus L. [6] and Phleum pratense [7]. (a) Estimates of the per capita competition coef cients (a) of eqn 1. Figures in parentheses are standard errors, obtained by re-analysis of the data presented in g. 2 of Goldberg & Landa (1991). (b) Ecological traits of each species (a) Effect of species i On species j [1] [2] [3] [4] [5] [6] [7] [1] 3.62 ) ) ( 0.53) ( 0.28) ( 0.20) ( 0.09) ( 0.18) ( 0.33) ( 0.22) [2] ( 0.22) ( 0.18) ( 0.17) ( 0.09) ( 0.17) ( 0.11) ( 0.05) [3] ( 0.70) ( 0.11) ( 0.27) ( 0.09) ( 0.12) ( 0.09) ( 0.08) [4] ( 0.65) ( 0.79) ( 0.27) ( 0.09) ( 0.19) ( 0.22) ( 0.09) [5] ( 0.35) ( 0.33) ( 0.21) ( 0.04) ( 0.23) ( 0.25) ( 0.03) [6] ( 0.84) ( 0.28) ( 0.21) ( 0.12) ( 0.51) ( 0.32) ( 0.20) [7] ( 0.30) ( 0.58) ( 0.13) ( 0.10) ( 0.20) ( 0.46) ( 0.10) (b) [1] [2] [3] [4] [5] [6] [7] Shoot allocation (%) Median days to emerge RGR (mg/mg/day) Seed mass (mg) Maximum weight (mg) Measuring competition Equation 1 is a re-arrangement of the classic hyperbolic competition model (Firbank & Watkinson 1985): w ij ˆ w m;i 1 a ii N i a ij N j 1 2 This equation may be re-written as: w ij ˆ w m;i 1 a ii N i e ij N j Š 1 3 The parameter e ij is de ned as the ratio of a ij to a ii and as such measures how many individuals of species i the competitive effect of each individual of species j is equivalent to, i.e. e ij is an equivalence ratio. Although simple, this formulation is justi ed by the general success of eqn 2 and close analogues in describing competition within and between plant species under a range of conditions and scales (Firbank & Watkinson 1985; Law & Watkinson 1987; Pacala & Silander 1990; Rees et al. 1996; Freckleton et al. 2000). Equations 2 and 3 are clearly the same model written in two different ways. Equation 2 is generally statistically easier to t, and eqn 3 is obtained through manipulation of the tted parameters of eqn 2 (e.g. Watkinson & Freckleton 1997; Freckleton et al. 2000). The advantage of viewing the competition model in the form of eqn 3 rather than eqn 2 is a matter of interpretation, and we show how this affects the interpretation of competition in this study below. In particular, a ij, the interspeci c coef cient in eqn 2, will be sensitive to changes in the intraspeci c coef cient (a ii ), whereas in eqn 3 there is no a priori reason to expect a ii and e ij to covary (Freckleton & Watkinson 1997a). Speci cally, since a ii is commonly de ned as the ecological neighbourhood of the species (Antonovics & Levin 1980; Watkinson 1980; Vandermeer 1984), we may expect this parameter, at a coarse scale, to be a function of plant size (Firbank & Watkinson 1985). Furthermore, since the equivalence coef cient represents a ratio of inter- to intraspeci c effects, an obvious analysis is to compare these values with differences between species traits. Estimates of the equivalence coef cients for all pairs of species are given in Table 2.

4 Predicting competition coef cients for plant mixtures 351 Table 2 Estimates of the per individual equivalence (e) of each species pair, estimated as e ij = a ij /a i (see eqn 2) using the values reported by Goldberg & Landa (1991) summarized in Table 1a. The species were Lolium perenne L. [1], Trifolium pratense L. [2], Trifolium repens L. [3], Rumex crispus L. [4], Chenopodium album L. [5], Amaranthus retro exus L. [6] and Phleum pratense [7] Effect of species i On species j [1] [2] [3] [4] [5] [6] [7] [1] 1.00 ) ) [2] [3] [4] [5] [6] [7] Analytical methods We aimed to determine how the equivalence ratio e ij correlates with the ecological traits (Table 1b) recorded by Goldberg & Landa (1991). We compared these with differences in measures of performance, i.e. the ratio of the trait of species j to that of species i. To analyse these associations, we performed regressions of the equivalence coef cients on all differences in ecological traits. We used the standard errors we estimated in our re-analysis of the data presented by Goldberg & Landa (1991) to generate sampling distributions for the equivalence coef cients. These were generated by a simple simulation procedure. For each coef cient, pseudo-random values were generated using the estimated coef cient and standard error, together with pseudo-random values drawn from a t-distribution with the appropriate degrees of freedom. These randomized values were used to analyse relationships between pairs of coef cients. RESULTS Intraspeci c competition There was no signi cant correlation between the intraspeci c coef cients (a ii, the leading diagonal in Table 1a) and any of the ecological traits measured in Table 1b. Although not statistically signi cant, it is perhaps suggestive that the largest value of a ii (3.62) was recorded for Lolium, the species with the largest mean weight per isolated plant (130 mg). For comparison, the next largest value of a ii was 1.12, less than one-third of a ii for Lolium, whilst the next largest maximum plant weight was 39 mg, similarly less than one-third of the maximum plant weight of Lolium. Correlates of interspeci c interactions There were signi cant associations (Fig. 1) between the equivalence coef cients, e ij, and all the differences in ecological traits (Table 1a) between the species other than RGR. The strongest correlation was with differences in maximum plant size. It should be noted that the correlation of e ij with difference in shoot allocation is negative, indicating that increased root allocation increased competitive ability. Since shoot allocation was measured on plants that were not subject to competition, and hence increased allocation to roots is not a consequence of shoot competition, it may be concluded that plants were competing for nutrients in this experiment. Although the strongest individual correlate of the equivalence coef cients was the difference in maximum size, maximum plant size is, however, closely related to the other variables: regression analysis indicated that the plant size may be accurately predicted from the other variables (r 2 ˆ 0.99; n ˆ 42; P < ). By excluding plant size from the analysis, it is thus possible to relate the equivalence coef cients to a combination of the other trait differences. The best tting relationship was between the differences in equivalence coef cients and differences in RGR and seed size (r 2 ˆ 0.77; n ˆ 42; P < ). Cross-species correlations In this section, we explore some of the wider consequences of the relationships described above. Speci cally, we explore the issue of how coef cients for pairs of species should relate to each other, re-examining the notion that competitive effects and responses are uncorrelated for these species (Goldberg & Landa 1991); in addition, we look at the numerical basis for the existence of hierarchies identi ed by Goldberg & Landa (1991). At a broad scale, a general reciprocal relationship exists between pairs of equivalence coef cients (Fig. 2a). The line in Fig. 2a represents the relationship e ij ˆ e ji ±1, and it is clear that most points fall close to this line. It should be noted that the data are re ected about the line y ˆ x, since the 42 pairs of coef cients are not independent. For the 21

5 352 R.P. Freckleton and A.R. Watkinson Figure 1 Correlates of competition coef cients. The competition coef cients are measured as the per individual equivalence of one species to the other (see text and Table 2 for details) and these are related to the difference between the ecological traits of the species. The effect of (a) maximal mean weight difference (y ˆ ± x; r 2 ˆ 0.72; P < ), (b) seed size difference (y ˆ x; r 2 ˆ 0.20; P ˆ 0.002), (c) emergence time difference (y ˆ ± 0.509x; r 2 ˆ 0.24; P ˆ ), (d) difference between species allocation to shoots (y ˆ ± 0.631x; r 2 ˆ 0.38; P ˆ 0.001) and (e) difference in RGR (y ˆ x; r 2 ˆ 0.03; P ˆ 0.146). independent pairings, we tted the regression. The maximum likelihood estimate of c (assuming an error distribution from the exponential family and t using a Rosenbrock pattern search; Rosenbrock 1960) was not signi cantly different from unity (c ˆ , P{c ˆ 1} > 0.05, d.f. ˆ 20). Table 3 presents tests of this hypothesis for each of the 21 independent pairs of coef cients. The tests were performed in the following way: for each of pseudo-random pairs of coef cients generated as outlined above, the product e ij e ji was generated. The null hypotheses that e ij e ji ˆ 1 was tested by recording the proportion of these pairs for which this product was greater or less than unity depending, respectively, on whether the estimated product was less than or greater than unity. Table 3 shows that the null hypothesis of reciprocity was accepted in 16 out of the 21 cases. It should be noted that the pattern of reciprocity was considerably stronger for the standardized equivalence coef cients than for the unstandardized a ij. For the equivalence coef cients, the correlation between e ij and e ±1 ji was r ˆ 0.76 (n ˆ 42, P < 0.001) compared with r ˆ 0.36 (n ˆ 42; P<0.01) for the correlation between a ij and a ±1 ji. This difference is discussed below. The nal analysis performed was designed to analyse the degree of transitivity of competition coef cients amongst the pairwise interactions. In particular, we hypothesized that, as a consequence of the close and predictable relationships between pairs of coef cients in Fig. 1, then for three species i, j and k, we should be able to predict the equivalence of species i to species k using the relation e ik ˆ e ij e jk. Figure 2b shows the mean value of each of the 42 e ik averaged across all possible pairwise combinations of the other equivalence coef cients. It is clear that the predictions are closely correlated to the values estimated in Table 2 (Spearman rank correlation, r s ˆ 0.85, n ˆ 42, P < 0.001), i.e. the competition coef cients are transitive. We also found similar relationships between coef cients estimated using four-, ve-, six- and seven-way transitive relationships, as would be expected given the transitivity of the three-way combinations (see gure legend for details). DISCUSSION Our re-analysis of the data of Goldberg & Landa (1991) shows: (i) that the outcome of competition in this system is well predicted by life-history traits; (ii) as a consequence, there are simple patterns of association between coef cients for pairs of species studied in this experiment; and (iii) similarly, there are predictable relationships between coef- cients across competitive hierarchies. Such predictions are

6 Predicting competition coef cients for plant mixtures 353 Figure 2 Relationships among equivalence coef cients. (a) Reciprocity between coef cients. The graph shows the per individual equivalence of species j to species i, plotted against the per individual equivalence of species i to species j. Note that the graph is re ected about the line y ˆ x, such that the open points are the reciprocals of the lled points. The line shows the reciprocal relationship y ˆ x ±1, and is not a tted curve. (b) Transitivity amongst competition coef cients. The per individual equivalence of species i to species k was estimated by multiplying the equivalence of species i to another species j by the equivalence of species j to species k. The points show the average coef cient estimated through all possible combinations of pairs of species. The line indicates the relationship y ˆ x. We similarly found signi cant correlations when analysing four-, ve-, six- and seven-way transitivity (Spearman rank correlations: four-way, r s ˆ 0.835; ve-way, r s ˆ 0.851; six-way, r s ˆ 0.835; seven-way, r s ˆ 0.843; n ˆ 42, P < in all cases). potentially important, since it has been shown that the effects of competition between plants on mean biomass, as measured by simple coef cients, may translate directly into impacts on community dynamics (Moloney & Chiariello 1998; Freckleton et al. 2000). The system studied here is clearly simplistic in that it represents competition between glasshouse grown plants over only a few weeks. In reality, most systems are, of course, considerably more complex. For example Freckleton et al. (2000) found that competitive interactions between grasses and legumes in naturally regenerating communities were considerably weaker than those found in this study, or where grasses and legumes had been sown together (Freckleton et al. 2000). This difference was attributed to different patterns of emergence in natural communities compared with the synchronous emergence pattern under experimental conditions. Where seedlings emerge within established plant communities, competition functions might be expected to depend not only on between-plant competition for resources, but also on the size structure and emergence pro le of the community. However, greenhouse studies conducted under controlled conditions allow us to specify hypotheses that can then be tested under more complex eld conditions (Freckleton & Watkinson 2000). Our main conclusions are that reciprocity and transitivity of coef cients represent reasonable null models from which to analyse the structure of competition matrices for plant communities. The theoretical basis for expecting coef cients to follow these patterns relates to short-term competition for limiting resources. If species i removes, per individual, R times as much resource as j, then the per individual equivalence of i to j will be R, if all other things are equal. Similarly, if species j removes, per individual, r times as much resource as species k, the ratio of resources removed by species i and k is rr, and hence the equivalence of species i to species k is rr. In the system analysed above, competitive ability correlated positively with allocation to roots (measured under non-competitive conditions), indicating that plants were competing in this way for soil resources. The structure of competitive interactions thus corresponds to simple models of competition for limiting resources (Tilman 1982, 1988), with the consequence that the structure of the competition matrix is readily predicted. On the other hand, if interactions are not dominated by resource competition, patterns of association between coef cients may be very different. For example if allelopathy is important, competition coef cients do not reciprocate and the outcome of competition can be complicated (e.g. Durrett & Levin 1997). More generally, niche differentiation between species will push coef cients away from reciprocity and destroy the simple patterns of association and transitivity we have identi ed above. Previously, the notion of competitive transitivity has been used to describe the consistency of competitive hierarchies (Keddy & Shipley 1989; Shipley 1993). In this sense, competitive transitivity has been used to describe simple topological properties of competition matrices. The results presented above indicate that the notion of competitive transitivity may potentially be taken a step further and used quantitatively to predict the numerical relationships between coef cients. For instance, in topological terms, transitivity

7 354 R.P. Freckleton and A.R. Watkinson Table 3 Tests of reciprocity of competition coef cients. The data of Goldberg & Landa (1991) were re-analysed to generate standard errors for the per capita competition coef cients (see Table 1a). A simulation procedure was then used to test for reciprocity (see text for details). (o, null hypothesis of reciprocity accepted, P > 0.05; *null hypothesis of reciprocity rejected, P < 0.05; **P < 0.01; ***P < 0.001) T. pratense T. repens Rumex Chenopodium Amaranthus Phleum Lolium o ** o o o ** T. pratense o o o o o T. repens o o o *** Rumex ** * o Chenopodium o o Amaranthus o predicts that, if e ij > 1 and e jk >1, then e ik >1. By contrast, in the data analysed above we showed that e ik ˆ e ij e jk. As we have stressed above, it is not plant biomass per se that determines competitive ability in this experiment: rather maximum mean plant size (in this study) is an approximate measure of the amount of resource a species removes and hence makes unavailable to other species. Factors such as seed size, seedling size and emergence time are important determinants of biomass. In this system, the analysis demonstrates that the effects of these variables on competitive ability is through the way in which they determine how much resource each individual is capable of removing. Maximum individual biomass in the absence of competition is thus a simple measure of, or proxy for, the amount of resource individuals remove, and is a direct function of the traits such as seed size, emergence time, RGR and allocation to roots. This strongly suggests that tests of theories relating competitive ability to ecological traits have to explicitly consider how ecological traits in uence individual and population level resource uptake. Gaudet & Keddy (1988) suggest that total stand biomass may be a correlate of competitive ability, but this does not separate individual and population level factors. The utility of the simple empirical approach to studying competition that is represented by models of the form of eqn 3 has been questioned by a number of authors. In particular, Tilman (1988) questions whether such approaches can be used to derive a mechanistic understanding of competition, whilst Grover (1997) recognizes that empirical approaches are at present limited in that it is not possible to link patterns of variation in competition coef cients, either between species or as resource levels are varied, to variation in competition for resources. The large number of single-species populations and increasingly multispecies dynamics that employ such approaches is evidence of their utility, particularly in predictive modelling. Our analysis has shown that it is possible to relate the two forms of model through only a few key measurements on isolated plants. As noted by Goldberg (1996), one of the major problems with deriving general conclusions concerning the determinants of competitive ability is that a range of measures of competitive intensity exist, e.g. the relative crowding coef cients of replacement series analyses (de Wit 1960), competitive equivalence from full additive series designs (Law & Watkinson 1987) and simple indices of competition based on comparisons of performance of plants grown in the presence and absence of neighbours (Silander & Antonovics 1982; Goldberg & Scheiner 1993; Grace 1995). The utility of these has been frequently and hotly debated (Connolly 1986, 1988; Snaydon 1991, 1994; Sackville Hamilton 1994; Freckleton & Watkinson 1997a,b, 1999; Peltzer 1999). The difference between the values and relative values of the unstandardized coef cients (Table 1a) and standardized coef cients (Table 2) highlights why distinguishing between different measures of competition is important. We may, for example, wish to compare coef cients across several species pairings in order to look for evidence of niche differentiation or for other ecologically signi cant patterns of coef cient values. Freckleton et al. (2000), for example, used this approach to claim evidence for niche differentiation between two grasses and a legume. The analysis presented here for a larger competition matrix makes it clear that distinguishing between different coef cients is of importance for measuring the strength of competition, and that some measures of competition yield clearer information than others. An important issue relating to these coef cients is that the outcome of competition in classic competition models that predict the long-term consequences of competition for coexistence depends on values of e rather than directly on the values of a. Although both a and e ij are required in order to predict community dynamics, it is conditions on the equivalence coef cients that determine whether coexistence may be possible or not. Speci cally, if e ij ˆ e ji ±1, then simple dynamic coexistence between species is impossible (irrespective of the value of a) in the absence of other factors such as heterogeneity, disturbance or

8 Predicting competition coef cients for plant mixtures 355 spatial processes. This is because, if e ij ˆ e ji ±1, the zero net growth isoclines, plotted as functions of population densities, for the two species will always be parallel and one or other of the species will always outcompete the other as in classic phase-plane analysis (e.g. Maynard Smith 1974). If this relation does not hold (i.e. e ij 6ˆ e ji ±1 ), then simple coexistence may be possible, although will depend on the values of other coef cients. Similarly, the pattern of competitive transitivity depends on how the coef cients have been de ned. Speci cally, if we accept that e ik ˆ e ij e jk (Fig. 2), then it cannot be true that a ik ˆ a ij a jk since a ik ˆ a i e ik ˆ a i e ij a j e jk. Whether or not the pattern of transitivity identi ed in Fig. 2 will hold for these unstandardized coef cients will depend on the degree to which a ii and e ij covary with each other. Transitivity of coef cients does not depend on reciprocity. Reciprocity may result from simple resource competition between pairs of species. By contrast, transitivity results when the mechanisms of competition are consistent across competitive hierarchies. For example, if e ij and e ji do not reciprocate, then there is no reason why the triplet e ij, e ji and e jk should be transitive with respect to e ik. It is important to note, however, that the strength of competition measured by a ii and e ij from experiments that consider plant weight as the dependent variable is not necessarily the same as the strength of competition in a population dynamic sense (Chesson & Huntly 1997). In the context of community dynamics, the intensity of competition is determined by the impacts of competition on the per capita rates of population change and how these covary with changing densities and environmental conditions, as well as the impacts of spatial and temporal processes on models of the sort analysed above (e.g. Pacala & Tilman 1994). It is important to distinguish the population and individual level measures of competitive ability and intensity (Tilman 1988; Chesson & Huntly 1997). Our analysis and interpretation of the competition coef cients suggest relatively straightforward tests of the degree to which individual and population level processes in uence the structure of communities: if individual plant level patterns of competition and resource consumption dominate, then competitive hierarchies should be a direct re ection of the per individual equivalence coef cients (e.g. Moloney & Chiariello 1998). If, on the other hand, population level processes are dominant, then individual level competition coef cients should be less important in determining competitive dominance; instead, the outcome of competition will be a function of some other demographic variable, such as reproductive rate or dispersal. Our re-analysis of the data of Goldberg & Landa (1991) shows that the outcome of resource competition (in the sense of competition for resources within the period of a growing season) under controlled conditions should be highly predictable from a small number of measurements on isolated plants. Under eld conditions, a wide range of other factors could modulate and modify these patterns. There are as yet few eld data with which these predictions can be compared. Freckleton et al. (2000) estimated coef cients for a three-species mixture. These coef cients did not reciprocate and were not transitive. This was because of niche differentiation between the legume and two grasses, as well as an effect on population growth of competitive effects that did not result from competition between plants for resources. This shows that the analysis presented here, in terms of reciprocity, presents a null hypothesis against which patterns of association between coef cients can be tested. It also indicates that the degree to which competition coef cients based on net population growth rates agree with predictions from experimental systems, where resource competition predominates, should present a powerful rst step in testing the importance of different processes in structuring plant communities. The advent of powerful new techniques for analysing the strength of competitive interactions within communities in situ (e.g. Pacala & Silander 1990; Rees et al. 1996; Law et al. 1997; Freckleton & Watkinson 2001) suggests an exciting potential for linking a range of previously disparate approaches. 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10 Predicting competition coef cients for plant mixtures 357 Watkinson, A.R. (1981). Interference in pure and mixed populations of Agrostemma githago. J. Appl. Ecol., 18, 967±976. Watkinson, A.R. & Freckleton, R.P. (1997). Quantifying the impact of arbuscular mycorrhizae on plant competition. J. Ecol., 85, 541±545. de Wit, C.T. (1960). On competition. Versl. Landbouwk. Onderz. Netherlands, 66, 1±82. Editor, F.I. Woodward Manuscript received 22 January 2001 First decision made 5 March 2001 Manuscript accepted 13 April 2001 BIOSKETCH Rob Freckleton's research has centred on techniques for combining ecological models with eld data in order to generate predictions of population and community dynamics that are spatially and temporally robust. His current research (with Paul Harvey's group and the University of Oxford and Mark Pagel at Reading University) is looking at comparative techniques for analysing the ecological processes underlying adaptive radiations and community assembly.