• above-ground biomass;
  • assemblage-level thinning;
  • Poisson distribution;
  • self-thinning;
  • species richness


  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References

1 A unimodal relationship between species richness and primary productivity is commonly reported. To explain this pattern, the mechanisms proposed in the many hypotheses are generally complex and almost all are without a strong empirical foundation. Here we evaluate the role of self-thinning in plant assemblages: assemblage-level thinning.

2 We developed a simple two-parameter model of species richness that predicts that plant species richness will be determined by a unimodal relationship between total plant density and above-ground biomass. This model provides a very narrowly defined set of testable quantitative predictions, and thus is the first falsifiable model of assemblage-level thinning. We fit this model to the species richness–above-ground biomass data from 14 empirical studies that are often cited as evidence of a general diversity–productivity relationship. In addition, we compared our model to two other models, one more flexible and one more constrained than our own.

3 We found that our model of species richness explained a substantial and statistically significant portion of the species richness observed in 11 of the 14 empirical studies of species richness–biomass patterns. Therefore, given the conservative nature of our model, and the number of previously published data sets explained by this model, we argue that assemblage-level thinning not only provides a viable and exceedingly parsimonious explanation, but may also be a widespread phenomenon.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References

Many authors have drawn attention both to the unimodal pattern of plant species richness as a function of above-ground biomass and to the decline in plant species richness in response to increasing soil fertility (Grime 1973, 1979; Tilman 1982; DiTomasso & Aarssen 1989; Rosenzweig & Abramsky 1993; Tilman & Pacala 1993; Rosenzweig 1995; see also the results of a recent NCEAS workshop at Nearly as many mechanisms have been proposed to explain these patterns of species richness as there are examples of the patterns (Tilman & Pacala 1993; Abrams 1995; Rosenzweig 1995), but for most of them there is little empirical support. The assemblage-level thinning hypothesis (Stevens & Carson 1999), which is supported by some empirical data, suggests that the unimodal pattern of species richness and the decline in species richness in response to fertilizer addition both arise as a probabilistic by-product of an underlying density–biomass relationship. In this study, we developed a model of the species richness–density–biomass relationship and then used 14 previously published data sets to assess the significance of assemblage-level thinning for known patterns of species richness.

The assemblage-level thinning hypothesis predicts that assemblages undergo thinning in response to soil fertility to the same extent as seen in monocultures (the Sukatchew effect; Sukatchew 1928 in Harper 1977). Thus, as productivity rises, surviving individuals within assemblages become much larger and, as these drive down survivorship of other individuals, total density declines. As a result, when equal area samples are compared, those in productive habitats contain fewer individuals than unproductive samples and thus, by chance alone, they will contain fewer species. Recently, we found that total density declined dramatically along an experimental productivity gradient and that this decline largely explained the observed decline in species richness (Stevens & Carson 1999). The only other model to explain this phenomenon (the no-interaction model; Oksanen 1996) contains a mathematical formulation that is so flexible it can fit a variety of patterns. As well as the relationship predicted by self-thinning, the no-interaction model could fit a positive, linear relationship between biomass and species richness over any range of above-ground biomass. Oksanen’s no-interaction model is thus difficult to falsify. Even though past efforts have focused attention on assemblage-level thinning (Oksanen 1996), we still have no idea of the ubiquity of this phenomenon or species richness–density–biomass relationships. Our falsifiable model allows us to address these questions.

Developing a general model of total plant density along productivity gradients

Total stem density in a plant monoculture increases as above-ground biomass increases above 0 g m–2 (Harper 1977). As productivity rises and robust individuals continue to increase in size, individuals will become crowded, and this is likely to result in thinning (Sukatchew 1928 in Harper 1977; Westoby 1984; White 1985; Weller 1987, 1990). Therefore, we would probably find a unimodal density–biomass relationship if the density of a monoculture is regulated primarily through resource competition, and providing we sampled such a monoculture along a sufficiently broad productivity gradient. We hypothesize that entire assemblages are regulated in the same manner. Even with no change in relative species abundances, this decline in density would result in a decline in the number of species per unit area along a productivity gradient as a result of the relationship between the number of individuals and the number of species (Gotelli & Graves 1996). Further, depending on the magnitude of the decline in density and on the number of species in the regional species pool, this effect is likely to persist in samples of large areas containing large numbers of individuals (Gotelli & Graves 1996).

We used a single continuous function to model the gradual transition from very uncrowded assemblages (including samples with zero biomass) to high biomass assemblages. We chose a simple non-linear Michaelis–Menten-type function because it has asymptotic properties that matched our biological expectations at both ends of a productivity gradient. Thus, like the original self-thinning rule, we chose a function that accurately describes an empirical pattern. Specifically, we used:

  • image(1)

where D(B) is total density (number of stems m–2), B is total above-ground biomass (g m–2), and k is a constant. Specifically, this model exhibits two critical properties long observed in monocultures. First, when biomass (B) is very low, D(B) approaches B/k, because 10–8B3 is close to zero. Thus, when biomass is very low, the model predicts a positive, approximately linear relationship between density and biomass. This linear relationship in low biomass, low density stands is observed in monocultures when initial density is varied (Harper 1977). Secondly, when biomass is high, D(B) approaches the same empirical self-thinning slope observed in monocultures of a wide variety of species (Gorham 1979; White 1980). The resulting constrained family of curves for D(B) is therefore consistent with empirical observations in uncrowded, low biomass monocultures as well in crowded, high biomass monocultures. Because D(B) is so constrained, it cannot fit all possible data sets, and thereby provides falsifiable predictions for density of plant assemblages in response to above-ground biomass. In contrast, Oksanen’s model is too flexible, does not make specific predictions, and thus is difficult to falsify.

Among the most fundamental determinants of species richness is the number of individuals sampled (reviewed in Gotelli & Graves 1996), although most models of species richness (e.g. canonical lognormal, broken stick) assume that the mechanisms that control species richness do not change with the number of individuals present (Gotelli & Graves 1996). There are, however, at least two different classes of mechanisms (density-independent mortality and resource competition) that control density and species richness along a productivity gradient. There is also a vast myriad of species-number functions available (Fisher et al. 1943; Preston 1948, 1962; May 1975; Coleman et al. 1982; Williams 1995; He & Legendre 1996; reviewed in Gotelli & Graves 1996), but we found no mechanistic justification for selecting any particular function. Therefore, to facilitate comparison between this model and Oksanen’s no-interaction model, we, like Oksanen (1996), chose the function of Fisher et al. (1943):

S  = αln[1 + (N/α)],(2)

where S is the number of species, N is the number of individuals in a sample, and α is an empirically derived constant (Fisher et al. 1943). Combining equations 1 and 2, we then have:

  • image(3)

where, as above, S is the number of species predicted, B is total above-ground biomass (g m–2), A is plot size (m2) and k and α are habitat-specific constants. We refer to equation 3 as ‘ALT’, the assemblage-level thinning model of species richness with parameters k and α.

Materials and methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References

The data sets

To assess how common assemblage-level thinning might be, we searched the literature for studies that found a negative or unimodal relationship between above-ground biomass and species richness. R.B. Waide & M. R. Willig ( nceas-web/results/projects/95WAIDE1) found that such relationships were common only when the productivity gradients were within a single community or among relatively similar communities. At biome or global scales, they found that a positive monotonic richness–productivity relationship predominated. We therefore restricted our analysis to data from within communities and among community types (sensuMoore & Keddy 1989), where the unimodal species richness productivity curve appears to be well substantiated. In addition, we considered studies of experimental fertility gradients (within communities) where species richness typically declines in response to increased soil fertility (DiTomasso & Aarssen 1989). These criteria resulted in our using 14 studies (listed in Table 1) that sampled species richness and above-ground biomass along productivity gradients (Table 1). There was a number of characteristics of particular data sets that required special attention prior to analysis, and colleagues that could possibly influence interpretation of results.

Table 1.  Tests of fit to a Poisson distribution for species richness. In a given model, if the deviance of species richness increases with the mean, the deviance/d.f. = 1.0 and the deviance follows a χ2 distribution. If the deviance increases faster than the mean, then the deviance/d.f. ratio is greater than 1.0 and the deviance departs from a χ2 distribution (*P < 0.05, ***P < 0.001)
   Model deviancesModel deviance/d.f.
StudynPlot m2ALTNointeractionMeanALTNointeractionMean
  1. a, among community gradient; w, within community gradient; n, natural productivity gradient; e, experimental productivity gradient, insect, herbivory-generated productivity gradient; v, plot size varied for biomass although species richness was measured in 1.00-m2 plots; NA, not applicable because biomass and species richness were measured per 500 ramets, not in a constant area.

Al-Mufti et al. (1977)a,n140.2573.341.395.2 5.64***3.17***7.32***
Carson & Root (in press)w,insect300.12523.623.546.10.810.811.59*
Day et al. (1988)a,n80. 0.45*0.48*1.11
Garcia et al. (1993)w,n500.25115.4107.5156.4 2.36***2.19***3.19***
Goldberg & Miller (1990)w,e200.5012.712.516.10.670.660.85
Moore & Keddy (1989)a,n2130.25327.2322.6540.5 1.54***1.52***2.55***
Shipley et al. (1991)a, n480.25105.093.0181.3 2.23***1.983.86
Stevens & Carson (1999)w,e720.04774.4110.5124.910.91***1.561.76
Tilman (1987)w,e360.3019.714.035.5 0.56*0.40***1.01
Wheeler & Giller (1982)w,e340.2552.548.498.6 1.59*1.47*2.99**
Wheeler & Shaw (199)a,n844.0272.2259.3447.4 3.28***3.12***5.39***
Wilson & Keddy (1988)a,n730.04192.8138.3193.0 2.68***1.92***2.68***
Wisheu & Keddy (1989)a,n880.25350.7260.9313.4 4.03***3.00***3.60***
Zobel & Liira (1997)a,n,v271.00 v187.2170.6235.2 7.20***6.56***9.04***
Zobel & Liira (1997)ramet (used only to assess type I err 27 or)NA 138.1 120.5 142.7 5.31*** 4.64*** 5.49***  

Plot size in these studies spanned two orders of magnitude, from 0.04 m2 (Wilson & Keddy 1988; Stevens & Carson 1999) to 4 m2 (Wheeler & Shaw 1991). For several studies we obtained data values from the papers or directly from the authors (Day et al. 1988; Goldberg & Miller 1990; Shipley et al. 1991; Stevens & Carson 1999; Carson & Root in press; Zobel & Liira 1997). For the majority of studies, we estimated species richness and above-ground biomass from figures presented in the published papers.

While most authors presented species richness and above-ground biomass for each sample collected, Al-Mufti et al. (1977), Day et al. (1988), Tilman (1987) and Zobel & Liira (1997) presented only means of these variables. In contrast to a previous analysis (Oksanen 1996), we used living biomass from Al-Mufti et al. (1977) rather than living biomass plus litter because litter does not play a role in the assemblage-level thinning hypothesis. In the Tilman (1987) data set, means alone resulted in very small sample sizes from each of four fields (n = 9 for each field). In order to have a large enough sample to analyse, we included the means from all four fields in a single analysis, rather than testing each field separately. Because total species richness varied drastically among the fields, we standardized species richness values only to the grand mean of all species richness values across the fields, resulting in n = 36. This is analogous to centring richness values to make different sites comparable (Jager & Looman 1987) yet would not affect the biomass estimate for peak species richness.

In a factorial experiment that manipulated three soil nutrients and water, Goldberg & Miller (1990) found no overall relationship, positive or negative, between total above-ground biomass and species richness. However, a subset of their data (no water addition treatment) did exhibit a negative richness–above-ground biomass relationship, and therefore we included the ‘no water’ subset in our study.

Although one part of the data set of Zobel & Liira (1997) was used in the comparative analysis, another part counted species and measured biomass based on samples of 500 ramets, rather than on plots of fixed area. Such a sampling regime would probably be immune to any effect of thinning on species richness, and a model of strictly assemblage-level thinning (ALT) should not explain a significant amount of variability in such a data set. Therefore, we used this portion of the data in Zobel & Liira (1997) to assess the tendency for ALT to infer a thinning relationship where none could be present (type I error). In contrast, we expected that the no-interaction model would fit this 500 ramet sample because the no-interaction model is not a model of ALT alone. We included this data set in all the analyses and tables, but refer to it only when assessing type I error.

The hypothesis tests


To evaluate the assemblage-level thinning hypothesis, our analyses compared ALT to two other models of species richness along above-ground biomass gradients. The first ‘model’ was simply the overall mean of species richness across all levels of biomass. If the ALT explained substantially more variability in species richness than the overall mean, then we concluded that thinning could play a role, or contribute to, variation in species richness along a biomass gradient. This comparison is analogous to testing whether the slope parameter in a simple linear model differs from zero. The comparison between ALT and the overall mean is necessary because neither ALT nor the no-interaction model has a constant term (e.g. an intercept or a mean). Therefore, standard Poisson regression tests (i.e. partial deviance tests; McCullagh & Nelder 1989; Neter et al. 1996) were trivial and, indeed, showed that all the parameters for both models were always significantly different from zero.

The other model we compared to ALT was Oksanen’s no-interaction model (Oksanen 1996). If assemblage-level thinning is the only process at work in controlling species richness along biomass gradients, then the no-interaction model would be overparameterized and would tend to overfit the model. In contrast, if factors in addition to thinning regulate species richness, then the no-interaction model would not contain too many parameters and would indeed be the appropriate model to describe the pattern. Thus, if the no-interaction model explained substantially more variability in species richness than ALT, we concluded that factors in addition to thinning contributed to variation in species richness. In contrast, if ALT explained as much of the variability in richness as the no-interaction model, we concluded that thinning alone could explain the pattern of species richness and above-ground biomass. Below we describe tests that select the appropriate model (the overall mean, ALT, or no-interaction) and thereby indicate whether assemblage-level thinning could account for the observed patterns of species richness and above-ground biomass.

Testing of a key assumption: an underlying Poisson error distribution

We fit the three models to available data assuming a Poisson distribution of species richness, with a unity link function and a Poisson error (Crawley 1993). This means that we assumed that the models would predict species richness directly rather than some transformation of species richness (e.g. log[S]). Because species richness is a positive integer and the variance tends to increase with the mean, we modelled it as a Poisson variable (Sokal & Rohlf 1981; Vincent & Haworth 1983; Crawley 1993; Neter et al. 1996). We used JMP v. 3.2.2 (SAS Institute Inc. 1995) to fit these models and provide significance tests of the model parameters. Because species richness is not likely to be a normal variable, we did not minimize either the error sum of squares or the negative loglikelihood of the normal model. Rather, we used JMP to minimize the Poisson –log[likelihood] loss function (–[S*ln[model]–model– ln[S!]], where S was species richness) (SAS Institute Inc. 1995). If species richness is distributed about these models as a Poisson variable, then their deviances should follow the x2 distribution. Therefore, the ratio of deviance/d.f. should equal 1.0. We tested this assumption with the x2 goodness-of-fit test (Neter et al. 1996).

Testing for the presence of assemblage-level thinning

To provide evidence that a particular model provided an accurate representation of species richness along biomass gradients, we used Akaike’s information criterion (AIC) (Hilborn & Mangel 1997) to account for both the fit of a model to data and the number of parameters in that model. AIC provides a measure of how well a model describes a biological process without overfitting (too many parameters) or underfitting (too few parameters) (Hilborn & Mangel 1997). For a given model (Ma) of species richness (Si), AIC is calculated as the sum of the negative loge likelihood (L[]) plus two times the number of parameters in the model (pa) (AICa = L[Si/Ma] + 2pa) (Hilborn & Mangel 1997). The model (ALT, no-interaction, or the overall mean) with the smallest AIC is regarded as the best model or, more specifically, as the model that provides the most accurate and most parsimonious description of the phenomenon. We also examined likelihood ratios among pairs of competing models (Reilly 1970; Haefner 1996) and Schwarz’s Bayesian information criteria (SAS Institute Inc. 1997) and found these two approaches to be in very close agreement with the use of AIC.

Assessing the amount of variation explained by each model

We also examined the magnitude of the variation in species richness explained by ALT and the no-interaction model. Even if a model explains a statistically significant portion of the variability in a data set, the proportion of variation explained provides a relative measure of model fit that allows comparison among different data sets (Haefner 1996; Hilborn & Mangel 1997). In contrast, the AIC used above varies in magnitude with the total variability in a data set, and therefore is not a relative measure, even though it is a preferred criterion for non-nested model selection. The proportion of the deviance (i.e. variation) explained by ALT or the no-interaction model relative to the overall mean (i.e. 1 –[residual deviance/total deviance from the overall mean) is the Poisson regression analogy to the least-squares coefficient of determination, r2 (i.e. r2 = 1 – residual sum of squares/total sum of squares; Neter et al. 1996). This measure of model fit was thus analogous to r2 in simple linear regression, but lacked a strong theoretical basis for establishing precise P-values (Agresti 1990).

Testing for local bias in the models

We defined local bias as a consistent over- or underestimation of species richness for any segment of the empirical biomass data. If thinning was not the only factor contributing to the pattern of species richness along biomass gradients, then we would expect ALT to under- or overestimate richness along some segment of the biomass gradient and this would been seen as high local bias. Alternatively, if the data contained a lot of noise or random variability, then ALT would show little local bias in spite of a poor fit to the data.

We assessed local bias in each model using a bootstrapped runs test, where a run was a set of 1 or more consecutive residuals of a given sign (+ ,) (Sokal & Rohlf 1981; Manly 1989). We calculated a local bias statistic (LB) that was the quotient of the observed number of runs divided by the number of runs in a random, unordered sample of the residuals. Because consecutive residuals should have an approximately equal probability of being positive or negative, our null hypothesis predicted that the number of observed runs was equal to the number of runs in a random unordered sample of the residuals (LB = 1.0; Sokal & Rohlf 1981). The alternative hypothesis (local bias existed) predicted that residuals were more serially correlated (less random) than expected by chance alone. Specifically, this meant that the number of observed runs should be less than the number in a random sample of residuals, and thus LB should be substantially less than 1.0. We used Resampling Stats v. 4.0.9 (Simon 1997) to write this resampling program. The P-value for this one-tailed test was merely the proportion of times (out of 10 000) that the resampled LB was greater than or equal to 1.0. Note that under the null hypothesis, the test of LB≥ 1 is equivalent to the test: [the number of runs under the null hypothesis]≥[number of runs observed]. LB has the benefit of an intuitive simplicity because if the observed residuals are random, they result in a mean LB of 1.0, and smaller values (< 1.0) indicate less randomness of residuals.

Checking the predicted densities of the models

As ALT and the no-interaction model depend upon their internal prediction of density (equation 1) to predict species richness (equation 3), then the predicted density should be consistent with commonly observed empirical estimates of density. Therefore, after we fit these models to each species richness–biomass data set, we plotted the density predicted by each of these models. The resulting densities should be consistent with densities observed in monocultures of herbaceous plants (Gorham 1979; White 1980).


  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References

Testing the assumption of a poisson distribution

Species richness should follow a Poisson distribution because it is a count variable and always positive (Sokal & Rohlf 1981; Vincent & Haworth 1983; Crawley 1993). However, we found that species richness was distributed as a Poisson variable in less than a third of the tests (Table 1). More commonly, species richness was overdispersed (deviance > d.f.). Consequently, in these cases species richness might have been modelled more accurately with a negative binomial distribution (Agresti 1990). However, in other cases species richness appeared to be underdispersed (Day et al. 1988; Goldberg & Miller 1990; Tilman 1987 data sets) and might have been better modelled with a normal distribution. If our goal had been to describe data accurately, then the exact form of any statistical model must be a matter of experimentation (Crawley 1993). However, data description was not our goal; rather we developed a single falsifiable model to provide a reasonably conservative test of the assemblage-level thinning hypothesis. Therefore, we preferred to impose the same assumptions on all data sets. Given that there were both data sets that were underdispersed and ones that were overdispersed relative to a Poisson distribution, we believe that this was the best single choice of possible distributions.

Tests of the role of assemblage-level thinning in affecting species richness along biomass gradients

Assemblage-level thinning appeared to contribute to the decline in species richness along biomass gradients in the majority of studies (Table 2). Specifically, ALT provided a better fit than the overall mean (i.e. had lower AIC) in 11 out of the 14 studies (Table 2). of these 11, ALT provided the best fit in three data sets (Day et al. 1988; Goldberg & Miller 1990; Carson & Root in press), while the no-interaction model provided a better fit than ALT in eight cases. This indicates that assemblage-level thinning alone could explain the observed patterns of species richness (Fig. 1) in three of the 14 data sets, but that thinning worked in concert with other factors to determine the pattern of richness in the remaining eight data sets (Table 2). In three data sets (Wilson & Keddy 1988; Wisheu & Keddy 1989; Stevens & Carson 1999) ALT provided no better fit than the overall mean, indicating that assemblage-level thinning, as quantified by ALT, was not likely to contribute to thinning. In these three data sets the no-interaction model provided a better fit than the overall mean, indicating that factors other than assemblage-level thinning created the richness–biomass pattern.

Table 2.  Test of assemblage-level thinning. Variation explained is the proportion of the deviance explained by each model, relative to the overall mean. Akaike’s information criterion (AIC) provides a measure of model selection equal to the deviance of the model plus 2.0 for each parameter in the model. The model with the lowest AIC value provides the most accurate and parsimonious model. Values in bold below show models better than the mean, and underlined values are the best of the three models. Zobel & Liira (1997)ramet is a per-ramet sample of species richness and thus should not be explained well by ALT
   Proportion of variation explained 1 − [D(model)/D(mean)]Akaike’s information criterion (AICa = L[Si/Ma] + 2pa) (smaller is better)
StudynPlot m2ALTNi-interactionALTNo-interactionMean
  1. a, among community gradient; w, within community gradient; n, natural productivity gradient; e, experimental productivity gradient; insect, herbivory-generated productivity gradient; v, plot size varied for biomass although species richness was measured in 1.00-m2 plots; ramet, biomass and species richness measured per 500 ramets.

Al-Mufti et al. (1977)a,n0.230.5777.347.397.2  
Carson & Root (in press)w,insect0.490.4927.629.548.1  
Day et al. (1988)a,n0.600.567.29.49.8  
Garcia et al. (1993)w,n0.260.31119.4113.5158.4  
Goldberg & Miller (1990)w,e0.210.2316.718.518.1  
Moore & Keddy (1989)a,n0.400.40331.2328.6542.5  
Shipley et al. (1991)a,n0.420.49109.099.0183.3  
Stevens & Carson (1999)w,e00.12778.4116.5126.9  
Tilman (1987)w,e0.450.6023.720.037.5  
Wheeler & Giller (1982)w,e0.470.5156.554.4100.6  
Wheeler & Shaw (1991)a,n0.390.42276.2265.3449.4  
Wilson & Keddy (1988)a,n00.28196.8144.3195.0  
Wisheu & Keddy (1989)a,n00.17354.7266.9315.4  
Zobel & Liira (1997)a,n,v0.200.27191.2176.6237.2  
Zobel & Liira (1997)ramet (used only to assess type I error)0.03 0.16 142.1 126.5 144.7   

Figure 1. Species richness as a function of above-ground biomass (g m–2). Each model (ALT, no-interaction, and the overall mean) were fit with a non-linear regression minimizing the model deviance (a, among community gradient; w, within community gradient).

Local bias is a measure of lack of fit and tended to be higher in ALT than in the no-interaction model (Table 3). However, local bias by ALT was not higher than in the no-interaction model where ALT provided the best fit (Day et al. 1988; Goldberg & Miller 1990 data sets; Carson & Root in press). Not surprisingly, local bias in ALT was significantly greater than expected by chance alone in the three data sets that ALT did not fit (Wilson & Keddy 1988; Wisheu & Keddy 1989; Stevens & Carson 1999). Thus, tests of local bias were consistent with the other two measures of model fit (AIC and percentage deviance explained).

Table 3.  The local bias statistic (LB) estimates the extent to which a model tends to over- or underestimate species richness within any subset of biomass values. As serial autocorrelation of the residuals tends toward zero, LB tends toward 1.0. Cases where local bias was greater than expected by chance alone are indicated as follows: †P < 0.1, *P < 0.05, **P < 0.01, ***P < 0.001
   Local bias (smaller is worse) Observed number of runs/mean of bootstrapped runs
StudynPlot m2ALTNo interactionMean
  1. a, among community gradient; w, within community gradient; n, natural productivity gradient; e, experimental productivity gradient; insect, herbivory-generated productivity gradient; v, plot size varied for biomass although species richness was measured in 1.00-m2 plots; ramet, biomass and species richness measured per 500 ramets.

Al-Mufti et al. (1977)a,n140.251.1151.1440.474**
Carson & Root (in press)w,insect300.1251.2011.2010.803
Day et al. (1988)n,a80.251.0241.1211.242
Garcia et al. (1993)w,n500.250.9741.0700.920
Goldberg & Miller (1990)w,e200.50.9100.9061.012
Moore & Keddy (1989)a,n2130.251.0010.9970.837*
Shipley et al. (1991)a,n480.250.9320.9650.847
Stevens & Carson (1999)w,e720.040.614**1.0011.028
Tilman (1987)w,e360.30.788†0.9480.351***
Wheeler & Giller (1982)w,e340.250.9981.0960.779
Wheeler & Shaw (1991)a,n844.01.0011.0740.814*
Wilson & Keddy (1988)a,n730.040.707**0.9440.802*
Wisheu & Keddy (1989)a,n880.250.619***0.646***0.660***
Zobel & Liira (1997)a,n,v271.00v0.9620.9681.037
Zobel & Liira (1997)ramet (used only to assess type I error)27 NA 0.962 1.126 1.003

Assessment of density predictions and comparison of alt and the no-interaction model

Fitting ALT to species richness data resulted in density–biomass relationships (i.e. thinning relationships) that were very similar among all the data sets (Fig. 2). ALT predicted that, regardless of how rapidly any given assemblage increased in density at low biomass values, all assemblages would thin at the same rate at a high biomass values. This thinning line (Fig. 2) was the same as that observed in a large number of studies of self-thinning in monocultures (D = 108B–2; cf. Gorham 1979). These results were not surprising, as we designed the model to exhibit these characteristics. None the less, we present these results to illustrate how our assumptions of assemblage-level thinning limited the variety of curves fit by the ALT model. This limitation occurred because the effect of k was large only when biomass was low (equation 1). In contrast, the no-interaction model could fit any peak density at any level of biomass (Fig. 2) because that peak was fit statistically to the data. This flexibility resulted in predicted densities of high biomass samples (e.g. number of stems 1000 g−1 m–2) that ranged over five orders of magnitude. This variation is not consistent with predictions of self-thinning in monocultures (Gorham 1979; White 1980) but may be consistent with other mechanisms controlling species richness. Therefore, the no-interaction model does provide a model related to ALT that can provide a very useful basis of comparison.


Figure 2. Predicted density as a function of above-ground biomass, comparing ALT and the no-interaction model. Both models were fit using the observed species richness and above-ground biomass values.

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Assessment of type i error

We analysed one data set in which thinning could not influence species richness because the estimates of species richness and biomass were based on samples of 500 ramets rather than quadrats of fixed area (Zobel & Liira 1997). We found that ALT explained only 3% of the variance in species richness (Table 2). In contrast, ALT explained 20% of the variation in the fixed area quadrat data from Zobel & Liira (1997), in a case where thinning could play a role in controlling richness. We believe this is strong evidence that assemblage-level thinning does contribute to the unimodal pattern of species richness in plots of fixed area (Table 2). In addition, the AIC showed very little difference between ALT and the overall mean (Table 2; AICALT = 142.1, AICMEAN = 144.7), indicating that the model provides very little improvement in fit beyond the overall mean. Other indicators of model fit not shown here (the likelihood ratio, the chi-squared test, Bayesian information criterion) showed even less support for choosing ALT over the mean using this 500-ramet data set. Unlike ALT, the no-interaction model explained a substantial portion of the variability (16%) in the 500-ramet data set (Zobel & Liira 1997). Further, the no-interaction model had an AIC value much lower than the overall mean (AICNI = 126.5, AICMEAN = 144.7), indicating that the no-interaction model provided a good fit to the 500-ramet data set. We conclude that the no-interaction model cannot be used by itself to test for the presence of assemblage-level thinning. ALT, however, can be used to test for the presence of assemblage-level thinning because the risk of type I error appears low.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References

The analyses presented here demonstrate that assemblage-level thinning is likely to be causing or at least contributing to unimodal and negative plant species richness–biomass relationships along many productivity gradients. Eleven of 14 data sets were consistent with the assemblage-level thinning hypothesis (ALT), and for three data sets assemblage-level thinning was the only mechanism required to explain the relationship between species richness and total above-ground biomass.

Species-specific interactions could possibly generate the same pattern in species richness predicted by these models, and this could lead to overestimates of the prevalence of assemblage-level thinning. To demonstrate conclusively whether species-specific interactions or assemblage-level thinning controls richness in any particular case, we would need the relative densities and the relative frequencies of each species (Stevens & Carson 1999). Measures of biomass or relative cover alone (Tilman 1987) do not provide the information necessary to determine the factors controlling richness (Stevens & Carson 1999). We must also point out that the slope of the density–biomass relationship may not be identical to the quantitative pattern observed in monocultures (i.e. the negative 3/2 thinning rule), and yet density may still constitute the predominant control of species richness along a biomass gradient (Stevens & Carson 1999). In this case, tests using the ALT restricted quantitative interpretation of this hypothesis may provide an overly conservative estimate of the prevalence and importance of assemblage-level thinning. The purpose of this paper was to assess the likelihood that thinning-related processes, which have been well studied at the population level (Harper 1977), might be generating species-richness patterns in assemblages. The survey of 14 studies performed here indicates that it is quite likely that assemblage-level thinning is frequently contributing to declines in species richness along productivity gradients.

While the no-interaction hypothesis (Oksanen 1996) has created controversy (Grime 1997; Marañón & Garciá 1997; Rapson et al. 1997), the evidence against it as a general phenomenon is not strong. In his original proposal, Oksanen (1996) used one prominent example of the unimodal species richness–biomass curve (Al-Mufti et al. 1977) to show that assemblage-level thinning was a plausible explanation for the pattern. In that paper, Oksanen argued that the small plot size used by Al-Mufti et al. (1977; 0.25 m2) resulted in the pronounced effect of density on species richness. As a response to this, Rapson et al. (1997) went back to several of the sites originally used by Al-Mufti et al. (1977) and resampled the same stands of vegetation. Rapson et al. (1997) used three plot sizes and showed that the unimodal richness biomass relationship was still clearly observed at larger plot sizes; from this they argued that Oksanen’s hypothesis was incorrect. Indeed, they pointed out that the high biomass, low diversity stands would have to be nearly 105 times as great in area in order to have the same number of species. However, the high biomass, low diversity sites (Urtica and Pteridium stands) and the low biomass, low diversity sites (woodland herbaceous species) were also likely to have low densities. In contrast, the moderate biomass, high diversity sites were grasslands, which probably had high densities. Stand density appears to co-vary with species richness at these sites, and therefore, consistent with our findings and those of Oksanen (1996), density is likely to contribute substantially to the observed unimodal pattern of richness.

The primary evidence in support of assemblage-level thinning comes in the form of direct tests. Goldberg & Miller (1990) and Stevens & Carson (1999), the only studies of plant species richness along experimental productivity gradients that measured density of individuals, found that (i) species richness was more strongly correlated to density than to any other variable, and (ii) density declined markedly with increased soil nitrogen. Goldberg & Miller (1990) showed that among all their treatments species richness was most closely related to density. However, they found that richness and density were unrelated to total above-ground biomass. Only in a subset of their data (no additional water) was there a significant decline in density with increasing total above-ground biomass. As a result of this decline in density, species richness also declined as biomass increased. Stevens & Carson (1999) showed that along an experimental productivity gradient, above-ground biomass was superficially related to species richness. However, once the effect of density on species richness was removed, biomass explained none of the variation in species richness. Thus, these two studies found that only when there was a strong relationship between density and biomass did we observe a strong relationship between species richness and biomass. Zobel & Liira (1997) showed that when they counted species on a per area basis rather than a per stem basis, the unimodal richness–biomass pattern was intensified, indicating that density contributed substantially to the unimodal pattern. Recently, in a common garden experiment, Goldberg & Estabrook (1998) showed that experimental increases in density led to increased rates of interspecific competitive exclusion, and caused species richness to decline relative to richness predicted by their null model. In spite of this, however, species richness (number of species 0.09 m–2) continued to increase in response to increasing density until density was more than four times higher than ambient levels. All of these studies (Goldberg & Miller 1990; Zobel & Liira 1997; Goldberg & Estabrook 1998; Stevens & Carson 1999) point to the overriding importance of total stem density in controlling species richness in sample plots of sizes typically used to study herbaceous plant assemblages.

Notably, ALT did not provide an accurate description of the species richness–biomass pattern in Stevens & Carson (1999), even though they showed that declines in density caused the decline in richness along a fertility gradient. Subsequent analyses (not presented here) showed that two factors caused a lack of fit by ALT. First, the species richness–biomass relationship, although statistically significant and unimodal in nature, was highly variable, and therefore no model could explain a lot of the variation in the relationship. Secondly, the slope of the actual density–above-ground biomass relationship was less than the slope predicted by self-thinning in monocultures. Therefore, because ALT was constrained by the –3/2 thinning rule, it did not fit the species richness–biomass relationship in Stevens & Carson (1999).

Oksanen (1996) suggested that equal-area samples along a biomass gradient could contain fewer individuals at high levels of productivity not only because of thinning, but at least in part because larger species may predominate in productive or fertile areas. In contrast, the assemblage-level thinning hypothesis only assumes that at high levels of biomass individuals of a single suite of species become larger, causing density and richness to decline (Stevens & Carson 1999). Despite this important distinction, both hypotheses (large species and large individuals) are consistent with earlier findings of Gorham (1979). Gorham (1979) examined 65 stands of 29 woody and herbaceous species, spanning five orders of magnitude in shoot density and eight orders of magnitude in shoot biomass. He showed that the empirical –3/2 self-thinning pattern accurately described the above-ground biomass–density pattern that existed when all species were examined simultaneously. Therefore, both these hypotheses (large species, and large individuals–ALT) predict that samples of equal numbers of individuals will show no unimodal richness-biomass pattern (for a review of size asymmetry in plant competition see Schwinning & Weiner 1998). Only ALT, however, predicts no change in species composition.

One difference between ALT and the no-interaction model is that ALT assumes that the magnitude of thinning (density-dependent mortality) increases gradually as total biomass increases, resulting in a smooth curve throughout the range of available biomass (Fig. 2). In contrast, the no-interaction model assumes that density-dependent factors operate only at biomass values above a particular biomass threshold (Oksanen 1996). As a result, the no-interaction model is a discontinuous function at peak biomass, always exhibits a distinct peak in species richness, and that peak can occur at any species richness–biomass combination (Fig. 2). The degree to which species richness actually exhibits a sharp peak in species richness along biomass gradients may determine which model is a better descriptor of the data. In habitats where density dependence operates only above a threshold, the no-interaction model may provide a better fit to available data. However, where density dependence becomes increasingly important as biomass rises, then ALT may be a better descriptor of species richness.

We wish to make clear that we do not believe that all species richness patterns are a function of an underlying density pattern. It seems likely that a number of mechanisms could operate on a variety of temporal and spatial scales to generate unimodal species richness patterns. We do believe, however, that patterns of total plant density along productivity or biomass gradients may contribute substantially to generating commonly observed patterns of plant species richness (Zobel & Liira 1997; Stevens & Carson 1999). Further, assemblage-level thinning represents one of the simpler explanations for the unimodal species richness productivity curve, and we have found that it is consistent with much of the available data.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References

We are indebted to J. Oksanen and S. Scheiner for encouragement and advice, and D. Goldberg, T. Miller, S. Scheiner, E. Siemann, J. Weiner and two anonymous referees for comments on various drafts of this manuscript. We would also like to thank B. Shipley and D. Goldberg for providing original data from their respective studies. This work was supported by the McKinley Research Fund, Pymatuning Laboratory of Ecology, and the Department of Biological Sciences, University of Pittsburgh.


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  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgement
  8. References
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Received 13 March 1998revision accepted 3 December 1998