Moving on from Metabolic Scaling Theory: hierarchical models of tree growth and asymmetric competition for light

Authors


Correspondence author. E-mail: dac18@cam.ac.uk

Summary

1.Coomes & Allen (2009) showed that the Metabolic Scaling Theory of Plant Growth (MST-PG) has no empirical support, because the only piece of confirmatory evidence was an analysis of a small and noisy data set from which it was impossible to draw strong inferences. Our re-analyses showed that MST-PG predictions were contained within very broad 95% confidence intervals, creating an illusion of close adherence to theoretical predictions. In a response to our paper, Stark, Bentley & Enquist (2010) acknowledge these shortcomings.

2.  We reasoned that MST-PG makes inaccurate predictions because asymmetric competition for light is not included in the theoretical model. We argued that asymmetric competition has its greatest impact on small trees within populations, such that the mean size-scaling relationship of the population has a greater exponent than that of trees growing without competitors (i.e. the slope predicted by MST-PG). Stark, Bentley & Enquist (2010) appear to dispute this logic and criticize the statistical approach we developed to test our hypothesis.

3.  Here we use Bayesian hierarchical models (BHMs) to re-examine three forest data sets, and find no support for MST-PG. Based on a large data set from New Zealand, the majority of tall tree species have scaling exponents greater than predicted by MST-PG, suggesting that asymmetric competition may have influenced the scaling relationships. Parameters for many small tree and shrub species were very poorly estimated, in part because their size ranges were narrow, so these species cannot be used to argue for, or against, MST-PG.

4.  We hypothesize that scaling relationships are most affected by asymmetric competition in high-productivity forests because, as many ecologists argue, competition for light is intense in such forests (the productivity-dependent scaling hypothesis). We estimate growth scaling exponents for Nothofagus forest sites which differed in above-ground biomass (AGB) production. As predicted, scaling exponents were greater in sites with high AGB production and close to MST-PG predictions in sites with low AGB production.

5.Synthesis. Competition from taller neighbours has strong effects on tree growth, particularly in forests growing in productive locations. We continue to advocate for the inclusion of asymmetric competition into general models of tree growth, mortality, recruitment and size structure.

Introduction

A central prediction of the Metabolic Scaling Theory of Plant Growth (MST-PG) is that tree diameter growth inline imagescales with diameter according to a power function:

image(eqn 1)

and that the scaling exponent (α) of that function is very close to 1/3 for all species within a community, and for the community as a whole (Enquist et al. 1999). A prediction of MST-PG is that differences among sites and species primarily affect the scaling coefficient, β, and not α. We believe that this theory is conceptually flawed because it takes no account of effects of competition for light, even though there is a large body of scientific evidence indicating that light competition plays a central role in forest dynamics. Numerous studies have related sapling growth to variation in the amount of light available to them (e.g. Pacala 1994; Wright, Canham & Coates 2000; Kobe 2006; Collet & Chenost 2006; Coomes & Allen 2009), some of which have carefully integrated competitive and size-related effects (MacFarlane & Kobe 2006). Other studies have related growth to indices of asymmetric competition obtained from neighbourhood analyses (e.g. Uriarte et al. 2004; Wyckoff & Clark 2005; Coates, Canham & LePage 2009; Coomes & Allen 2007b). Asymmetric competition also has been an axiomatic concept in the development of theories underpinning forest ecology. It provides an explanation for the evolution of height because having a slight height advantage over neighbours allows a plant to capture the lion’s share of a vital resource (Tilman 1988; Falster & Westoby 2003). In self-thinning theory, it is the relatively small trees that lose out in the battle to capture resources and die, generating a characteristic pattern of stem size growth and density loss over time (Westoby 1984). Coexistence theories have also evoked asymmetric competition in their arguments (e.g. the stratification theory of Kohyama & Takada 2009), and secondary succession models often focus on differences in the ways trees respond to light (Pacala et al. 1996; Purves et al. 2008). We believe that there are compelling reasons to include competition for light in scaling models.

Asymmetric competition for light has, by definition, its strongest impact on the growth of smaller trees, and it is for this reason that we believe it steepens the slope of the allometric growth curve of plant populations (Fig 1a). Our extension to the MST-PG hypothesis is that the taller trees within a population – those unhindered by competition for light – follow the growth trajectory predicted by MST-PG theory, whilst shaded trees have growth curves dependent upon the ever-changing size structure of the forest. Coomes & Allen (2007a) extended the MST-PG to include competition for light as follows:

image(eqn 2)

where BL is the basal area of taller neighbours, and competitive effects are defined by parameters λ2 and λ3. This model assumes that a tree without taller neighbours follows a power-law growth curve, because equation 2 reduces to a power function when BL = 0, and we reasoned that α  1/3 as predicted by MST-PG. However, relatively small trees within populations have their growth reduced by asymmetric competition: their growth rates depend on the evolving size-structure of neighbouring trees and, consequently, the individual growth trajectories do not necessarily follow power-law curves. We argue that the mean growth curve of all trees in the population is approximately power-law in form and has an exponent > 1/3 because of the effects of asymmetric competition. The functional form we chose for incorporating competition is based on the assumption that (a) the light flux reaching a small tree depends nonlinearly on the leaf area of taller trees around it (we used the Beer–Lambert law to describe this) and (b) its growth depends nonlinearly on light flux (we used the Michealis–Menten function to describe this).

Figure 1.

 Conceptualization of how competition affects the scaling of tree growth with stem diameter: (a) asymmetric competition for light has its strongest impact on small trees (arrows), so the mean growth curve (solid line) is steeper than the growth curve of trees without competitors (dashed trees); (b) symmetric competition has similar effects on trees of all sizes (arrows), so the mean growth curve has the same slope as that of trees without competitors; (c) assuming that competition for light is strong on high-productivity sites, and competition for below-ground resources is strong on low-productivity sites, it follows that scaling slopes will be steepest on high-productivity sites.

Coomes & Allen (2007a) have already tested the asymmetric competition hypothesis. We fitted equation 2 to a long-term growth data set from New Zealand Nothofagus forests, exploiting natural variation in crowding within and among 250 permanent plots to quantify competitive effects. Using nonlinear mixed-effects modelling, we found that asymmetric competition had important influences on growth. We estimated that α = 0.19 ± 0.029 for those trees growing without taller neighbours, while the mean curve in forest stands had α = 0.52 ± 0.023 (Coomes & Allen 2007a). Thus, asymmetric competition steepened the scaling exponent, as we had hypothesized. Stark, Bentley & Enquist (2010) and Enquist, West & Brown (2009) fail to mention this or other related studies (e.g. Muller-Landau et al. 2006), when making their case for asymmetric competition having no influence on scaling relationships.

In this paper, we develop these ideas further, arguing that population-level scaling exponents should vary systematically with productivity. Ecological theory suggests that competition for light is most intense at sites where plant productivity is highest, because at these sites plants allocate resources to leaf and stem production (Tilman 1988) and produce denser canopies (Coomes & Grubb 2000) as a result of mineral nutrients, particularly N, limiting leaf production (Baribault, Kobe & Rothstein 2010). It follows that asymmetric competition for light will have its greatest effect on scaling exponents at sites supporting high productivity (Fig. 1a). Forests on poorer soils, or in drier regions, allocate more resources below-ground and have more open canopies, and below-ground competition becomes increasingly important (Coomes & Grubb 2000). Being larger than one’s neighbours is often thought to have relatively little benefit in the battle for nutrients (Schwinning & Weiner (1998) review evidence for this), so small and large trees are similarly affected by competition (Fig. 1b). Therefore, we hypothesize that the scaling of tree populations depends on above-ground productivity: if productivity is low then asymmetric competition is weak and α should be close to 1/3 whereas if productivity is high, we expect strong asymmetric competition to result in a steeper slope (Fig. 1c). We call this the productivity-dependent scaling hypothesis (see also Pretzsch & Biber 2010).

Stark, Bentley & Enquist (2010) have responded to Coomes & Allen (2009) by criticizing the approach we used to model growth. They seem to agree that data in Enquist et al. (1999) were misinterpreted, and that the paper provides no empirical support for MST-PG, but argue that the methods of Coomes & Allen (2009) were equally flawed and conclusions unreliable. We respond directly to the points raised by discussing the merits of Standardized Major Axis (SMA) line-fitting, Ordinary Least-Squares (OLS) regression and Bayesian hierarchical modelling (BHM), emphasizing in doing so the advantages of hierarchical modelling when dealing with complicated multi-site or multi-species data sets. We then return to the fundamental question of how competitive interactions affect growth scaling, analysing three large data sets using BHM and providing evidence that asymmetric competition does indeed have important effects on growth scaling. Finally, we test the productivity-dependent scaling hypothesis by analysing the growth of Nothofagus trees in 250 permanent plots using BHM, and find support for it.

Materials and methods

Data sets

Three data sets are revisited in this paper. The San Emilio data set comprises 2277 stems of 45 species, recorded within an area of approximately 20 ha in a lowland Costa Rican forest (Enquist et al. 1999). It contains two diameter measurements for each stem, taken 20 years apart. The minimum diameter is 10 cm and measurements were made to the nearest centimetre in 1976 and to the nearest millimetre in 1996. The Nothofagus data set comprises 13 389 stems of one species, recorded within 250 permanent 20 × 20 m plots that sample about 9000 ha of mountainous terrain in New Zealand (Coomes & Allen 2007a & b). It contains two diameter measurements for each stem, recorded 19 years apart. The New Zealand NZ-NVS (National Vegetation Survey) data set comprises 120 842 stems of 73 species, recorded within 1670 permanent 20 × 20 m plots, and provides an overview of the nation’s forests (Russo, Wiser & Coomes 2007). It contains two diameter measurements for each stem, with the interval between measurements varying from 3 to 28 years. The minimum diameter in the NZ-NVS and Nothofagus data sets is 3 cm, and all measurements were made to the nearest millimetre.

Comparing analytical approaches

Stark, Bentley & Enquist (2010) tested two modelling approaches – OLS regression and SMA line-fitting – by creating simulated data sets with known parameter values, and observing whether the modelling approaches provided unbiased estimates of those known parameter values. They took the initial diameter measurements (D0) from the San Emilio data set and generated simulated data sets of diameters 20 years later (D20) by applying the following function, which is the integrated form of the power function:

image(eqn 3)

where α and β values were drawn from probability density functions with known means and variances (thereby introducing intraspecific ‘biological’ variation) and e values were drawn from a normal distribution centred on zero (this is described as ‘measurement’ error). Adding measurement error in this way assumes that Dt was measured with error but that D0 was measured with certainty. In reality, of course, D0 and Dt would have been measured with a similar degree of uncertainty. It also assumes that measurement error is constant with tree diameter, whereas it usually increases. We created our own simulated data sets, using the same approach as Stark, Bentley & Enquist (2010) to generate ‘moderate levels’ of intraspecific variability, except that we added uncertainty to D0 as well as Dt and allowed measurement error to scale with tree size. Specifically, we took the 2277 D0 values from the San Emilio data set and calculated Dt values assuming α ∼ N = 0.25, σ = 0.02), β ∼ LN(log = log(0.17), logσ= 0.02), and assuming both Dt and D0 were measured with error distributed as N = 0, σ = 0.05 D0). The simulated data are plotted in Fig. 2. We fitted models using SMA line-fitting (the approach is explained in Coomes & Allen 2009), OLS regression (we used the nls package in R to fit nonlinear functions) and BHM (see next section). The parameter estimates from these three approaches were compared.

Figure 2.

 Curves fitted to simulated growth data (crosses) using Standardized Major Axis line-fitting (red dashed line) and Ordinary Least-Squares regression (green dot-dashed line). The simulated data were created from a deterministic growth curve (blue solid line) by adding biological and measurement error. Initial diameters (D0) are identical to those in the San Emilio data set.

Bayesian hierarchical modelling

Bayesian hierarchical modelling provides a method for estimating biological variability and process error when fitting the model shown in equations 4–6:

image(eqn 4)
image(eqn 5)
image(eqn 6)

where j is the species (or plot) of the tree, and α, β, inline image and inline image are hyperparameters. The model samples the species- or plot-specific parameters αj and βj using an adaptive Metropolis algorithm but constrains them using a hierarchy to be drawn from common distributions. We assumed α to be normally distributed and β to be log-normally distributed (equation 6; cf. Stark, Bentley & Enquist 2010), and sampled them using a Gibbs sampler from a normal distribution. Variance parameters inline image and inline image were sampled in the same way from an inverse gamma distribution (i.e. with standard conjugate priors). The species-dependent process error of the response variable, inline image, was assumed to be normally distributed as both size-independent and size-dependent (i.e. process error increases with tree size); models allowing process error to increase with tree size were much better supported than those assuming invariance (e.g. a drop in DIC of 6609 for the San Emilio data set). The algorithm fits Dt using an adaptive Metropolis algorithm (Gelman, Roberts & Gilks 1996), and the hierarchical parameters are simultaneously estimated using a Gibbs sampler adapted from code provided by Lee (1997), written in C and compiled using MS Visual Studio 2008. We used 100 000–5 000 000 iterations of the algorithm for the burn-in phase and 50 000–250 000 for the sampling phase, which were sampled every 100 iterations to develop the posterior distributions of α, β and σ.

When modelling the simulated data, we used the Metropolis algorithm to fit a non-hierarchical model. When modelling the San Emilio data set, we (a) used the Metropolis algorithm to fit a power function to each species (i.e. non-hierarchical models), and (b) used hierarchical modelling to estimate mean cross-species α as well as variation in α and β among the 45 species. In the case of the NZ-NVS data set we ran four types of models: (a) using the Metropolis algorithm to fit a power function for each species (i.e. non-hierarchical models); (b) estimating variation among the 73 species using a single hierarchy; (c) splitting species into shrubs, subcanopy trees and canopy trees (see Russo, Wiser & Coomes 2007 for definitions) and running separate hierarchical models for each group; and (d) estimating variation among the 1670 plots. An explanation as to why we ran these models is given below. In the case of the Nothofagus data set we estimated the distribution of α and β values among our 250 permanent plots using a plot-based hierarchical model.

Testing the productivity-dependent scaling hypothesis

We tested the hypothesis that growth curves will be steeper in productive sites using the Nothofagus data set, by correlating posterior estimates of α with plot-level estimates of above-ground biomass (AGB) growth. The AGB of each stem was calculated from its diameter and height using a Nothofagus species conversion formula given in Harcombe et al. (1998). The plot-level AGB growth was calculated by summing the gain in AGB of all trees present in plots in both 1974 and 1993. We focused our analyses on the 116 mature forest plots, omitting stands that were undergoing major disturbance, or were in the early stages of regeneration or thinning, because these stands had narrower size ranges and/or many damaged trees so were less suitable for fitting scaling functions. Coomes & Allen (2007b) provide an explanation of how plots were categorized into these development stages.

Results and discussion

Comparing analytical approaches

Both Enquist et al. (1999) and Coomes & Allen (2009) used SMA line-fitting; the earlier of these papers tests whether scaling exponents differ from 1/3 whilst the later paper extends the approach in order to provide estimates of scaling slopes and their confidence intervals. In a response paper, Stark, Bentley & Enquist (2010) showed that SMA line-fitting produced upward-biased estimates of α when moderate amounts of measurement error and biological variation were introduced into simulated data (see also Fig S2 of Coomes & Allen 2009). Applying SMA line-fitting to our simulated data set, we found that α was indeed overestimated (0.33 vs. 0.25), whereas β was underestimated (0.14 vs. 0.17). Interestingly though, the predicted growth curves from OLS regression and SMA line-fitting are not very different from one another (Fig. 2), because a family of similar-looking curves can be produced when β negatively covaries with α. This raises an intriguing point. In the simulated data sets we know the functional form of the growth relationship and find OLS regression performs better in estimating the function’s parameters. In reality, we know that the growth curves of trees do not conform to power functions, because tree growth is affected by suppression and release from competitors (e.g. Wright, Canham & Coates 2000), and by damage from pathogens, insects, herbivores and falling branches. Unbiased estimates are produced by SMA line-fitting when residual variation is orthogonal to the fitted line, and may be a reasonable approach when there is little prior understanding of the form of the uncertainty and no desire to test alternative models (D. Falster, pers. comm.).

Our regression analysis of the simulated data set (n = 2277) confirms the conclusions of Stark, Bentley & Enquist (2010): OLS regression provides unbiased estimates of α and β when the data set is large (α = 0.248 vs. 0.250, β = 0.175 vs. 0.170). Ordinary Least-Squares regression is a special case of Maximum Likelihood Estimation (MLE) that assumes uncertainty is adequately encapsulated by a single error term that is normally distributed and size-invariant. Given that our simulated data set departs from these assumptions, it is not immediately obvious that OLS regression provides unbiased estimates, and this may explain why Stark, Bentley & Enquist (2010) found that OLS regression provides biased estimates of β when sample sizes are small (i.e. n < 50). A major advantage of OLS regression over SMA line-fitting is that it can be used to model the influences of multiple factors on growth, and to fit nonlinear functions (see Coomes & Allen 2009).

We recognize the advantage of using hierarchical approaches to model growth and other demographic processes (e.g. Turnbull et al. 2008; Lines, Coomes & Purves 2010). The major advantage of hierarchical modelling is that multiple species can be fitted jointly in a community, decomposing variation into components due to various kinds of sampling error and to real species differences. Hierarchical models can be fitted with MLE methods (e.g. Coomes & Allen 2007b) but BHM has a demonstrated ability for simple, efficient estimation of parameters, including confidence intervals and error propagation (Wyckoff & Clark 2005; Turnbull et al. 2008). Our analyses of the simulated data set using BHM showed that it provided unbiased estimates of the deterministic part of the growth curve: α = 0.254 (95% CI = 0.217–0.288) and β = 0.173 (95% CI = 0.156–0.192). It provided an estimate of the variance structure of the error term: σ(e) = 0.84 + 0.070 D0. The first term captures the biological variation in α and β while the second term encapsulates the process error we introduced in the simulation (including the increased variance with diameter).

Problems with small samples and narrow size ranges

Coomes & Allen (2009) showed that the only study claiming to provide empirical support for MST-PG (i.e. Enquist et al. 1999) should be disregarded because statistical analyses had been misinterpreted. Stark, Bentley & Enquist (2010) have agreed with our arguments, stating that: “given the small sample sizes for many species in the San Emilio data set … and the wide confidence intervals associated with MLE α estimates … it seems likely that, as suggested by Coomes & Allen (2009), the signal-to-noise ratio in this data set is low (meaning that) there is not enough power to detect differences between estimated and predicted allometric slopes … these results indicate that we cannot argue for or against the ‘optimized’ MST predictions or the importance of asymmetric competition for the tree community in the San Emilio data set.” Tree growth data are highly variable and large sample sizes are required to fit scaling functions with any degree of confidence; the San Emilio data set is simply too small to draw strong conclusions (Fig. 3). In fact, only 4 of the 45 species had any detectable relationship using OLS regression (Table S2 of Coomes & Allen 2009). The SMA line-fitting approach of Enquist et al. (1999) created an illusion of a close relationship between growth and size because essentially D02/3 values were plotted against themselves with small numbers added, an approach guaranteed to create misleadingly high r2 values. For example, Enquist et al. (1999) reported r2 values of 0.703, 0.903 and 0.857 for the three species shown in Fig. 3, even though there is no correlation between growth and size for these species.

Figure 3.

 Lack of relationship between stem diameter growth (Dt − D0)/20 and initial size (D0), for three species drawn from the San Emilio data set. Size is only one of many factors affecting growth, and correlations coefficients (r) are very low even for species with sample sizes (n) over 50.

A central statistical theorem is that confidence intervals narrow as sample sizes increase, when samples are drawn at random from a population. The same pattern is evident when comparing scaling exponents of different species, even though the exponents are estimated from sampling different populations of organisms. For example, exponent estimates for scaling relationships between metabolic rates and body mass of animals (taken from Glazier 2005) are funnel shaped with respect to sample size, and the 95% confidence intervals get narrower as sample size increases (Fig. 4a). When scaling exponents of the 73 New Zealand tree species are plotted against sample size, we find a similar pattern (Fig. 4b). Bayesian Hierarchical Modelling should reliably predict the mean α value if some species are poorly sampled, because BHM places greater weight on species that are sampled most when estimating hyperparameter values (Gelman, Roberts & Gilks 1996).

Figure 4.

 Scaling exponent estimates (with 95% confidence intervals) of species in relation to sample size and size range of the individuals. The top row represents metabolic rate vs. mass of animals (a and c, with the Metabolic Scaling Theory prediction of ¾ shown by the horizontal line) whilst the bottom row represents diameter growth vs. diameter relationships of 73 species of New Zealand tree (b and d, with MST prediction of 1/3 shown by the horizontal line). The scaling exponents of New Zealand trees were obtained by fitting separate models for each species, using a Metropolis algorithm. Only species with at least 100 stems are shown in (d).

Having a large range of body sizes is also critical for accurate estimation of scaling exponents, but this issue has received little attention in the scaling literature. Glazier (2005) provides the minimum and maximum body sizes of animals in his review of metabolic scaling, from which we calculated log (size range) as log (maximum size/minimum size), which we henceforth refer to as ‘size range’. When size range of animals is plotted against estimated scaling exponents, the funnelling of exponent estimates and narrowing of credible intervals is very clearly evident (Fig. 4c). This effect is also evident in the NZ-NVS data sets (Fig. 4d): here the size of the smallest trees is close to 3 cm for all species, so size range differentiates among the maximum sizes of species. We only show species with at least 100 stems in Fig. 4d, and yet many of those with narrow size ranges have no discernable growth–size relationships (i.e. wide credible intervals that overlap with zero). Recent scaling analyses have taken sample size into consideration (e.g. Isaac & Carbone 2010) but not size range. This topic is worthy of further, more rigorous, exploration.

Support for the asymmetric competition hypothesis

Hierarchical modelling of species

Early claims of invariant growth relationships (e.g. Enquist et al. 1999) have given way to the notion that MST-PG might ‘predict a basin of attraction, around which actual biological networks cluster’ (Price, Enquist & Savage 2007). The new question is whether the mean cross-species α value is accurately predicted by MST-PG. Our contention is that α values will be greater than predicted for many species because of asymmetric competition, and that the mean should also be greater. Russo, Wiser & Coomes (2007) used MLE to fit power functions to the 73 species in the NZ-NVS data set, and reported a cross-species median of 0.33, a number that Stark, Bentley & Enquist (2010) and Enquist, West & Brown (2009) interpreted as providing support for the ‘extended’ MST-PG. However, the results cannot be interpreted in this way, as Russo, Wiser & Coomes (2007) warned. The majority of species in NZ-NVS (48 of 73) are shrubs and subcanopy trees which have poorly estimated growth curves because of the narrow size range sampled (see previous section). Furthermore, we see an upward drift of scaling exponents with size range in the NZ-NVS data set (Fig. 4d). To explore whether these factors influence scaling exponent predictions, we fitted separate hierarchical models for shrubs, short trees and tall trees; note that there is no evidence of a relationship between size range and scaling exponent within these groups (Fig. 4d).

Using the three-category hierarchy approach, scaling exponents were significantly greater than zero for 22 of 24 canopy trees, but for only 10 of 31 subcanopy trees and none of the 17 shrubs. We think it makes little sense to include the ill-defined growth curves of the shorter species when calculating the ‘basin of attraction’. For the tall trees, the posterior mean α was 0.394 (95% CI = 0.163–0.628); 12 of the species had exponents significantly greater than 1/3, nine had exponents indistinguishable from 1/3, and three had exponents <1/3. Thus, our asymmetric-competition hypothesis was supported for the majority of tall tree species. The credible intervals of the posterior mean include 0.333 but are very wide. This means that the MST-PG hypothesis cannot be rejected, but certainly is not endorsed. In contrast, all 31 understorey trees and all 17 shrubs had α values significantly <1/3 and posterior means that were significantly <1/3: many of the species spend their entire life in the forest understorey and never escape asymmetric competition for light. Russo, Wiser & Coomes (2007) warned against interpreting the cross-species median so simplistically, given the poor fit of growth curves for shrubs and subcanopy trees, and our BHM highlights the problems that arise from doing so.

Whole-community growth curves

MST-PG does not accurately describe the average growth curve – the curve estimated without any consideration of species identity – of the natural forests we have analysed. For New Zealand’s forests, BHM allows us to account for variation in α and β among permanent plots and assess whether doing so has any effect on the posterior mean α. For both the NZ-NVS and the Nothofagus data α was significantly > 1/3: for NZ-NVS α was 0.37 (CI = 0.35–0.38), and for the Nothofagus data set α was 0.42 (CI = 0.38–0.46). We note however, that power laws provide poor fits to these growth curves. Russo, Wiser & Coomes (2007) found that growth increased with size until about 20 cm diameter and then levelled off; a power law provides a reasonable fit to the smaller trees, but not to the whole data set. Similar conclusions were reached by Muller-Landau et al. (2006) who analysed over 1.7 million trees in 10 tropical plots. Foresters and forest ecologists have traditionally used flexible functions, which allow growth to reach an asymptote or peak and then decline with size (Richards 1959; Vanclay 1995; Canham, LePage & Coates 2004).

Support for productivity-dependent scaling

We found support for the hypothesis that growth curves have greater scaling exponents on productive sites (Fig. 5). There was a strong positive correlation between α values, estimated by BHM of the Nothofagus data set, and above-ground biomass production. Interestingly, plots on the lowest productivity sites have α values which cluster around 1/3, suggesting that MST-PG may perform well when asymmetric competition is weak. These analyses are preliminary, as we will explicitly incorporate productivity and competition in a growth model in planned research. However, even our basic analysis illustrates that interplot variability is large, and correlated with an important site attribute.

Figure 5.

 Scaling exponents of the mean growth curves of 112 Nothofagus stands plotted against above-ground biomass growth rate (kg m−2 year−1). In support of our productivity-dependent scaling hypothesis, α values were greatest in high-productivity stands and approached 1/3 in low-productivity stands.

Concluding thoughts

There is abundant evidence that competition for light has profound effects on tree growth; not accounting for asymmetric competition (e.g. Enquist, West & Brown 2009; Stark, Bentley & Enquist 2010) thwarts further progress in elucidating the ecological mechanisms underlying tree growth. That said, ecologists are still a long way from fully understanding how above- and below-ground competition affects size-dependent processes of individuals and populations (Muller-Landau et al. 2006; Coomes 2006). Here, we have shown that population-level growth curves often exceed 1/3, but this only provides indirect support for our asymmetric competition hypothesis. Neighbourhood analyses of growth would provide compelling evidence because they actually quantify the impacts of competition and provide estimates of the growth trajectory of trees unencumbered by asymmetric competition (e.g. Coomes & Allen 2007a; Pretzsch & Biber 2010).

We have focused attention on growth in this article, but there is also evidence that competition for light affects mortality and recruitment of trees, and the size structure of forests (Coomes 2006), so other components of MST-PG need further consideration. For instance, the MST-PG theory of size structure assumes that each stratum of a forest canopy has equal access to light (the ‘energy equivalence’ assumption), but this is unrealistic given that the upper layers of leaves actually capture much of the incoming light. We also recognise that MST-PG is inaccurate in other respects, most notably in overestimating the growth rates of large trees (Muller-Landau et al. 2006; Russo, Wiser & Coomes 2007), suggesting that other ‘extensions’ are necessary.

Modelling of forest dynamics is playing an increasingly important role in predicting climate change science and the prediction of carbon storage in schemes aimed at reducing emissions from deforestation and degradation (Purves & Pacala 2008). The philosophy of developing models from fundamental principles, enshrined in MST-PG, is noble and has encouraged forest ecologists to build and test scaling models that include biologically important features such as resource competition. Although MST-PG is a fascinating macro-ecological theory, its omission of tree interactions and resources severely limits its utility in predicting tree and forest growth and therefore carbon storage and sequestration. Nevertheless, a refinement of MST-PG to more explicitly incorporate competition could hold great promise in modelling tree and forest growth across gradients of forest productivity.

Acknowledgements

We thank Daniel Falster for sharing his thoughts on appropriate, and inappropriate, uses of SMA line-fitting, and Drew Purves and Rich Kobe for helpful discussions. We are grateful to staff at Landcare Research for providing access to the NZ-NVS data base, and to innumerable data collectors. This work was supported by a NERC grant awarded to D.A.C., and by funding from the New Zealand Foundation for Research, Science and Technology.

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