## Introduction

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

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:

where *B*_{L} is the basal area of taller neighbours, and competitive effects are defined by parameters λ_{2} and λ_{3}. This model assumes that a tree without taller neighbours follows a power-law growth curve, because equation 2 reduces to a power function when *B*_{L} = 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).

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.