## Introduction

Mass–density relationships are at the core of quantitative and theoretical ecology (Damuth 1981; Gaston & Blackburn 2000). When a crowded population of plants grows and develops, intense competition leads to mortality, in the process known as density-dependent mortality or ‘self-thinning’ (Yoda *et al*. 1963; Morris 1999). Early work in the area (Yoda *et al*. 1963) showed that the relationship could be described as a power law of the form M = *K* N^{d}, usually analysed as log M = log *K* + *d* log N, where M is the mean mass of survivors, N is their density and *K* and *d* are parameters. For biological (Westoby 1984) and statistical (Weller 1987a) reasons, it is preferable to analyse the total biomass of survivors (B = N M) rather than M, in which case B = *K* N^{c} or log B = log *K* + c log N, where *c* = *d* + 1.

There has been much debate about whether the exponent of the mass–density relationship (*d*) is a universal constant, and if so, what its value is (Weller 1987a; Lonsdale 1990; Enquist, Brown & West 1998). Early studies and simple geometric models of self-thinning predicted *d* = −3/2 (Yoda *et al*. 1963; Miyanishi, Hoy & Cavers 1979), and this was called the ‘Self-thinning Rule’ (Westoby 1984). More recently, Metabolic Scaling Theory (West, Brown & Enquist 1997; Enquist, Brown & West 1998), based on the fractal nature of organisms' internal transport mechanisms, predicts that *d* = −4/3. This hypothesis has received support from broad interpopulation and intertaxa comparisons (Franco & Kelly 1998; Enquist & Niklas 2001; Niklas, Midgley & Enquist 2003). Field data suggest that there may not be a single self-thinning power law with the same exponent for all stands, but rather that the value of the exponent can vary with species (Pretzsch 2006), environmental factors (Deng *et al*. 2006) and even biotic interactions (Zhang *et al*. 2010). Furthermore, plants' allometric growth, specifically the change in plants' shape as they grow, has been reported to have an important influence on the slope of the self-thinning line (Miyanishi, Hoy & Cavers 1979; Weller 1987b; Wang *et al*. 2004; Dai *et al*. 2009).

It has been difficult to test these alternative hypotheses definitively because (i) experimental data on self-thinning are resource demanding, as each harvested population yields only one data point, (ii) forest data on mass are rarely based on harvests but rather on estimates from static linear measurements, which are biased when applied to allometric growth (Weiner & Thomas 1992), and (iii) the large natural variation means that distinguishing between the slopes −1/2 and −1/3 statistically requires more data than are usually available.

All models assume that the self-thinning trajectory is independent of the initial density, that is, stands with different initial densities will converge on the same trajectory (Fig. 1), although we know of no studies that specifically tests this. If plant allometry influences self-thinning but is also altered by competition, it is reasonable to predict that crowded stands grown at different initial densities may thin along different lines. At higher initial densities, plants will interact earlier, when they are smaller, so changes in allometry due to competition will also begin when plants are smaller.

While many studies of self-thinning have focused on the value of and variability in the M–N exponent (Weller 1987a,b; Lonsdale 1990; Deng *et al*. 2006; Dai *et al*. 2009), there has been much less discussion about variability in and the biological significance of the intercept log *K* (White 1981; Lonsdale & Watkinson 1983). The parameter *K* has the unit g m^{−3}, reflecting biomass packing by plants (White 1981), which could be due to the density of plant tissues or growth form. If self-thinning is driven by interactions in three-dimensional space, such as physical space filling or competition for light, log *K* will vary with the density of the material. If we imagine two species that are identical except for the density of their tissues or their architectural compactness, they will have the same thinning exponent but different coefficients. Log *K* has been shown to vary considerably among (Weller 1989) and within (Westoby & Howell 1986; Duarte & Kalff 1987) plant stands. It has a range of values restricted to one order of magnitude: for a wide variety of species, log *K* lies between 3.5 and 4.4 (when M is measured in g plant^{−1} and N in plants m^{−2}), with only a handful of examples transgressing a value of log *K* = 4.4 (White 1980). Some of this variability can be attributed to quantifiable differences in plants' growth forms (Lonsdale & Watkinson 1983; Norberg 1988) or environmental conditions such as light intensity (Dunn & Sharitz 1990) and soil fertility (Morris 1999). For example, there is evidence that grasses tend to have higher log *K* values than dicotyledons and that coniferous trees have higher log *K* values than deciduous trees (Lonsdale & Watkinson 1983), which appears consistent with the hypothesized role of tissue density and architectural compactness.

We grew crowded experimental populations of *Fagopyrum esculentum* at three initial densities to ask the following questions:

- Does initial density affect the self-thinning trajectory, or do all stands converge on the same trajectory?
- Do biomass–density relationships among populations of one species reflect self-thinning trajectories within populations?
- Are the self-thinning exponents more consistent with the Self-thinning Rule or the predictions of Metabolic Scaling Theory?
- How does height–diameter allometry and the resultant plant shape change over density and time as stands develop?

Our specific hypotheses are as follows: (i) self-thinning trajectories differ among different initial densities, (ii) interpopulation patterns are fundamentally different from self-thinning trajectories and (iii) the height–diameter allometric relationship is plastic in response to increasing intraspecific competition and changes during growth, influencing plant shape and the biomass–density relationship.