## Introduction

The roles of individual- and population-level processes, vs. extrinsic environmental variables, in structuring plant communities has long been a topic of interest in ecology (Grime 1977). In particular, the role of competition in organizing desert plant communities is often questioned. This is because constituent population often exhibit episodic germination, recruitment and mortality due to fluctuations in environmental variables (Fowler 1986). Some have argued that environmental fluctuations prevent xeric plant population from ever reaching equilibrium with resource availability, thereby minimizing the role of competition (Fowler 1986). Previous studies have assessed the strength of competitive interactions among desert plants by investigating the spatial configuration of individuals and their root systems (e.g. Chew & Chew 1965), by experimentally manipulating water and nutrient regimes (e.g. Sharifi *et al*. 1988), or by removing individuals and assessing treatment effects on survivors (e.g. Fonteyn & Mahall 1981). In general, these studies support the importance of competition, but the consequences of such interactions for the structure and dynamics of water-limited ecosystems have yet to be explicitly quantified.

Plant-allometry theory may provide a framework for quantifying how competition among plants influences the structure and dynamics of water-limited ecosystems. The theory links biological metabolism to ecosystem dynamics based on the size-dependence of individual-level resource use and architecture (Enquist, Brown & West 1998; Enquist *et al*. 1999; West, Brown & Enquist 1999; Enquist & Niklas 2001, 2002; Niklas & Enquist 2001; Allen, Gillooly & Brown 2005; Kerkhoff *et al*. 2005). Predictions of the theory are supported by comparisons within- and among-species that span nearly 20 orders of magnitude in size (Enquist *et al*. 1998; Enquist & Niklas 2002; Niklas & Enquist 2001). Successful application of the theory to xeric plant populations, where canopies do not overlap and water availability limits plant abundance and metabolism, would simultaneously support the importance of competition, and quantify its effects on the structure and dynamics of water-limited ecosystems.

Here we assess the role of competition in water-limited ecosystems by synthesizing plant-allometry theory with empirical data collected from the desert creosote bush, *Larrea tridentata*. This evergreen shrub is distributed throughout the Mojave, Sonoran and Chihuahuan deserts of North America (Barbour 1969), and often represents a substantial portion of the standing biomass and net primary production (Chew & Chew 1965). It is unusually drought-tolerant because of its ability to sustain photosynthesis under the water-limited conditions that occur over most of the year (Ogle & Reynolds 2002), and because of its ability to shed above-ground biomass during periods of water stress (Chew & Chew 1965).

### review of plant-allometry theory

We begin by reviewing previous work in plant-allometry theory, which has yielded equations that link individual metabolic rate (eqns 2, 5 and 6) to size-dependent changes in plant architecture (eqns 1, 3 and 4), population- and community-level abundance (eqns 7 and 8), and ecosystem-level net primary production (eqn 9).

The metabolic rate of a plant is equal to its gross rate of photosynthate production, *B* (g year^{−1}). Metabolic rate varies with body size, *M* (g), according to a power function of the form *B* ∝ *M*^{3/4}. This so-called ‘allometric’ relationship of metabolic rate to body size has long been known for animals (Savage *et al*. 2004), and has more recently been demonstrated for plants (Niklas 1994; Enquist *et al*. 1998; Niklas & Enquist 2001). West, Brown & Enquist (1997) derived a model that attributes this 3/4-power scaling exponent to the geometry of biological distribution networks, including vascular systems of plants. The model is derived based on three assumptions: (i) the biological network is fractal-like so that it fills space, (ii) the energy required to distribute resources through the network is minimized, and (iii) the final branches of the network are size-invariant terminal units. The guiding principle underlying these three assumptions is that natural selection has served to optimize energy use by organisms subject to fundamental physical and geometrical constraints.

Following assumption (iii), one size-invariant terminal unit of a plant's distribution network is the leaf. Here size-invariance means that leaf-level traits (e.g. photosynthetic rate per leaf) are assumed not to vary with plant size. This assumption in no way disagrees with observations that leaf-level traits vary substantially among species (Wright *et al*. 2004). It may also be violated, and yet still be reasonable for deriving predictions, provided that the size-dependence for total leaf mass is large relative to size-dependent changes in leaf-level traits. Allometry theory predicts that total leaf mass, *M*_{L} (g), should vary with plant size as (West *et al*. 1999; Enquist & Niklas 2002; Niklas & Enquist 2002):

where *l*_{o} is a normalization constant independent of plant size (g^{1/4}). Given the assumption of size-invariance for leaf-level traits, the metabolic rate of a plant can be expressed as the product of total leaf mass, *M*_{L} and, the size-invariant rate of photosynthesis per gram of leaf tissue, *P*_{L} (West *et al*. 1999):

where *b*_{o} = *l*_{o}*P*_{L} is a normalization constant independent of plant size (g^{1/4} year^{−1}). Equation 2 quantifies the relationship of metabolic rate (*B*) to total plant mass (*M*), leaf mass (*M*_{L}) and the photosynthetic rate per gram of leaf tissue (*P*_{L}).

Equations 1 and 2 can be extended to yield predictions on above- vs. below-ground biomass allocation by imposing three additional assumptions (Enquist & Niklas 2002; Niklas & Enquist 2002): (i) stem length is isometric to root length, (ii) densities of stems and roots are constant over ontogeny, and (iii) hydraulic cross-sectional areas of stems and roots are equivalent due to conservation of mass flow through the plant. Given these assumptions, total below-ground root mass, *M*_{R}, should be proportional to total above-ground ‘shoot’ mass, *M*_{S}:

In this expression, *r*_{o} and *s*_{o} are both dimensionless constants and *M*_{R} + *M*_{S} = *M*, so

Thus, the root : shoot ratio, *r*_{o}/*s*_{o}, is predicted to be independent of plant size.

Equations 1 and 2 can also be extended to predict the size-dependence of growth, *dM*/*dt* (g year^{−1}). The relationship between the gross rate of carbon fixation, *B*, and the amount of fixed carbon allocated to biomass production, *P*, is characterized by the carbon use efficiency ɛ = *P*/*B*. If this carbon use efficiency is approximately independent of plant size (*c.* 0·5, Gifford 2003) and if a relatively constant fraction of *P*, α, is allocated to growth, then the predicted size-dependence for growth is (Enquist *et al*. 1999):

where *g*_{o} = α·ɛ·*b*_{o} (g^{1/4} year^{−1}). Integrating eqn 5 and rearranging terms yields (following Enquist *et al*. 1999):

Equation 6 predicts a linear relationship between the fourth roots of plant mass at times *T*_{2}, *M*^{1/4}(*T*_{2}), and *T*_{1}, *M*^{1/4}(*T*_{1}), with a slope of 1 and an intercept of *g*_{o}(*T*_{2} *– T*_{1})/4.

The stoichiometry of the photosynthetic reaction is fixed, so the rate of resource use by a plant is constrained to be proportional to its metabolic rate. Therefore, if a population comprised of *J* individuals in an area of size *A* (ha) is at equilibrium with the supply rate of limiting resources in the environment, *R* (g ha^{−1} year^{−1}), then the total rate of metabolism for the population is constrained such that

where is the sum of the metabolic rates for all *J* plants comprising the population, *N* = *J*/*A *is population density (individuals ha^{−1}), is an average for plant size and ω is a parameter that characterizes the relationship between the metabolic rate of a plant and its rate of resource use (Enquist *et al*. 1998, 2003; Allen *et al*. 2005). For example, if water availability limits creosote abundance, then *R* is the supply rate of water to the creosote population (g H_{2}O ha^{−1} year^{−1}) and 1/ω is the water use efficiency (g photosynthate g^{−1} H_{2}O).

Rearrangement of eqn 7 yields an expression for population density (Enquist *et al*. 1998):

Here <*M*> is the average body size for all *J* plants comprising the population . The approximation symbol is required because unless all individuals are of the same size. The product of the average metabolic rate per plant, , and population density, *N*, yields an expression for net primary production,* n* (g ha^{−1} year^{−1}) (Allen *et al*. 2005; Kerkhoff *et al*. 2005):

This equation follows directly from the assumption that plant populations exploit all available resources. The predicted changes in abundance, *N*, in relation to resource availability (∝*R*) and plant size (eqn 8), therefore provide benchmarks for assessing whether plant populations are regulated by competition.