Our model is similar to that described previously (Chu et al. 2008), which itself was an extension of Weiner et al.’s (2001)‘zone-of-influence’ model. In the model, each individual obtains resources from a circular zone, and neighbouring individuals compete for resources where these zones overlap (Weiner et al. 2001). The area occupied by a plant, A, represents the amount of resources potentially available and is related to the plant’s biomass, B, as , where c is a constant (c = 1.0 in all the simulations presented here). An individual’s potential growth rate (i.e. in the absence of neighbours) is defined by the equation:
- ( eqn 1)
where Bmax is the maximum (asymptotic) plant mass, and r is the initial (maximum) growth rate (in units of mass area−1 time−1).
Neighbouring plants compete for the resources where their areas (A) overlap. The realized growth rate of the plant is described by the equation:
- ( eqn 2)
where Ac, the effective area of a plant, is calculated as the area it covers minus that part of the area lost to neighbours. After Weiner et al. (2001), we used a discrete approximation of continuous two-dimensional space, divided into a fine grid, to obtain the overlapped area.
Abiotic stress is included in the model with a parameter s, which ranges from 0 (no stress) to 1 (maximum stress), and which decreased the growth rate of all plants in a simple linear fashion (Molofsky & Bever 2002). Based on current understanding of facilitation, we assume facilitation only occurs when there is stress (s > 0), and it acts to ameliorate the negative effects of stress on plant performance (Callaway 2007). We assume that facilitation is a function of the total of all areas of overlap with neighbouring plants, Af. Thus, the average Af of plants within a population is positively related to population density and to mean plant size. The realized growth rate of the plant is modelled as
- ( eqn 3)
In our simulations, we consider the effects of competition, environmental stress and facilitation on populations with and without mortality. To explore the size inequality in populations without mortality, we assume that plants cannot have negative growth rates, but continue to live and maintain the maximum size they achieve (Weiner et al. 2001). We looked at a wide range of population densities (4.5–9.5, on a natural logarithmic scale), and we use the coefficient of variation (CV) of mass as the measure of size variation within the population (Weiner et al. 2001). To study self-thinning, we simulated a high initial population density (ln density = 9.1, equivalent to a density of 9000 individuals m−2). We also simulated other initial densities and the results were similar to those presented below. Dead individuals do not compete with or facilitate other plants and are excluded from the statistical analyses (Weiner et al. 2001).
Due to the association of harsh conditions and positive interactions, it is difficult to separate their effects on performance in field experiments, but easy to do so in simulations. To investigate the effects of stress without facilitation, we set Af = 0 in eqn 3, and the realized growth rate becomes
- ( eqn 4)
To explore the effect of the size symmetry of competition on the model, we consider three degrees of competitive size symmetry, expressed through the parameter p. They reflect three different ways of dividing areas of overlap among competing individuals (parameter b in eqn 4 and table 1 in Weiner & Damgaard 2006) to determine a plant’s effective area (Ac): P = 0.0 for complete symmetry (areas of overlap are divided equally among all overlapping individuals, irrespective of their sizes), P = 1.0 for size symmetry (areas of overlap divided according to the relative sizes of the overlapping individuals) and P = 5.0 for size asymmetry (larger individuals get a disproportionate share of areas of overlap). To have mortality occur under symmetric as well as asymmetric competition, we assume that individuals die if their actual growth rate falls below 2% (Stoll et al. 2002).
The simulations were stochastic, there was random normal independent variation in initial size (B0 = 1 mg; SD = 0.1), the initial growth rate was set up as r0 = 1 mg cm−2 t−1 (SD = 0.1), and the maximum individual mass was asymptotic (Bmax = 20 000 mg; SD = 2000). We take a ‘wraparound’ (torus) approach to avoid edge effects (Grimm & Railsback 2005). Individuals were distributed randomly in space. For populations without mortality, we collected data from simulations every five time steps after the first 10 steps, and we present here the results at 50 time steps. For self-thinning, the data collected was dependent on the mode of competitive (160, 120 and 90 steps for complete-symmetric, size-symmetric and size-asymmetric competition respectively), because it has been shown that growth and competition occur more slowly under symmetric than asymmetric competition (Stoll et al. 2002). All simulations were conducted on landscapes with the size of 120 × 120 grids and were performed in NetLogo (Wilensky 1999).
To test some predictions of the model, we conducted an experiment in an alpine meadow located in the eastern part of the Qing-Hai Tibetan Plateau, China (33°58′ N, 101°53′ E; 3500 m a.s.l.; 5° slope). The experiment was similar in design to an earlier experiment, which focused on mean plant size (Chu et al. 2008). The average annual temperature is 1.2 °C and precipitation is 620 mm year−1 at the study site. The vegetation is dominated by sedges, most notably Scirpus pumilus Vahl and Kobresia macrantha Boeck, and by grasses such as Elymus nutans Griseb (Wang et al. 2008). Soils are classified as alpine meadow soils (Gong 1999). We chose E. nutans as our experimental species because its high capacity for clonal growth produces clear density effects. It is also a dominant species at the study site, and previous studies conducted there have found that it is strongly facilitated by neighbours (Wang et al. 2008). Modules (culms) of this species occur individually or in tufts, both of which can be whole genets or connected by rhizomes. We consider both individually occurring culms and tufts as individual ramets (Scrosati 2000).
A total of 18 1 × 1 m plots were randomly selected in a relatively homogeneous area within the site in 2006, with a 50-cm walkway between plots. The site was an Avena sativa field in previous years and was tilled before the experiment. The plots were seeded on 28 June with a varying number of E. nutans seeds (collected locally in September 2005), to obtain six different ramet densities, from 700 to 2800 individuals m−2. Before sowing, a thin layer of soil was sieved over the plots to provide a surface as smooth as possible with minimal spatial heterogeneity. For the random initial distribution, seeds were mixed with sand and sown with a sieve. Each density level was replicated three times. One plot was destroyed by voles (Microtus oeconomus). To avoid edge effects, we collected data only from a 25 × 25 cm subplot within each plot for measurements. Above-ground biomass was harvested from the subplots in early September 2008, after a full growing season but before the arrival of low temperatures. Below-ground biomass cannot be measured with any degree of confidence in E. nutans. Individuals were counted within each subplot, individually harvested and dried at 80 °C until constant weight. We analysed the relationships among size variability, measured as the CV of mass, population density and mean individual mass, although low sample sizes did not permit statistical testing of density effects.