## Introduction

We have a quantitative understanding of plant competition only at the population level. Density-yield relationships, such as the ‘law of constant final yield’ and patterns of self-thinning, have been successfully modelled by researchers (Silvertown & Charlesworth 2001). One of the primary goals of population ecology is to explain population phenomena in terms of the behaviour of individuals. Foresters and plant ecologists have made numerous attempts to model the performance (e.g. size, growth or reproductive output) of individual ‘target’ plants as a function of measures of local crowding (Stoll & Weiner 2000) such as the number, distances and sizes of neighbouring trees (Aaltonen 1926; Opie 1968; Bella 1971; Weiner 1984; Woodall *et al*. 2003; D’Amato & Puettmann 2004). A complete description of an individual plant's competitive environment would include the number, size, distance, genotype and angular dispersion of all its neighbours (Mack & Harper 1977). Most studies have looked at only a small subset of these factors. While there have been some limited successes, attempts to account for target plant performance as a function of measures of local competition have been frustrated by several obstacles, which are briefly discussed below.

Static (‘snapshot’) description of neighbour relationships (e.g. Goldberg 1987) can make only a limited contribution to our understanding of competition among individual plants. Showing, for example, that larger individuals within a crowded population tend to have small neighbours, while relatively small individuals tend to have large neighbours, is not very informative, as plant size and neighbour size are not independent in any sense (Mitchell-Olds 1987). It is not clear if large individuals are large because they have small neighbours, or if the neighbours are small because their neighbour, the target plant, is large. All such a static relationship shows is that the total biomass of target + neighbours is limited, i.e. competition is occurring. To contribute to our understanding of the dynamics of competition, models of local competition need to be dynamic, looking at changes in target plant behaviour as a function of neighbourhood conditions at the beginning of one or more growth intervals.

The simplest dynamic models look at the size or reproductive output of an individual plant as a function of the number of neighbours within a given radius (‘neighbourhood distance’) from the subject individual (e.g. Silander & Pacala 1985). If there has been no density-dependent mortality, then one can argue that the number of individuals around a target plant may be considered independent of the target plant's performance (e.g. Benjamin 1993). The number of individuals within a fixed radius is insufficient information, however, and a whole growing season is too large a time-step, to account for much variation among individuals. There is usually large variation in the sizes of neighbours and therefore their effects. A target tree with 10 small seedlings within 2 m will grow almost unaffected by them, whereas 10 large trees within this distance will certainly have a major effect on the target tree's growth. Also, the effect of a neighbour decays with its distance, and the scale and, perhaps, the form of the decay function changes over the course of plant growth, so it is not surprising that the number of individuals within a fixed distance does not provide much information about the degree of competition experienced by an individual over its life. The relationship between local density and target size is often triangular: plants with many neighbours are almost always small, but plants with few neighbours can be large or small (Goldberg 1987; Stoll & Weiner 2000). We can conclude only that high local density can constrain plant growth. This triangular pattern could occur because factors other than competition limit the performance of many less crowded individuals, but it could also occur because local density is too crude a measure of competition, e.g. some plants have few but large and/or close neighbours.

Another complicating factor is the size dependence of plant growth itself. Plant growth during competition can be in large part a function of the plant's size, not just its local competitive environment. Plant growth is sigmoidal (Weiner & Thomas 2001): the relative growth rate decreases, while the absolute growth rate increases and then decreases as a plant grows. Plant size and neighbour size are often confounded in studies of local competition.

It has rarely been possible to separate the effects of neighbour size from those of neighbour distance, even in experimental studies, because, after a period of growth and competition, these two factors become confounded. For example, if plants are grown at high density, then after a period of growth they will be smaller than if they had been grown at a lower density. Not only is the number of neighbours around an individual target plant higher at higher density, but after competition has started, both the target's size and its neighbours’ sizes are smaller than they would be at lower density.

The development of local competition models has also been constrained by the reliance on data from unmanipulated populations in the field. Plants are almost always crowded in nature, and this means that naturally occurring combinations of the different components of local competition, such as neighbour number, size and distance, will be very limited and biased. Experiments that create new combinations of these variables are therefore necessary. To obtain a quantitative understanding of local competition we need to reduce the effects of neighbours to their components, vary these experimentally (Purves & Law 2002), and look at competition over shorter time intervals. A reductionist approach requires that we look at one or a few of these variables at a time, while holding other factors constant. Here we attempt to separate the effects of target and neighbour plant size to study the effects of the latter.

One reasonable hypothesis is that the effect of a neighbour is a function of its size, here defined as above-ground biomass (Goldberg & Werner 1983). Simple ‘per-unit-biomass’ effects can be used as a null hypothesis in comparing the effects of different species. It can be useful to distinguish the effect of the absolute size of a neighbour from that of its size relative to the target plant. If competition is ‘size asymmetric’, then we expect the per-unit-size effect of neighbours larger than the target to be greater than for neighbours smaller then the target (Fig. 1). The fit of a local competition model was significantly improved in two out of three cases when the per-unit-size effects of neighbours smaller than the target plant were discounted (Thomas & Weiner 1989).

We ask the following questions:

- 1Is the growth rate of an individual plant experiencing strong competition from neighbours more determined by its own size (size-dependent growth) or by the size of its neighbours (competition)?
- 2What is the form of the relationship between neighbour biomass and target plant growth?
- 3Is the per-unit-biomass effect of neighbours the same for neighbours larger and smaller than the target plant (size-symmetric competition) or is the effect per unit of biomass greater when neighbours are larger than the target plant (size-asymmetric competition)?