Much of the best quantitative data measuring effects of competition comes from studies of plants. A paper by Palmbald (1968), for example, expressly deals with population size results, rather than the more usual measures of 'yield‘ in studies of agricultural plant competition. Palmbald studied a variety of weeds, most annual, so that seed output was a measure of population growth. What he found was differing kinds of response in different species, but a tendency for the weedy annuals to have plasticity in reproduction, and perennials to have plasticity in growth parameters.

Capsella bursa-pastoris (shepherd’s purse) is fairly typical of annuals. Germination and pre-reproductive mortality show only slight responses, but the size of individuals and the number of seeds produced per individual are strongly affected by density. Seed production falls dramatically, from 23,000 at 1 plant per pot to 210 at 200 per pot, that is by a factor of about 100. Close to parallel is the decline in biomass per plant. Thus, the allocation of biomass to produce seeds is relatively constant, measured as the number of seeds per gram individual plant biomass. The data follows:

Sowing density % germination % mortality % reproducing % vegetative Dry weight (g) Dry weight/plant Seeds/repro ducing individual Total seeds

There are two things going on: One is measured at the individual level, where growth, size, and reproduction change approximately in proportion to density over fairly wide ranges. The other is reproductive output per unit area. This is the measure important to agriculture. A farmer wants to maximize his yield per acre, while minimizing his costs (here seeds planted). Since seminal papers in the 1950's it has been clear that there is a maximum production per unit area. Below some 'threshold' density the total production increases toward that maximum; above it plasticity (an almost universal characteristic of plant growth and reproduction) reduces the growth, etc. of individuals and maintains that maximum. Plants 'compensate' for density changes, and the result is the 'law of constant final yield' (Kira,et al. 1953,Yoda et al. 1958).

Bromus grown in pots Corn grown in a field

Do plants exactly compensate for density? The answer is no. Plants 'undercompensate' at low densities, and totals increase. At higher densities they frequently 'overcompensate', i.e. totals do not remain exactly constant, but decrease slightly. That’s what was evident in the data for Capsella. The growth of Bromus follows what would be called compensation, but seed production at very low plant densities cannot reach the maximum yield. The growth of corn shows ‘overcompensation’, in that total grain yield declines at very high densities.

Also note the impact of different fertilizer levels. The maximum yield increases in Bromus when additional nitrogen fertilizer is added. In general, the maximum yield is set by environmental conditions, both biotic (e.g. weed occurrence and density) and abiotic (climate, soil nutrients). Compensation develops over time. When plants start growth, the biomass density is low, and total biomass in pots increases with further growth (which could be described as 'undercompensation' if yield were measured at each time by the total biomass of plants). However, as they grow, biomass density increases, until is approaches the maximum. If we look at yield-density curves over time, they begin as straight lines, yield increasing in proportion to density.

At mid-season the relationship maintains the same slope at low density, but curves over to flatten at the maximum when plants are at a higher density. Late in the season the curve has reached its aymptote at all densities, and is flat.

There is more than one way to achieve that constant final yield. Plasticity seems to imply that all individuals are reduced in size. That isn't always the case, at least over large range in density. The density response is usually the result of mortality of individuals. Self-thinning is the 'rule' in most dense plant populations. Thus, the compensation we observe between individual plant weight and density, a slope of -1 in the log-log plot, is what will be observed at low density, when all individuals can achieve sufficient growth and size to survive. In dense populations, even while total biomass continues to grow, some individuals die.

The relationship between log mean plant weight and log density is steeper (due to the mortality). This relationship has a characteristic slope of -3/2. Over time survivors become larger, but mortality decreases population density. Total plant weight increases because plant weight of survivors is increasing more rapidly than density is decreasing (at least until the biomass carrying capacity is reached, but beyond that the slope should be -1).

The yield-density relationship initially proposed by Yoda: w = C x d -a or log w = log(C) - a log(d) Since a seems always to be about 3/2, this relationship has come to be called the -3/2 power law. Some basic points about the characteristics of self-thinning: 1)Plants increase in mean size (weight) over time; they grow. 2)No density-dependent mortality occurs until populations reach the self-thinning line (increase in individual plant biomass indicated by 'rise' from the x axis). 3)Mortality begins earlier in dense populations than in sparse ones. They reach the self-thinning line earlier).

4) Plants of the same mean weight are younger in sparse populations than in dense ones (the growth rate is higher in populations at lower density). 5) Populations at any density eventually reach a stage where weight increments and mortality are balanced; the slope then shifts to -1, total plant weight no longer increases. 6) a = -1 is reached by denser populations more rapidly. Descriptively: At any initial density individuals grow (at rates related to density, but without density-dependent mortality); graphically the trajectory rises essentially vertically from the x axis at the planting density. Growth eventually brings mean plant weight to (or near) the self-thinning line. The trajectory then turns to follow the self-thinning line (slope - 3/2).

Survivors are growing. Mean plant weight and total biomass are increasing, but density is decreasing as self-thinning occurs. Total plant biomass per unit area is still increasing. Eventually that growth (still following the self-thinning line) brings the population to the carrying capacity of the environment. At that point the trajectory shifts to a line having a slope -1, and growth is compensated by corresponding mortality to hold total biomass constant.

Why is the slope of the self-thinning line -3/2? It is believed due to limitation in the biomass which can be sustained by the amount of light captured (or total photosynthesis). The derivation is straightforward. If we, for purposes of argument, think of plants as living blocks, then the total biomass (or weight w) of a plant is proportional to the cube of its linear dimensions (its volume): w  l 3 but the area (or surface s) occupied (and therefore light capture potential) is proportional to the square of linear dimensions: s  l 2 Combining these two relationships: s.5  w 1/3 s  w 2/3

Assume all these individuals are equal. Then when growth has brought the population to the self-thinning line, mean surface area per individual will be inversely proportional to density, or: s  1/d and substituting 1/d for s, we get: w  1/d 3/2 or w  d -3/2

Self-thinning means that some individuals die; it's not random which plants die. As density and growth lead to self thinning, the size distribution of individuals within the population changes. What may well start as a symmetrical, normal distribution does not remain that way. The larger individuals (due to earlier germination, larger seed size, or other factors) capture a more than equal share of resources and tend to grow more rapidly. A 'hierarchy' develops. There are a few large individuals ('dominants') and many small ones ('suppressed'). The distribution of sizes is positively skewed. But then self-thinning leads to mortality of the smallest individuals, and therefore (at least in even-aged stands) can reduce the degree of skewing.