1. Self-thinning Constant final yield at low to moderate density is achieved through Plasticity; all individuals are reduced in size. That isn't always the case, at least over large range in density. At high density the 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, and 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).

3. The yield-density relationship initially proposed byYoda: 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). 1)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).

4. 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 (constant final yield). 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.

5. 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

6. 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.

7. 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.

8. The same phenomena (both individual plasticity and hierarchy) are also observed in animal populations.An example: Branch (1975) found almost exactly parallel responses to density in the limpet Patella. There is individual plasticity in size, leading to a biomass carrying capacity of 125g/m 2. Size distributions at low densities are dominated by large animals, but at high densities there are only a few large animals and most are small. The small animals are immature juveniles, so that as density increases the number of gamete producing adults decreases. Gamete production (and birth rate) initially rises at very low densities, reaches a peak at some intermediate density, then declines as density further increases. Earlier we called this the Allee effect, and presented it as the result of mate-finding at low densities. Now you can see that there may be other causes for the same relationship between birth rate and density.

9. What does this mean for birth and death rates? Over a fairly wide range of densities, death rates may not change much, but birth rates (whether due to mate finding or gamete production) may pass through 'equilibrium' with death rates twice due to the parabolic density response curve. That produces the low density 'instability'. What does initial size difference do to plant size and skewing? The difference grows if we consider exponential growth: w t = w 0 e rt since the same factor is multiplying an increasing biomass as plants increase in size with time. Consider the biomass of a pair of plants which begin (t = 0) at 1 and 2 grams respectively. If they have the same growth rate, r ( let's set it at 1 arbitrarily), and we look at the biomasses after 5 time units, the first plant weighs 148 grams, and the second 296. The same factor (ratio of biomass) accounts for a much larger actual difference in biomass.

10. There is also a possibility of differences in relative growth rate (the r in an individual growth equation). Plant growth is typically measured in terms of grams/gram per day, i.e. corrected for the size of the plant. If all individuals have the same physiological characteristics, then their relative growth rates are equal. However, in measurements on real populations RGRs were not equal; there was a significant correlation between plant size and relative growth rate. Therefore, those plants which are larger are also growing disproportionately faster. How do survivors respond to competition? Is the whole plant, i.e. each organ (roots, stems, leaves, flowers) reduced in the same way, or do plants allocate energy in ways which change with competitive stress? The resources for which plants compete are light from above and nutrients and water from below. There is, therefore, the potential for separable root and shoot competition. Above ground, competition changes the morphology of competing plants; the stem is elongated to enhance light capture in comparison with competitors, and leaves are typically 'thinner'.

11. Competition among Animal Species There is an older approach, based on niche overlap, and a newer approach based on such factors as disturbance frequency. First, the older method: 1)Thomas Park's studies of competition in flour beetles, Tribolium confusum and T. castaneum (1948, 1954, 1962) Park measured the fundamental niches of both species, and found them to be essentially completely overlapping with regard to abiotic conditions. He then placed the species in competition under a variety of climatic conditions. Under competition, the realized niches of the two species diverge.

12. Conditions under which consistent overlap of niche measures occurred were quite limited.Under those rare conditions, what resulted was termed 'competitive indeterminacy'; which species would win in competition could not be predicted.

13. Relative frequencies of success for the two species: Climate Percentage wins T. confusum T. castaneum hot-moist 0 100 temperate-moist 14 86 cold-moist 71 29 hot-dry 90 10 temperate-dry 87 13 cold-dry 100 0 T. confusum wins under dry conditions, and clearly does better when its cold. T. castaneum wins when its moist, unless it is cold.

14. It is experiments like this that result in the competitive exclusion principle, which states that if differentiation of the realized niches of species is precluded, either by the adaptive limits of the species or the character of the environment, then one of the competitor species will exclude the other. Does competitive exclusion occur with the variability and inconstancy of field conditions ? The classic experimental study which demonstrated spatial competitive exclusion in the field is Connell's studies of barnacles Balanus balanoides and Chthamalus stellatus (Connell 1961).

16. While the settlement rates of larvae over the entire elevation gradient were very similar, the natural distributions of adults were markedly different. Adults of Balanus rarely occur above the mean high water mark for neap tides, and abundance increases below the mean tide line. Chthamalus adults, however, are rarely found below the mean high water of neap tides, but may survive in the spray zone above the spring high water mark. The reason for this separation of realized niches is competition. Balanus grows faster, and either wedges out or grows over the top of Chthamalus, excluding it from areas where the Balanus can grow vigorously. In those areas Chthamalus is excluded, but it persists where its tolerance of desiccation permits its continued occupancy in the region. Balanus is far less tolerant of dessication. In addition to effects of competition on survivorship, there are also effects on growth. Those few Chthamalus that survive in lower regions are much smaller than uncrowded ones.