Self-thinning in populations of Trifolium repens – a greenhouse study

Introduction

Self thinning is the result of intraspecific competition in densely planted populations. The response to density is the mortality of a fraction of the population and changes in the distribution of biomass among surviving members. There are two prevalent views of expected relationship between shifting (declining) density and the average biomass of survivors: one, the classical view, is based on a simple model of plant geometry, the other, a more modern view, is based on a generalization of scaling laws broadly observed in both plants and animals.

The geometric model of self-thinning

The geometric model suggests that plants acquire their basic resource – sunlight for photosynthesis producing carbon- and energy-rich molecules – based on the exposed surface area of the plant. However, the plant must distribute the resource throughout a three-dimensional volume. The area exposed to sunlight scales as the square of the plant’s diameter. The volume scales as the cube of that same diameter. The density of plants in an area scales as the inverse of the area occupied (shaded) by an individual. We can assemble these relationships using the following equations:

Biomass  l3

Surface area  l2

Then surface area  biomass3/2

If all individuals are of equal size, then surface area  1/density, and

Biomass (mean per individual)  density-3/2

This relationship is referred to as the -3/2 power law. Since eventually all area is covered by plants, there is a maximum biomass and density that can be achieved, called a constant final yield, and designated by C (like a carrying capacity). The power law is usually written:

w = CN-3/2

or, when log transformed

log w = log C -3/2 log N

What this means is that if we plot the log of individual plant biomass against log density, we should see a slope of -3/2 for the regression line.

The transport scaling model

A more recent view finds a different scaling law. It posits that the power law should reflect the way both plants and animals distribute resources within their bodies. Both groups use a branching network of tubes (though physiological mechanisms and what is transported differ) to supply resources throughout their bodies. It is interesting to note that the terminal branches of most of these systems all have about the same diameters. There are a few suggested rules for these branching systems: 1) the system must reach throughout the organism; all cells need the resources, 2) the terminal branches are the same diameter, independent of the body size of the organism, 3) these systems need to minimize energy loss in transfer. As a consequence, the summed diameter of each successive (daughter) layer of branching should at least approximately equal the diameter of the parent branch. As a corollary of this, a relationship can be derived that suggests a scaling law between metabolic rate and mass with an exponent of 0.75, and a parallel law relating mass per plant to density with an exponent of -3/4. However, we would usually plot the independent variable, density, on the X axis, and the dependent variable, average plant mass, an the Yaxis. Such a graph would have a slope, as we normally measure it, of -4/3. Enquist, et al. (1998), who developed these ideas, showed that the observed slope for an enormous range of plants was much closer to-4/3 than -3/2.

The Distribution of Individual Biomasses under Self-Thinning

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.

Assume that we start with a normal distribution of seed size. Typically, larger seeds germinate more rapidly (assuming that imbibation of water is not a limiting factor). The early start on growth gives them an advantage, and the hierarchy mentioned above 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. The distribution of plant weights moves slowly back in the direction of a normal distribution. In these short term studies, we’ll never get back to anything like a normal distribution. The question is whether we will be able to see the hierarchy develop, and mortality of the smallest reduce the degree of skew over these few weeks.

So that you can see the pattern of hierarchy, the figure below shows one experiment from the literature:

Methods

Seeds of Trifolium repens were planted in small flats on commercial topsoil at a moderately high density as uniformly as possible. Flats will be lightly watered on alternate days. Once seedlings are well established, plugs of fixed size will be removed from random locations within the flats on a weekly basis. Plants will be individually weighed, and density determined from the number of plants occurring within each plug. You/we will plot the relationship between mean plant weight and density and also the distribution of plant weights in samples for each date. In calculating mean plant weight, you/we will also determine how skewness in the distribution of weights changes over time.

Bibliography

Weiner, J. and O.T. Solbrig. 1984. The meaning and measurement of size hierarchies in plant populations. Oecologia, 61:334-336.

Enquist, B.J., J.H. Brown and G.B. West. 1998. Allometric scaling of plant energetics and population density. Nature, 395:163-165.