We proceed, therefore, by examining possible bases for the general trend, and then enquiring why different species or populations might display their own variations on this common theme. Two broad types of explanation for the trend have been proposed. The first (and for many years the only one) is geometric; the second is based on resource allocation in plants of different sizes.

The geometric argument runs as follows. In a growing cohort of plants, as the mass of the population increases, the leaf area index (L, the leaf area per unit area of land) does not keep on increasing. Instead, beyond a certain point, it remains constant irrespective of plant density (N). It is, in fact, precisely beyond this point that the population follows the dynamic thinning line. We can express this by writing:

where X is the mean leaf area per surviving plant. However, the leaf area of individual plants increases as they grow, and so too therefore does their mean, X. It is reasonable to expect X, because it is an area, to be related to linear measurements of a plant, such as stem diameter, D, by a formula of the following type:

where a is a constant. Similarly, it is reasonable to expect mean plant weight, P, to be related to D by:

where b is also a constant. Putting Equations 5.26-5.28 together, we obtain:

This is structurally equivalent to the -3/2 power law in Equation 5.23, with the intercept constant, c, given by b(L/a)3/2.

It is apparent, therefore, why thinning lines might generally be expected to have slopes of approximately -3/2. Moreover, if the relationships in Equations 5.27 and 5.28 were roughly the same for all plant species, and if all plants supported roughly the same leaf area per unit area of ground (L), then the constant c would be approximately the same for all species. On the other hand, suppose that L is not quite constant for some species (see Equation 5.26), or that the powers in Equations 5.27 and 5.28 are not exactly 2 or 3, or that the constants in these equations (a and b) either vary between species or are not actually constants at all. Thinning lines will then have slopes that depart from -3/2, and slopes and intercepts that vary from species to species. It is easy to see why, according to the geometric argument, there is a broad similarity in the behavior of different species, but also why, on closer examination, there are variations between species and no such thing as a single, 'ideal' thinning line.

Furthermore, contrary to the simple geometric argument, the yield-density relationship in a growing cohort need not depend only on the numbers that die and the way the survivors grow. We have seen (see Section 5.10) that competition is frequently highly asymmetric. If those that die in a cohort are predominantly the very smallest individuals, then density (individuals per unit area) will decline more rapidly as the cohort grows than it would otherwise do, and the slope will be shallower, especially in the early stages of self-thinning. This idea is supported by a comparison of self-thinning in normal Arabidopsis thaliana plants with self-thinning in mutants that overexpress phytochrome A, greatly reducing their shade tolerance, and making competition amongst them more asymmetric (Figure 5.33a).

It seems possible, too, to use departures from the assumptions built into Equations 5.26-5.29 to explain at least some of the variations from a 'general' -3/2 rule. Osawa and Allen (1993) estimated a number of the parameters in these equations from data on the growth of individual plants of mountain beech (Nothofagus solandri) and red pine (Pinus densiflora). They estimated, for instance, that the exponents in Equations 5.27 and 5.28 were not 2 and 3, but 2.08 and 2.19 for mountain beech, and 1.63 and 2.41 for red pine. These suggest thinning slopes of -1.05 in the first case and -1.48 in the second, which compare quite remarkably well with the observed slopes of -1.06 and -1.48 (Figure 5.33b). The similarities between the estimates and observations for the intercept constants were equally impressive. These results show, therefore, that thinning lines with slopes other than -3/2 can occur, but can be explicable in terms of the detailed biology of the species concerned - and that even when slopes of -3/2 do occur, they may do so, as with red pine, for the 'wrong' reason (-2.41/1.63 rather than -3/2).

Figure 5.33 (a) The relationship between total biomass and density in two wild-type strains of Arabidopsis thaliana plants (□ and ■) and a phytochrome A overexpressing mutant strain (•) 15, 22 and 33 days after sowing (bottom to top). Data points give means (±1 SE, n = 3). In each case, the strains were sown at two initial densities; solid black regression lines are shown in each case for the higher density. The steeper dotted black line has a slope of -1/2 (indicative of -3/2 self-thinning) and the shallow line a slope of -1/3 (indicative of -4/3 self-

thinning). Model trajectories for asymmetric (-) and symmetric (---) competition are also shown. The mutant strain exhibited shallower thinning lines, indicative of more asymmetric competition. (After Stoll et al., 2002.) (b) The species boundary line for populations of red pine, Pinus densiflora (slope = -1.48) from northern Japan. (After Osawa & Allen, 1993.)

complications of the geometric argument

Figure 5.33 (a) The relationship between total biomass and density in two wild-type strains of Arabidopsis thaliana plants (□ and ■) and a phytochrome A overexpressing mutant strain (•) 15, 22 and 33 days after sowing (bottom to top). Data points give means (±1 SE, n = 3). In each case, the strains were sown at two initial densities; solid black regression lines are shown in each case for the higher density. The steeper dotted black line has a slope of -1/2 (indicative of -3/2 self-thinning) and the shallow line a slope of -1/3 (indicative of -4/3 self-

thinning). Model trajectories for asymmetric (-) and symmetric (---) competition are also shown. The mutant strain exhibited shallower thinning lines, indicative of more asymmetric competition. (After Stoll et al., 2002.) (b) The species boundary line for populations of red pine, Pinus densiflora (slope = -1.48) from northern Japan. (After Osawa & Allen, 1993.)

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