Ecology. Michael Begon

area) will decline more rapidly as the cohort grows than it would otherwise do. 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. We see this in a study by Osawa and Allen (1993), who estimated a number of the parameters in these equations from data on the growth of individual plants from two species: 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 slopes that they observed: −1.06 and −1.48 (Figure 5.38). 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. They also show 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 5.38 The species boundary line for populations of red pine,Pinus densiflora, from northern Japan (slope = −1.48).

      Source: After Osawa & Allen (1993).

      self‐thinning in sessile animals

      Animals must also ‘self‐thin’, insofar as growing individuals within a cohort increasingly compete with one another and reduce their own density. And in the case of some sessile, aquatic animals, we can think of them, like plants, as being reliant on a resource falling from above (typically food particles in the water) and therefore needing to pack ‘volumes’ beneath an approximately constant area. It is striking, therefore, that in studies on rocky‐shore invertebrates, mussels have been found to follow a thinning line with a slope of −1.4, barnacles a line with a slope of −1.6 (Hughes & Griffiths, 1988), and gregarious tunicates a slope of −1.5 (Guiñez & Castilla, 2001). There is, however, nothing linking all animals quite like the shared need for light interception that links all plants.

      5.9.5 A resource‐allocation basis for thinning boundaries

      This need to include all types of organisms in considerations of self‐thinning is reflected in studies seeking alternative explanations for the underlying trend itself. Most notably, Enquist et al. (1998) made use of the much more general model of West et al. (1997) that we discussed in Chapter 3, which considered the most effective architectural designs of organisms. We saw there that the rate of resource use per individual, u, or more simply their metabolic rate, should be related to mean organism body mass, M, according to the equation:

(5.30)

      where a is a constant and the value ¾ is the ‘allometric exponent’.

      −4/3 or −3/2?

      They then argued that we can expect organisms to have evolved to make full use of the resources available, and so if S is the rate of resource supply per unit area and N_max the maximum density of organisms possible at this supply rate, then:

      (5.31)

or, from Equation 5.30:

      (5.32)

      But if the organisms have arrived at an equilibrium with the rate of resource supply, then S should itself be constant. Hence:

      (5.33)

where c is another constant. In short, the expected slope of a population boundary on this argument is −4/3 rather than −3/2. Similarly, the figure in biomass–density relationships (equations 5.24 and 5.25) would be −1/3 rather than −1/2.

Enquist and colleagues themselves considered the available data to be more supportive of their prediction of a slope of −4/3 than the more conventional −3/2, though this had not been the conclusion drawn from previous data surveys. Indeed, as we saw in Chapter 3, the idea of ¾ being a consistent or universal allometric exponent, and a slope of −4/3 therefore being expected, has itself been called increasingly into question (e.g. Glazier, 2005). Nonetheless, Begon et al. (1986) had found a mean value of −1.29 (close to a value of −4/3) for self‐thinning in experimental cohorts of grasshoppers, and Elliott (1993) a value of −1.35 for a population of sea trout, Salmo trutta, in the English Lake District. On the other hand, studies on house crickets, Acheta domesticus, have shown that the allometric exponent in in their case is not ¾ but 0.9 (Jonsson, 2017), and the estimated slope of their self‐thinning line −1.11 (the exact reciprocal of the allometric exponent; Figure 5.39a).

Figure 5.39 Self‐thinning lines vary in their support for the metabolic theory. (a) Self‐thinning in house crickets, Acheta domesticus, plotting mean weight against density on log scales. Replicate populations were established with between five and 80 newly hatched nymphs and followed until all survivors had hatched into adults. Lines join points from the same replicate, with the exception of the regression line fitted to self‐thinning populations from the three highest densities only (slope ± 95% CI, −1.11 ± 0.05). (b) Self‐thinning in common buckwheat, Fagopyrum esculentum, plotting log biomass against log density. Three initial densities, 8000 (green), 24 000 (blue) and 48 000 (red) individuals m^–2 were harvested after 22, 32, 42, 54 and 64 days. The dashed line is fitted to all data combined (slope, 95% CI: −0.38 (−0.30 to −0.47)), and the solid lines are fitted to the individual initial densities (slopes, 95% CIs: −0.45 (−0.36 to −0.55), −0.47 (−0.40 to −0.55) and −0.50 (−0.43 to −0.59) for 8000, 24 000 and 48 000, respectively).

      Source: (a) After Jonsson (2017). (b) After Li et al. (2013).

Moreover, when experimental populations of common buckwheat, Fagopyrum esculentum, were grown at a range of densities, the best estimate for the slope of the biomass–density relationship overall was −0.38 ( Скачать книгу