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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 (Figure 5.39b), very similar to the value of −0.33 predicted by the metabolic theory (and significantly different from −0.5, predicted by the areal argument). But if separate lines were fitted for each of the three initial densities, the slopes were −0.45, −0.47 and 0.50, all consistent with the areal argument and significantly different from −0.33 (Figure 5.39b). The different intercepts of the three lines (plants sown at higher initial densities had greater biomass) seemed to reflect an effect of initial density on growth form and perhaps on the degree of asymmetry in the competitive process (Li et al., 2013). This suggests, in turn, that light interception may drive patterns in individual populations while metabolic constraints set limits in a species overall.
What seems clear is that we have moved further from, not closer to, anything that could be called a self‐thinning ‘law’. But this represents progress in the important sense of acknowledging the range of forces acting on growing, competing cohorts of individuals, and recognising, too, that the details of a species’ morphology or physiology may influence the way in which those forces act and the slopes of the resulting relationships. The patterns we observe are likely to be the combined effect of a range of forces, even if in some cases one of those forces may dominate – metabolic constraints in mobile animals, light interception in many plants; light interception in individual populations, metabolic constraints in a species overall. Universal rules have their attractions but Nature is not so easily seduced.
APPLICATION 5.4 Density management diagrams
The precise nature of self‐thinning and species boundary lines, and of the forces shaping them, are important and interesting issues, but from the point of view of managing growing, crowded, single‐species cohorts, those details are arguably less important than a simple recognition of the fundamental patterns that underlie all variants of these lines – that as cohorts grow and compete, there are boundaries in density‐biomass or density‐mean size space beyond which they cannot go, and trajectories that they tend to follow. This underpins, for example, one particular approach to the exploitation of commercially important growing cohorts: the construction and use of density management diagrams (DMDs) (Jack & Long, 1996). We conclude here, therefore, by looking at the use of a DMD for Norway spruce, Picea abies, in central‐southern European montane regions (Figure 5.40), focusing on general principles rather than going into detailed calculations. Data had been compiled from France, Germany, Italy, the Czech Republic, Romania and Bulgaria on the density of trees and their average diameter in a total of 1609 plots, selected as having at least 80% Norway spruce and being markedly ‘even‐aged’ (a strongly unimodal and non‐skewed diameter distribution). These were then used to estimate a species‐boundary – in this case, a linear combination of density and individual size on a log–log plot (Figure 5.40) that encompasses all but the 2% of plots with the greatest biomass. (Using this rather than the absolute maximum biomass prevents the line being overly influenced by a few rogue populations.) Other analyses of the data were able to estimate the speed and direction with which populations are likely to proceed through this size‐density space.
We can see that the DMD encapsulates what the data can tell us about self‐thinning in Norway spruce. The use of the DMD in managing any particular Norway spruce population (or type of population) then proceeds as follows: (1) the starting position of the population on the DMD is identified; (2) the target position is also identified, along with the likely trajectory to it in an unmanaged population; (3) the trajectory and time‐scale of an alternative route to the target is estimated, usually based on a managed reduction in density, designed to prevent or delay the onset of competition‐related mortality (Vacchiano et al., 2013). In this case, examples include managing future timber production (illustrated in Figure 5.40), ensuring mechanical stability against wind damage (where susceptibility is greatest in stands with slender trees that arise when the intensity of competition is high), enhancing the protective powers of stands against avalanches and rockfalls (similar to windfirmness, but with the additional need to avoid gaps between trees that may ‘release’ an avalanche), and minimising vulnerability to spruce bark beetle attack (achieved by delaying canopy closure).
For the timber production in Figure 5.40, the target tree diameter, for commercial purposes, is 40 cm. From an initial population of trees 10 cm in diameter, growing at
