Figure 5.36 Crowded plant populations typically approach and then track self‐thinning lines. Self‐thinning in Lolium perenne sown at five densities: 1000 (), 5000 (), 10 000 (), 50 000 () and 100 000 () ‘seeds’ m⁻², in (a) 0% shade, where the observations after 35 days are circled in red, and (b) 83% shade. The lines join populations of the five sowing densities harvested on five successive occasions. They therefore indicate the trajectories, over time, that these populations would have followed. The arrows indicate the directions of the trajectories, i.e. the direction of self‐thinning. For further discussion, see text.

      Source: After Lonsdale & Watkinson (1983).

      We can see that mean plant weight at a given age was always greatest in the lowest density populations (illustrated, for example, after 35 days (circled points) in Figure 5.36a). It is also clear that the highest density populations were the first to suffer substantial mortality. What is most noticeable, though, is that eventually, in all cohorts, density declined and mean plant weight increased in unison: populations progressed along roughly the same straight line. The populations are said to have experienced self‐thinning (i.e. a progressive decline in density in a population of growing individuals), and the line that they approached and then followed is known as a dynamic thinning line (Weller, 1990).

      The lower the sowing density, the later was the onset of self‐thinning. In all cases, though, the populations initially followed a trajectory that was almost vertical, reflecting the fact that there was little mortality. Then, as they neared the thinning line, the populations suffered increasing amounts of mortality, so that the slopes of all the self‐thinning trajectories gradually approached the dynamic thinning line and then progressed along it. Note also that Figure 5.36 has been drawn, following convention, with log density on the x‐axis and log mean weight on the y‐axis. This is not meant to imply that density is the independent variable on which mean weight depends. Indeed, it can be argued that the truth is the reverse of this: that mean weight increases naturally during plant growth, and this determines the decrease in density. The most satisfactory view is that density and mean weight are wholly interdependent.

      ’the –3/2 power law’

      Plant populations (if sown at sufficiently high densities) have repeatedly been found to approach and then follow a dynamic thinning line. For many years, all such lines were widely perceived as having a slope of roughly −3/2, and the relationship was often referred to as the ‘−3/2 power law’ (Yoda et al., 1963 ; Hutchings, 1983), since density (N) was seen as related to mean weight by the equation:

(5.22)

      or:

(5.23)

      where c is constant. (In fact, as we shall see, this is even further from being a universal law than the ‘law’ of constant final yield, discussed previously.)

Note, however, that there are statistical problems in using Equations 5.22 and 5.23 to estimate the slope of the relationship in that is usually estimated as B/N, where B is the total biomass per unit area, and so and N are inevitably correlated, and any relationship between them is, to a degree, spurious (Weller, 1987). It is therefore preferable to use the equivalent relationships relating overall biomass to density, lacking autocorrelation:

(5.24)

      or:

(5.25)

      5.9.2 Species and population boundary lines

      If we now turn to the third type of study, listed in Section 5.9, we can note that although, again, combinations of density and mean weight for a particular species are being used to generate a relationship between them, it is not a single cohort that has been followed over time, but a series of crowded populations at different densities (and possibly different ages). In such cases, it is more correct to speak not of a thinning line but of a species boundary line – a line beyond which combinations of density and mean weight appear not to be possible for that species (Weller, 1990). Indeed, since what is possible for a species will vary with the environment in which it is living, the species boundary line will itself subsume a whole series of population boundary lines, each of which defines the limits of a particular population of that species in a particular environment (Sackville Hamilton et al., 1995).

      dynamic thinning and boundary lines need not be the same

      Thus, a self‐thinning population should approach and then track its population boundary line, which, as a trajectory, we would call its dynamic thinning line – but this need not also be its species boundary line. The light regime, soil fertility, spatial arrangement of seedlings, and no doubt other factors may all alter the boundary line (and hence the dynamic thinning line) for a particular population (Weller, 1990 ; Sackville Hamilton et al., 1995). Soil fertility, for example, has been found in different studies to alter the slope of the thinning line, the intercept, neither, or both (Morris, 2002).

      thinning slopes of −1

      The influence of light, in particular, is worth considering in more detail, since it highlights a key feature of thinning and boundary lines. A slope of roughly