Density (plants/m 2 )

Figure 2.4. Analysis of a model with three constraints on population growth. Parameters go, m, f, and p are as in Figure 2.2, while Bm~x

=

105, Wmax

=

108, and el through e5 all equal 2.

Dotted l1nes are dB/dt

=

0 isoclines for each constraint considered independently. The maximum individual weight, the carrying capacity, and the spatial constraint are labelled "wmax", "Bmax", and

"f,p", respectively. The dashed line is the true isocline for all three constraints and the solid line is a model solution for 450 years. Regions I, II, and III are, respectively, portions of the plane where model behavior is dominated by the spatial

constraint, the carrying capacity, and the maximum individual weight.

including forestry (Reinke 1933, Curtis 1971) and plant ecology (Whittaker and Woodwell 1968), and such relationships provide the basis for widely used methods of nondestructive sampling of plant populations (see references in Hutchings 1975). When traced to this rather commonplace origin, the power equation form of the

self-thinning rule is unremarkable.

The slope of the model thinning line is determined only by the power, p, of the area-weight relationship and equals the classic thinning rule slope of

B =

-l/2 only if p

=

1/3. Plant growth is

typically not isometric (Mohler et al. 1978, Furnas 1981, White 1981} sop is not generally l/3, and the model predicts that thinning slopes should vary systematically with plant allometry.

This hypothesis has not been supported in previous reviews and experimental tests {Westoby 1976, Mohler et al. 1978, White 1981), so the causal mechanisms formalized in the model developed here have been discredited as possible explanations for the self-thinning rule. However, some new analyses presented in Chapters 5 and 6 show that measured self-thinning slopes do vary significantly from the idealized value of the thinning rule, and that these deviations can be correlated with differences in plant allometry, thus verifying a major prediction of the model developed here.

Even for this very simple model, the exact value of the self-thinning constant, K, depends on all the model parameters and can not be interpreted as a function of a single measurement, such as biomass density in space (as discussed in White 1981 and Lonsdale and Watkinson 1983a). Biomass density in space is a component of

the parameter f, but its relationship to log K is confounded by other components off, such as initial plant shape and the degree of tolerable overlap between plants, and by the dependence of log K on the allometric power p. Since thinning slopes do vary with plant allometry (Chapter 6), direct comparisons of thinning intercepts of experimental thinning lines are not clearly interpretable because the comparison of the constants among power relationships is not meaningful unless the relationships have the same power {White and Gould 1965). Experimental interpretation of K will, then, require a careful statistical analysis relating K to several factors,

including plant allometry, initial plant shape, the density of biomass per unit of occupied space, and some measure of allowable overlap between plants, such as shade tolerance.

The position of the model self-thinning line is also affected by the rate constants g0 and m, but these effects are relatively

small over a large range of reasonable parameter values. The primary effect in the log B-log N plane of g0 and m is to

determine the rate and direction of approach toward the thinning line. By concentrating on the thinning line, these rate dynamics are deemphasized in favor of a focus on the limitations imposed by available growing space and plant geometry.

The model also predicts that the parameters of the

self-thinning line are not species constants invariant to changes in all environmental factors except illumination (as proposed by Yoda et al. 1963, Hickman 1979, Hozumi 1980, White 1981, and Hutchings 1983). Any factor that could affect the density of biomass in

occupied space or the degree of allowable overlap between plants would also affect the thinning constant K, while environmental factors affecting plant allometry would also change both K and the self-thinning slope. The remarkable abilities of plants to vary their sizes, shapes, canopy densities, etc. in response to

environmental and competitive factors are well-documented (see references in Harper 1977), and allometric relationships for a species also vary significantly among sites (Hutchings 1975). Peet and Christensen (1980) reported that the thinning characteristics of a pine stand could be permanently altered if the initial density is very high and speculated that some aspects of tree geometry were fixed by the initial growing conditions. Such permanent effects of initial density would be another cause for variation in thinning line parameters among populations of a species. The prediction that thinning line parameters are not species constants is tested in Chapters 4 though 6.

Enhanced Model

The model with additional constraints on total biomass and individual plant weight can explain most general features of population dynamics in the log B-log N plane. The four phases of population growth that were discussed in Chapter l and diagrammed in Figure 1.1, page 9 are reproduced by the model in Figure 2.4, with the sharp corners of idealized behavior replaced by gradual

transitions from one growth phase to another. The four phases are explained by the model as the sequential operation of three

..

"

constraints on population growth: the spatial constraint, the carrying capacity, and the maximum individual size. The transition from adherence to the self-thinning rule to constant biomass at the carrying capacity, shown by movement from region I to region II of Figure 2.4, has been discussed by White and Harper (1970) and Hutchings and Budd (198la). Experimental evidence of this transition has been reported by Schlesinger and Gill (1978), Lonsdale and Watkinson (1983b), Watkinson (1984), and Peet and Christensen (1980). The lasi of these papers also reported that thinning trajectories eventually become even less steep than the constant biomass line, as shown in the transition from region II t6 '

region III in Figure 2.4.

The modified model can also explain the ambiguous results of experiments comparing the self-thinning lines of deeply shaded populations to those of populations grown under better

illumination. Some investigators have concluded that reduced illumination lowers thinning lines and shifts the slope in the log B-log N plane from -1/2 to 0 (White and Harper 1970, Kays and Harper 1974, Lonsdale and Watkinson 1982), while others have

observed no change in slope when illumination is decreased (Westoby and Howell 1981, Hutchings and Budd 198lb).

The enhanced model (equations 2.20 and 2.21) is used here to investigate the effects of decreased illumination on the log B-log N trajectory, but the constraint on individual weight is eliminated to simplify the analysis. It is unlikely that plants would reach their greatest potential size during short experiments under reduced

illumination, so this constraint would be irrelevant. A reduction in illumination could shift the positions of the two remaining constraining lines in the log B-log N plane. As a first

approximation, a certain percentage reduction in illumination might be expected to reduce the carrying capacity by the same percentage.

It is more difficult to estimate the effect of a light reduction on the spatial constraint line. Lower illumination can stimulate

plants to emphasize height growth rather than radial growth (Harper 1977), so that the allometric power p relating area covered to weight might decrease, resulting in a steeper thinning line.

However, illumination changes may also alter the density of biomass in occupied space and change the thinning intercept (Lonsdale and Watkinson 1982). Since species differ widely in their shade

tolerances, meristem placements, and other important physiological and morphological factors, the actual effect on the spatial

constraint line should vary among species. Regardless of these particulars, the carrying capacity will become the dominant constraint over a wider range of plant densities under reduced illumination as long as the spatial constraint line does not drop too drastically.

The consequences for the observed log B-log N trajectory of a drop in total supportable biomass with reduced illumination are shown in Figure 2.5. In this particular example, the carrying capacity drops by one log unit when illumination is reduced by 90%, but the spatial constraint line is not affected. With the light reduction, the lowered carrying capacity dominates model behavior

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