CHAPTER 7

THE OVERALL SIZE-DENSITY RELATIONSHIP Introduction

This cnapter considers the overall relationship between size and density among stands of different species and evaluates its relevance to the self-thinning rule. This relationship has been observed by plotting log

w

against log N for stands of species

ranging from herbs to trees (Gorham 1979), and by plotting

self-thinning lines for many species on a single log w-log N plot (White 1980). In both cases, the data are grouped around the line log

w =

-3/2 log N + 4. This trend is referred to here as the overall relationship to distinguish it from the self-thinning lines of individual populations.

The existence of the overall relationship has been interpreted as strong evidence for the self-thinning rule. It has been claimed to show that: (1) The model for intraspecific thinning of Yoda et al. (1963f also applies to interspecific weight-density

relationships (Gorham 1979). (2) The same self-thinning law applies to all plants (Hutchings and Budd 198la, Hutchings 1983). (3) The slope and position of the thinning line are insensitive to plant geometry (Furnas 1981, Hutchings and Budd 198la). (4) The empirical generality of the self-thinning rule is beyond question (White

1981). The causes of the overall relationship are examined here to determine if the existence of the relationship actually supports such interpretations.

Models of the Overall Size-density Relationship Heuristic Model

The overall relationship between size and density can be deduced from basic principles using a heuristic model for

fully-stocked stands of different plant species. Assume that all the plants in a given stand are the same size and shape and that each plant occupies a cylindrical volume of space called its volume of influence (VOl). Plant weight and the volume and dimensions of this cylinder are related by

w = v d = (n R2 h) d (7. 1)

where w is plant weight (in g), vis the size of the VOl (in m3), d is the density of plant material in the VOl (in g/m3), and R and h are the radius and height, respectively, (in m) of the VOl. The ground area occupied by each plant is n R2 and if all ground

area is covered the density of plants per ~² is simply

(7.2) The shape of the plants in a stand can be represented by the height to VOl radius ratio, T

=

h/R. Since h

=

^T R, equation 7.1 can be rewritten as

(7.3) which can be combined with equation 7.2 to give

w

=

[T d I In] N- 312 (7.4) for a particular stand.

If plants in all stands had exactly the same values of T and d, then the overall relationship among all stands would reduce to

w

⁼ K N- 312 where K = T d I ln. Actually, ^T and d will vary

among stands. Gorham•s (1979) data can be used to estimate a range for T. Assume that VOis are cylindrical and that the base area,

a,

of a VOI is liN, then substitute and solve

a =

n R2 to obtain

R =

^(TI N)- 1

1

2• The estimated

R

values can then be divided into the reported average heights to yield T, which ranges from 2 to 12 for Gorham•s 65 stands. Measurements of biomass per unit of canopy volume can provide estimates of d, the density of biomass in

occupied space. For trees, the measurements range from 600 to 40000 glm 3, with most falling between 4000 and 13000 glm3 (White 1980). Lonsdale and Watkinson (1983a) reported values between 1500 and 5200 glm³ for three herbaceous species. Combining the ranges for T and d with the log-transformed version of equation 7.4,

log

w

^{= -312 log N +} log(T d I In) {7.5) yields equations for the minimum and maximum average weights

possible at any plant density. The minimum values of 2 and 600 for T and d, respectively, give log w

=

-312 log N + 2.83, while the respective maximum values of 12 and 40000 give the parallel line log

w =

-312 log N ⁺ 5.43. These two lines define a region of the log w-log N plane enclosing all fully-stocked stands of identically

sized plants. Gorham's {1979) line, log

w =

-1.49 log N + 3.99, is parallel to the boundary lines in the approximate center of the enclosed region. The principal axis line through a random sample of

points distributed uniformly between the model boundary lines would have a slope of -3/2 and would be near Gorham's line.

Improved Model

The assumption that all plants in each stand are the same size and shape is clearly invalid, so this section analyzes a more

complex model which represents each stand as a population with a lognormal distribution of individual weights. Within each stand, base area is represented by a simple power function of weight.

Lognormal weight distributions have been repeatedly observed in plant stands (Koyama and Kira 1956, Obeid et al. 1967, White and Harper 1970, Hutchings and Budd 198la). Power functions relating plant dimensions to weight have been validated and applied by both ecologists (Whittaker and Woodwell 1968, Hutchings 1975) and

foresters (Bruce and Schumacher 1950). The weight distribution of a stand is specified by the lognormal distribution parameters ~ and a, and the mean weight is

(Johnson and Katz 1970). The area occupied by an individual is obtained by assuming that all plants fill a cylindrical VOl to a density of d g/m3, and that the height to radius ratio of a plant

of average weight is T. Solving w

=

n

R

^{3 T} d for

R

then gives the base radius of a plant of average weight

R_

= [ w J

{1/3)

W TI T d '

and the base area of the plant of average weight is simply n

R

²

Across an entire stand, R is related to w by

where c and p are constants, so that base area a is

( 7. 7)

(7.8)

(7.9) Equations 7.7 and 7.9 can now be combined to fix the value of c at c

=

Ia I (In wp) and yield a general formula for the base area of any plant

2p 2

which simplifies to a ⁼ c1 w , where c1

=

n c • Since

ln w ii normally distributed with mean ~ and standard deviation a (Johnson and Katz 1970) and c1 and p are constants,

ln a

=

ln (c1 w2P)

=

_{ln c1} ⁺ 2 p ln w is normally distributed with mean ln c1 ⁺ 2 p ^~ and standard deviation 2 p a. The expected value of a is

(7.11)

which simplifies to

(7. 12) Assuming N

=

1 I

a,

equation 7.12 gives

(7. 13) the plant density for a stand at 100% cover specified by parameters ,, a, d, and p.

The overall log w-log N relationship for this model was

estimated by using Monte Carlo techniques to select parameters for 500 plant stands, then calculating the resulting densities

(equation 7.13) and fitting a relationship between log

w

and log N.

Parameters ' and a were actually derived from two parameters

with more direct biological interpretations: the average weight

w

and p, which measures the range of weights within a population.

Individual weights in a crowded plant population commonly span about two orders of magnitude (White 1980). This information was applied to the distribution of ln w by assuming that the ninety-ninth

percentile weight is approximately ^p times the first percentile value, where p ranged from 10 to 1000 to allow for deviations from the reported value of 100. Since ln w is normally distributed, the ninety-ninth percentile value is 4.652 standard deviations above the first percentile value and e4•652a

=

^p. With this information, ' and a can be calculated sequentially from a= p/4.652 and

' =

ln

w-

0.5 a2•

Values of the input parameters log T, p, log d, log

w,

and log p were chosen from uniform random distributions on the

intervals given in Table 7.1. Statistics for the 500 derived values

of~' a,

R, h, a,

N, and Bare given in Table 7.2, along with values of a constant ^K calculated from ^K ~ B

IN.

Since

w

was chosen ~priori and random parameter variations were imposed in deriving N,

w

is a truly independent variable and the correct method for estimating an overall linear relationship between log wand

log N is regression of log N against log w. The regression line was log N ~ -0.654 log w + 2.799 (r 2 ~ 0.96, P < 0.0001, 95% CI for

slope~ [-0.67, -0.64]), which can be rearranged to give

log

w

^~ -1.53 log N + 4.28 {95% CI for slope~ [-1.56, -1.49]), or equivalently, log B ~ -0.53 log N + 4.28 (95% CI for slope ~

[-0.56, -0.49]). Figure 7.1 shows the 500 hypothetical plant stands and this fitted relationship after transformation to log B-log N form to facilitqte comparison with real plant data.

Discussion

The models show that the linear overall relationship between log size and log density results from the simple geometry of packing three-dimensional objects onto a two-dimensional surface. The slope of -3/2 in the log w-log N plane follows directly from assuming a cylindrical shape for the VOI. The volume of any cylinder is a cubic function of the base radius while the base area is a function of the same radius squared, so that the ratio of these two powers is 3/2. This result is robust to changes in the basic cylindrical

Table 7.1. Parameter Ranges for Monte Carlo Analysis of the Improved Model.

Parameter limitsa Parameter M1nimum Maximum

log

w

-2.00 7.00 log ^T 0.30 1.08

p 0.05 0.45

log d 2.78 4.60 log x 1.00 3.00

Anti log 1 imitsb Minimum Maximum

0.01 107

2 12

600 40000

10 1000

aparameter values for 500 plant stands were chosen from uniform random distributions on the indicated intervals.

bThese columns give the antilogs of the limits of parameters for which uniform random distributions of the log values were used.

Table 7.2. Ranges of Variation of Variables Derived in the Monte Carlo Analysis of the Improved Model.

Percentiles

St. Min. Median Max.

Variablea Mean Dev. 0 5 50 95 100

I;; 4.94 6.07 -5.48 -4.25 4.80 14.5 15.9

(J 1.00 0.28 0.496

o.

551 1.01 1.43 1.48

R 0.657 1.21 0.00238 0.00557 0.126 3.08 8.99

log R -0.889 0.881 -2.62 -2.25 -0.90 0.49 0.95

h 3.57 6.68 0.0105 0.0278 0.598 18.1 42.0

1 og fi -0.185 0.891 -1.98 -1.56 -0.22 1.26 1.62

a

6.38 24.0 0.0000188

o.

000112 0.0598 33.6 258.7

log

a

-1.24 1. 76 -4.73 -3.94 -1.22 1.53 2.41

N 1680 5632 0.00387 0.0298 16.7 8893 53270

log N 1.24 1.76 -2.41 -1 .53 1.22 3.95 4.73

B 31110 72750 34.1 117 3927 142500 727600

log B 3.62 0.98 1.53 2.07 3.59 5.15 5.86

K 35350 44340 1042 2150 18590 127100 294400

log ^K 4.24 0.55 3.02 3.33 4.27 5.10 5.47

astatistics are calculated for 500 plant stands with input parameters chosen randomly from the ranges in Table 7.1.

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