Ecological Modelling, 45 (1989) 237-242 237 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

THE ' 3 / 2 POWER LAW': A C O M M E N T O N T H E S P E C I F I C

C O N S T A N C Y O F K

LUiS SOARES BARRETO

lnstituto Superior de Agronomia, Tapada da Ajuda, 1399 Lisboa Codex (Portugal)

(Accepted 18 November 1988)

ABSTRACT

Barreto, L.S., 1989. The ' 3 / 2 power law': a comment on the specific constancy of K. Ecol. Modelling, 45: 237-242.

The author analyses the specific constancy of K, the constant in the ' 3 / 2 power law'. To support his analysis, a new mathematical expression for K is established and used. He concludes that K, for a given species, varies with site quality: the better the site quality, the higher is K. He also states that what characterizes the ' 3 / 2 power law' is the specific constancy of the ratios: initial density (when the stand enters the "3/2 power line')/final or asymptotic density; initial standing volume/final standing volume: initial mean tree volume/ final mean tree volume; and the properties derived from this constancies. In a brief epistemological comment, the author discusses the apparent determinism implicit in the ' 3 / 2 power law'.

INTRODUCTION

I n th is p a p e r w e wi l l a p p r o a c h t h e c o n s t a n c y o f K - t he c o n s t a n t in t he

' 3 / 2 p o w e r l aw ' - fo r a g i v e n spec ie s . F o r th is p u r p o s e w e wil l e s t a b l i s h a

n e w e q u a t i o n fo r K a n d we wi l l u se s i m u l a t e d d a t a f r o m a p r e v i o u s p a p e r

( B a r r e t o , 1988e) . W e wi l l t r y to s h o w t h a t K c h a n g e s w i t h d e n s i t y . In

n a t u r a l s t a n d s , h i g h e r d e n s i t i e s ( l o w e r s i te q u a l i t y ) a r e a s s o c i a t e d w i t h l o w e r

v a l u e s o f K .

THE ' 3 /2 POWER LAW'

A b r i e f a n d e l e g a n t w a y to e s t a b l i s h t h e ' 3 / 2 p o w e r l a w ' is b y u s i n g the

m e t h o d s o f d i m e n s i o n a l a n a l y s i s . I f L is t he l i n e a r d i m e n s i o n , v the m e a n

t ree v o l u m e , a n d p t h e n u m b e r o f t r ee s ( d i m e n s i o n l e s s ) p e r a r e a un i t , t h e n :

[v] = C (1)

[ p ] = L 2 (2)

0304-3800/89/$03.50 © 1989 Elsevier Science Publishers B.V.

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If we admit that there is a relationship between v and p, assuming the form:

v = K p x (3)

we have:

- 2 x = 3 (4) x = - 3 / 2 (5)

and then we have the equation for the ' 3 / 2 power law':

Vp 3/2 = K (6)

Equation (6) can be written as:

Vp '/2 = K (7)

where V ( = vp) is the standing volume. An alternative deduction of equation (7) can be found in Barreto (1988g).

The ' 3 / 2 power law' implies that the intraspecific competit ion became dominant and the trees are not affected by interspecific competition. But if they are, we will find (equation 3):

v = C p ( l ) a p ( 2 ) b . . . p ( n ) " (8)

where C is a dimensionless constant and the p ( i ) ' s are the densities of the several plant populations present, including the trees.

From dimensional analysis and equation (8) we obtain:

a + b + c + --- + n = - 3 / 2 (9)

It is not possible to obtain the values of a, b . . . . . n from equation (9). The ' 3 / 2 power law' is only deductible when intraspecific competit ion is dominant. After the trees get taller they are no longer competing for space and light with other plants and, probably, ecological isolation mechanisms occur to separate the niches, attenuating interspecific competition.

We elaborated an ecological interpretation of the ' 3 / 2 power law' in a previous booklet. We proved that the ' 3 / 2 power law' is equivalent to saying that the final (asymptotic) density, p ( f ) , and the stand mean height are affected by the same factors that globally express the site quality (Barreto, 1987). This may be expressed by a mathematical relationship between the stand height at age t(0) (when it enters the ' 3 / 2 power law') h(0) and the final density, p ( f ) . Let us assume that the mathematical form of the relationship is as follows:

Q = h(O) p ( f ) (10)

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From equation (10), the ' 3 / 2 power law' (equation 6) can be deduced (Barreto, 1987). Khilmi (1962) admitted also that Q, as K, is constant for a given species.

AN ANALYSIS OF THE ADMITTED CONSTANCY OF K

Let us write the logarithmic form of the ~3/2 power law', as it is generally graphed in the In v- ln p space:

l n v = l n K - 1 . 5 1 n p (11)

The main criticism of the ' 3 / 2 power law' had been focused on the value of the power of p, in equation (6) (e.g. Zeide, 1987), this is the regression coefficient in equation (11). We think this is a wrong target. A value difference of 1.5 only means that (due to man-made or natural causes) the stand had moved along more than one ' 3 / 2 power line', during its life. In our opinion what deserves to be scrutinized is the admitted constancy of K for each species.

It was shown (Barreto, 1987) that we can follow the trajectory of a stamt that moves along more than one ' 3 / 2 power line', In the space In l , - In p, when the stand ' jumps ' from one ' 3 / 2 power line' to another, the next line has, obviously, an interception point with the In v axis that is different from the previous one, that is, In K ( K ) had changed to a lower value.

Given h the average tree height, H stand dominant height and d the average tree DBH, the following relationships can be stablished (Barreto, 1988e):

hp ' /2= C( h ) (12)

Up '/2 = C( H) (13)

dp ~/2 = C( d ) (14)

where C( . ) are constants. If:

v = d2hf× 3.14159/4 (15)

where f is the form factor, then:

K = p3/Zd2hF (16)

K=p3/2F C(h) p-1/2 C(d) 2 p - , (17)

K = F C ( h ) C ( d ) 2 (18)

where F = 3.14159/4 f .

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T A B L E 1

Mar i t ime pine: values of constants C(s)

SQ C(h) C(H) C(d) K

G r a d e D 24 409 435 5.66 4939 20 404 435 5.43 4778 16 397 435 5.09 4461 'Average silviculture' 24 480 521 6.31 7341 20 468 (0.2) 517 6.01 6896 16 467 (0.6) 512 5.58 6214

It was already shown that f is constant for a given stand following the ' 3 / 2 power line' and increases with density (Barreto, 1988b). Then, K is constant for a given species only if C(h) and C ( d ) decrease with density. In a previous paper (Barreto, 1988e) we simulated six stands of Pinus pinaster strictly following the ' 3 / 2 power law' and we calculated the values of C(h), C(H), C(d) and K. We reproduce table 3 of the referred paper in Table 1.

First, we numerically check equation (18). For Grade D, SQ 24, f - - 0.480 (Barreto, 1988b, table 1). Then, using the values in Table 1:

K = 0.480 X 409 X 5.662 X 3 .14159/4 = 4939.5

As it can be seen, in Table 1, K = 4939. Now, let us see what happens to C(h) and C(H), in Table 1. In fact, they

decrease with density. But if we admit so, we must also admit that K decreases with density (from better to poorer site quality).

If we pay attention to equation (7), it is evident that if K is constant for a given species, we only have to predict the evolution of stand density to immediatly obtain the standing volume. This is too much determining and nature's simplicity.

It is our judgement that, with the evidence available to us, the farthest we can go is to admit, for a given species, that K is constant for each site quality (the better the site quality, the higher is K) .

Also, if we quote tables Lv and LVI in Khilmi (1962, p. 128) we observe that the constancy of Q for a given species can hardly be admitted. A pattern similar to that of K is much more acceptable (Q decreases with site quality).

Thus, we are much more prone to accept that what characterizes the ' 3 / 2 power law' it is not the specific constancy of K, but, for a given species, the constancy of p(O)/p(f), V(O)/V(f) and o(O)/v(f), as we already showed (Barreto, 1987) being p(0) the density at age t(0), p(f) as previously defined, and similarly for standing volume, V, and mean tree volume, o.

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F I N A L C O M M E N T S

Comparing the orthodoxies of Forest Mensuration and the Theory of Forest Production, some of our developments may look like something aparl from the paradigm, in Kuhn's sense. Nevertheless. it is our judgment thal the present results, added to those already published (Barreto. 1987, 1988a. b, c, d, e, f) form a unified approach, ecologically sound and coherent, to the dynamics of self-thinned, even-aged, pure stands.

Our results allow us to make stand-specific predictions of several parame-- ters of growth that are consistent with the basic relevant concepts of ecology, namely those associated to a population under intensive intraspe-- cific competition.

Now, we have available a coherent body of theory that can support and clarify the analysis of empirical data (sometimes contradictory) and the scheduling of silvicultural operations.

As can be seen, the cornerstone of our theoretical findings are the constancy of p(O)/p(f), V(O)/V(f) and t,(O)/c(f) for a given species, established in our seminal booklet (Barreto. 1987), in this line of research.

This constancy and others that it implies, apparently could put the dynamics of self-thinned even-aged, pure stands inside a very tight determin- istic jacket. We do not agree with this position, for two main reasons.

As there are other causes of mortality (due to natural stochastic events) the stand can move to another ' 3 / 2 power line' and have a different trajectory. On the other hand, although there is a highly probable trajectory for the stand as a whole, the individual trajectories of the trees are highly unpredictable, from the beginning of the ' 3 / 2 power line'.

In a certain sense, Heisenberg's 'principle of uncertainty" is also applica- ble to self-thinned, even-aged, pure stands. The predictions we made using the implications of our results based in the ' 3 / 2 power law' may be seen as 'probabil ist ic laws', as in quantum theory.

As, due to discontinuities introduced by quantum theory, an orbit of an electron can be regarded as a series of detached positions and not a continuous line, the ' 3 / 2 power line' must be not seen as a continuous trajectory of the stand but as a series of the most frequent and probable positions that the stand occupies during its life. Some events can deviate the stand from the ' 3 / 2 power line' for a while (discontinuities) but later, if its 'ampli tude ' in Westman's (1985) sense is not exceeded due to homeostasis, that is, the necessity to take advantage of all available resources by individu- als striving under intensive intraspecific competition, the stand returns to the same ' 3 / 2 power line'. For this reason, the constancy of basal area, relative spacing and LAI during the stand's life is theoretically relevant.

If we accept the comments we had just issued, namely ' the principle of

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uncer ta in ty ' appl ied to ecological systems, we must regard the s imulat ion of

forests using the ' t r ee model ' a p p r o a c h with some caut ion. Also, the predic t -

ions of s tand models with a very high tempora l resolut ion (annual values, as

in mode l Z I C A - 8 6 - Barreto, 1987) must be regarded caut ionari ly.

REFERENCES

Barreto, L.S., 1987. Um novo m&odo para elabora~ao de tabelas de produ~ao. Aplica~ao ao pinhal. Servi~o Nacional de Parques, Reservas e Conserva~ao da Natureza, Lisboa, 52 pp.

Barreto, L.S., 1988a. The basal area of stands following the '3 /2 power law'. Centro de Estudos Florestais, INIC, Lisboa, 3 pp.

Barreto, L.S., 1988b. The form factor of stands following the '3 /2 power law'. Centro de Estudos Florestais, INIC, Lisboa, 4 pp.

Barreto, L.S., 1988c. Current increments of stands following the '3 /2 power law'. Lisboa: Centro de Estudos Florestais, INIC, Lisboa, 3 pp.

Barreto, L.S., 1988d. The maximum current increment of stands following the '3 /2 power law'. Centro de Estudos Florestais, INIC, Lisboa, 3 pp.

Barreto, L.S., 1988e. The relative spacing of stands following the '3 /2 power law'. Centro de Estudos Florestais, INIC, Lisboa, 11 pp.

Barreto, L.S., 1988f. Stands following the '3 /2 power law'. Maximum mean annual increment and LAI. Centro de Estudos Florestais, INIC, Lisboa, 5 pp.

Barreto, L.S., 1989g. Model SANDRA-88 to predict the growth of populations following the '3 /2 power law'. Centro de Estudos Florestais, INIC, Lisboa, 4 pp.

Khilmi, G.F., 1962. Theoretical Forest Biogeophysics. National Science Foundation and Department of Agriculture, Washington, DC, 155 pp.

Westman, W.E., 1985. Ecology, Impact Assessment, and Environment. Wiley, New York, 532

PP. Zeide, B., 1987. Analysis of the 3/2 power law of self-thinning. For. Sci., 33: 517-537.