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ADVANCES IN ECOLOGICAL RESEARCH, VOLUME 14 The Self-Thinning Rule MARK WESTOBY I. Introduction . 11. The Significance of the Rule 111. Present Knowledge of the Rule . . A. Summary History of the Rule . B. Relation to the Competition-Density (C-D) Effect C. Present Empirical Basis of the Rule . A. The Dimensional Argument . B. Dimensional Prediction of Exceptions C. Allometric Relations . D. GrowthModels . V. Relation to Population- Structure . IV. Explanations of the Rule . . VI. Problems for Research . VII. Summary . References . . 167 . 170 . 175 . 175 . 177 . 179 . 203 . 203 . 204 . 206 . 208 . 209 . 218 . 219 . 220 I. INTRODUCTION The self-thinning rule describes plant mortality due to competition in crowded even-aged stands. Thinning is mortality imposed on crops; self- thinning was therefore the label applied to mortality imposed by the crop on itself. The rule is best understood with respect to a graph of log biomass (log B) per unit area vs log density (log N) of survivors (Fig. l), called a B-N diagram by Westoby (1981). The characteristic equation of the rule B = CN-’12 (1) (2) defines a straight “thinning line” of slope - 1/2 on the B-N diagram log B = log C - ‘/2 log N Since log mean weight of survivors W = B/N, an alternative formulation is the traditional W = CN-3‘2 (3) defining a thinning line of slope -3/2: 167 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved. lSBN 0-12-013914-6

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Page 1: [Advances in Ecological Research] Advances in Ecological Research Volume 14 Volume 14 || The Self-Thinning Rule

ADVANCES IN ECOLOGICAL RESEARCH, VOLUME 14

The Self-Thinning Rule MARK WESTOBY

I. Introduction . 11. The Significance of the Rule

111. Present Knowledge of the Rule . .

A. Summary History of the Rule . B. Relation to the Competition-Density (C-D) Effect C. Present Empirical Basis of the Rule .

A. The Dimensional Argument . B. Dimensional Prediction of Exceptions C. Allometric Relations . D. GrowthModels .

V. Relation to Population- Structure .

IV. Explanations of the Rule .

.

VI. Problems for Research . VII. Summary .

References .

. 167

. 170

. 175

. 175

. 177

. 179

. 203

. 203

. 204

. 206

. 208

. 209

. 218

. 219

. 220

I. INTRODUCTION

The self-thinning rule describes plant mortality due to competition in crowded even-aged stands. Thinning is mortality imposed on crops; self- thinning was therefore the label applied to mortality imposed by the crop on itself. The rule is best understood with respect to a graph of log biomass (log B) per unit area vs log density (log N) of survivors (Fig. l), called a B-N diagram by Westoby (1981). The characteristic equation of the rule

B = CN-’12 (1)

(2)

defines a straight “thinning line” of slope - 1/2 on the B-N diagram

log B = log C - ‘/2 log N

Since log mean weight of survivors W = B/N, an alternative formulation is the traditional

W = CN-3‘2 (3)

defining a thinning line of slope -3/2:

167

Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

lSBN 0-12-013914-6

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168

4

31

MARK WESTOBY

t Log N

Fig. 1. The B-N diagram. The line of slope - 1/2 is the self-thinning line. Arrows indicate the trajectories that stands follow at different biomass-density combinations. See text for further discussion.

log W = log C - 3/2 log N (4)

on a graph of log W vs log N. What happens to a plant stand can be represented as a trajectory on the

B-Ndiagram. Stands of small plants, near the bottom of the B-Ndiagram, tend to accumulate biomass until they approach the thinning line (arrows in Fig. 1). Then they suffer mortality in a relation to biomass accumulation such that they travel along the line. The thinning line therefore represents a sort of dynamical upper equilibrium condition. We are accustomed to upper equilibria N = constant, describing a carrying capacity for popula- tions described by N only, and sometimes B = constant, for populations described by B only. As is suitable for plant populations, which are inad- equately described by B alone or N alone, the carrying capacity is a joint function of B and N, Eq. (1) (Westoby, 1981). Plant stands are of course subject to reductions in B (with or without concomitant reductions in N) which are not due to processes of growth and competition in the stand; these can be considered crowding-independent disturbances, analogous to density-independent mortality in models of populations described by N alone. Following such disturbances, intrinsic growth and mortality proc- esses resume in the stand; the trajectory represented by the arrows in Fig. 1 reasserts itself from the new B-N condition of the stand.

The rule has three notable features (details and evidence are considered later in this article). (1) Time does not enter into the rule; mortality is a function only of biomass accumulation. The innumerable complexities of how species and site affect the rate of biomass accumulation over time are not dealt with. It is this exclusion that allows the rule’s beguiling simplicity. (2) Because mortality is driven by the rate of accumulation of biomass, mortality is slower when conditions for growth are worse, a counter- intuitive result. This applies both to slower growth on infertile soils (see

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THE SELF-THINNING RULE 169

Section III,C,4) and to slower net growth due to removal of biomass by clipping (Westoby and Brown, 1980) or grazing (Dirzo and Harper, 1980). (3) Not only does the thinning line have a slope of about - 112 [ - 3/2 on the basis of Eqs. (3) and (4)] for most studied species under most condi- tions, but its location on the B-N diagram (i.e., its B intercept at a given N ) varies remarkably little between species and between growing condi- tions. White (1980), summarizing literature estimates of thinning lines, found a six-fold range of intercepts, which appears very small against the background of the 10I6-fold variation in mean plant weight over which the thinning lines have been fitted (Fig. 2, and J. White, personal communi- cation).

This article has two components, which the reader will find rather dif- ferent in tone. One component is a review of what is known, what appears to be known, and what definitely is not known about the self-thinning rule. This review comes second in the article, (Sections 111-V), but can equally well be read first. In it I adopt the orthodox scientific attitude and am suspicious

lo-' 100 101 lo2 lo3 lo4 lo5 Plant density (rn-* 1

Fig. 2. The lines are visual or regression estimates of the thinning line for 31 stands of different species; the estimates were tabulated by White (1980) in his Table 2.2. Some of the forest tree lines are taken not from field data but from yield tables, i.e., smoothed interpolations of field data. From Fig. 2.9 of White (1980).

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170 MARK WESTOBY

about data and conservative toward the conclusions that can be drawn from them. The first part of the article argues that the rule shows promise of linking areas of ecological knowledge which are at present disconnected, and so may come to occupy a central position in ecology. In making this case I set aside doubts about the several features of the rule, and give a personal view of the structure of ecological knowledge, without defending that view against other possible views.

A great deal of new material has appeared since the substance of this paper was drafted in 1982, and I have done my best to include, or at least mention, new material I have become aware of during 1983. However this paper should not be taken as providing comprehensive coverage beyond 1982.

In particular, there has recently been a proliferation of theory relevant to the rule (e.g., Charles-Edwards, 1984b; Hara 1984a,b,c; Hozumi 1983; Lonsdale and Watkinson, 1983; Morris and Myerscough, 1984; Perry, 1984; Zeide, 1984). I hope one of these contenders, or some combination of them, may in due course prove to be a firm framework on which the facts can be hung for display. But for the moment these theories remain untested except as a posteriori accounts of data. In my opinion the self- thinning rule remains an empirical phenomenon rather than a body of the- ory, and this article treats it that way.

11. THE SIGNIFICANCE OF THE RULE

Ecology-biology at the population level of organization and above- has two main parts, at least as it is taught to undergraduates in English- speaking countries, and as reflected in widely used general ecology texts (Odum, 1971; Colinvaux, 1973; Ricklefs, 1980; Krebs, 1978). These are population and evolutionary ecology and ecosystem function.

Population ecology goes with evolutionary ecology because natural se- lection is a demographic process. The vigor of this field rests on two solid foundations: first, the logically necessary truth that changes in population density are the result only of births, deaths, immigrations, and emigrations; and second, that patterns in those demographic parameters, and many oth- ers, are likely to be intelligible only when seen as products of evolution. Extensions of population ecology to deal with whole communities have had equivocal success, however. The most recent high-water level was marked by some of the papers in Cody and Diamond (1975). Since 1975 we have seen a concerted attack on the evidence for large-scale structuring of com- munities by competition (Connor and Simberloff, 1979; Simberloff and Connor, 1981; Lawton and Strong, 1981; Simberloff and Boecklen, 1981).

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THE SELF-THINNING RULE 171

In addition, May (1974, 1975, 1976, 1977) has shown for models of single populations and Noy-Meir (1981) for models of three-population combi- nations that a very wide range of behavior is possible, so it is hard to imag- ine very definite predictions being made by models which represent large networks of interacting populations.

Ecosystem function-energy flow and biogeochemical cycling-aims to fill in complete budgets for particular ecosystems and to make comparative generalizations. This field is characteristically empirical, and does not have a theoretical structure unless one counts the principles of thermodynamics and chemistry.

These two major parts of ecology barely articulate with each other. Among the few points of contact are (1) species richness, for which some explanations involve both energy flow and models of population interac- tions, (2) numbers of species in trophic levels, for which Pimm (1980) has offered explanations in terms of population interaction, as alternatives to the traditional energy flow explanations, and (3) succession, which figures in global generalizations about ecosystem function (Odum, 1969; Margalef, 1968), but in which there have recently been developments based on pop- ulation dynamics (Horn, 1975; Connell and Slatyer, 1977; Noble and Slat- yer, 1980). However, none of these points of contact is as yet well founded enough to be put forward confidently in introductory texts.

My case here is that the self-thinning rule offers hope of articulating pop- ulation ecology with ecosystem functions. Most obviously, the rule has “sockets” of the right type; it relates the dynamics of numbers on the one side to the dynamics of biomass accumulation in the all-important first trophic level on the other. But beyond this, the self-thinning rule suggests how the elements of the two major parts of ecology should be rearranged to allow them to articulate with the rule, and indirectly with each other.

The flows of energy and nutrients through ecosystems are usually de- scribed as though the ecosystems were in steady state, for example in pyr- amids of biomass or numbers, and in budgets of fluxes for nutrient cycles at equilibrium. In reality, though, most ecosystems are accumulating plant biomass most of the time. Annual net primary production has a positive average in all ecosystems except those supported by organic matter trans- ported from elsewhere. Table 1 gives a classification scheme for how the accumulating plant biomass is dissipated; the classification uses as char- acters whether the biomass is dissipated continuously or intermittently, and the actual disposition of the biomass. The point that emerges from this table is that in the majority of the world’s vascular plant ecosystems, biomass is accumulating most of the time, rather than being grazed or decomposed in balance with net primary production. In these ecosystems much or all of the accumulated biomass is dissipated in episodes, as it is torn loose by

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172 MARK WESTOBY

Table 1 Regimes of Biomass Dissipation

Fate of biomass Intermittent‘ Continuous

Exported Aquatic macrophytes (often)= Phytoplankton Rocky intertidal with storms Coral reefs Mangroves

Intertidal marshes

Grazed

Decomposed in situ

Burned

Arctic tundra Grasslands under rotational

grazing

Phytoplankton Certain grasslands Saltgrass goose

pastures

Deserts Aquatic macrophytes Tropical rainforests following

cyclones or landslides Tropical rainforests in still water

Alpine and some arctic tundra

Most temperate and some Fire-tolerant

Many grasslands Mediterranean shrublands

tropical forests eucalypt forests

Whether the process is intermittent or continuous obviously depends on scale; at the scale of individual plants death is always intermittent. Here I consider scales corresponding to a plant population, about 10’- los individuals.

bExcept insofar as net primary production fluctuates. ‘For example, kelp beds by storms, or Eichhorniu when a water body is purged by flood.

flood or storm waves, mowed by herbivores which intermittently reach high densities due to population fluctuation or behavioral concentration, decom- posed following mass kill due to drought or cyclone, or burned, sometimes following a mass kill (White, 1979; Ford, 1982). The self-thinning rule is informative about this pattern of biomass rising over a period, then col- lapsing; the system repeatedly follows the characteristic up-and-across tra- jectory on the B-Ndiagram. The rule is relevant not only to crop situations, but also to a majority of the world’s ecosystems.

By the same token, many of the world’s dominant plant populations, considered in terms of numbers, do not have continuous fluxes of recruit- ment and mortality, and the resulting stable age distribution reflecting the age-specific survival curve. Rather, their dynamics are made up of bursts of recruitment, followed by a period of growth and competitive mortality in the roughly even-aged cohort that results. Subsequent seedlings are at a great competitive disadvantage. After an interval, the average and variation

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THE SELF-THINNING RULE 173

of which are important properties of an ecosystem, there follows some dis- turbance which removes most of the biomass and allows a new burst of recruitment. Again, the B-N diagram is more applicable than are classical demographic approaches.

Even if the B-N diagram is applicable to the dynamics of dominant plant species in a majority of the world’s ecosystems, this is only a small minority of all the species present in the ecosystems; how can it be said that the B- N rule might have predictive power about the organization of the complex webs of interaction which are ecosystems? Most ecologists have liked to believe that ecosystems are intricate and delicate machines with undia- grammed innards, with which we dare not tinker. I would argue that the importance of interactions between each pair of species in a complex web has been overestimated in understanding how ecosystems are organized. The importance of the physical structure provided by the biomass of those few species that dominate space, and contribute most of the organic stand- ing crop, has been underestimated. Subtle interactions, hard to study in- dividually and hard to model collectively, have been overvalued, and obvious features of ecosystems such as the quantity and spatial arrangement of plant standing crop have been undervalued.

The history of two strands of ecological thought illustrates why I think the importance of structure has been underrated. First, consider species packing. A long tradition of explaining coexistence of competitors led up to intense efforts, during the 1960s and 1970s, to understand variation in local species richness of animal guilds. There was great interest in niche overlap with respect to food and other resources, and in the possibility that permissible overlap (“limiting similarity”) varied between continents or be- tween latitudes for reasons of enviromental equability, predation pressure, or taxonomic history. Nevertheless, by far the loudest note sounded by all the empirical results (as opposed to the theoretical discussion) was that the most powerful predictor of local species richness was the physical structure of the habitat, e.g., foliage height diversity for birds (MacArthur, 1965; Recher, 1969), shrub shape diversity for lizards (Pianka, 1973), and archi- tectural complexity of host species for folivorous insects (Lawton and Strong, 1981). Schoener (1974) summarized 81 studies of the niche axes along which coexisting members of animal guilds were differentiated; mi- crohabitat was important far more often than food or time of activity.

A second strand of ecological thought is the concept of the keystone spe- cies. The metaphor of the keystone is itself instructive about the attitudes of ecologists to the ecosystems they study. The ecosystem is likened to a delicate arch, each species held up by all the others. Among the species, one is particularly irreplaceable, even though not particularly conspicuous or abundant.

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174 MARK WESTOBY

Krebs (1978) discussed three examples of keystone species, the presence or absence of which can radically change an ecosystem. They are (1) the starfish Pisaster ochraceus; removing them from northeast Pacific rocky intertidals allows the mussel Mytilus californianus to monopolize space (Paine, 1974); (2) lobsters; removing them from Canadian east coast sub- tidals allows much higher densities of sea urchins, which in turn can convert beds of kelp and other macroalgae to virtually bare rock (Mann and Breen, 1972) [sea otters can have a similar effect via sea urchins (Simenstad et al., 1978; Duggins, 1980)]; (3) African elephants, which can convert dense scrub or woodland into open grassland (Laws, 1970; Caughley, 1976). What all these examples have in common is that they are cases in which a predator is capable of removing the dominant, habitat-structuring species con- tributing most of the ecosystem’s biomass. What is curious about the ap- plication of the keystone species concept to these examples is that it is the predator, or in the lobster/sea otter case a predator of the predator, that has been called the keystone. Why not the mussels, the kelp, or the trees and shrubs? Presumably because this would be too obvious; ecologists have preferred to see ecosystems as constructed intricately, delicately, and in- scrutably. It is clearly these taxa that, with their large biomass, structure the habitat and control what other species are present. The natural and direct approach is to focus attention on these taxa, and on any processes (including predation) which affect the quantity and distribution of the bi- omass they contribute.

My view of the interrelations among the different areas of ecology I have mentioned is diagrammed in Fig. 3, which shows graphically why I believe the self-thinning rule to be central. The physical environment, particularly climate but also substrate, is the prime determinant of net primary pro- duction (Lieth, 1975), and thence of the rate of increase of biomass between episodes of biomass destruction. Given a rate of biomass accumulation, the self-thinning rule allows us to understand the mortality process. But to an understanding of how the average physical environment affects normal pri- mary production, we need to add a more systematic knowledge (see Table 1) of the regimes-frequency, intensity, season, mechanism-of biomass destruction or export in dominant photosynthesizers. Together these deter- mine how long a period of competitive growth and mortality continues. The biomass dissipation regime also determines the circumstances under which recruitment, whether of seedlings or of vegetative regrowth, occurs. Over evolutionary time the nature and intensity of competitive mortality and the possibilities for recruitment shape the type of plants which are present as dominants: their woodiness, height at maturity and longevity, the insect- and disease-resistance of their stem tissues, their ability to shoot from the base. In this way the self-thinning rule links ecosystem function with evo- lutionary demography. The type of plant which is selected can in turn in-

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THE SELF-THINNING RULE 175

Fig. 3. Schema of interrelations among different areas of ecological knowledge, showing cen- tral position of the self-thinning rule. Arrows indicate where one type of knowledge is im- portant in predicting another group of phenomena. Solid arrows show effects in ecological time, broken arrows effects over evolutionary time. Shaded groupings indicate major disci- plines within ecology.

fluence the regime of biomass dissipation (Ford, 1982), e.g., by flammability (Mount, 1964; Mutch, 1970; Comins, 1981), or by the possibility that there is a predator (“keystone”) capable of preventing its biomass from accu- mulating at all. In most ecosystems, however, herbivory can subtract only a small part of net primary production (Ricklefs, 1980).

The other most important link of the self-thinning rule is that given the growth form of the plants involved, the linked dynamics of biomass and numbers go far toward characterizing the changing physical structure of the environment from the point of view of the animal inhabitants. Available information suggests this physical structure is the most important deter- minant of animal community properties (which is not to say it may not have its effect by mediating processes of competition or predation within the animal community).

111. PRESENT KNOWLEDGE OF THE RULE

A. Summary History of the Rule The history of the rule and its relation to the competition-density (C-D)

effect (see below) have been detailed in English by White (1981) and in Japanese by Hozumi (1973). Briefly, the rule was developed and proposed

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176 MARK WESTOBY

by Japanese foresters and plant biologists in a period of very fruitful work on density effects in plant populations beginning in the 1950s and contin- uing to the present. The rule was first proposed by Tadaki and Shidei (1959) but was raised to the status of a major generalization by Yoda et a/. (1963). [Harper (1977) called it the Yoda rule.] Harper (1967) and White and Har- per (1970) introduced the rule to the Western literature. Since then there have been a series of reviews citing the rule as a leading principle of plant demography (Harper and White, 1971, 1974; Harper, 1977; White, 1980). Major strands of work since 1970 have been (1) experimental studies on herbaceous plants (discussed below under various headings), (2) gathering of data accumulated in the forestry literature and their expression in terms of the rule (White, 1974, 1975, 1980, 1981, and unpublished), (3) applica- tion of the rule to forest management (Tadaki, 1963; Ando, 1968; Drew and Flewelling, 1977, 1979; West and Borough, 1983), and (4) efforts to integrate plant growth models with the empirical relationships of the rule and the C-D effect (notably Hozumi, 1977, 1980; Aikman and Watkinson, 1 980).

Throughout this history, the usual formulation of the rule has been w = Kp - 3/2 ( 5 )

where w is the average weight of surviving plants, p is the density of sur- vivors, and K is a parameter. The formulation

I use here is mathematically equivalent. It may reasonably be asked whether this change of notation is not gratuitous and unnecessarily confusing. It may be confusing, but I believe it is a useful change of formulation, for the following reasons. First, the use of N rather thanp emphasizes that the rule is a principle of population dynamics. I hope that it will soon become impossible to write a chapter on population dynamics in a general ecology text without giving some account of the rule, and moreover using basic models for the dynamics of N which are compatible with the rule. Second, I believe the important linking role of the rule between population ecology and ecosystem function ecology is best seen when biomass, rather than mean weight, is used as a variable. Third, one of the most characteristic features of the self-thinning process is that there is a broad frequency distribution of individual weights, usually with two or more orders of magnitude dif- ferences between the largest and the smallest plants. The formulation of Eq. ( 5 ) focuses attention on the average plant, but average plants hardly exist. For example, in the formulation of Eq. ( 5 ) mortality of small plants can produce an increase in average weight, even if none of the surviving plants has grown at all.

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THE SELF-THINNING RULE 177

B. Relation to the Competition-Density (C-D) Effect

Interest in the effects of sowing density on yield goes back a very long way in agriculture and forestry. We must begin by setting aside formulas proposed in which the yield (Y, in grams per square meter) is due to re- productive activity of the crop (Bleasdale and Nelder, 1960; Holliday, 1960; Willey and Heath, 1969), rather than being all or most of the above-ground biomass. Particularly in annual crops, the transition to fruit, seed, or tuber production may be delayed if individual plants are small, with the result that yield at a set harvesting time may be lower at very high densities. This effect is not known to occur with respect to biomass yield.

It was recognized early that over a wide range of sowing densities, and after a sufficient period of growth, the effects of high density are absorbed by slower growth of each plant such that

(7) log W = log k - 1.0 log N

or

B = k

the “law of constant final yield.” Kira et al. (1953) generalized this rela- tionship to

(9)

where a and k rise as growth goes on, a from 0 to 1, and the relationship spans an increasing range of sowing densities. The C-D effect is a rela- tionship at one point in time among a set of stands sown at different den- sities. It represents “constant yield” only among stands at one time; yield actually rises over time. Figure 4 shows how a set of stands goes through a series of C-D effects as they grow.

In contrast, the self-thinning rule describes the trajectory followed over time by a single stand after it has begun to suffer mortality from crowding; the trajectory of the stand sown at highest density in Fig. 4 is an example. This distinction from the C-D effect is of the greatest importance, and fail- ure to understand it has been responsible for a good deal of confusion in the Western literature. Probably the problem has arisen because in principle stands established at different densities do thin along the same line [the “thinning line” described by Eq. (6)], although they arrive at the line at different places (biomass-density combinations) and times. In consequence data from late harvests of stands sown at different densities can be used to fit the self-thinning line, provided the harvests are taken after the stands have started to suffer substantial mortality due to crowding. This has been done in several seminal papers (Yoda et al., 1963; White and Harper, 1970)

log W = log ki - a log N

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178 MARK WESTOBY

r

+- C W

Q -

m 2

Low High Log density

Fig. 4. The relationship between the C-D effect and the -3/2 thinning rule. after Fig. 2.7 of White (1980). Subscripted t indicates successive time periods. Inset: 1, the relationship among different times for a stand at one density, before competitive mortality begins; 4, the rela- tionship among stands at different densities at one time, before competitive slowing of growth begins; 3, the C-D effect, slope 45"; 2, the -3/2 power law of self-thinning, slope 56".

and in examples used in influential reviews (Harper, 1977). Perhaps because of this, some authors have fitted thinning lines to all data from late har- vests, including some data points for which it is not clear that mortality due to crowding has substantially begun (shade treatment of Kays and Har- per, 1974; Hickman, 1979; Miyanishi et d., 1979; Panetta, 1979).

Usually the result of incorrectly including data points from stands that have not begun to thin is to flatten the estimated slope of the thinning line from - 3/2 toward - 1, because it is data from stands sown at low density that tend to be mistakenly included. This issue has interacted with discus- sion about possible explanations of why the slope should be -3/2, since any exceptions to the -3/2 slope are clearly useful tests of proposed ex- planations; this point is discussed further below. It is also true that even when data follow the trajectory of a single stand over time, there is a prob-

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THE SELF-THINNING RULE 179

lem in deciding which data points to include. Again, the proper test is that there should have been substantial (say, 20%) mortality, which appears to be due to crowding rather than to some extraneous cause such as a path- ogen outbreak. Mistakenly including data points from stands which have not yet begun to thin gives too steep an estimate of slope for the thinning line.

West and Borough (1983) have recently published data on Pinus radiata stands established at a range of densities; their data suggest a different kind of relationship between the C-D effect and the self-thinning rule. When all live plants were included, the C-D relationship had a slope which stayed near - 1 as the stands grew older. However when only dominant plants, defined as those which showed substantial continuing growth, were in- cluded, the slope of the C-D relationship approached - 1.5 as growth went on and more and more plants were nondominant.

C. Present Empirical Basis of the Rule

1. Empirical Base White (1980) gathered together published thinning lines, and others de-

rived by him from forestry data, and expressed them on a constant basis of grams per plant and plants per square meter. There are a total of 3 1 such lines. The average slope is close to -3/2, although there is some scatter (Fig. 2). Gorham (1979) assembled data points for the most crowded (i.e., highest biomass-density combination) stands of 29 species. A line fitted to these points-a between-species estimate of a thinning line presumed to be in common to all the species-also had a slope close to -3/2 (Fig. 5). In a similar way, the set of thinning lines diagrammed by White (1980; Fig. 2) shows a more impressive fit to the -3/2 slope when taken as a whole, than when each line is considered individually.

It is sometimes argued that the self-thinning rule can be “generalized” by inserting a parameter, which could vary depending on circumstances, in place of the - 3/2 exponent. In my opinion such a step would not generalize the rule but would dissipate its strength to the point that it would not be worth calling a rule. If the rule has force across all species (Figs. 2 and 5) it can only have one slope for this purpose. If individual species were to thin along slopes other than -3/2 depending on their properties or on the field situation, the extrapolated lines would necessarily diverge. If the slope varied among species, there could only be a between-species rule provided each within-species thinning line was short, i.e., applied over a small range of N a n d W.

Some random variation in the slope of thinning lines would be expected, of course, even if the true slopes were -3/2 in all instances. Many slopes

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180 MARK WESTOBY

6

5 m - 4 m

- - .c ._ $ 3 t - 2

5 1

d o

'D

0

m

-1

-2 10-1 100 10' 102 103 104

Shoot density (m-*)

Fig. 5. Relationship of shoot dry weight to density among 65 crowded stands of 29 plant species, after Gorham (1979). Reprinted by permission from Nature (London) 279, p. 148. Copyright 0 1979 MacMillan Journals Ltd.

have been estimated from less than 10 data points, sometimes 3 or 4. Par- ticularly when data are obtained from destructive harvests of small quadrats or pots as is usual in studies of herbaceous species, the standard errors on each harvest are commonly very large, although taking logarithms tends to make the data scatter less discomforting. Overall it is at present impossible to tell in what proportions the variation in slope among observed thinning lines is attributable to differences among species, differences among envi- ronmental conditions, or experimental error. Experiments are needed that are designed to give a strong statistical test for differences between two species grown at the same time and in the same place.

The same propositions apply to the other parameter of the rule, the in- tercept log C. When the data are expressed in standard units, C is the shoot dry weight per plant and per square meter at a density of one per square meter. White (1975, documented in White, 1980) first standardized units and showed that the intercepts mostly lay within the range log C = 3.5- 4.4, very narrow compared to the 1016-fold range of mean plant weight over which the rule appears to apply (Figs. 2 and 5 and J. White, personal com- munication). Subsequent extension of the data base suggests log Cis as high as 5.0 in some examples (J. White, unpublished). The predictive power of this relative constancy of log C was demonstrated when apparently outlying observations for Chenopodium album (Yoda et al., 1963) proved to have been measured on a wet weight basis, although this was not noted in the original source (White, 1980).

Variation in the estimate of the intercept is correlated with variation in

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the estimate of the slope. If there is experimental error in the slope, this will also be reflected in the estimated intercept, particularly if the reference density of 1.0 m-’ lies outside the range of the experimental data. Some of the variation observed by White may be due to this effect; however, since we do not know to what extent variation in slope itself may be systematic rather than random, this is hard to assess. If in comparing two or more thinning lines one is willing to take it as proven that they have the same slope (e.g., Westoby and Howell, 1981), that slope can be fitted through the means of each thinning line. The thinning lines will then be separated by the same amount independent of the density (or biomass) at which they are assessed; under these circumstances I shall refer to the thinning lines as at different levels.

Although the observed thinning lines are very tightly bundled relative to the span of W and N over which the rule operates (Fig. 2), the bundle is in fact of substantial width. From Fig. 2, there can be about sixfold dif- ferences between biomass when comparing two thinning lines at any given density. If even some of this is systematic rather than random variation, it is clearly of more than trivial importance. It may be best to speak of a thinning band rather than a thinning line. There are in fact reasons to be- lieve that some of the variation is systematic; these reasons will be discussed when specific environmental and other factors are dealt with below.

2. Problems of Fitting

Two problems of fitting a thinning line to empirical data-the interaction of slope and intercept, and deciding when stands have arrived at the line- have already been mentioned. Here it will be useful to comment briefly on the several other problems.

When stands follow trajectories which approach the line from below, “substantial crowding-dependent mortality’’ can be used as a test for their having arrived at the line. But what about stands which approach the thin- ning line from above? This is not a common problem, since by definition stands do not follow trajectories which cross the thinning line from below to above; the line can be thought of as dividing a zone (below the line) where all biomass-density combinations are possible, from a zone (above the line) of biomass-density combinations which cannot be sustained. Nevertheless, it is possible to enter this zone by three means: (1) sowing seeds at densities so high that the resulting seedling biomass is above the line before the seedlings come to depend on photosynthesis rather than on stored reserves in the seeds (Yoda et al., 1963; Stone, 1978; Hickman, 1979; Westoby and Howell, 1981, 1982; Lonsdale and Watkinson, 1982); (2) growing stands under full daylight and then transferring them to shade (Westoby and Howell, 1982), where the thinning line is at a lower level (see

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Section III,C,5); and (3) recruiting juveniles into a stand of larger plants which is already on the thinning line. This last is only a relevant case if one considers the self-thinning rule may be applicable to mixed-age stands; tra- ditionally the rule has been considered restricted to even-aged stands.

Each of these cases is taken up below; here I am concerned only with the problems they create in fitting thinning lines. The sort of data which result are indicated schematically in Fig. 6. It is usually impossible to demonstrate with formal statistics that such a trajectory does not consist of thinning along a single line of overall slope flatter than - 1/2, rather then consisting of an approach from above to a thinning line, followed by thinning along a line of slope - 1/2. This issue has arisen sharply in interpreting effects in deep shade (see below). The best experimental solution is to have an independent estimate of the location of the thinning line from a separate stand of the same species, grown under the same experimental conditions but sown at a lower density, so that it approaches the thinning line from below and the usual mortality test can be applied. It is possible that there are other features associated with the behavior of stands above the thinning line; for example, they may suffer mortality faster in real time than during normal self-thinning, i.e., travel along the thinning line. However, this re- mains to be proved, and in the meantime it would of course be circular to use faster mortality as a criterion of being above the thinning line.

There can be difficulties in deciding when mortality in stands has been crowding dependent. [I use the term crowding to mean closeness to the thinning line, i.e., some joint function of density and biomass; crowding

cp

8 J

\ o \

O \ o 0 \

0 0 0

O O 0 \ 0

Log N

Fig. 6. Hypothetical data such as might be expected to result from successive harvests of an experiment establishing stands above the thinning line; it would take very many data points or very small standard errors on each to allow travel along the thinning line to be distinguished from convergence with the line from above.

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dependence is analogous to density dependence in populations described by density N only (Westoby, 1981).] Plant stands, like other populations, can suffer from more or less catastrophic episodes of crowding-independent mortality due to extraneous causes such as windthrow or pathogen out- break. Depending whether the mortality falls at random, selectively on small plants, or selectively on large plants, the stand travels in different directions on the B-N diagram (Fig. 7). It then resumes the accumulation of biomass. In practice even during normal self-thinning, plants which are being com- petitively suppressed are commonly heavily attacked by fungal and bacterial populations; they are killed ultimately by competition but proximately by pathogens, particularly in herbaceous stands. As is well known, many fungi which are saprophytic on dying tissues of vigorous plants become patho- genic on weak plants. The presence or absence of the fungus can therefore not be used as a test. In practice I have assumed that plants are dying of crowding so long as the mortality does not open up gaps in the canopy; data from stands which have been opened up are discarded. Other exper- imenters rarely comment on this problem, but it is probably present. Pub- lished data commonly show sharp mortality with little gain of biomass over one time interval, followed by biomass accumulation with little mortality over the next time interval (e.g., Hiroi and Monsi, 1966). However, this sort of jerkiness can easily result from experimental error when stands are being measured by destructive harvests of replicates.

A particular kind of crowding-independent mortality occurs when “big- . bang” (semelparous) reproducers senesce after reproduction. Such senes- cing stands naturally do not obey the rule. Beyond this, it has generally been assumed that once reproduction begins and the pattern of biomass

I

Log N

Fig. 7. Consequences on the B-N diagram of an episode of crowding-independent mortality: A, selective on small plants; B, random; C, selective on large plants.

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184 MARK WESTOBY

allocation changes, with leaves perhaps not increasing either in quantity or in height despite continued biomass accumulation, the self-thinning rule lapses (e.g., Dirzo and Harper, 1980). The key to the matter is the disso- ciation of biomass increase from canopy expansion and elevation; repro- duction as such does not disqualify stands from obeying the rule, as shown by many forest tree populations. There do not seem to have been any stud- ies of species which are big-bang reproducers but not annuals.

More generally, there is an issue of which plant parts should be included in the estimates of biomass. Most data from experiments on herbs use com- plete harvests of shoots; data on forest trees usually are based on nondes- tructive measurements such as diameter at breast height (DBH) and stem height, converted by standard forestry methods to merchantable timber vol- ume. White (1980) further tried to correct for different weights per unit volume in different tree species. Data to correct for the weight of minor branches and foliage are rarely available. These would usually add 15-35% for trees of 20 cm DBH (Keays, 1971), a correction which is small on a logarithmic scale, and relative to the other sources of error. This correction would of course be more important for small than for large trees.

These partial measurements affect the intercept of the fitted thinning line, but do not affect its slope unless the proportion of the plant which is measured changes during stand development. Even then, very substantial changes would be needed to make much difference to slopes on the loga- rithmic B-N diagram. White and Harper (1970) and Westoby and Howell (1981) found that including root weights in experiments on herbs made no very substantial difference to the thinning slope; similarly Mohler et al. (1978) found that using total weight rather than trunk weight changed the estimated thinning slope only from - 1.52 to - 1.47 for Prunus pennsyl- vanica.

In more recent years experimenters interested in effects under deep shade or low nutrients (Westoby and Howell, 1981; Furnas, 1981; Morris, 1980; Lonsdale and Watkinson, 1982) have made a point of measuring roots, since those environmental factors are well known to affect root-shoot ra- tios. No clear generalization has emerged. Most available theory to explain the - 3/2 slope (discussed below) treats the thinning process as one of com- petition among shoots, with roots having their effects only indirectly by an interaction with shoot growth; on this basis the size and height of the can- opy would be the prime variables, with others such as shoot weight or whole plant weight acting as surrogates.

Until recently thinning lines were always fitted by regression of plant weight, or biomass, on density, except for a few authors (e.g., Kays and Harper, 1974) who gave only visual fits. Mohler et al. (1978) pointed out that the assumptions of regression were not met, with the result that a

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slightly different line can be obtained depending on whether W is regressed on N, N on W, or B on N, and depending on the size of the residual var- iance. Mohler et al. (1978) recommended, and used, the principal axis of a principal components analysis, and Lonsdale and Watkinson (1982), Hutchings and Budd (1981a), and West and Borough (1983) have taken similar steps. However, the major body of assembled data against which new data can be compared (White 1980) relies on regression or visual fit- ting.

R. E. Furnas (personal communication) has emphasized that there is a problem in deciding at what stage of data reduction to take logarithms. Standard procedure is to average arithmetically over plots or pots which are conceived as subsamples of one stand at one time, and then to plot each such average on a logarithmic scale and fit the thinning line to the loga- rithms. The problem is in deciding when two samples should be considered to represent different stand conditions, rather than replicates of one stand. For example, consider two pots of similar soil sown at the same density, but in which biomass accumulates faster in one than in another due to a different location in the greenhouse. If these are considered as replicates, the effect is to average them arithmetically, but if they are treated as sep- arate data points, using their logarithms in regression has the effect of av- eraging them geometrically. Similarly, it could be argued that different plots in forestry trials should be treated as separate stands, rather than as rep- licates. Or conversely, it could be argued that data from plots or pots with different fertilities should be treated as replicates when their biomasses and densities are of the same order, given that a joint thinning line is to be fitted to data from both fertilities, as is common.

Most authors have not attempted to calculate their data in different ways with respect to this problem, so it is not yet known whether it makes any substantial difference to the conclusions. The greater the positive skew in the population of observations, the greater will be the difference between arithmetic and geometric mean. Cousens and Hutchings (1983) found it necessary to use the geometric mean of replicate Ascophyllum nodosum stands to make this species conform to White’s (1980) estimate of the thin- ning line’s location. In principle I believe that any two samples should be treated as replicates provided conditions affecting their growth have not been systematically varied. To subdivide populations into separate stands on the basis of normal heterogeneity within them would lead to a version of the rule which could only be applied to very local populations.

B. Zeide (personal communication) raises a related point in attacking the use of the arithmetic mean to describe average plant weight, and claiming that the rule itself is an artifact of this inappropriate choice of variable. He recommends the power average, which is the plant weight corresponding

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186 MARK WESTOBY

to the arithmetic mean of the areas occupied by plants. The averaging pro- cedure is not, of course, an issue in the B-N formulation of the rule. In effect, Zeide is arguing that the apparent empirical strength of the rule would dissolve if the raw data were recalculated on a B-N basis, or using a power average for W. This remains to be seen.

3. Mixtures of Species On present evidence, the rule does apply to mixed-species stands taken

as a whole, but not to each component taken separately. The evidence con- sists of four experiments on herbs (White and Harper, 1970; Bazzaz and Harper, 1976; Furnas, 1981; Malmberg and Smith, 1982; Fig. 8 shows the sort of outcome), and a statement by White (1974, and personal commu- nication) that the rule applies to some mixed-species forest stands; however, details of the forestry evidence have not yet been published. It would seem that in mixed stands, which individuals survive and which individuals die is very largely determined by which species the individual belongs to, rather than by its size relative to others, as in single-species stands.

Present evidence is, of course, very limited on this point; the rule’s interest and importance will be far greater if it proves to apply reliably to mixed- species stands. Most particularly, present evidence deals only with stands where both species are similar in growth form and successional status. Data on mixtures of radically different growth forms, e.g., trees and herbs, would be hard to interpret because total biomass would be so dominated by the trees’ contribution. Variation in herb biomass would disappear against the background of measurement error in the tree biomass. It would be of great interest to have data for a B-N diagram from a mixture of trees with one

200 400

Number of surviving plants

Fig. 8. Self-thinning trajectories of the components of a mixed stand, results from less fertile soil. Lines S and L are regression slopes for trajectories of Sinupis ulbu and Lepidium sutivum, respectively. The broken line shows -3/2 slope. After Fig. 2 of Bazzaz and Harper (1976).

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fast growing and light demanding, and the other slower growing but shade tolerant. It is a common assertion in silviculture that certain species, e.g., Abies balsamea, persist in the understory below faster growing species, growing only slowly but suffering little mortality. If true, this would imply a B-N diagram for a Picea glauca-Abies balsamea mixture, for example, somewhat as indicated in Fig. 9.

4. Effects of Soil Fertility

Until recently it had been agreed that stands on soils of higher nutrient status thin along the same trajectory as on less fertile soils; they accumulate biomass faster, but the relation of mortality to biomass accumulation was considered the same (Yoda et al., 1963; White and Harper, 1970; Bazzaz and Harper, 1976; White, 1980). Indeed, this surprising phenomenon was important in drawing attention to the generality of the self-thinning rule (Yoda et al., 1963). It was, and is, counter-intuitive that the growing condi- tions provided by the soil would not affect mortality rates directly, but only via biomass accumulation, and particularly that better growing conditions would generate higher mortality rates.

However, three items of data have come forward to challenge the uni- versality of this generalization (Furnas, 1981 ; Morris, 1980; J. White, per-

10 1 o2 1 o3 1 o4 1 o5 Number of survivors (ha-'

Fig. 9. Hypothetical B-Ntrajectories in a mixed forest stand of a light-demanding and a shade- tolerant species. (O), Total; (0), light-demanding component; (&, shade-tolerant compo- nent. Numbers indicate successive sampling occasions.

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188 MARK WESTOBY

sonal communication). Morris (1980) and Furnas (1981) reported that the slope of the thinning line is flattened from - 3/2 to - 1 when the soil en- vironment is made sufficiently difficult. On the other hand, J. White (un- published) has found, from published yield tables for stands of certain forest tree species, that the thinning line is lowered (i.e., intercept reduced but slope not changed) on plots of lower site index. Site index is a measure used by foresters to describe growth potential; it is usually the canopy height achieved after some standard number of years. It presumably reflects soil conditions to a large extent. Data are presented in Figs. 10, 11, and 12. It is important to realize that comparable data sets for other species do not show any lowering of the thinning line.

Several interpretations of the results of Morris (1980) and Furnas (1981) are possible:

1 o3

10'

1 o2 1 o3 Density (acre-' )

Fig. 10. Thinning lines for Pinus ponderosu on sites of different quality. Better sites have higher site index (SI). Data calculated by J. White (personal communication) from results of Meyer (1938).

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THE SELF-THINNING RULE 189

10'

10'

", r - m

- 5

f

2 C m

1 oo

I

1 0' 1 o3 Density (acre-' )

Fig. 11. Thinning lines for Piceu excelsa on sites of different quality; higher lines are from better quality sites. Data calculated by J. White (personal communication) from results in Murphy (1917).

1. There is in fact a change of slope when soils are limiting. 2. The actual effect they observed was as for the forestry cases: a low-

ering of the thinning line rather than a flattening of the slopes; a single data point at lower right of the thinning diagram could be the difference between this and the first interpretation.

3. A flattened slope results under conditions so extreme that plants are killed by those conditions directly. In general such conditions would pro- duce decline in biomass and in density simultaneously, but there must exist some marginal level of the environmental condition at which growth in the survivors just balances loss of biomass in those that die, so that a slope of - 1 (0 on biomass scale) results. In effect, mortality is crowding indepen- dent. This argument has been made with respect to the effects of shading by Westoby and Howell (1981, 1982). The situation is complicated in the case of nutrients, because at very low soil nutrient level, e.g., sand and

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190 MARK WESTOBY

50

SI 40

1 o2 1 o3 Density (acre-’ )

Fig. 12. Thinning lines for Populus tremuloides on sites of different quality. Better sites have higher site index (SI). Data calculated by J . White (personal communication) from results of Kittredge and Gevorkiantz (1929).

water in Morris (1980), the nutrients input in seed must be a large propor- tion of all the nutrient in the system; very probably in this treatment there was a decline in the total nutrient present over the course of the experiment.

With respect to the forestry data collected by White, it is extremely puz- zling that a lowering of the thinning line should be found in some species but not in others. The only remotely plausible hypothesis I can think of to explain this is that some species grow in different shapes on different site types, while other species simply grow faster or slower. For the effect to be in the right direction, those species that (hypothetically) changed shape would have to do more competing (occupy more “biological space”) at a given stem volume under infertile conditions. This question could be ex-

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amined for the species in Figs. 10, 11, and 12. Morris and Myerscough (1984) have recently distinguished “altered-form” and “altered-speed” hy- potheses of growth and competition, and discussed how resource levels should affect outcomes under the two hypotheses.

5. Effects of Shading Until recently it has been thought that in deep shade, stands thin along

a line flatter than - 3/2 (or - 1/2), ranging down to - 1 (or 0). This con- clusion was based on data of White and Harper (1970) on Raphanus and Brassica and on their analysis of data on Helianthus presented by Hiroi and Monsi (1966). An experiment on Loliurn by Kays and Harper (1974) also seemed to show this effect, although no thinning line was statistically fitted, and there was little or no thinning of the stands sown at the lowest densities. These and most subsequent experiments have not applied constant light fluxes, but have used burlap, plastic strips, or proprietary shade-cloth to exclude some percentage of daylight. Naturally no such experiments have been carried out on forests. The absolute light income in each treatment varies through the day from day to day, and most importantly there may be a substantial overall increase or decrease over the course of an experi- ment (usually 8-30 weeks) as the seasons change. W. M. Lonsdale (personal communication) has pointed out that if the slope of the thinning line does respond to shading, the line might accordingly be expected to change slope as the experiment went from spring to summer, for example, and a curved trajectory might result.

J. White (personal communication) has pointed out that a misprint in Fig. 4(a) of White and Harper (1970) allowed the evidence of Hiroi and Monsi (1966) to seem stronger than it actually was. At White’s suggestion I re-present the data (Fig. 13). While the data points for 23% daylight have an overall slope, shown in the fitted regression line, of - 1.08, only the data from the starting density of 1600 m-2 (circles) contribute to this de- viation from - 3/2. The data points from starting densities of 400 and 100 m-2 are just as compatible with a -3/2 slope.

The supposed change of slope in deep shade has been remarked on by most commentators on the rule, despite its limited empirical support, be- cause a good test of any explanation of the -3/2 slope would be whether the explanation could also account for this apparent exception. Attempts at explanation will be discussed below. However, Westoby and Howell (1 98 1, 1982) and Hutchings and Budd (198 1 a) reported experiments sug- gesting that the effect of shade was rather to lower the level of the line without changing its slope. Over the same period, Furnas (1981) and Lons- dale and Watkinson (1982) reported experiments suggesting a flatter slope in the shade, as previously thought. The contrast between these two pos-

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192 MARK WESTOBY

100 500 1000 3000 Survivors (m-2 1

Fig. 13. Self-thinning under shade in Hefiunthus unnus. Data points are all from stands which had begun to thin under 23% daylight, at starting densities of 1600 (0), 400 (A), and 1 0 0 m-z (0). A regression line fitted to all these data collectively, slope - 1.08, is shown. The other solid lines are regressions fitted to stands thinning under 60 and 100% daylight. The broken line shows a slope of - 1.5. Redrawn from Fig. 4a of White and Harper (1970); derived from data of Hiroi and Monsi (1966).

sibilities is not so complete as might be thought. Clearly when the light income is so low that positive net photosynthesis is not allowed even for isolated plants, there must be a declining biomass at the same time as mor- tality, i.e., a positive slope (biomass basis). At full daylight there is a - 112 slope. Somewhere in between there must exist a light income at which the stand thins along a slope of 0. The question is whether a zero slope is found only transitionally in a narrow range of light incomes (Fig. 14, func- tion A), as suggested by the data of Westoby and Howell (1981), or is found over a fairly broad range of degrees of shading (Fig. 14, function B).

Figure 15 represents what seem to be the alternative logically possible outcomes in deep shade; some of these were discussed by Westoby and Howell (1982). Figure 15a shows a literal interpretation of a simple change in slope of the thinning line. This is very unlikely, as it implies that at high N a greater biomass can be supported in deep shade than in full light. This, not surprisingly, has not been observed; White and Harper (1970), Kays and Harper (1974), and Furnas (1981) represented their data as a fit to only that part of Fig. 15a to the left of the two thinning lines. The obvious mod- ification is Fig. 15b.

Figure 15c shows a pattern in which stands in a particular degree of deep shade suffer mortality with no net change in biomass irrespective of the biomass-density combination from which they start; that is to say, there is

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+1.0-

,+0.5

s 0 2

L

.- CI u-

iii 0 -

-0.5

193

-

- I 1 I 1 1 1 0 20 40 60 80 100

Percentage full sunlight

Fig. 14. Two alternative relationships between the degree of shading and the slope of the self- thinning trajectory; see text for further discussion.

no thinning line in the usual sense, rather the stands follow trajectories of zero slope from any starting point. The contrast with Fig. 15b shows up one important property of Fig. 15b, which is that the zero slope appears only after a particular level of biomass has been accumulated, and is not a property of the degree of shading only. While many of the available data can be reconciled with Fig. 15b, a preliminary period of thinning along a - 1/2 slope has not been observed before the thinning along a zero slope. However, most of the experiments supporting this model have five or less harvests to describe the deep-shade trajectory, so two separate slopes would be indistinguishable, except that the average slope would not be 0 but some- where between 0 and - 1/2.

A problem with Fig. 15b is this: if a zero-slope trajectory (no accumu- lation of biomass) is produced by an interaction of biomass level with de- gree of shading, rather than by a given degree of shading irrespective of biomass, why does this same principle not apply to thinning in full light? There should exist a level of biomass accumulation which cannot be ex- ceeded by a given species even in full light. Lonsdale and Watkinson (1982) have suggested that this is in fact the case. Although this suggestion is a fairly radical reformulation of the accepted version of the self-thinning rule, it is not incompatible with the available data. This point is taken up in the next section. Fig. 15d shows the simplest version of this model.

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194 MARK WESTOBY

f

Fig. 15. Six possible outcomes of the effect of deep shade on the B-N diagram; see text for discussion of each case. Solid line shows full-light thinning line; broken line shows deep-shade thinning line; arrows indicate trajectories in deep shade.

The suggestion of Westoby and Howell (1981) that shading lowers the level of the line without changing its slope is diagrammed in Fig. 15e. Wes- toby and Howell (1981, 1982) pointed out that this model does not exclude all the others; there must exist a degree of shading at which mortality goes on with little net biomass accumulation, as in Fig. 15c or perhaps Fig. 15b or d. The question is whether this is a transitional phenomenon (as indicated in Fig. 14, function A) or occurs over a broad range of degrees of shading (Fig. 14, function B). Data given by Westoby and Howell (1981) (Fig. 16) are closer to Fig. 14, function A.

There is a practical problem in analyzing data to distinguish between Fig. 15e and the other models, which has already been alluded to in Section 111,C,2. Stands established at high N and/or in full light, then grown in shade, are expected from the model of Fig. 15e to follow trajectories of zero or even positive slope while approaching the shade thinning line from above. The problem is that without an a priori estimate of the position of the shade thinning, the approach to the thinning line cannot easily be dis-

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THE SELF-THINNING RULE 195

1 .o

.- 0.5 - m .- C .F 0 5

m 8 -0.5 5

T

-1 .o 20 40 60 80 100

Percentage full sunlight

Fig. 16. Slope f 95% confidence limits of regressions of log shoot biomass of Beta vulgaris on log density of survivors as a function of relative light flux (percentage of full light for the summer experiment). Regressions were based on values from harvest 3 onward (100 and 9% relative light flux treatments) or from harvest 4 onward (other treatments). From Westoby and Howell (1981).

tinguished from travel along it. For example, consider the trajectories fol- lowed by stands in 18% of summer daylight in Fig. 17 and in “shade throughout” in Fig. 18. Westoby and Howell (1981, 1982) interpreted these as cases of Fig. 15e, and fitted thinning lines to later harvests only. But it could be argued that thinning lines of flatter slope should have been fitted through the whole of each trajectory. In the case of Fig. 17, this would have produced a version of Fig. 16 more like function B in Fig. 14.

Another notable feature of the model of Fig. 15e is that it provides one possible source of variation in the level of thinning lines (see Fig. 2). Wes- toby and Howell (1981) found that stands of Beta vulgaris grown in winter thinned as though they had been grown in summer, but with a lower light income; and in fact, winter light income was about 55% of that in summer (see Figs. 16 and 17). The range of levels of the thinning line created by shading was about six-fold, which is about the same as the range observed in measured full-light thinning lines (Fig. 2). In principle, it might be ex- pected that the level of biomass which can be supported at a given density would be affected by light income during the growing season, by leaf area per unit biomass, and by the amount of light interception per unit leaf area. On this last point, Lonsdale and Watkinson (1983) have found that an erect- leaved species had a higher thinning line than a horizontal-leaved species grown under the same experimental conditions. Some interesting patterns with respect to foliage form also emerge from the data summarized by White (1980) in his Table 2.2 and his Fig. 2.9 (Fig. 2 of this article). Species can be ranked with respect to the levels of their thinning lines (biomass or mean

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196 MARK WESTOBY

1000 -

320 -

N-

‘E

F 4 loo-

0

In -

-

Density (rn-2)

Fig. 17. Trajectories followed by self-thinning stands of silver beet seedlings under different light fluxes in a space of log shoot biomass against log density of survivors. Each point rep- resents one harvest, the mean of four replicates. (O), Condition before shading; (A), 100% of summer daylight; (A), 55%; (0), 37%; (O), 25%; 0, 18%; (W), 9%. From Westoby and Howell (1981).

weight at a given density). Because of the interaction of estimated slope and estimated intercept, I have made these rankings (Table 2) visually from Fig. 2 in three separate zones of density, rather than ranking the species based on their estimated intercept at 1 m-2. Among trees (1-3 x m-2) and small trees or shrubs (1-3 rn-3 there is a very clear tendency for needle-leaved species to have higher thinning lines than broad-leaved spe- cies. Similarly, Morris and Myerscough (1983) have recently observed a thinning stand of the needle-leaved sclerophyll shrub Banksia ericvolia which transgressed the outer boundary of the thinning line proposed by White (1980). This does not necessarily mean, of course, that the leaf shape as such is important; the division between needle-leaved and broad-leaved species corresponds closely to the evergreen/deciduous and the softwood/ hardwood division. Moreover the comparison is not controlled for light income or for any other factor that might be relevant.

Among White’s data on herbs (1-3 x lo3 m-2) there is no evidence that erect-leaved species have higher thinning lines. However, Lonsdale and

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THE SELF-THINNING RULE 197

./ 1000 5000 10,000 50,000

Density of survivors (m-2)

Fig. 18. Trajectories followed by self-thinning stands of Trifolium subterraneum grown in a glasshouse under different light treatments. (0), Condition before shading; (0), full daylight throughout; (m), shade throughout; (O), transferred from full light to shade; (0). transferred from shade to full daylight. Most points represent the mean of two harvests, but one harvest only for condition before shading, and for the last points of full daylight throughout, shade- to-full daylight, and full daylight-to-shade treatments. From Westoby and Howell (1982).

Watkinson (1983) have assembled further data from grasses which generally have thinning lines at higher levels than those reported by White. They show that erect-leaved or needle-leaved species have more biomass per unit can- opy volume. If canopy volume rather than weight is used on the y axis, the thinning lines for different species are brought closer together.

Finally, Hutchings and Budd (1981b) and Lonsdale and Watkinson (1982) have proposed a model (Fig. 150 which combines the features of Fig. 15c with those of Fig. 15d.

6. Maximum Size of Plants Taken at face value, the self-thinning rule predicts that stands of any

species can continue ad infinitum to decrease in density of survivors and increase in biomass and mean weight of survivors. This is unreasonable; the world does not contain dandelions the size of sequoias, nor sequoias proportionately larger. The thinning line for any given species cannot be considered as prolonged indefinitely to the upper left of the B-N diagram. The question is, where does the line of - 1/2 slope, in full daylight, stop, and what happens there? Here are some possibilities:

1. It is clear in all self-thinning data that the rate of travel of a stand along the thinning line slows as the stand develops. A constant rate of

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198 MARK WESTOBY

Table 2 Ranking of Level of Thinning Lines in Three Different Density Zones"

Density Rank Species Foliage

1-3 x 10-3m-2 1 2 3 4 5 6 l a 9 10 11 12 13 14 15 16 11

1 2 3 4 5

1 2 3 4 5 6 I 8 9 10 11

1-3 m-'

1-3 x lo3 m-'

Abies sachalinensis Tsuqa heterophylla Pinus monticola Abies concolor Pseudotsuga menziesii Chamaecyparis thyoides Pinus strobus Pinus ponderosa Pinus contorta Liquidamber styraciflua Picea mariana Quercus spp. Populus tremuloides Castanea dentata Ainus rubra Populus deltoides Carya spp.

Abies sachalinensis Chamaecyparis thyoides Prunus pennsylvanica Populus tremuloides Corylus avellana

Trifolium pratense Brassica napus/Raphanus sativus Helianthus annuus Erigeron canadensis Triticum sp. Medicago sativa Carex lacustris/C. rostrata Plantago asiatica Amaranthus retrofexus Ambrosia artemisiiifolia Erigeron canadensis

Needle Needle Needle Needle Needle Needle Needle Needle Needle Broadleaf Needle Broadleaf Broadleaf Broadleaf Broadleaf Broadleaf Broadleaf

Needle Needle Broadleaf Broadleaf Broadleaf

Horizontal Horizontal/erect Horizontal Horizontal Erect Horizontal Erect Erect Horizontal Horizontal Horizontal

'Thinning lines as reported by White (1980).

movement of the point representing the stand up the B-N diagram would represent a constant relative growth rate of biomass, while relative growth rates are well known to decline as growth goes on. Perhaps this slowing of the trajectory reaches a point at which progress along the line is negligible; stands accumulate no more biomass, but also suffer no more mortality due to competition.

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THE SELF-THINNING RULE 199

100

50 e c

k

- 5 5 '

n

g 10

z 2

0

C

1 .

0.5

2. There could come a stage at which individual plants cannot accu- mulate more weight except in proportion to the weight losses of dying plants; i.e., mortality continues but there is no increase in stand biomass, and the thinning line, the boundary of possible biomass-density combinations, bends over to a zero slope. This possibility appears in the models of Fig. 15d and f, and with respect to shaded treatments in Fig. 15b.

3. Substantial crowding-independent mortality may always intervene be- fore stands stop traveling along the thinning line. Even though episodes of such externally induced mortality, producing changes in biomass and den- sity as in Fig. 7, occur at random with respect to the position of the stand on the B-N diagram, the probability that a stand will travel beyond a cer- tain point before such an episode may be vanishingly small.

The second effect (2) is shown in data for oak and beech taken from forestry yield tables by White and Harper (1970) (Fig. 19). These data should be seen as suggestive rather than as unqualified, since the measure is mer- chantable timber volume rather than biomass, and the data are not from a particular experiment but are interpolations in a generalized pattern. It is interesting to note that the point at which biomass stops accumulating in

.

.

'

.

I I I I

50 100 500 1000 Density of thinned trees (acre-' 1

Fig. 19. Relationships between density and mean volume per tree for beech and oak in fully stocked stands for up to 150 years of age. The data have been recalculated from Forestry Management Tables (Bradley et al., 1966) which predict yields obtained in accordance with the thinning practices advocated. Beech, yield class 100 (0); oak, yield class 40 (0). Fitted lines indicate a change of slope from - 1.5 to - 1 . After White and Harper (1970).

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200 MARK WESTOBY

Fig. 19 corresponds roughly to the time at which maximum canopy height is reached in oak and beech (Bradley et al., 1966).

The three possibilities listed above are not alternatives. Evolution could operate to make all three correct accounts of the situation, in the following sense. Plants in a competitive race are under natural selection to grow quickly; height growth, in particular, gives a disproportionate advantage for light capture in the future. But the nature of the selection also depends greatly on the expected length of the race, that is, the size (and height) which surviving plants will reach before external sources of mortality intervene. Annuals can expect to grow only over 1-4 orders of magnitude of plant weight, trees over 7-10 orders of magnitude. Plants which expect (i.e., have been selected by the statistics of past experience) to need to make continued height growth beyond 10, 30, or even 50 m if they are to survive continued competition from others which have reached that height must invest in stems of a different, and more expensive, type even in early growth (Whitmore, 1975). It is clear empirically that annuals have higher relative growth rates than short-lived perennials, which in turn have faster growth than long-lived perennials, other things being equal (Mooney and Gulmon, 1979). Some reasons are also clear why this should be so; structures which will live longer and reach higher must be built more densely per unit volume, and be pro- tected with defensive chemicals. The detailed costs of tissue types of dif- ferent longevity and load-bearing capacity are an underinvestigated subject. In any event, each plant species has implicit in its anatomy and morphology a maximum height, which varies widely between species, and which is shaped by past selective experience of how long the competitive race will last before factors other than within-species crowding dominate mortality.

What the - 1/2 slope means is that more biomass can be supported in stands of fewer, larger plants. This is not because there is more foliage, intercepting more light (White, 1977; Hutchings and Budd, 1981a; contra Westoby, 1977); rather the same amount of foliage is deployed further up- stream in the descending light flux, and accordingly there is a greater weight of supportive tissue. The foliage, meanwhile, is concentrated into fewer individuals. It seems reasonable to suppose that the - 1/2 coefficient rep- resents the cost in structural tissue of the height gain needed to allow a successful individual to capture the percentage of the fixed canopy weight previously occupied by its neighbors that die. This point will be discussed further in Section IV. Its significance here is that one might expect the - 1/2 relationship to stop applying when the maximum height of the species is reached. Beyond maximum height, a zero slope might reasonably be ex- pected to apply, as in Fig. 15d or f. (This is by no means a logical necessity, however; equally, disproportionately more structural tissue could be needed to support a broader canopy at a given height.)

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THE SELF-THINNING RULE 20 1

The three possible outcomes at the upper left of a thinning line of - 1/2 slope, for any given species, could then be interrelated in the follow- ing sense: the maximum height and weight of a species are molded by ev- olution such that it is unusual for a species to spend substantial time at those maxima. In the absence of disturbance, a species which stopped height growth could expect to be overtopped by a species which did not, and con- sequently to suffer mortality at rates not dependent on its own degree of crowding. Species should therefore be selected to continue height growth up to times by which a biomass-destroying and height-reducing disturbance would usually have supervened. In the uncommon event that a substantial period is spent at maximum height without either disturbance or overtop- ping, one of two things might happen. If crowns of a single stem system can expand further laterally, thinning of zero slope could result. If they cannot, the stand would become stationary on the B-N diagram.

7. Genets and Ramets Thus far I have assumed that the plant “individuals,” in terms of which

the density of survivors is counted, are units the whole of whose canopy rests on a single point of support at ground level. What happens in species where genets are at least potentially made up of several-to-many such ra- mets? (I exclude here the usage of ramet which would apply it to separate branches or even leaves within one aerial stem system.) One view (Harper, 1977, 1981; Hutchings, 1979) is that the rule does apply to genets of such species, and that it does not apply to their ramets. However, this seems to be because ramets do not usually grow enough to travel along a thinning line, rather than because the thinning line does not represent an outer boundary for them. Apparent competitive thinning has been observed in tillers of Lolium perenne (Lonsdale and Watkinson, 1982), and in shoots of coppicing Castanea sativa (Ford and Newbould, 1970). However, com- petitive thinning among ramets would only be expected to be observable over a rather narrow range (Hutchings and Budd, 1981b). Consider Fig. 20. An upper limit to the span over which ramet thinning can be ob- served is set by the maximum size characteristic of ramets of the species, obviously smaller for tillers of Lolium perenne than for stems of Castanea sativa. Beyond this size, by definition, genets expand by multiplication of ramets rather than by increasing the size of the ones they already have. If, then, a stand is sown at a genet density lower than the minimum crowded ramet density (trajectory A in Fig. 20), it is unlikely to undergo ramet thin- ning. If sown more or less at the minimum crowded ramet density, most species will probably concentrate on growing one or a few ramets to full size, before multiplying ramets (trajectory B); even if a species could be found which multiplied ramets tenfold, for example, (trajectory C) when

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202 MARK WESTOBY

Log ramet density

Fig. 20. B-N diagram representing trajectories of ramets rather than genets, if the thinning rule applies to them. The various trajectories shown (see text for discussion) indicate why ramet thinning should be hard to observe.

sown at this density, the resulting ramet thinning line would be observable only over one log cycle of density, which is marginal for fitting a thinning line. Hutchings (1979) and Mook and van der Toorn (1982) gave several examples like trajectory B in Fig. 20; these trajectories are readily inter- preted as being constrained under the thinning line, even though they do not travel along the line far enough for a regression to be fitted. If a stand were sown at even higher genet, and ramet, density (trajectory D), the re- sulting thinning line would usually be interpreted as a genet thinning line, although it is a ramet thinning line, too. In order to observe a ramet thin- ning line over a substantial span, it would be necessary to generate a stand with few genets, many ramets, and little biomass (trajectory E). This could be done by clipping a grass stand to encourage tillering, and then letting it grow without clipping. This experiment has not been done; however, the data of Ford and Newbould (1970) on coppiced chestnut are analogous.

8. Mixed-Age Stands The rule is at present derived entirely from data on even-aged stands. It

would be of great interest to know whether mixed-age stands follow dif- ferent trajectories on a B-N diagram, and if so in what respect.

There is consensus among plant demographers that a plant’s size, relative to its neighbors, is more important than its chronological age in determining its prospects for growth, survival, and reproduction (Harper, 1977; Ra- botnov, 1969). Accordingly, I believe the best question to ask is whether trajectories on a B-N diagram are affected by the frequency distribution

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THE SELF-THINNING RULE 203

of plant sizes (population structure), and if so how. From this perspective, the even-agedness or otherwise of the stand becomes one of several influ- ences on population structure. What little is known about the role of pop- ulation structure is considered below in Section V.

IV. EXPLANATIONS OF THE RULE

A. The Dimensional Argument

Yoda et a/. (1963) already put forward a dimensional argument which might explain the -3/2 slope. It runs as follows: (1) volume, or weight (W), of a plant is proportional to the cube of some linear dimension P

W a P3

while (2) the ground area s occupied by the plant is proportional to the square of the linear dimension

s a P2

If (3) the density of survivors (N) is inversely proportional to the average ground area occupied

Noc s-1

then

w a ~ - 3 ’ 2

quod erat dernonstrandum. This argument has been repeated many times since. It is popular because it produces the right answer, and also because a phenomenon which produces a straight-line relationship on a log-log graph is inherently likely to have a dimensional explanation. The argument is certainly not literally true, however. Yoda et a/. (1963) themselves called it “a crude approximation.” White (1981) has documented at length that weight is not proportional to the cube of any linear dimension, most par- ticularly not to any dimension which relates to a radius of the area occu- pied; White’s conclusions are considered further in Section IV,C.

Yoda et a/. (1963), White (1981), and Hutchings and Budd (1981b) also supposed it was necessary to assume that thinning started when the stand was 100% occupied (by the ground areas s of the plants). However, M. Slatkin (unpublished) has built a model in which plants are represented by expanding hemispheres randomly distributed on a surface; when a smaller hemisphere is overlapped to a given degree by a larger, it dies. This model produces a - 3/2 relationship, between mean hemisphere volume and num- ber of survivors, before all the surface is covered by plants.

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204 MARK WESTOBY

B. Dimensional Prediction of Exceptions

A good test for any explanation of the -3/2 slope, or indeed of the intercept C, is whether it predicts known exceptions, or still better, new exceptions which can then be shown experimentally to exist.

The first interest in this possibility was the comment made by White and Harper (1970), and repeated by Kays and Harper (1974) and Harper (1977) among others, that the supposed flattening of the slope to - 1 in deep shade indicated that the rule must reflect competition by shading in the canopy, rather than root competition.

Following White’s (1975, 1980) demonstration that the intercept C varied very little between observed thinning lines, I tried to relate variation in C to shape in a set of Trifolium subterraneum varieties (Westoby, 1976). The stems grow radially along the ground from a central tap root; height is determined by petiole length, which varies between varieties. No effect was detectable. In the same paper, I pointed out that if plant height is constant through the thinning process, the obvious modification of the argument in Section IV,A predicts a slope of - 1 rather than of -3/2. Given that Tri- folium subterraneum canopies do not rise during thinning, but that a - 1.5 slope was actually observed, I thought at the time this argument was a re- ducfio ad absurdum of the simple dimensional reasoning. Since then I have realized that although the top of the stand’s canopy did not rise, mortality would have been removing from the stand individual plants whose canopies were generally deployed lower than those of survivors, so that the mean canopy height of survivors was probably increasing during thinning.

Since shading is well known to increase the amount of leaf area per unit shoot weight of plants, I have argued (Westoby, 1977) that the supposed change of slope in low light was a phenomenon of shoot weight which would not appear if shoot leaf area were used as a measure instead. White (1977) refuted this argument, pointing out two deficiencies. First, the data on leaf area-weight ratios came from plants of different sizes to those in the thin- ning data with which they were combined. Second, the argument implies that stands increase continuously in total leaf area (leaf area index, LAI) as they do in biomass, whereas it is well known that LA1 reaches a maxi- mum while thinning continues. This second point seems to me decisive. Admittedly in the subsequent experiment of Westoby and Howell (1981), we found that LA1 of Beta vulgaris continued to increase throughout, reaching 14 in full light by the end of the experiment, a figure which is unusually high, but not unprecedented for an erect-leaved species. Simi- larly, in Ceanothus stands studied by Schlesinger and Gill (1978, 1980), irradiance under the canopy continued to decline during thinning. This is not general, however, as is shown by Hutchings and Budd (1981a) and by continuing thinning in tree stands for which the LA1 does not continue to

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THE SELF-THINNING RULE 205

increase. Dewsberry (1977) generalized the mathematical formulation of my (Westoby, 1977) account.

Since 1977 the situation has not been clarified with respect to explana- tions of a - 1 slope in low light; rather, the phenomenon itself has appeared equivocal. Westoby and Howell (1981, 1982) and Hutchings and Budd (1981a) did not find it, but on the other hand Furnas (1981) and Lonsdale and Watkinson (1982) reconfirmed the change of slope; the data are dis- cussed in Section III,C,5.

Most recently, Lonsdale and Watkinson (1983) have sought to generalize the rule by treating it as a limit on the biomass which can be packed into a unit of canopy volume. This limit in turn must depend ultimately on light interception. Presumably the canopy can be deeper or denser either where erect leaves allow more light to be transmitted through a given leaf area index, or where shade tolerance allows leaves to make a photosynthetic profit in lower light. Zeide (unpublished manuscript) has discussed toler- ance of forest trees in these terms, and has shown that species which are tolerant of growing under other species do not necessarily persist well under canopies of their own species; this implies that the properties of the shading canopy are more important than the light adaptation of the leaves of the smaller plants.

Miyanishi et al. (1979) thought they had found a case like that in the reductio ad absurdurn argument of Westoby (1976), with Portulaca olera- cea growing laterally but not in height, and the resulting thinning line hav- ing a - 1 slope. To explain this they argued that the - 3/2 relationship was only a special case of a family of relationships, with slopes ranging from - 1 to infinity. The slope would depend on the dimensions in which the growing plants expanded, with -3/2 being produced when the plants ex- panded vertically and laterally in the same proportions. White (1981) showed that the actual example of Miyanishi et al. (1979) was a case of the C-D effect.

Furnas (198 1) reformulated the dimensional argument so that it referred to the dimensionality with which resources were distributed, rather than to the dimensionality of the plant’s body. His theory argued that - 3/2 thin- ning was “the consequence of the following three approximations which are considered to be independent of time and growing density. (1) The bi- omass [used by Furnas to mean individual weight] w of the plants is pro- portional to the amount of limiting resource exploited. (2) The amount of limiting resource exploited is proportional to the volume in the environment accessible to the plants. (3) The volume accessible to the plants is propor- tional to p-3’2 [p being plant density].”

Furnas treated light income as inherently two dimensional and the soil environment as three dimensional, unless it was exceptionally shallow. This led him to interpret full-light stands as limited by the soil environment. His

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206 MARK WESTOBY

theoretical argument has the virtue of by-passing the allometries among plant parts, which intractably refuse to accommodate the older dimensional account. Furnas’ account makes the following predictions, some of which differentiate it from others: (1) In low light the slope should change from -312 to - 1. (2) The level of the line should rise in more fertile soils. (3) In soils so shallow as to be effectively two dimensional the slope should change from - 3/2 to - 1.

Furnas’ experiments tended to confirm these predictions, but by his own account experimental problems and large variances made the confirmations weak. Prediction (3) is new, data on predictions (1) and (2) have already been discussed. Prediction (1) is generally thought to be true, but may not be; conversely, prediction (2) has been generally thought wrong, but there are some recent observations suggesting it may sometimes be true.

If any of the above dimensional arguments are true, plants growing in water might be expected to behave differently on the B-N diagram from those on land. J. Cecins and B. Stone (unpublished; student projects at Macquarie University) have tried to investigate self-thinning in the floating plants Salvinia and Eichhornia, on the premise that their growth would be limited to two dimensions. This proved naive; surviving plants typically rode up over those which died, forcing them down in the water. In addition, genets of both genera fell apart quickly as they grew, and the numbers sur- viving could not be counted.

Large algae are attached to a solid substrate, but their photosynthetic tissues are supported above them by the lifting force of the water rather than by a stem which resists gravity. Schiel and Choat (1980) showed that some algae grew better in high-density stands than at low density, and ar- gued that algae did not in general behave in the same way with respect to density as terrestrial plants. However, Cousens and Hutchings (1983) showed that the data of Schiel and Choat (1980) lay inside the thinning boundary as defined by White (1980) and so did not contradict the self- thinning rule.

I have often heard it suggested that the rule might apply to sedentary animals, although no tests appear to have been published. Furnas’ refor- mulation suggests the rule should apply to animal species of whatever kind. He provided data indicating a mean - 3/2 slope for a cohort of trout (Sal- velinusfontinalis) and a mean - 1 slope for planarians fed on tubificid worms. His interpretation that the resource was three dimensional in the one case and two dimensional in the other had a distinctly a posteriori air.

C. Allometric Relations

The explanations discussed above often imply some allometric relation- ship among plant parts. White (1981) reviewed in detail the actually ob- served relationships for trees. He showed that either weight W or volume

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THE SELF-THINNING RULE 207

V are related to stem diameter d at 1.3 m by an exponent which is definitely less than 3. For many species

W a d2.5 (21)

Similarly if the combination of linear dimensions d2h is used, where h is plant height, then

W a (d2hr (22)

where x is always less than 1. White then focused on the ground area covered by a vertical projection

of the crown, C,. He quoted various sources to argue (1) that while C, totalled over all trees per unit ground area varies between different forest types, it is constant through the time-course of thinning for any given for- est, and (2) that there is an allometric relationship

C, a dZa (23)

which describes how plant shape changes during development. The 2 is in- truded so that a is the allometric coefficient relating crown diameter to stem diameter. From argument (1) above and proportionality (23)

N a CA-* a

and then from proportionality (21)

(24)

Under the self-thinning rule proportionality (24) would have an exponent of -312. In other words, for relationships (21) and (23) to generate the self-thinning rule, a should be about 2.513, or at least within the range 2.512.6-2.513.4, i.e., about 0.95-0.74. White then showed that many values of a do in fact lie in this region.

White’s purpose in gathering these data was to show that the traditional but naive W oc 13, N a P - 2 does not fit the facts. He stopped short of putting forward the allometries supported by his data as an alternative ex- planation. Nevertheless, his data certainly suggest W a d2.5, N a d’.6 as an alternative. The effect of his argument is to make the -312 exponent, so redolent of dimensional meaning, appear a purely accidental outcome of other exponents which have no simple explanation, and perhaps no the- oretical one.

The most conspicuous difficulty with this type of explanation is that the two allometries involved show distinct signs of being more predictable within a species or a forest type than between species, particularly between species of radically different growth form. However, if anything, the rule applies better between species rather than within one species (Fig. 2). Another prob- lem is that exceptions would be expected; for example White (1981) cites bodies of data for two species with a in proportionality (23) about 1.6, rather

W a N-2.512”

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208 MARK WESTOBY

than 0.8. These species would then be expected to thin along lines of quite different slope, about -0.8 rather than - 1.5, unless the relationship (21) were different enough from the usual to compensate. Species aberrant by this amount have not been observed. Apparently aberrant tree stands have mostly had slopes steeper, not shallower, than -3/2. Furnas (1981) seized on this observation made by White and Harper (1970), and suggested that heartwood accounted for the extra weight at a given density. Furnas used data from trees other than those from which the thinning data came to suggest that if sapwood alone was counted, this would restore the thinning lines to the expected -3/2 slope. However, since White and Harper (1970), many tree thinning lines of -3/2 slope have been documented; the same transformation applied to them would make their slope flatter than - 1.5.

D. Growth Models There have been some efforts to mesh the ideas, variables, and equations

dealing with growth of individual plants or stands with the self-thinning rule. Such models would be valuable inter alia in describing the time course of thinning, which the rule itself does not. The models of Hozumi (1977, 1980, 1983) and Hara (1984~) take the thinning rule as given. The models of Aikman and Watkinson (1980), Charles-Edwards (1984a,b), Morris and Myerscough (1984), and Perry (1984) predict the -3/2 rule or some of its properties from assumptions about the growth of individual plants, at least under some parameter combinations. Aikman and Watkinson do so by in effect using the traditional dimensional argument. They assume that weight accumulation is proportional to growth in three dimensions, and that mor- tality is driven by the extent to which the surface on which the plant grows is covered. In my judgment, then, although this model is useful, it does not help to explain the -3/2 exponent.

The models of Charles-Edwards, Morris and Myerscough, and Perry do not appear to use the simple dimensional argument. These models offer hypothetical explanations for the -3/2 exponent in that some of their pa- rameter combinations generate it and others do not. The models predict either that certain parameter values are the only ones found, or that the -3/2 exponent is not general. It would seem to me that all these models cannot be right, and perhaps none of them are. This is an area where only time and experimentation will tell.

It is worth noticing that these models all aim to predict within-species thinning trajectories. However, the - 3/2 exponent is much more clear-cut as an across-species relationship; there remains some room for doubt as to how consistently individual species follow that slope.

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THE SELF-THINNING RULE 209

V. RELATION TO POPULATION STRUCTURE

The empirical base of the rule is data from even-aged stands. In tradi- tional life tables and population models age is the index used to characterize individuals, and the predictor of their chances of survival and reproduction. On this basis, even-aged stands would be considered as a homogeneous group of individuals. This is far from true however; thinning stands come to include individuals spanning a wide range of size, and their size is the most important predictor of their fate and reproductive success. Classical demographic models handle plant populations clumsily at best; new models are just beginning to be worked out (Werner and Caswell, 1977; Hubbell and Werner, 1979).

It was recognized early that crowded plant stands developed an “L- shaped” frequency distribution of plant weights (Fig. 21). In early work, this distribution was thought to be evidence in itself of competitive inter- action among individuals. Koyama and Kira (1956) showed that this was not so, and that small random variation among individuals in initial weight, or in relative growth (RGR), or both, produced an “L-shaped” distribu- tion. If the random variation is gaussian, a lognormal distribution results, which can be L-shaped on an arithmetical scale. Figure 22 shows graphically how this comes about. A constant RGR generates a straight-line increase in weight on a log scale, the slope being the RGR. From the same starting point, a bundle of lines with gaussian variation in slope fans out; a slice across this bundle at some later time gives a gaussian distribution of log weights, that is, a lognormal distribution of weights. When weights are log- normally distributed, one-dimensional measures of size such as DBH may often be normally distributed; such measures are commonly the form in which forestry data are collected.

The lognormal is therefore a baseline: the null hypothesis, which would result in the absence of mortality or of interaction among individuals.

Tagetes Tagetes Sinapis L ycopersicon ~~ A l , ~ erecta alba

E 3 P

LL

Fig. 21. Equal-interval frequency histograms of plant weight for four species harvested 6 weeks after planting. From Fig. 5 of Ford (1975).

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MARK WESTOBY

Sampling time

Time

Fig. 22. Schematic diagram of how gaussian variation in relative growth rate can give rise to lognormal variation in weight. See text for further explanation.

Oddly, many distributions actually observed during thinning seem to roughly approximate the lognormal (Fig. 23) or the normal distribution of stem diameter (Fig. 24).

Two main effects operate in developing distributions in even-aged stands. First, large plants suppress small plants more than vice versa (Cannell et al., 1984). The result is a “hierarchy of dominance and suppression”

40

c P .- : 20 c P

O“ 10

e r

0.1 1 10 100 Mean volume (ft3) per tree

Fig. 23. Volume distributions in even-aged stands of Pinusponderosu. Age in years is indicated on each curve. After White (1980), drawn from data of Meyer (1938).

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THE SELF-THINNING RULE 21 1

0

50

40

I I 30 G m Q C

0 .- m

e ; 20

c

10

10 20 30

DBH (inches)

Fig. 24. Diameter distributions in even-aged stands of Pinus ponderosu. Age in years is in- dicated on each curve. From White (1980), drawn from data in Meyer (1938).

(White and Harper, 1970), in which smaller plants are at an accumulating disadvantage, and finally die. This differentiation of RGR has the effect of broadening the distribution. Second, mortality of smaller individuals truncates the distribution from the left.

When growth depends on the size of the individual relative to others in the stand, the consequences for distributions can be understood in terms of distribution-modifying functions (DMFs) (Westoby, 1982). These are functions relating the increment of a size measure to the same size measure, across all individuals in the stand at a point in time. For example, the in- crement of DBH can be graphed against DBH, or RGR against In weight. When these scales are chosen, simple relations result between the shapes of DMFs and the changes which they generate in the shape parameters of the frequency distribution of plant sizes, described using the same measure (Fig. 25). Suppose the DMFs are described by polynomials in x. DMFs de- scribed only by a constant (a term in xo) affect only the first moment (mean) of the distribution. DMFs with a term in X I affect the second moment (stan- dard deviation); those with a term in xz affect also the third moment (skew); those with a term in x 3 affect also the fourth moment (kurtosis). Hara (1984a,b) has developed a formal system of theory on this: his G(t , f ) functions correspond to DMFs.

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212 MARK WESTOBY

Distribution- modifying function

D M F ’

I

Size measure

Fig. 25. Schematic of distribution-modifying functions (DMFs) and their effects on the shape parameters of frequency distributions. All DMFs with positive increments increase the mean of a distribution. DMF 1 does not affect shape parameters; DMF 2 affects standard deviation but not skew or kurtosis; DMF 3 affects standard deviation and skew but not kurtosis; DMF 4 affects standard deviation, skew, and kurtosis. After Westoby (1982).

Few data have yet been expressed in the form of DMFs. Data on thinning stands of herbs are obtained by destructive harvest of replicates, so that data on growth of particular individuals come mostly from the forestry literature. Figure 26 shows DMFs from Mohler et al. (1978), West and Bor- ough (1983), and West (1980) and from data of Ford (1975 and personal communication). West (1981) reported data similar to those in West (1980) but for pure Eucalyptus obliqua stands rather than for Eucalyptus regnans with an admixture of E. obliqua. On the evidence of the data in Fig. 26, most DMFs are of the third type (DMF 3) in Fig. 25; that is, it takes a term in x2, but not in x 3 , to describe the DMF. The smaller plants in the stand all have much the same growth rate, which is negligible or very small. Among the larger plants, in which the biomass increment of the stand is concentrated, growth depends sharply on relative size, with larger plants doing better. The data of Mohler et al. do not show any clear-cut deviation from DMF 2 (Fig. 26a). There are hints of DMF 3, and in the data for a 6-year-old stand even a hint of DMF 4, but the data are not of sufficient resolution to show whether these patterns are real. A DMF of type 3 pro- duces skew to the right in the distribution, and this tendency is often seen.

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THE SELF-THINNING RULE 213

The tendency is often less than might be expected from the DMF alone, because at the same time the peak to the left of the distribution is being eroded from the left by mortality. Figure 26d also shows data from Ford (1975) on mortality of Picea sitchensis. These show that while the smallest 50-70070 of the stand all had much the same growth rate, mortality was concentrated in the smallest 20-30%.

There is substantial discussion in the literature about bimodal frequency distributions during thinning. First it should be made clear that bimodality as such is not a testable property of distributions. While many model dis- tributions (null hypotheses) have two modes, these are either single distri- butions with intense platykurtosis, not different in kind from those with one mode, or else they are distributions compounded of two separate dis- tributions, each with a defined mean, variance, and contribution to the compound distribution. To test for a compound distribution, an a priori estimate of the expected parameters of the component distributions is needed. Consider what happens in a broad flat (platykurtic) distribution. A finite sample from this is likely to have two (or more) modes; the smaller the sample, or the more platykurtic the distribution, the more likely is the sample to have two modes.

Reports of observed bimodal distributions therefore rely on two observed peaks in a finite sample, and sometimes on a test for significant platy- kurtosis. The main data are due to Ford and Newbould (1970), Ford (1975), Mohler et al. (1978), and West and Borough (1983). In addition there have been reports of bimodality of height (Ford, 1975; Hutchings, and Barkham, 1976; Bradbury, 1981) and of seedling weight (Rabinowitz, 1979), presum- ably the consequence of some sort of density dependence during germina- tion, rather than during growth. These data are indecisive in themselves, for the reasons given above, but belief in the phenomenon of bimodality has been reinforced by a series of models which predict the development of bimodality (Diggle, 1976; Gates, 1978, 1982; Aikman and Watkinson, 1980; Ford and Diggle, 1981). In principle it needs a DMF like DMF 4 (Fig. 25) to generate bimodality in a stand (Westoby, 1982). The models which generate bimodality do so because they depict competition among plants as one-sided, i.e., large plants affect small plants disproportionately more than small affect large. It is qualitatively clear, and can be quanti- tatively shown (Westoby, 1982) that such models are likely to generate DMFs like DMF 4. This leaves open the question whether DMFs like DMF 4, and bimodal distributions in even-aged stands, are typical or even com- mon in the real world.

So much for the influence of the self-thinning process on population structure; what of the converse influence of population structure on the self-thinning process? This is one of the most important, yet unexplored, problems associated with the self-thinning rule. On the face of it the self-

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b

2 4 6 8 10 12 Stem diameter (cm)

o o o 0

0

0

0 0 0 1: 0

0

0

1 I 1 I

10 20 30 40 Diameter (cm)

Fig. 26. DMFs: (a) Pnmuspennsylvanica at 4 years (O), n = 31; 6 years (0), n = 26; and 14 years (a), n = 18. Replotted from Fig. 2 of Mohler et ul. (1978). (b) Pinus rudiutu, from West and Borough (1983). (c) Eucalyptus regnans with a proportion of E. obliqua, from plot R12 of West (1980). Stand was 56 years old at start of 4 years over which diameter in- crement was measured. The line Y = 0.67 - 0.0098 x + 0.00032 x2 is the best fit by weighted least-squares regression. (d) Picea sitchensis. from data of Ford (1975 and personal com- munication). Points not connected by lines are DMFs from plantations at 6 ft initial spacing at Clocaenog (0) and Gwydir (A), increment measured over about 5 years. Points connected by lines show the corresponding mortality over the full measurement period for Clocaenog (0) and Gwydir (a); at higher diameters there was no mortality. To be DMFs, data in (d) should strictly be in units of cm year - I rather than cm cm - year - I ; the shape, however, would be very similar.

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THE SELF-THINNING RULE 215

100 0 - -d

- - 60 0 - E 40

A - - H 20 - A 4 0

: E >

I I I I I

thinning rule is a property of even-aged stands; these provide the data base from which the rule has been empirically derived. Certainly, though, the rule does not depend on members of the stand being all of the same size; in self-thinning stands individuals commonly span two orders of magnitude of weight. It is generally thought in plant demography that age is an im- portant predictor of a plant's fate only insofar as it affects the plant's size, size being the essential parameter. Very little is known as to whether stands with population structures other than those which typically arise during growth of even-aged stands will thin along trajectories other than those predicted by the rule.

Two sorts of population structure bear consideration. First, could stands arise in which all individuals were of the same size or nearly so, with an overall biomass-density combination placing them on the self-thinning line, and what trajectory on the B-N diagram would be expected of such stands?

I doubt that it is possible under field conditions for stands to reach the thinning line without substantial variation in size among individuals. As

0.05

0.04

0.03 '; - , 0.02 5 0.07 5

a u

- 0

- 0.8 - %-

0.6 0

L

3 .- g o

a

l- a

a a

20 40 60 DBHOB (cm)

3 5 10 20 30 DBH (cm)

Fig. 26. (continued)

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216 MARK WESTOBY

has already been discussed, even in the absence of competition, logarithmic growth of individuals magnifies small differences in growing ability or in initial size. On top of that, the thinning line cannot be approached without competition, and it is in the nature of competition for light, at least, that larger plants do disproportionately well. The resulting positively sloped DMF will rapidly enlarge even small initial differences between plants. Stone (1978) established stands of Raphanus safivus at very high sowing densities such that thinning resulted as soon as seedlings had exhausted the metabolic reserves in the seeds. He manipulated frequency distributions of seedling weight by sieving seeds into different size categories; among other treat- ments was one with a narrow distribution of seed sizes relative to the con- trol, which was a commercial mixture of sizes. The narrow distribution of seed sizes did produce a narrower distribution of seedling sizes, but vari- ation among seedlings was still very substantial, and subsequent growth and thinning quickly eradicated differences between this treatment and the con- trol.

Second, and most important for the rule, how might stands with two distinct modes of plant size behave on the B-N diagram? The usual for- mulation of the self-thinning rule, with its limitation to even-aged stands, implies that such stands would not obey the rule. On the other hand it has also been argued that even-aged self-thinning stands have bimodal distri- butions of plant size (see above). If in fact the rule is applicable to bimodal or multimodal-sized stands of the sorts that can result from recruitment at two or more times, its value will obviously be even greater than is thought at present.

It may seem extraordinary that information on this possibility is not available from forestry. The problem with forestry data is that stand mea- surements are usually restricted to plants above some minimum size, since it is known that smaller individuals are not likely to be part of the final harvest. Conversely, where studies are carried out on seedlings, it is unusual for the overstory to be fully characterized, and I know of no case where the overstory was characterized and was itself undergoing self-thinning.

Figure 27 shows what seem to be the reasonable possibilities. Figure 27a shows the two starting conditions which will be considered; A is a stand of larger plants, at a biomass-density combination below the thinning line, and to this is added a cohort of new recruits, moving the stand to A' on the line; B is a stand of larger plants on the thinning line, and recruitment of smaller individuals moves it to B'. Because the recruits are two orders of magnitude or more smaller than the larger plants, the immediate increase in biomass due to the recruitment is insubstantial. Figure 27b shows the expected trajectories from A' and B' if these mixed-size stands obey the rule; the trajectories are like those seen for even-aged stands. Figure 27c

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THE SELF-THINNING RULE 217

d I

I I

Fig. 27. Schematic B-Ndiagrams of the possible effects of adding a cohort of small individuals to a cohort of larger individuals in a stand. (a) The nature of the addition: A-A', the resulting mixed-age stand has a biomass-density combination on the thinning line (dashed); B-B' , the resulting mixed-age stand is beyond the line. (b-e) Resulting trajectories under various alter- native hypotheses; see text for further explanation.

shows the expected trajectories if the recruits are killed by competition faster than small individuals within even-aged stands. Figures 27d and e show possible trajectories if the recruits are better able to survive, in the face of competition from the cohort of larger plants, than smaller individuals within a single cohort would be.

As already indicated, we have no data to support one or the other of the possibilities in Fig. 27. Reasonable arguments from general principles could be made to support any of the possibilities. If, for example, the fate of small plants within a stand were to depend only on their size ranking relative to others, and were not affected by how much smaller they were absolutely, then Fig. 27b would be expected, and generally the self-thinning rule would be very insensitive to the shape of the frequency distribution of plant sizes.

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218 MARK WESTOBY

If plants smaller by two or more orders of magnitude of weight than the dominants were suppressed more severely than those less than two orders of magnitude smaller, Fig. 27c would be expected. Third, if plants two or more orders of magnitude smaller were more likely to persist than those nearer in size to the dominants, Figs. 27d or e would be expected. It seems implausible that this third outcome could happen as a result of the two size classes using different resources; if this were a possibility, the spread of size classes in even-aged stands would also be expected to produce trajectories other than those predicted by the rule. (Notice, though, that if the two cohorts form very distinct layers in the canopy, the lower cohort may be well illuminated near the edge of the stand; edge effects may be particularly important in experiments which generate such conditions.) Conceivably, Figs. 27d or e could arise if individuals which started their life in deep shade, or which were so far behind the dominants as to have no prospects of reach- ing the canopy, tended to operate on a “survival” metabolic strategy rather than on a competitive strategy. It has often been argued that seedlings of those forest species which typically recruit in the shade of stands of their own adults or those of other species are able to persist for long periods, making only very slow growth but not dying, until death of the adults shad- ing them releases them into more active growth. This has been argued, for example, for Abies balsamea and for species of tropical rain forests. There is reason to doubt this, however; actual demographic studies of such seed- lings show high mortality rates (e.g., Hett and Loucks, 1968; Hett, 1971; Connell, 1979) and suggest that the continuous presence of large numbers of such seedlings in forest understories is due to continual recruitment more than to high persistence.

VI. PROBLEMS FOR RESEARCH

Although the rule seems at first glance to rest on a solid empirical base, there are many inconsistencies and gaps in our knowledge, as the length of the above discussion shows. My own feeling is that the pattern of trajectory boundaries which underlie the rule probably looks like Fig. 28. (This pat- tern has already been hinted at in Figs. 15f and 19.) The slopes of - 1/2 and 0 would probably be common to the two parts of the limiting line for all species/environment combinations, but the level of the part with - 1/2 slope would vary within a small range, and the breakpoint from - 1/2 to 0 would vary greatly. It is conceivable that some species have thinning slopes which deviate from -1/2, but not so much so that extrapolation over a relevant range of N would carry the species’ trajectory boundary outside the thinning band of overall slope - 1/2.

Although there is very little direct support for the pattern of Fig. 28, my

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THE SELF-THINNING RULE 219

Log N

Fig. 28. Best guess at the pattern which underlies the self-thinning rule. There is a line for each species/environment combination, and this line represents an outer boundary of sus- tainable biomass-density combinations; in the absence of mortality or biomass loss which is crowding independent, stands tend to rise toward the line and then travel along it. Each line is made up of a section with roughly - 1/2 slope, changing at a breakpoint to a section with 0 slope. In the ordinary course of events, stands rarely travel past the breakpoint. The sectors of - 1/2 slope vary somewhat in level (biomass at a given density), but all lie within a “thinning band” spanning the full range of sizes of plant shoots. The width of this band is only about one order of magnitude, small relative to its length; the band characterizes the between-species aspect of the rule.

opinion is that the model makes physiological sense, and that it can accom- modate the available empirical results, bearing in mind that stands should rarely travel beyond the breakpoint and along the line of zero slope. I there- fore judge that the outstanding problems are (1) to understand variations between species and environments in the level, and perhaps slope, of the sloping part of thinning lines, and (2) to characterize the transitions between sloping part and horizontal part and understand variation in these. These two problems converge in the issue of how self-thinning is affected by deep shade, which was discussed at some length above.

The other two outstanding problems about the rule are (3) how it can be applied to mixed-species stands, and (4) whether it can be applied to mixed- age stands, i.e., the influence of population structure on thinning trajec- tories.

VII. SUMMARY

The self-thinning rule is one of a complex of propositions about how plant stands grow in the absence of crowding-independent mortality. Cen- tral to the rule is the equation B = CN- 1’2, which defines a line of slope - 1/2 on a B-N diagram, a graph of log biomass versus log density. There seems to be such a line defining an outer boundary of sustainable B-N

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220 MARK WESTOBY

combinations for each species in each environment; stands follow trajec- tories on the B-N diagram which rise vertically, then travel along the line. In addition, the set of lines for all species-environment situations seems to fall within a comparatively narrow band which is itself characterized by the equation. This “thinning band” is known to span about a 101O-fold range of density of survivors and a 10’-fold range of biomass per unit area, but the band’s width represents less than a 10-fold range of biomass at a given density. It is not yet clear to what extent variation among observed thinning lines in their position within the thinning band is due to the species, due to the environment, or stochastic. Outstanding research problems are to un- derstand this variation, to confirm that the rule applies to mixed-species stands and generalize about what happens to the components of mixed stands, and to elucidate whether, and how, self-thinning trajectories are affected by the population structure of the stand.

I believe the self-thinning rule may come to occupy a central place in our understanding of ecosystems, because it links the dynamics of numbers to the dynamics of biomass in those species which dominate space and struc- ture habitats in many ecosystems. Our knowledge of average net primary production rates needs to be transformed into an understanding of regimes of biomass accumulation and dissipation. These processes are sometimes in steady-state balance, but more often they alternate. The self-thinning rule predicts the demographic, i.e. , natural selection, consequences of phases of biomass accumulation. The rule is therefore essential to an evolutionary and comparative interpretation of the life histories of plants and the bio- mass structures they produce. The physical structure provided by the dom- inant contributors of biomass, in turn, is the most important factor affecting the animal communities found at a given place.

ACKNOWLEDGEMENTS

J. D. Bell, E. D. Ford, R. E. Furnas, A. C. Grice, M. J. Hutchings, W. M. Lonsdale, E. C. Morris, A. R. Watkinson, P. W. West, J. White, and C. Zammit made valuable com- ments. D. A. Charles-Edwards, E. D. Ford, T. Hara, W. M. Lonsdale, D. A. Perry, P. W. West, and B. Zeide have shown me interesting manuscripts in advance of publication. F. A. Bazzaz, E. D. Ford, E. Gorham, P. W. West, and J. White allowed me to use some of their figures. I should particularly like to thank J. White for enlightening correspondence over several years.

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