Bwl. Rm. (1967), 49, p p . 207-264

GRADIENT ANALYSIS OF VEGETATION"

BY R. H. WHITTAKER

Department of Population and Environmental Biology, University of California, Irvine

(Receiwed I May 1966)

CONTENTS

I. Introduction . . . . . (I) The concept of gradient

analysis . . . . . (2) Brief history . . . . (3) Preliminary definitions . .

11. Direct gradient analysis . . . (I) A transect along a single

gradient . . . . . (2) Ordination . . . . (3) Pattern analysis . . . (4) Hyperspaces and evolution .

111. Indirect gradient analysis . . (I) Similarity measurements . .

(2) Matrices and quantitative classification . . . .

(3) Plexuses . . . . (4) Early Wisconsin gradient

analysis . . . . . ( 5 ) Wisconsin comparative ordina-

tion . . . . . (6) Factor analysis . . .

IV. Conclusion and Summary . . (I) Choice of techniques . . (2) Concepts . . . . (3) Perspective . . . .

V. References . . . . .

233 239

141

243 248 254 254 254 255 256

I. INTRODUCTION

(I) The concept of gradient analysis Gradient analysis is a research approach for study of spatial patterns of vegetation.

It seeks to understand the structure and variation of the vegetation of a landscape in terms of gradients in space of variables on three levels-environmental factors, species populations and characteristics of communities. This article reviews gradient analysis both as a group of techniques for analyzing and describing vegetation and as a source of new theoretical understanding of natural communities. It may be fair to say that gradient analysis has changed the conception of vegetation as much as research on the genetic basis of variation and evolution has changed the concepts of plant species. In both cases the change involved shift of emphasis from classification of the objects of study to analysis of kinds and degrees of relationship among these objects.

Gradient analysis and classification are alternative approaches to the vegetation of a landscape. If, for example, one stands on a viewpoint in the Southern Appalachian Mountains in the autumn, one sees a complex and varicoloured mantle of vegetation covering the mountain topography. Different kinds of plant communities are marked

tory, Upton, Long Island, N.Y., under the auspices of the U.S. Atomic Energy Commission. * A contribution from the ecology programme Biology Department, Brookhaven National Labora-

208 R. H. WHITTAKER out in the pattern by differences in the forms and colours of their trees. There is a marked relationship between kinds of vegetation and kinds of topographic position in the landscape. In the valley at one’s feet one may observe these plant communities: (u) a mixed and multicoloured broad-leaved ‘cove forest’ in the valley bottom, (b) a dark-evergreen hemlock forest on the moist, lower north-facing slope just above the cove forest, (c) oak forests marked by red and yellow foliage, on many open slopes above the hemlock forest, ( d ) more open oak heaths in which a layer of evergreen shrubs may be seen through the separated crowns of oak trees and which occupy most of the upper slopes of the valley, and ( e ) pine forest on the dry, upper south-facing slope, with pines forming an open canopy above an open shrub layer. Much the same sequence may be observed in other valleys of these mountains.

The student of vegetation seeks to construct systems of abstraction by which relationships in this mantle of vegetation may be comprehended. The traditional approach is through classification of plant communities into community types. The five kinds of vegetation given above constitute a system of community types by which much of the vegetation of a given elevation belt in the mountains may be classified. As regards the three levels of study, each community type may be characterized by: (a) environment-kinds of topographic positions or ranges of magnitudes of environ- mental factors in which communities of the type occur, (b) species populations-which species are usually present, in what numbers of individuals, in communities of the type, and (c) over-all community characteristics (such as structure of the vegetation, total numbers of species present, total mass of organic material and rate of production) which are shared, within some ranges of values, by communities of the type.

In gradient analysis as an alternative, the five vegetation types are treated as parts of a single continuum from cove forest to pine forest. Relations of the three levels of study now appear in a different light. ( a ) There exists an environmental gradient, along which many characteristics of soils and climates change, from moist valley to dry open slope. (b) Species populations are distributed, each according to its own physiological responses, along this gradient. (c) The different combinations of species along the gradient are recognized as community types by ecologists, and these com- munity types are related to one another along gradients of community characteristics. The community types are the ‘colours’ which man recognizes in the vegetational spectrum (Brown & Curtis, 1952); but the spectrum may also be studied in terms of parallel (or otherwise related) gradients of environmental factors, species populations and community characteristics.

This article will develop in greater detail the meaning of gradient analysis in terms of techniques, concepts and theory. As is often the case, the method, as a broadly conceived research approach, has evolved by a complex interplay of suggestion and substantiation among techniques, concepts and theory. These need consequently to be treated in parallel, rather than as if one had simply resulted from the other. Develop- ment of gradient analysis will be considered through a series of stages or phases. In each phase a more detailed ‘discussion’ of techniques and results is followed by a ‘ conclusion’ stating interpretation in terms of concepts and theoretic implications. I t is hoped thus to serve the interests both of readers interested in details of method for

Gradient analysis of vegetation 209 possible research application and of those interested primarily in surveying the meaning of gradient analysis by way of the conclusions.

The phases will be arranged in two parallel sequences; for two, complementary, approaches must be distinguished within gradient analysis even though there has been extensive exchange of ideas between them and their results are convergent. In the first of these approaches vegetation samples are arranged and studied according to known magnitudes of (or indexes of position along) an environmental gradient which is accepted as a basis of the study. This approach, to which the term gradient analysis was originally applied (Whittaker, I~SI), may be termed direct gradient analysis. In the other approach vegetation samples are compared with one another in terms of degrees of difference in species composition and on the basis of these degrees of difference are arranged along axes of variation. The axes may or may not correspond to environ- mental gradients ; but if they do correspond, the approach to environmental gradients is indirect or inferential. The approach may consequently be termed indirect gradient analysis. My own work has been primarily in direct gradient analysis for the purpose of inquiry into the theory of vegetation structure and classification ; extensive develop- ment of techniques of indirect gradient analysis has occurred in the School of Wisconsin of Curtis and his associates. The direct approach will be discussed first and the indirect approach second, but the reader should recognize that these two streams run parallel in time.

( 2 ) Brief history In earlier vegetation studies it was generally taken for granted that vegetation

‘consisted’ of the community types into which it was classified. These community types were assumed to be well-defined natural units which were part of the structure of vegetation (and not simply part of the structure of a classification) and which generally contacted one another along narrow boundaries called ‘ ecotones’. It was thought that research methods had, of necessity, to be based on these units of which vegetation consisted. So fully accepted was this theory of the structure of vegetation that it had no name as a theory; recently (Whittaker, 1956,1962) it has been designated the ‘ community-unit theory’.

In American ecology the dissent from this theory was first effectively stated by Gleason (1926) in a paper on ‘The individualistic concept of the plant association.’ Gleason advanced two central ideas which may be restated as follows. ( I ) The principle of species individuality-each species is distributed in relation to the total range of environmental factors (including effects of other species) it encounters according to its own genetic structure, physiological characteristics and population dynamics. No two species are alike in these characteristics, consequently, with few exceptions, no two species have the same distributions. (2) The principle of community continuity-communities which occur along continuous environmental gradients usually intergrade continuously, with gradual changes in population levels of species along the gradient. Gleason’s ideas met with intense opposition (Nichols, 1929 ; Clements, Weaver & Hanson, 1929). Gleason restated his views in another paper (1939)’ and later, when the course of time had begun to make the field ready for them, his ideas were supported by Cain (1947) and Mason (1947). During the two decades

I4 Biol. Rev. 42

21 0 R. H. WHITTAKER between 1926 and 1947 these ideas were latent, uninvestigated and largely forgotten.

In the summer of 1947 a study was carried out in the Great Smoky Mountains of Tennessee which was designed to test the community-unit theory and individualistic hypothesis (Whittaker, 1948, 1951, 1956). Results supported Gleason’s ideas. Vegeta- tion was conceived as primarily a complex continuum of populations, rather than a mosaic of discontinuous units. The method of research which dealt with vegetation in terms of continuity and gradient relationships was termed ‘gradient analysis. ’ These results were set forth in a thesis (Whittaker, 1948, see 1956), the logic and evidence of the test of the community-unit theory were developed in a paper (1951, see also 1956), and the findings extended to animal communities (1952).

During the same period research on the forests of Wisconsin was independently carried out by J. T. Curtis and his associates. Results, which strikingly paralleled my own in their demonstration of species individuality and community continuity, were also published in 1951 and 1952 (Curtis & McIntosh, 1951 ; Brown & Curtis, 1952). Work by Ellenberg (1948, 1950, 1952) in Germany during the same period used related methods and obtained similar results, though these were not applied to testing the individualistic hypothesis. Work on different bases led Matuszkiewicz (1947, 1948), Motyka (1947), Major (1951), Walter & Walter (1953), Goodall (1953a, 1954a, b) and Beard (1955) to approaches emphasizing continuity. As is so often the case, discoveries for which the time was ripe were made by a number of scientists working independ- ently. As is so often the case too, others had anticipated the discoveries to a greater extent than was realized at the time.

The Americans had not known, until it appeared from a search of the European literature (Whittaker, 1953), that the statement and rejection of Gleason’s ideas in the United States had been paralleled in France in statements of Lenoble (1927, 1928) and their rejection by phytosociologists (Allorge, 1927 ; Braun-Blanquet, 1928 ; Pavillard, 1928), and that much the same ideas had been expressed in Russia by Ramensky (1924). Ramensky’s formulation of the two principles stated above and their implica- tions for research was clearer than Gleason’s, and his conception of the vegetational mantle was much like that developed by myself in the Great Smoky Mountains. Unlike Gleason and Lenoble, Ramensky (1930) went beyond argument to extensive research on species distribution and community relationships. It is Ramensky, rather than Gleason, Lenoble, Ellenberg, or the recent Americans, who should be recognized as the originator of gradient analysis.

History of schools of ecology and expressions of the community-unit theory have been reviewed in more detail elsewhere (Whittaker, 1962). Despite the prevalence of the community-unit theory, there were a number of other antecedents of gradient analysis; among them should be mentioned the notable studies in Iceland and Den- mark by Hansen (1930, 1932), the work on ecological series in Russia by Keller (1925- 26), Sukatschew (1928, 1932) and others, and studies of forest site-type series by Cajander and Ilvessalo (1921) and others. ’

Gradient analysis of vegetation 21 I

( 3 ) Preliminary definitions A number of terms may best be defined in advance. A particular, limited area of vegetation which seems homogeneous-the area is

limited so that there is no marked, progressive change within it toward a different kind of vegetation-is a plant community. The sum of the environmental factors-or, better, the pattern or gestalt of those interrelated factors-which affect the plants in that community constitute its environmental complex (Billings, 1952). Both plant community and environmental complex are part of a broader system comprising a natural community (of plants, animals and saprobes) and its environment. This open system of community-and-environment is termed an ecosystem (Tansley, 1935 ; Evans,

When information about a plant community (species present, kind of soil, etc.) is obtained, that information constitutes a vegetation sample. Usually a sample is taken from an area of specified size and shape; such a sample area is a quadrat. A collection of samples which represent the vegetation of a landscape or part of a landscape, and which are used for gradient analysis or classification, will be termed a set. Subsets, or smaller numbers of samples from the set which are grouped together for any purpose of gradient analysis or classification, will be termed groups. If an ecologist chooses to formulate a class concept defining a group, the group also represents a (lower level) community type. When samples of a group have their characteristics averaged, the resulting average or synthetic vegetation sample is a composite sample. Composite samples may serve to represent either community types or positions along a gradient in gradient analysis.

A vegetation sample consists of, at least, information on environment and a list of species present, usually with indications of their relative importance. Occurrence refers to the mere presence or absence of a species in the sample area. An importance value is some expression of the massiveness, conspicuousness, activity or interest of a given species in the community. A variety of importance values have been used for different kinds of organisms and research purposes, among them, for terrestrial communities (cf. Curtis & McIntosh, I950), the following. Density is the number of individuals of a species per unit ground surface area (or other spatial measurement). Coverage is the percentage of ground-surface area in a community above which foliage of a given species occurs. Basal area is the area occupied by cross-sections of tree stems at 1-4m. above the ground surface (or other basal measurement) per unit ground surface area. Frequency is the percentage of small subquadrats within a larger sample quadrat in which a given species has been observed. Constancy is the percentage of the larger sample quadrats of a group or set in which a given species has been observed. Presence is the percentage of samples in which a species occurs when the samples are not quadrats of the same size. Species biomass is the total mass (usually, dry weight of organic matter) of a species present at a given time per unit ground surface area. Species production is an expression of the amount of organic matter (dry weight) produced, or energy bound, per unit ground surface per unit time.

1956; Odum, 1959).

14-2

212 R. H. WHITTAKER If an importance value for a given species is divided by the total of the same kind

of importance values for all species in the sample, the resulting percentage is a relative importance value. If two or more kinds of importance values are added together (often in the form of relative importance values) the result is a synthetic importance value. The most important one, two or few species in a community or sample are termed its dominants. Their dominance implies that they have the highest value for some import- ance value in the community or that they are judged most important in community structure and function. Character species are species (which may or may not be im- portant) whose distributions are centred in or largely restricted to a given community type; their relative restriction to it is termed j d e l i t y .

11. DIRECT GRADIENT ANALYSIS

(I) A transect along a single gradient ( a ) Discussion

In a first and simplest application of gradient analysis, vegetation samples are taken at equal intervals along an environmental gradient-for example, 50 m. elevation intervals up a long, even mountain slope. The samples will usually include measure- ments of plant populations in quadrats; I use samples combining counts of trees and shrubs in 0.1 hectare (20 x 50 m.) quadrats with counts of herbs in twenty-five I m. square subquadrats within the 0.1 ha quadrat, and coverage measurements. A field transect-that is, a series of such samples taken as one moves along a gradient in the field-can effectively show how densities of some major plant populations change along the gradient. Clearer results are possible when several samples represent each interval, and their population data are combined so that much of the sample-to-sample irregularity is averaged out. Five samples may, for example, be combined into a composite sample to represent each IOO m. interval along an elevation gradient. Most of my work in gradient analysis has been based on such composite transects. Field transects with one sample per interval were taken along specific elevation and topo- graphic gradients to control the possibility that averaging groups of samples might blur population discontinuities, or otherwise alter apparent population relations from those existing in the field.

Results from a composite transect of the elevation gradient on southwest-facing slopes bearing pine forests in the Great Smoky Mountains are shown in Fig. I (Whit- taker, 1948, 1956). Species populations do not have sharp boundaries along the gradient at points which might correspond to sharply defined physiological limits of tolerance. Each species has a central mode or peak of maximum density, and decreases in density gradually with increasing departure from this peak in both directions; the curves appear to be binomial or Gaussian in form (Gause, 1930; Whittaker, 1951, 1952; Brown & Curtis, 1952). As asserted by Ramensky (1924) and Gleason (1926), no two species have closely parallel distributions. No boundaries separate the three community types an ecologist is likely to distinguish along this gradient-Pinus virginiana forest at low, Pinus rigida heath at middle, and Pinus pungens heath at high elevations.

Gradient analysis of vegetation 213 The population curves and community gradients are responses to the ‘elevation

gradient ’, but elevation is a variable without relevance to the physiology of plants. Along the elevation gradient many factors of environmental complexes-temperature and growing season, precipitation and humidity, wind velocity, atmospheric pressure, evaporation-change concomitantly. The elevation gradient is a complex climatic gradient for which elevation itself is merely one useful index of relative position. Neither elevation nor any other particular gradient can be accepted, without experi- ment, as ‘cause’ of a population distribution. Rather, it is accepted that a complex- gradient of many characteristics of environment exists and population distributions are observed in relation to one another along this complex-gradient.

I I I I I I I I I I I

: 8 0 & 2 1 60 40

20

400 600 800 1000 1200 1400

elevation (m.)

Fig. I . Results from a composite transect of the elevation gradient, Great Smoky Mountains, Tennessee. Samples from dry, south-facing slopes bearing pine forests and pine heaths are grouped by IOO m. intervals. Above, species populations in percentages of stems over I cm. d.b.h. (diameter at breast height, 1.4 m. above the ground) in the sample (Whittaker, 1956): a, pinus virginiana; b, P. rigida; C , P. pungens; d, Quercus marilandica; e, Q. coccinea. Below, trends in community characteristics: f, species diversity of all vascular plant strata in sample quadrats (in percentage of a maximum of forty-four for a cove forest sample, data of Whittaker, 1965) ; g, tree-stratum above-ground net annual production (in percentage of a maximum of 1200 g./mZ./year for a cove forest sample, data of Whittaker, 1966); h, coverage of the pre- dominantly ericaceous shrub stratum (percentages of ground surface above which foliage OCCUT~, data of Whittaker, 1956). Transect data have been smoothed for all curves exceptfand g.

Many of the gradient relations observed must remain observations only-they are essentially correlations for which the context of interlinkage and causation is in- adequately known. Most of them are neither simply coincidental nor causal; they are expressions, in two particular measured variables, of the many reciprocal influences and cyclic processes relating communities and environments in ecosystems. In some cases, however, when the correlation is strong and when there is reason to regard the magnitude of one gradient as functionally consequent on the magnitude of another, the two gradients may be regarded as ‘cause’ and ‘effect’ in a sense appropriate to this context (Whittaker, 1962). It is reasonable, for example, to regard the gradient of decreasing net production of Fig. I as an effect of the gradient of decreasing tempera- -tures and lengths of growing season with increasing elevation.

214 ( b ) Conclusion

R. H. WHITTAKER

Results of broad significance emerge from so simple a technique as following changes in species populations and community characteristics along an environmental gradient :

( I ) Populations of species along continuous environmental gradients typically form bell-shaped, binomial curves, with densities declining gradually to scarcity and absence on each side of a central peak (Fig. I) .

(2) Species are not organized into groups with parallel distributions along the gradient. Each species is distributed in its own way, according to its own population response to environmental factors that affect it (including effects of other organisms), as stated in Ramensky and Gleason’s principle of species individuality.

(3) Because of the tapered form of population distributions of species, composition of communities changes continuously along environmental gradients (if the gradients are uninterrupted and the communities undisturbed). Community types are class concepts abstracted from the continuum of community variation along environmental gradients.

(4) Environments and communities together constitute ecosystems. Since a transect relates a community gradient to an environmental gradient, a transect is an approach to studying a gradient of ecosystems or, to use a term of Clements’ (1936), an ecocline. The environmental aspect of the ecocline, a gradient of environmental complexes in which innumerable factors of environment vary together through space, may be termed a complex-gradient (Whittaker, I 956) in distinction from a factor-gradient for one measurable characteristic of environment.

(5) The corresponding community gradient may be termed a coenocline (Whittaker, 1960). Within the coenocline or whole-community gradient particular characteristics of communities change, forming gradients or trends of such community characteristics as coverage, productivity and species diversity, in which the adaptations of communities to changing conditions along the environmental gradient are expressed (Fig. I).

(6) Study of transects thus permits the relating to one another of gradients on the three levels of study-environment (factor-gradients), species populations (binomial curves and their modifications), and communities (gradients of community composition and trends of community characteristics).

(2) Ordination ( a ) Discussion In a second phase of gradient analysis an environmental gradient may be accepted

as given, but there is no satisfactory environmental index (such as elevation) by which samples may be arranged in sequence. In such cases the samples can usually be arranged by their own characteristics. The process of arranging samples (or species) in relation to one or more gradients or axes of variation is ordination (Goodall, 19543, as a translation of Ramensky’s Ordnung, in German versions of his articles, 1924,1930).

Together with elevation, the ‘topographic moisture gradient’ has a major effect on distribution of plants in mountains. This gradient in a given valley may extend from a moist ravine bottom outward (at essentially the same elevation) to a north-facing slope

Gradient analysis of vegetation 215 and around the ridge to northwest-, west-, and southwest-facing slopes. Great contrasts in moisture conditions characterize the extremes of the gradient, the moist ravine and dry southwest slope. Although it is not feasible in most field-work to measure moisture factors of both soil and atmosphere and integrate them into expres- sions of position along the gradient, there are ways in which vegetation samples can be arranged:

(a) Topographic position may be used as a crude index of moisture conditions. Samples are arranged in a composite transect of which the intervals may be deeper ravines, shallower canyons, sheltered lower slopes and open slopes of varying exposures from northeast and north through northwest, east, and west, to south and southwest. Techniques for more exacting treatment of topographic moisture gradients are discussed by Loucks (1962).

(b) Species distributions in the composite topographic transect may be used for a second arrangement of the samples. Species are grouped by relative positions along the gradient; for the topographic moisture gradient the groups may be: (0) mesics (species with their population modes at or near the moist extreme), (I) submesics (centred in less moist situations than the preceding, but in the moister half of the transect), (2) subxerics (centred in the drier half of the transect but not near the dry extreme), and (3) xerics (populations centred at or near the dry extreme). The numbers given are applied as weights to data on species composition of samples. Thus a ravine forest sample might include IOO trees of which 60 are of mesic species, 30 of submesic, 10 of subxeric, and none of xeric species. Multiplied by weights and divided by the unweighted total number-(60 x 0 + 30 x I + 10 x 2 + o x 3)/100 = 0.50-a weighted average of species composition results, as an index of position along the gradient.

Weighted average techniques were developed independently by authors including, at least, Ellenberg (1948,1950,1952), Whittaker (1948,1951,1956), Curtis& McIntosh ( I ~ s I ) , and Rowe (1956). I use two independent weightings-in forests one with weights applied to densities of tree stems and another with weights applied to frequencies of herbs and shrubs (Whittaker, 1960). When each sample has been thus twice weighted, samples as points are plotted against the two scales as axes of a scatter-figure (Fig. 2). The samples are now grouped, using the two weightings as checks on each other, along the oblique axis of the scatter figure so that five samples represent each interval of a composite transect.

(c) As a third means of arranging samples, (Whittaker, 1960; cf. Bray & Curtis, 1957) they may be compared with composite samples representing the extremes of the gradient. Composition is averaged for a group of samples from most mesic (moist) ravines and for a group of samples from most xeric (dry) southwest slopes, to form two composite end-point samples. Each sample in the full set is compared with both these end points by percentage similarity or some other measurement expressing the degree to which a sample differs from the end-point samples. (Meaning of such measurements will be discussed later.) Samples may now be arranged in sequence from those strongly related to the samples from the mesic extreme and with little in common with the xeric extreme, through various intermediate compositions, to samples strongly related to the xeric extreme with little in common with the mesic

216 R. H. WHITTAKER extreme. The technique may sometimes be aided by use of a composite sample for the mid-point of the gradient (open east-facing slopes, or samples with equal similarity to the mesic and xeric end-point samples) (Whittaker, 1960, fig. 3).

herb and low shrub weighting

Fig. 2. A double weighted-average ordination of Sonoran desert samples from upper (low indexes) to lower (high indexes) valley plain or bajada below the Santa Catalina Mountains, Arizona (Whittaker & Niering, 1965). The scale on the ordinate applies weights to occurrences of tree and arborescent shrub species in 0.1 ha quadrats, that on the abscissa to frequencies of herbs and shrubs in metre-square subquadrats. Points representing samples are grouped by fives into ten steps of a composite transect. Samples which are deviant in relation to this ecocline from upper to lower bajada lie to one side of the axis. Samples above the axis on the left are from less xeric upper bajada disturbed by grazing, those below the axis on the right are from desert washes.

The effectiveness of different techniques for ordinating the same set of samples has been measured (Whittaker 1960; Bray 1961 ; Loucks, 1962). A number of approaches are possible, among them:

(u) It is assumed that increased effectiveness of ordination will be expressed in reduced dispersion of species distributions through the intervals of the transects. T h e numbers of intervals through which species distributions extend are averaged for the transects being compared, using the same lists of species which are represented by sufficient measurements to be useful and occur in several but not all steps of the transect (W-hittaker, 1960; cf. Bray, 1961).

(b) I t is assumed that more effective ordination will be expressed in closer approach to binomial pattern of importance values along the gradient and hence greater consistency in the decline of importance values away from the peak or modal values. Differences between successive transect steps which decline in sequence away from the peak importance value are summed; from this sum are subtracted the amounts by which importance values violating these trends would have to be reduced to render them consistent with the trends. These values are averaged for species in the transects being compared (Bray, 1961 ; cf. Whittaker, 1960).

( c ) It is assumed that more effective ordination will result in closer mean similarities of adjacent samples in the sequence into which samples have been ordinated. A

Gradient analysis of vegetation 217 statistical test, ‘x’ test) comparing the means of similarities to given samples of the three samples closest to them in the sequence with the mean similarity value for all comparisons of samples in the set was applied by Loucks (1962).

The tests used by myself (1960) showed that weighted averages gave a more effective ordination than either arrangement by topographic position or comparison with end- point samples. Bray’s (1961) tests did not show significant difference in effectiveness between weighted average and sample similarity ordinations. Both Bray’s (1961) and Loucks’s (I 962) tests showed that vegetation characteristics (weighted averages or sample comparisons) give more effective ordinations than such factor-gradient magnitudes as light intensity and soil water-retaining capacity.

Ordination by characteristics of the vegetation itself is not only possible, it is in some cases more effective than ordination by any available measurement of environ- ment. A normal research perspective suggests that one should first use measurements of environment to arrange samples along a gradient, and second study the vegetational response to the gradient as expressed in composition of those samples. It is a legitimate alternative, however, first to ordinate samples by measurements of the vegetation, and second to observe the way environmental factors change in the ecocline which the ordination represents. When an ordination based on species distributions is used to study species distributions the approach is circular (Whittaker, 1956). It is usually possible, however, to use a field transect or other independent data to establish that patterns of species distribution in the ordinated sequence of samples are not artifacts of ordination.

The ‘ topographic moisture gradient’ may further illustrate the meaning of the ecocline to which these techniques are applied. The complex-gradient and coenocline are not merely parallel but coupled, for each contributes to the determination of the other. A topographic moisture gradient in mountains is occupied by vegetation. This vegetation contributes, along with weathering, to the development of soil characteristics which affect water penetration into the soil, subsurface flow, and availability to plants; above-ground vegetation affects, by transpiration, re-evaporation of rain from leaves and microclimatic effects on wind and humidity, the water available to the com- munity and the evaporative stress on plant foliage in the community. Moisture conditions affecting plants are determined not merely by precipitation and topo- graphy, but by function of ecosystems which develop in relation to precipitation and topography. The bell-shaped distributions of plant populations represent the responses of populations not only to ‘moisture’ but to all factors of the ecocline, which (in relation to genetic structure and physiological characteristics of species populations) affect the dynamics of species populations. In studying the ecocline as a system of functionally interrelated gradients of environmental factors, species populations and community characteristics, it is reasonable to use for ordination those characteristics of either environments or communities which best combine effective expression of position in the ecocline with relative ease of measurement (Whittaker, 1954b).

Composition of plant communities often provides the most sensitive and most easily measured expression of position in the ecocline. The upper panel of Fig. 3 illustrates, for a larger number of species than Fig. I , change in community composition along

218 R. H. WHITTAKER an ecocline. For the continuum of changing proportions among species populations we may use the term compositional gradient (Bray & Curtis, 1957). The composi- tional gradient is one of a number of possible abstractions from the coenocline (the trends illustrated in Fig. 3 bottom panel, and the sequence of community types to be discussed are others); among these the compositional gradient is most useful for ordination,

aJ 9

2

Y

C

a

1 2 3 4 5 6 7 8 9 10 11 1 2 1 3 , rnesic xeric

moisture gradient

Fig. 3. Results from composite transects of the topographic moisture gradient, Great Smoky Mountains, Tennessee. Above, smoothed population curves of major tree species in percent- age of stems over I cm. d.b.h. in samples, 460-760 m. elevation (Whittaker, 1951). Samples were grouped in thirteen steps by weighted-average ordination. Middle, three types of popula- tion curves (Whittaker, 1956): a , more sharply peaked (Twgu canadensis, 760-1070 m.); b, more broadly dispersed (Acer rubrum, 76c-1060 m., see also Pinus rigida, Fig. I ) ; c, bimodal (Quercus alba, 460-760 m., see also Figs. 4 and 7). Bottom, trends in community characteristics, 970-1370 m. elevation: d, species diversity of all vascular plant strata in sample quadrats (in percentage of a maximum of forty-four for a cove forest sample, data of Whittaker, 1965); e, total above-ground net annual production (in percentage of a maximum of 1200 g./m.2/year for a cove forest sample, data of Whittaker, 1966);f, shrub-stratum coverages (percentages of ground surface above which foliage occurs, data of Whittaker, 1956). The peak in curvefand dip in curve d at transect steps 3-4 reflect high coverage of Rhododendron maximum and low species diversity in forests of Tsuga canadensis, the peak density of which is in the same transect steps at that elevation.

Weighted averages use distributional relations of species to provide expressions of relative positions along the compositional gradient. For this purpose the species must be grouped and assigned numbers for weights. The groupings are clearly arbitrary; the same species can as easily be grouped into three or five groups as into the four used for the ordination on which Fig. 3 was based. Because the species groups are based primarily on location of population modes along the gradient, they have been termed cornmodal groups or commodiu (Whittaker, 1956). Ellenberg’s (1950) term, which will be used here, is ecological groups. Weighted averages are probably the most useful means of ordination by the vegetation itself, along a gradient accepted as given (Whittaker, 1954b).

Gradient analysis of vegetation 219

The diversity of species distributions in Fig. 3 may be noted. Not only are the population peaks scattered along the gradient, the curves show a range of variation of forms (middle panel) from sharp-peaked binomial curves of relatively narrow ampli- tude, through broader, flatter, distributions of wide dispersion, to bimodal curves having two peaks along the gradient. Some of the widely distributed species show morphological gradients or clines, as evidence of genetic complexity of their popu- lations along the gradient. Populations of wide-ranging forest species, like those of the prairies (McMillan, 1960, 1965), may include gradients of genetic composition and physiological characteristics in adaptation to the environmental gradients along which they occur. Other, bimodal populations show partial separation of ecotypes- sub-populations with different genetic, physiological and sometimes morphological characteristics, adapted to different environments-within the species population. Width and pattern of population along a gradient are in part expressions of genetic structure of the population (Whittaker, 1954b, 1956). Bimodal species reduce the effectiveness of both weighted averages and measurements of sample similarity for ordination, and species recognized to have bimodal distributions may be excluded from the computations.

Despite its continuity the coenocline can be divided into community types. The coenocline illustrated in Fig. 3 includes the five community types along the moisture gradient referred to in the Introduction. A sequence of community types along an environmental gradient is an ecological series, and this term and concept have a long and significant history in Finnish and Russian ecology (Whittaker, 1962) from their origin in work of Cajander (1903) and Keller (Dimo & Keller, 1907; Keller, 1925-26).

By means of ecological series, two major approaches to natural communities and their environments-classification and gradient analysis-may be co-ordinated. On the one hand results from a gradient analysis may be summarized in terms of com- munity types familiar to ecologists. On the other hand, community types from a study based on classification may be arranged in sequence in relation to an environ- mental gradient. Thus the remarkable study of Hansen (1930) arranged community types along gradients of elevation and snow cover and observed the trends in life- form composition and other characteristics in these ecological series. Trends of forest production and other community characteristics in ecological series of forest site-types have been studied in both the Finnish school of Cajander (Cajander & Ilvessalo, 1921 ; Ilvessalo, 1922; Palmgren, 1928; Kujala, 1945) and the Russian of Sukatschew (1928, 1932; Sozava, 1927, Sambuk, 1930; Sokolowa, 1935). In the approach of Poore (1956, 1962; cf. Ratcliffe, 1959; Gimingham, 1961; McVean & Ratcliffe, 1962; Ramsay & DeLeeuw, 1965 a, b) lower-level community types, termed noda, are conceived as reference points in the largely continuous, multidimensional pattern of vegetational variation. Relations of noda in the pattern are expressed in terms of ecological and successional series in relation to environmental and successional gradients. In these studies, community types (or the composite samples representing them) have been ordinated; and when the ways in which environmental gradients, species distributions and community trends relate the community types to one another are observed, the approach combines classification and gradient analysis.

220 R. H. WHITTAKER

(b) Conclusions ( I ) Variations from the typical bionomial distribution along a gradient (Fig. 3)

express genetic complexity of the populations. Some species show widely dispersed population curves and morphological evidence of genetic heterogeneity. Other species show partial or full separation of subpopulations or ecotypes with different adaptive centres. Results of a gradient analysis may thus be closely linked with genecology.

(2) These species populations form a complex, flowing population continuum along the environmental gradient. This gradient of populations, as distinguished from the community gradient or coenocline of which it is one aspect, may be termed a compositional gradient. The species occurring along an environmental gradient can be arbitrarily grouped into sets of species having their population centres or modes close together, these sets are ecological groups.

( 3 ) When it is not feasible to arrange vegetation samples into a transect by measure- ments of environment, they may often be arranged by measurements applied to the vegetation itself. The arrangement of samples (or community types or species) in relation to one or more gradients is ordination. Most ordinations not based directly on measurements of environment use measurements expressing relative position in a compositional gradient to arrange the samples. (4) A sequence of community types along an environmental gradient forms an

ecological series. The community types may be either products of an approach through classification, or arbitrarily bounded segments of a coenocline. In either case the ecological series represents a means by which treatment in terms of both community types and the continuities by which these community types are related, results from classification and gradient analysis as different modes of abstraction, may be combined.

( 3 ) Pattern analysis ( a ) Discussion

Means are needed for analyzing communities in relation to two or more gradients. Much of the observed variation of vegetation on mountain landscapes is related to two complex-gradients, elevation and the topographic moisture gradient, and it is natural to treat these as two axes of analysis. The elevation gradient may be used as ordinate and the topographic moisture gradient as abscissa of charts on which popula- tion levels are plotted and outlined with contours (Fig. 4). Source of the background pattern of vegetation will be discussed. In two dimensions, distributional figures for species become binomial solids (Fig. 4, Quercus prinus), any transection of which cuts a binomial curve. These figures may be conceived as hills with peaks at the population optima for species, and slopes declining in all directions with increasing departure from these optima.

No two species have population figures which fully parallel one another, and the population modes are scattered in the pattern of environments and communities represented by the diagram-quite according to the principle of species individuality. Structure of the vegetation pattern may consequently be conceived in terms of some hundreds of these population figures superimposed, forming a complex population

Gradient analysis of vegetation 22 I

Quercus pr inus stems -

Z 4500 -- z

z 0 ;4003-

3500-

3000-

2500-

i

Quercus alba stems -

Quercus borealis and var. maxima stems

i

Fagus grandifolia stems

6000-

9 I TA8LE

MOUNTAIN PINE

HEAlH

PITCH PINE

HEATH

VlRGiNIA PINE

F3fii,T

Fig. 4. Population charts for four tree species, Great Smoky Mountains, Tennessee (Whittaker, 1956). Data are percentages of tree stems over I cm. d.b.h. in composite samples of approxi- mately 1000 stems.

222 R. H. WHITTAKER continuum (Whittaker, 1951, 1956), any transection of which is a compositional gradient.

Genetic complexity appears in the figures for some species (Fig. 4). Two ecotypes, separated by elevations in which the species is nearly absent, appear within the population of Quercus alba. Q w c u s borealis shows an extended cline connecting a more mesic low-elevation population ( Q . borealis var. maxima) with a less-mesic high-elevation population (Q. borealis var. borealis). The population of Fagusgrandifolia includes three ecotypes, one of them (‘gray’ beech, Camp, 1951) dominant in the beach gap forests of high elevations, another (‘white’ beech) in beech-hemlock ravine forests of low elevations, and the third (‘red’ beech) a subordinate tree in cove forests between I IOO and 1400 m.

The ridges of these population figures are displaced toward the left from higher toward lower elevations. Such displacements express a familiar distributional relation- ship. When a species population is followed from a more humid to a drier climate, the centres and limits of species populations shift along the topographic moisture gradient toward its mesic end (Boyko, 1947; Billings, 1952; Walter &Walter, 1953; Whittaker, 1960). This shift toward the mesic may be observed both from higher elevations toward the drier climates of lower elevations within a mountain range (Whittaker, 1956, 1960; Whittaker & Niering, 1964, 1965) and along a geographic gradient from more humid to more arid climate (Walter & Walter, 1953 ; Whittaker, 1960). Similar displacements appear when vegetations on parent materials forming soils with widely different properties are compared within the same climate (Whittaker, 1954a, 1960).

As dominant species shift along the topographic gradient from one climate to another, the community types they characterize must shift. In the Siskiyou Mountains the response of one community type, the mixed evergreen forest (dominated by Pseudotsuga menziesii and broad-sclerophyll trees), to the climatic gradient from humid coastal climates inland to drier continental climates was observed (Whittaker, 1960). Along this climatic gradient the mixed evergreen forest shows continuous shift in topographic position, progressive increase to, followed by decrease from, maximum importance in relation to other community types, and gradual change in floristic composition. In the middle part of this climatic gradient mixed evergreen forest is prevailing climax type, but its prevalence gives way continuously into the area of redwood prevalence near the coast and that of oak woodland prevalence near the inner end of the range.

If vegetation is to be compared between different climates, effects of the topographic gradient within each of these climates must be controlled. There are, then, a number of possible approaches to comparison (Whittaker, 1956, 1960; Whittaker & Niering, 1965): (a) Coenoclines as units and population distributions in them are compared between different climates, elevations or parent materials. If these climates, elevations or parent materials form a sequence along a gradient, vegetation patterns are being approached through a transect of transects. (b) Topographic coenoclines may be treated in continuity with one another in a chart representing vegetation and species distributions in relation to two gradients (Fig. 4). (c) Coencoclines as expressions of climate may be compared in terms of average composition of the samples representing

Gradient analysis of vegetation 223

them, or composition of mid-point samples, or extent of change in community composition along the gradient. ( d ) Comparisons involving three gradients are more difficult, but it is possible to compare along one gradient whole vegetation patterns in relation to two other gradients. Fig. 5 represents two vegetation patterns along the parent-material gradient in the Siskiyou Mountains.

elevicton SISKIYOU MOUNTAINS ff m VEGETATION PATTERN ON DlORlTE

7000 - I I I I I l r l l I

2ooo Trvga rnercensiana

SUBALPINE FORESTS 4000 -

Abler onc color

Sclerophyll-Pseudocrug:,

rawnes ridges. summits draws - lower slopes

5 openslopes

NE EN€ E ESE SE SSE S SSW NNE N NNWNW WNW W WSW SW

mesic xeric

SISKIYOU MOUNTAINS ele"atl0" ft. m. VEGETATION PATTERN ON SERPENTINE

mesic xeric

Fig. 5. Mosaic charts of mountain vegetation on quartz diorite (left) and peridotite and serpentine (right), Siskiyou Mountains, Oregon (Whittaker, 1960). The patterns are based on 270 samples on diorite, 160 samples or records on serpentine, plotted by elevation and topo- graphic position; the lines mark out in the continuous patterns community types of the author's classification.

It is possible for someone intimately acquainted with a vegetation pattern to ,comprehend much of its design in terms of distributions of species populations, but it is not easy to convey this comprehension to others. Simpler means of abstraction are needed and these are likely to depend on more conventional use of community types as units. In one technique the vegetation samples are plotted, each as an individual point, on a chart with elevation and topographic position as axes. Each sample has been classified as a community type, or as intermediate to two community types. Boundaries are drawn around community types outlining the ranges of elevation and topographic positions they occupy and revealing the manners in which they relate to one another in the pattern as a whole. Fig. 5 and the background pattern of Fig. 4 represent such 'mosiac charts' (Whittaker, 195 I , 1956, 1960; Whittaker & Niering, 1965; cf. Waring & Major, 1964). Alternative approaches (see also Gams, 1961) include the construction of compound ecological series, such as that of Sukatschew (1932) for Russian pine forests shown in Fig. 6 (cf. Sukatschew, 1928; Matuszkiewicz, 1947; Gams, 1961), schematic arrangements of community types without the labour of mosaic chart construction (Poore & McVean, 1957; Ratcliffe, 1959; Wace, 1961;

224 R. H. WHITTAKER Johnson & Billings, 1962; Bliss, 1963; Hartl, 1963; Ellenberg, 1963; Florence, 1964), and the community-type plexus to be discussed below.

(b) Conclusion The major concepts of gradient analysis developed above each undergoes transfor-

mation from unidimensional to multidimensional form when more than one gradient is considered :

(I) The ecocline or ecosystemic gradient becomes in more than one direction an ecosystemic pattern or landscape pattern. The complex-gradient in one becomes in several dimensions an environmental pattern. The coenocline in one becomes a com- munity pattern in more than one direction.

Piceeta hylocomiosa r-------------

pic. vacciniosa I I I I

Piceeta sphagnosa

I -- -__ --_- -- --

I Pic. sphagnosum ti\ .--. quercet.:

Reihe C

n Piceeta herbosa

Fig. 6. A compound ecological series for Russian spruce forests (Sukatschew, 1932, fig. 34). Spruce forests with oxalis (P. oxal. = Piceeta oxulidosum) are the central or climax type, from these radiate series: series (Reihe) X toward drier, nutrient-poor soils; series B toward wetter soils with stagnant water and poor nutrient conditions, and bog forests ; series C toward drier soils with more favourable nutrient content and forests mixed with oaks (Piceetu quercetosum) ; and series D toward wet soils with moving water; series E connects the latter with the bog forests.

(2) The binomial curves of species populations become, in two dimensions, binomial solids, modes of which represent adaptive peaks (in terms of population function, not simply of individual physiology) in relation to the pattern of ecosystemic environments that the population encounters (Fig. 4). Genecological complexity of some species populations appears in the form of two or more modes, partially or wholly separated from one another in relation to the environmental pattern.

( 3 ) The compositional gradient in one dimension becomes a complex population continuum in more than one dimension. This population continuum may be conceived in terms of many binomial solids superimposed on one another in relation to the environmental pattern. Trends of community characteristics along gradients become patterns of these characteristics in relation to more than one gradient (Whittaker, 1952, 1956; Whittaker & Niering, 1965).

Gradient analysis of vegetation 225

(4) The ecological series along one gradient becomes a community-type pattern when community types are arranged in relation to more than one gradient (Figs. 5,6). Although such patterns represent high levels of abstraction at which most detail on species populations is lost sight of, they are a fruition of gradient analysis which can bring into communicable form some most important relations among environmental gradients, dominant species populations, and the community types they characterize.

(a) Discussion (4) Hyperspaces and evolution It may be appropriate now to state a theory of the population structure of vegetation

based on community continuity and species individuality, and exceptions thereto. One may choose to recognize in the landscape certain complex-gradients as major

directions of variation of environments, gradients along which many factor-gradients vary and along which environmental complexes of particular points in the landscape are continuously related. These n complex-gradients for a landscape may serve as axes for an n-dimensional abstract space or hyperspace (cf. Goodall, 1963).

At a given place in the landscape a community develops through successional time. Through the course of succession species populations may be replaced by other species populations, and in most cases the environmental complex is increasingly modified by the communities developing in relation to that environmental complex. A succession is thus an ecocline of communities-and-environments changing through time. The succession leads to a self-maintaining community or climax, adapted to the conditions of the environmental complex (as modified) and to sustained utilization of the resources of environment in an ecosystem of relatively stable, steady-state function (Whittaker, 1953). Although the climax communities in two places with different environments may be less widely different than the successional communities which preceded them, their convergence is only partial. Differences in environment are expressed in differences in composition of climax communities ; differences in environ- ments along a complex-gradient are expressed in a climax coenocline. Differences of environments in the complex environmental pattern of the landscape are expressed in a pattern of climax communities (Whittaker, 1953), and the pattern of actual vegetation is further complicated by disturbance and successional communities.

Pattern and hyperspace concepts emphasize the continuity of vegetation, but actual vegetation is a complex mixture of continuity and discontinuity (Whittaker, 1956). Some of the discontinuities are from causes extrinsic to the plant communities, such as topographic discontinuities, contrast of adjacent soil parent materials, and differences in disturbance effects. In general, the more disturbed the vegetation is the more likely it is to appear discontinuous. Some relative discontinuities in vegetation are neither simply extrinsic nor intrinsic. A prairie-forest border, for example, may be made abrupt by effects of fires which are in part set by man but also are part of the normal environment of the fire-adapted grasslands. There are also some relative discontinuities which are intrinsic to vegetation; these have been inadequately studied but deserve brief discussion.

When cove forests of the Great Smoky Mountains are studied in elevation transects from 450 m. up to 1350 m., average composition of the forests changes gradually and

15 Biol. Rev. 42

226 R. H. WHITTAKER continuously. At elevations around 1350-1400 m., however, especially in south- facing canyons and concave slopes, the population density of beech (Fugus grundifoliu) increases rapidly as one climbs (Fig. 7). Mesic deciduous forests of higher elevations are dominated by beech. The beech forests thus appear as a ‘zone’ of vegetation in marked contrast with the cove forests of all lower elevations, separated from them by a relatively abrupt transition (Whittaker, 1956). Similar results were obtained at a far point of the world from the Smokies, in transects from mixed forest into southern beech (Nothofagus nrenxiesii) in New Zealand (Mark, 1963; Scott, Mark 8, Sanderson, 1964)-

3 0 1 e f h L

600 800 1000 1200 1400 1600

elevation (m)

Fig. 7. Population curves in a composite elevation transect of mesic forests, Great Smoky Mountains, Tennessee (Whittaker, 1956). Above, ‘plateau ’ distribution of Fagus grandifolia: a, grey; b, red, and c, white ecotypes (see also Fig. 4). The boundary of the grey beech popula- tion, 1350-1500 m., is sharper in some field transects in south-facing canyons (Whittaker, 1956, fig. 13) , less sharp in transects in north-facing canyons. Below, other major tree species: d, Acer rubrum; e, Tsuga canadensis; f, Halesia monticola (bimodal); g, Tilia heterophylla; h, Acer spicatum; i , Aesculus octandra; and j , Betula allegheniensis (bimodal). Data are percentage of stems over I cm. d.b.h. in composite samples at IOO m. elevation intervals.

Since all major tree species of middle-elevation cove forests in the Smokies extend upward into the beech forests, the partial discontinuity is not a product of competitive exclusion of other species by beech. In certain subalpine shrub communities in other mountains closed patches of the different species form a mosaic; and in each unit of the mosaic one species is strongly dominant and the other shrub species are largely excluded. Evidence from the broad overlap of species populations in general, however, is against the application to plant communities of the concept of sharp boundaries of competitive exclusion for certain animal species occupying closely similar niches (Gause, 1934; Hairston, 1951 ; Whittaker, 1965). As an alternative interpretation it may be suggested that one species, in this case Fugus, has a strong competitive advantage over other species in some range of the environmental gradient. Throughout that range of the gradient this species forms 6900/ , of the canopy trees of the forests. Since it can never form more than 100% or some value below this, its population figure is flattened or vertically truncated, compared with the binomial distributions of other species. Such distributions have been termed ‘plateau ’ distributions (Whittaker

Other observations affect the evaluation of plateau distributions as exceptions to ‘956).

Gradient analysis of Vegetation 227 vegetation continuity. (a) I have never observed plateau distributions in communities of mixed dominance (and a majority of natural communities are such). (b) Relative discontinuities of species populations were in the Smokies limited to the borders of the beech forests and the grassy balds, two communities of relatively ‘special’ or ‘extreme’ environments which form a small part of the vegetation pattern. (c) Studies of moun- tain vegetation in the western United States, which has been described in terms of the life-zones of Merriam (1898), indicate that plant populations in these zones are fully continuous (Whittaker, 1960; Whittaker & Niering, 1964, 1965). Data presented by Daubenmire (1966) suggest (though they do not show the forms of the population curves) that in eastern Washington Artemisia tridentata and Festuca idahoensis may form plateau distributions in vegetation in which other species are continuously distributed. In the study of Beschel & Webber (1962), continuity of vegetation was shown by distribution curves for dominant species and by statistical tests (rank comparison) in swamp forests in which belts dominated by different species could be recognized from the air.

The question of discontinuity may be differently stated in relation to hyperspaces. In the hyperspace of which complex-gradients are axes, positions of samples (or centres of species distributions) may be located by measurements of environmental variables or indexes chosen to represent each complex-gradient. Alternatively, positions may be located by relative similarities of sample composition in a composi- tional hyperspace of which compositional gradients are axes. One may then ask the following questions. (a) Do vegetation samples of a set form natural clusters, separated by space with relatively few samples, in compositional hyperspace? (The samples must be of adequate number, taken from the landscape by means of choice which exclude subjective preference toward under-representation of transitional samples). (b) Do species populations form natural clusters, relatively separate from other clusters, in environmental hyperspace (or compositional hyperspace, or a hyperspace of directions of correlation among species distributions) ?

Statistical approaches to these questions are difficult and results of any technique bearing on them require cautious interpretation. In general, the scattering of samples and species in hyperspaces, in studies of indirect gradient analysis to be described, are in agreement with the continuity of sample composition and scattering of species centres along gradients observed in direct gradient analysis. Goodall (1954b) found that small samples from the Australian mallee formed two fairly distinct clusters representing the ridge and hollow communities. These clusters appeared to be due, however, to the fact that the ridge and hollow environments covered greater areas than intermediate environments. In the beech transect discussed from the Smokies, the beech forest samples form a distinct cluster when compared by percentage similarity (expressing their sharing of dominant species), but not when compared by coefficient of community (expressing relative similarity in total community composition). It is likely that further work will reveal more frequent partial clustering of samples for varied reasons-difference in relative area of habitat types and corresponding com- munity types, extrinsic and sometimes intrinsic vegetation discontinuities, the manner in which strong dominance effects percentage similarity and related measurements,

15-2

228 R. H. WHITTAKER and the tendency toward spottiness in coverage of a vegetation pattern by a limited sample set, which are reasons mostly consistent with vegetational continuity.

In studies of species association (e.g. Bray, 1956; Vries, 1953; Dagnelie, 1960; Looman, 1963 ; Ramsay & De Leeuw, 1964; Beals, 1965) species do not form clearly distinct clusters, though it is possible to use the weak clusters which are suggested as bases of ecological groups. A technique applied to transects (Whittaker, 1960) suggests weakly defined clusters of species at the mesic (ravine) end of certain gradients, but not of other gradients. The clusters are limited to wet-soil streamside species, and these species are dispersed in their distributional relations to elevation. Forest- grassland borders have high species diversities, with numbers of species and ecotypes centred in the transition (‘edge effect’ of Odum, 1959). Analysis of species distribu- tions shows that species are differently distributed within the transition and in relation to other environments and communities (Whittaker, 1956; Bray, 1956, 1960, 1961 ; Patten, 1963). Such transitions are not lines along which communities meet but are themselves steeper community gradients.

Few indeed are the ecological generalizations free from exceptions and limitations. Available evidence suggests, however, that intrinsic relative discontinuities in vegeta- tion, and sample and species clusters, are of restricted occurrence and implication as limitations on the principles of community continuity and species individuality. Some authors have speculated toward the idea that species evolve as multi-species associations and hence may evolve toward formation of natural clusters of associated species characterizing relatively discrete community types (Du Rietz, 1921 ; Allee et al. 1949; Dice, 1952; Goodall, 1963). It is thus necessary to ask the evolutionary mean- ing of the scattering of species centres along environmental gradients which is actually observed.

The question converges with that of how species evolve in relation to one another within the community. The position of the species in the community, its particular way of relating to other species, environment and space within the community, and seasonal and diurnal time, is its niche. If niche characteristics are assumed to be related to one another along gradients, then these gr.idients as axes define a niche hyperspace (Hutchinson, 1957). The principle of Gause (1934; Odum, 1959; Hardin, 1960; Wallace & Srb, 1964) states that no two species in a stable community can occupy the same niche, competing for the same environmental resources in the same part of intracommunity space at the same time. Species consequently evolve toward avoidance of competition by differentiation of niche, by such division of niche hyperspace that each species occupies (or is centred in) a different part of that hyperspace (Hutchinson, 1957; MacArthur, 1960; Whittaker, 1965). Patterns of relative importances of species-lognormal distributions and other forms-linking dominant with subordinate and rare species result from the manners in which niche space and environmental resources are divided among species (MacArthur, 1960; Whittaker, 1965). The community is an assemblage of niche-differentiated species which conceivably, if ecologists understood enough, might be ordinated in a niche hyperspace. The relative richness of a particular community in species per unit area is the community’s aIpha species-diversity (Whittaker, 1960; MacArthur, 1965).

Gradient analysis of vegetation 229 Species can avoid competition also by occupying different habitats; that is, different

positions along environmental gradients and in environmental hyperspace. Implica- tions of species interactions for their distributions do not really support the assumption that they should evolve toward natural clusters of species with closely similar distribu- tions (Whittaker, 1962). Instead, an extension of the principle of Gause implies that species should evolve toward dispersion of their distributional centres in environ- mental hyperspace (Whittaker, 1965). Since the species which occur together are also niche-differentiated, their populations do not form boundaries of mutual exclusion but overlap freely. The structure of compositional gradients-broadly overlapping population distributions mostly of binomial form, with centres scattered along the environmental gradient-is thus a consequence of species evolution toward both niche and habitat diversification.

400 I T 300 8 200

2 100 5 -

400 4- $ 300

a

(I) 200 100

1 2 3 4 5 6 7 a 9 10 rnesic xeric

moisture gradient

Fig. 8. Contrasts of beta diversities of vegetation along topographic moisture gradients. Above, moderately low beta diversity (less change in composition along the gradient, with widely dis- persed population curves) at 460-470 m. elevation, Siskyou Mountains, Oregon (Whittaker, 1960). Below, high beta diversity (narrower population curves, greater change in composition along the gradient) at 1830-2140m., Santa Catalina Mountains, Arizona (Whittaker & Niering, 1965). Half-change values expressing relative change in composition in the ten-step transects were 1.1 for the tree stratum of the Siskiyou transect, 3.4 for that of the Catalina transect. Species, above, a, Taxus brm$olia; b, Chamaecyparis lawsoniana; c, Castanopsis chrysophylla; d, Abies concolor; e, Pseudotsuga menziesii; f, Lithocarpus dem$ora ( x 0 . 5 ) ; g , Quercus chrysolepis ; h, Arbutus menziesii. Species, below, a, Abies concolor ; b, Quercus rugosa ; c, Pseudotsuga menziesii; d, Pinus ponderosa ; e, Arbutus arizonica ; f, Quercus hypoleucoides ( x 0.5) ; g, Pinus chihuhuana ; h, Quercus arizonica ; i, Arctostaphylospringl~; j , Pinuscembroides ; k , Garrya wrightii; 1, Qumcus a o r y i .

The implication of different degrees of species-habitat diversification for vegeta- tional gradients is illustrated in Fig. 8. A larger number of species with narrower habitats (i.e. narrower distributional amplitudes or dispersions) are accommodated along the topographic moisture gradient in the lower coenocline. There is consequently a greater degree of floristic and compositional change along the gradient in the lower coenocline, a higher beta diversity (Whittaker, 1960, 1965). MacArthur (1965) has found that alpha diversities of bird communities are (for a given degree of structural diversity of the plant community) similar in temperate and tropical climates ; but that

230 R. H. WHITTAKER continental tropical bird communities show higher beta diversities. In studied cases of mountain vegetation in the western United States (Whittaker, 1960, 1965; Whit- taker & Niering, 1965)) both alpha and beta diversities, and consequently the landscape species-diversity, or gamma diversity resulting from both these, increase from maritime to continental climates.

Environmental and niche axes may be conceived as forming together a combined environment-and-niche hyperspace in which each species has its own distinctive position. A major trend of species evolution is toward reduction of competition by diversification in the positions of species in this hyperspace; from this trend result the gamma diversities of landscapes and, still more broadly, the richness in species of the living world. Some species have two or more population centres in this hyperspace; ecotypic populations thus considered are experiments in combinations of genetic possibility and adaptive opportunity, within the species. By gradual shift in genetic composition of populations and by evolution of ecotypic differentiation, species may change through evolutionary time their positions in the hyperspace and their distribu- tional and niche relations to other species. Natural communities consequently do not evolve by a phylogeny of divaricate descent. Species may change their associative combinations with one another in evolutionary time (Mason, 1947; Whittaker, 1957), and the evolutionary relations of community types are reticulate.

(b) Conclusion ( I ) From the environmental pattern of the landscape one may abstract an environ-

mental hyperspace, axes of which are the n complex-gradients recognized in the landscape. From the landscape pattern one may also abstract a compositional hyperspace, with compositional gradients as axes; these axes may, but need not, correspond to those of the environmental hyperspace. Representations of compositional hyperspace provide the co-ordinate systems in relation to which samples (and species) are ordinated in indirect gradient analysis, as discussed in following sections.

(2) Species individuality and community continuity may be differently stated in relation to these hyperspaces. Centres of populations of species and their ecotypes are primarily scattered, rather than grouped into clusters in environmental hyperspace. Samples taken by objective procedures are primarily scattered, rather than grouped into clusters representing distinct community types, in compositional hyperspace. Excep- tions to this continuity of sample composition occur, however.

(3) Species evolve toward niche differentiation, by which direct competition within the community is avoided. They evolve also toward habitat diversification, toward occupation of scattered positions in environmental hyperspace, so that plant species are in general not competing with one another in their population centres. The niche differentiation implies that species are partial competitors whose distributions may overlap broadly, forming the population continua along environmental gradients revealed by gradient analysis. (4) From this evolution toward niche-and-habitat diversification there results the

population structure of vegetation patterns. Numerous plant species, with population centres scattered along environmental gradients, each with binomial distributions

Gradient analysis of vegetation 231 broadly overlapping those of other species, freely and variously combine into com- munities which predominantly intergrade with one another, forming a complex and potentially continuous but variously interrupted population pattern. From this popula- tion structure of vegetation result the dficulties of classification which have vexed phytosociologists. Gradient analysis is an effective alternative approach to its investiga- tion and understanding.

(a) Discussion

111. INDIRECT GRADIENT ANALYSIS

(I) Similarity measurements

The essential basis of indirect gradient analysis is the use of comparisons between samples, or between species, in such ways as to cause gradient relationships to emerge from the data. Measurements of relative similarity of sample composition, or of relative similarity of species distribution, are used to arrange samples or species along axes which may correspond to environmental gradients. The approach may con- sequently be described as comparative ordination. The underlying gradient relationships of species populations and environmental factors make the ordination possible, The ordination in turn makes possible the following of gradients of environments, species populations, and community characteristics along the axes of ordination.

There are many possible ways of expressing relative similarity of two community samples. These measurements have been reviewed in some detail by Dagnelie (1960; see also Greig-Smith, 1964; and citations of Goodall, 1962). Among them:

(a) The earliest and simplest measurement, the coeficient of community of Jaccard (1902)~ compares samples in terms of the number of species they share among the total number of species occurring in one or both, without regard to the importance values for species. CC = c/(a + b - c), in which c is the number of species occurring in both samples, a the total number in one sample and b the total number in the other. A number of variants have been used, most widely among them Srarensen’s (1948)

(b ) Other measurements, computed from importance values for species, express the relative similarity of samples in quantitative composition. Measurements of this sort have been independently devised and applied by a number of authors. A simplest version, which may be termed percentage similarity (Odum, 1950), sums the percentage of species composition shared by two samples in the form,PS = IOO - 0-5 El a - b( = 2 min. (a, b), in which a and b are, for a given species, the percentages of importance values in samples A and B which that species comprises. This measurement was used (with quadrat frequencies as importance values) by Gleason (1920). It has been applied to samples of animal communities by Renkonen ( 1 9 3 9 Agrell(1941)~ Odum (1950) Pearson (1963) and others; its computation and characteristics have been discussed (Whittaker, 1952, Whittaker & Fairbanks, 1958). The form, PS = , / [ E ( U - ~ ) ~ ] is suggested by Vasilevich (1962) and Orloci (1966). Related measurements were used by Kulczytiski (1928), Raabe (1952)~ Barkman (1958) and Morisita (1959; Ono, 1961).

(c) The Wisconsin variant of percentage similarity (Bray & Curtis, 1957) is based on a two-step conversion of importance values into percentages. First, in each horizontal

cc = 2c/(a+b) .

232 R. H. WHITTAKER row (for a given kind of importance value for a given species in all samples in which it is represented) all values are converted to percentages of the maximum value in that row. Second, importance values are converted into percentages vertically; in each column the values become percentages of the total of the values (as already horizontally converted) in that column. Columns are now directly compared, with percentage similarities computed by PS = Cmin. (a, b), summing the smaller of the two percentages for species.

(d) Goodall (1953b), Hughes & Lindley (1955) and Groenewoud (1965~) have used the discriminant function (Fisher, 1936) and the related generalized distance (D2) of Mahalanobis (1936) for sample comparison.

(e ) Coefficients of correlation, or of rank comparisons, may be computed for the importance values of species in the two samples (Motomura, 1952; Beschel & Webber, 1962; Ghent, 1963).

Statistical tests of the probability that two samples are the same are of interest in some cases, but they are not appropriate to comparative ordination of samples from different communities. These samples are in fact different, and what is needed is an expression of the degree to which they differ. Such measurement is provided by coefficient of community, percentage similarity and their variants. The two measure- ments have complementary merits and limitations. Kontkanen (1950) and Whittaker (1960; Whittaker & Fairbanks, 1958) have found use of both measurements for the different evaluations they provide is often desirable. The Wisconsin variant of percen- tage similarity to some extent combines advantages of the two measurements. Quantita- tive data are taken advantage of, but the horizontal conversion to percentages has the further advantage of permitting different kinds of importance values, and data on minor, as well as major, species to be effectively used in the same computation.

Some results from direct gradient analysis may clarify the meaning of similarity measurements. Coefficients of community and percentage similarities have been computed for samples which are one unit, two units, three units, etc., apart along transects; the resulting curves are plotted in Fig. 9. Of these curves it may be observed: (a) In the upper parts of the curves the logarithms of similarity measurements decrease in proportion to distance along the environmental gradient. (b) Extrapolated back to zero distance along the gradient, the curves intersect the ordinate, not at rooyo but at lesser values. Such must be the case, for two samples from the same community give similarity values below 100%. Such ‘internal associations’ computed for replicates of foliage insect samples (Whittaker, 1952) ranged from 50 to 80%; com- parison of replicate vegetation samples (Bray & Curtis, 1957) gave a mean value of 82%. Extrapolations to zero units in vegetation transects have yielded values of 60-94% for different strata and different kinds of vegetation studied by the author. (c) In analogy to the half-life, a half-change unit based on reduction of sample similarity to one-half the internal association may be used as an expression of relative distance along compositional gradients (Whittaker, 1956, 1960; Whittaker & Niering, 1965). (d) Sooner or later the curves depart from the straight line and drop rapidly to zero similarity, as they must when distance along the gradient is great enough for no two species to be shared by the samples compared. ( e ) Slopes of these curves, and hence

Gradient analysis of vegetation 233 rates of change of community composition along the gradient, may be quite different for different fractions of the same communities along the same gradient.

The other face of the same problem is the measurement of relative distributional similarity of species. Sample similarity measurements compare vertical columns in tables in which the columns represent samples (or composite samples). Measures of distributional similarity of species (see also Cole, 1949; Goodall, 1952, 1962; Morisita, 1959; Dagnelie, 1960; Greig-Smith, 1964) compare horizontal rows, in which are recorded occurrences or importance values for species, in the same table. Approaches to such measurement include:

(a) Comparison of the actual number of samples in which the two species both occur with the number of joint occurrences to be expected from the product of their frequencies of occurrence (Forbes, 1907, 1925; Dice, 1945; Cole, 1949, 1957; Fager, 1957, 1963). Species association in this case becomes, most simply, A = an/bc (Forbes, 1907); in more effective forms A = (ad-bc)/[(a+b)(a+c)] and related equations of Cole (1949, applied by McIntosh, 1957; Omura & Hosokawa, 1959; Vasilevich, 1961 ; Cook & Hurst, 1962), or A = a/J(bc)-o.5$1 (Fager & McGowan, 1963). In these, a is the number of samples containing both species, b is the number in which the first species occurs and c the number in which the second species occurs (so assigned that b < c), d is the number of samples in which neither species occurs and n is the total number of samples.

80 60 40

1 I 1 I I I I I I I I I I 1

0 200. 400 600 800 1000 1200 sample separation in meters of elevation

I I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10

transect intervals

Fig. 9. Decline in mean sample similarity measurements with increasing sample separation along the elevation gradient in the Great Smoky Mountains (Whittaker, 1956, 1960, fig. 18). The upper curve is for coefficients of community, the lower for percentage similarities, on the same logarithmic scale on the ordinate. The abscissa indicates the separation of the samples being compared in terms of elevation in metres and transect intervals (of 122 m.).

(b) Statistical measures of departure of actual co-occurrence in samples from expected co-occurrence by chance, by application of x2 or fourfold point correlation coefficient to z x z contingency tables (Nash, 1950; Goodall, 1 9 5 3 ~ ~ ; Vries, 1953; Vries, Baretta & Hamming, 1954; Hopkins, 1957; Gounot, 1959; Welch, 1960; Agnew,

234 R. H. WHITTAKER 1961; Cook & Hurst, 1962; Dagnelie, 1962a; McDonough, 1963; Frei, 1963; Greig-Smith, 1952, 1964; Ramsay & De Leeuw, 1964; Beals, 1965; Mueggler, 1965; Yarranton, 1966).

(c) Application of coefficient of correlation, or preferably rank correlation, to the importance values for the two species in samples or columns where both occur (Iljinski & Poselskaja, 1929; Stewart & Keller, 1936; Tuomikoski, 1942; Dawson, 1951 ; Kershaw, 1961 ; Williamson, 1961a; Debauche, 1962; Dagnelie, 1962~) .

( d ) The simple percentages of samples in which both species occur together- ' percentage co-occurrence ' (Whittaker & Fairbanks, 1958). Two versions of percentage co-occurrence, analogous to the coefficient of community variants of Jaccard and Serensen, are PC = a / ( b + c - a ) (Agrell, 1945; Ellenberg, 1956; Whittaker & Fairbanks, 1958) and PC = za/(b+c) (Bray, 1956) in both of which a is the number of samples in which both species occur, b is the number in which the first species occurs and c the number in which the second species occurs.

( e ) To express separately the association of each of a pair of species with the other, the association index of Dice (1945) uses the ratio of the number of samples in which both species occur to the number in which the other species occurs, hence a/c for association of the first species with the second, and a/b for the second with the first (Gardner, 1951; Hegg, 1965). (0 Measurement by the formula used for percentage similarity of samples, applied

to horizontal rows of the table (when data in these rows have been converted to per- centages of the horizontal sums)-' percentage similarity of distribution ' (Whittaker & Fairbanks, 1958).

(g) Distance by which the modes of species populations are separated along a compositional gradient or in a compositional hyperspace (Bray & Curtis, 1957).

The statistical measurements (b) and (c), and measurements based on spatial distances between individual plants in a given community (Goodall, 1965), are appropriate for study of association between species in samples taken from a single community or from a narrow range of environments. Measurement of relative distribu- tional similarity in samples from a wide range of environments and communities is a different problem. Since it is almost always the case that two species are differently distributed; the degree of improbability that their distributions are the same is not the question to be asked.

What is to be measured is the degree of overlap of species distributions (Whittaker & Fairbanks, 1958). Jleasures (d) and (e ) provide simple expressions for this purpose. All these measures are, however, strongly affected by choice of samples; they express distributional relations of species in a set of samples, not in the field. Measurements in form cf) may be most appropriate in theory, but are also most difficult to obtain. There is, from the nature of the problem, no fully satisfactory measurement of distributional similarity of species in a set of samples from different communities. Despite their evident limitations both the non-statistical measures (a), (d ) and ( e ) and, especially for narrower ranges of communities, the statistical measures (b ) and (c) can in practice give useful indications of relative distributional similarity of species.

Gradient analysis of vegetation 235 (b) Conclusion

( I ) Most indirect ordinations are based on measurements of relative similarity of community samples or of species distributions in samples. Principal measures of sample similarity include the percentage of species shared by two samples (coefficient of community) and percentage of the quantitative composition of the samples that is shared (percentage similarity).

(2) These measures may be understood as approaches toward indication of ecological distance, the extent to which samples are distant from one another along compositional gradients, or in compositional hyperspace.

(3) Analogous measurements can be applied to distributional similarity of species. Statistical measurements through x2 or correlation are appropriate to study of distributional relations of species in a given community or a narrow range of environ- ments. For study of species distribution in samples from a wider rangc of environments and communities, simpler expressions reflecting degree of distributional overlap of species populations are appropriate.

(4) The manner in which species populations relate to one another within com- munities and along environmental gradients limits the effectiveness of these measure- ments. Nevertheless, used with skill, perceptiveness of the ecological factors which affect them and consciousness of their limitations, they can give results of considerable scientific interest discussed in the following sections.

( 2 ) Matrices and quantitative classiJcation (a) Discussion

The result of computing similarity values between all pairs of samples in a set is a matrix, such as illustrated in Table I . In a matrix which has not been ordered (Table I

has been ordered) the high and low similarity values are scattered. Much information of interest is present in the thicket of numbers, but techniques are needed by which significant relations may be made to emerge from the thicket.

A first step is the ordering of samples in the matrix to reveal groupings and gradient relations (Kulczytiski, 1928; Renkonen, 1938; Matuszkiewicz, 1948; Guinochet, 1955; Guinochet & Casal, 1957; Bray 1956; Clausen, 19573; Macfadyen, 1963; Balogh, 1958; Whittaker & Fairbanks, 1958; Falitiski, 1960; Agrell, 1963). One sample which is in some sense ‘extreme’ for the set (it may have lowest total of its similarity values with all other samples) is chosen for first position. That sample which has highest similarity measurement with the first is listed second, that sample remaining which has highest similarity with the second is listed third, and so on. By this means (in some cases repeated with different choices of first sample) many, or all, of the highest similarity values are manoeuvred down to the oblique axis. The numbers in the matrix may be replaced with symbols, or degrees of darkening of squares, to form a ‘trellis diagram’ in which the arrangement of high similarity values is visually apparent.

In some cases an effective ordination results; for the samples have been arranged in sequence from the first sample chosen, representing one extreme of the gradient, to

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Gradient analysis of vegetation 237 the sample least related to it, representing the other extreme (Bray, 1956; Dix & Butler, 1960). Such arrangement along a gradient is in part the case in Table I, in which choice of a most oligotrophic lake as the first sample produces an approximate ordination of lakes of increasing solute content from oligotrophic, through eutrophic, to saline (Whittaker & Fairbanks, 1958). It is not necessarily the case, however, that the axis of an ordered matrix will represent a single environmental complex-gradient. Other treatment, beyond the ordered matrix, is often necessary to reveal adequately the directions of interrelation among samples.

The matrix may for, one thing, be used for classification as well as for ordination. In Table I, groups of samples related by high similarity values have been boxed into triangles along the axis. Gaps between the groups on the axis may have no meaning more fundamental than that samples intermediate to these groups did not happen to be taken. The groups along the axis constitute community types which can be character- ized in terms of species composition, environmental characteristics and levels of percentage similarities relating the member samples. It is thus possible to turn the techniques of similarity measurement and matrix treatment to the purposes of quantitative classification of communities.

Extensive work of a Polish school of ecology is based on such an approach. An earlier Polish ecologist, Kulczyhski (1928) was one of the pioneers in application of similarity measurements and use of matrix ordering, ideas for which he gives credit to Czekanowski (1909). Application of these techniques led Motyka (1947) and Matusz- kiewicz (I 947, 1948) to early achievements in comparative ordination and rediscovery of the principles of species individuality and community continuity. A series of articles applied matrix techniques to the classification of Polish vegetation and related the units of classification to those of the school of Braun-Blanquet (Matuszkiewicz, 1950, 1952, 1955, 1958; Motyka, Dobrzahki & Zawadzki, 1950; Motyka & Zawadzki, 1953; Izdebski, 1963a, b; Faliriski, 1958, 1960).

S~rensen (1948) also has used similarity measurements for quantitative classifica- tion of samples. The S~rensen and Jaccard coefficients of community have been used for classification by Dahl (1957) and others (Nordhagen, 1943; Evans & Dahl, 1955; Ellenberg, 1956; Hosokowa, Omura & Nishihara, 1957; Looman & Campbell, 1960). Barkman (1958) compared a number of similarity measurements for classification of epiphyte associations into higher units of the system of Braun-Blanquet. Groenewoud (1965 a) has used the D2measure of Mahalanobis (I 936) to seek clustering of samples in his set. Dahl(1957, cf. Matuszkiewicz, 1948; Barkman, 1958; Ramsay, 1964) found coeffi- cients of community useful aids to classification although it was not possible to specify particular magnitudes to define community-types on the different levels of a hierarchy.

The same techniques of similarity measurements and matrix treatment can be applied to ordination and classification of species. Bray (1956) computed percentage co-occurrence of species in samples from an ecocline between oak forest and prairie in Wisconsin. Co-occurrence values were compiled into a matrix, and the matrix was ordered to arrange species in sequence by relative positions along this ecocline. The species sequence was then divided into three ecological groups, and weights for these ecological groups were used for weighted average ordination of the samples. This

238 R. H. WHITTAKER sample ordination was strongly correlated with one independently obtained by ordering a matrix of percentage similarity measurements for samples.

A number of applications of species-association measurements to community classification are possible:

(u) Ecological groups of distributionally related species may be marked out in an ordered matrix of distributional similarity measurements or may be recognized from correlation analysis. Each of these (because distributional centres of its species are close together) should characterize a community type by occurrence of its species at relatively high importance values in samples from that type. In the system of Braun- Blanquet (195 I) ecological groups become groups of character-species (and in some cases of differential-species) defining associations and other community types (Ellen- berg, 1950, 1952; Gounot, 1959; Looman, 1963; Doing, 1963; Hegg, 1965). Quantita- tive definition of community types thus may be stated in terms of occurrence, or importance, of species of a given ecological group in samples. These community types may be classified into higher-level units by distributional similarity values linking member species of their ecological groups, by their sharing species of ecological groups of wider-ranging species, or by similarity measurements comparing composite samples for the community types.

(b) A contrasting premise may be used for definition of homogeneous community types. It may be assumed that, if an appropriate level of quadrat size for samples has been chosen, existence of correlations among species implies that populations of some species are responding in parallel to environmental factors affecting different samples of the set. Samples are consequently so grouped as to eliminate measured correlations among species within each group (Goodall, 1953a, 19j4b; Greig-Smith, 1964). The resulting groups should be homogeneous communities or community types represent- ing narrow spans of compositional gradients and presumably of environmental gradients (Goodall, 1953a; Hosokawa, 1955; Hosokawa et ul. 1957; Rayson, 1957; Williams & Lambert, 1959; Looman, 1963). These homogeneous types may in turn be further grouped or classified.

(c) Williams & Lambert’s (1959, 1960; Grieg-Smith, 1964) ‘association analysis’ divides the sample set first by the presence or absence of that species which has the highest sum of association values with all other species. The resulting two groups may be in turn divided by the species having highest sums of association values with other species within them, and the resulting four groups similarly divided, and so on; a dichotomous hierarchy is thus generated. Williams & Lambert (1961 ; Lambert & Williams, 1962; Lambert & Dale, 1964; see also Gittins, 1965~) have applied related techniques to classification by sample similarities and to interrelation of results from species-association and sample-similarity treatments.

Quantitative techniques have had extensive recent development in the classification of organisms (Sokal & Sneath, 1963; Heywood & McNeill, 1964; Sokal, 1965, and others). Possible techniques for quantitative classification of ‘ individuals ’ with ‘ attributes’ are closely parallel for taxonomy, where the individuals are organisms (or populations) with lists of characteristics or measured values of those characteristics and ecology, where the individuals are community samples (or composite samples)

Gradient analysis of vegetation 239 with lists of species or importance values for species. Degrees of discontinuity with one another vary widely among taxonomic groups, but community samples are taken from predominantly continuous vegetation patterns in which centres of species distributions are scattered. There are many ways of classifying a given set of samples, or a given set of species. Different similarity measurements may group the samples or species of a set differently and the same measurement applied to the different strata or fractions of the same communities may also group the samples for these communities differently. Quantitative techniques cannot solve the fundamental problems of classifying samples from continuously intergrading communities (Whittaker, I 962) ; but they have possibilities of interest-especially in the exploration and comparison of different ways of grouping samples that the ecologist’s intuition alone cannot explore- which remain to be exploited.

(b) Conclusion (I) Similarity values may be computed among samples of a set and compiled into a

sample-similarity matrix, a table of values comparing each sample with every other sample. Values for relative similarity of distribution of the species occurring in a set of samples may be similarly compiled into a species-association matrix. The matrices embody much information about relations of species and samples to one another, but these relations are not easily apparent.

(2) A simplest means of rendering some of these relations apparent is the ordering of samples, or species, in the matrix so that high similarity values lie along the oblique axis of the matrix (Table I). In some cases the samples (or species) may thus be arranged in a sequence representing a compositional gradient and an environmental complex-gradient. Such is the essence of comparative ordination-the use of similarity comparisons between samples or species to arrange them along a compositional axis. In many cases other techniques than ordering of the matrix are needed for effective ordination.

(3) One may also recognize and delimit groups of high similarity values along the axis of an ordered matrix (Table I). A group of samples related by high similarity values then become a community type. A group of species related by high distribu- tional similarity values are an ecological group and representation of species of this ecological group may be used to define a community type. A number of other tech- niques for quantitative classification of communities are possible.

(a) Discussion It is natural to represent distances by lines on a diagram. When a matrix includes a

small number of samples, these may be arranged in a diagram called a plexus, in which lengths (or widths, or numbers, or distances without connecting lines) express relative similarity of samples (cf. Curtis, 1959, p. 492; Gimingham, 1961 ; Ono, 1961 ; Beak & Cope, 1964; Ramsay, 1964; Barkman, 1965; Ramsay & De Leeuw, 1965b). The Polish school of Matuszkiewicz (1947), Falihski (1960) et al. have published a number of such diagrams, which they term ‘dendrites’. Thresholds may be sub- tracted from the similarity values or the values may be otherwise transformed. Even

(3) Plexuses

240 R. H. WHITTAKER after transformation, the similarity values will not in most cases fit on to a plane surface, and it is often impossible to use lines scaled by similarity values to connect a given sample with samples other than the 2, 3 or 4 with which it is most closely related.

As the number of samples increases, it becomes increasingly difficult to construct a sample plexus and to interpret the meaning of that which is constructed. Solution to the difficulty may involve these steps: (a ) The samples are grouped into community types and importance values for species within each of these community types are averaged to obtain composite samples ; (b ) Similarity values are computed between the composite samples in all possible directions and are compiled into a second-order matrix; ( c ) High similarity values from this matrix are used to construct a second- order plexus relating community types, rather than individual samples. Such a second- order plexus (Fig. 10, left) is a community-type pattern, equivalent for indirect ordina- tion (with community types arranged in relation to one another in a compositional hyperspace) to the mosiac chart and compound ecological series described above for direct ordination.

5

disamointment D. ashlandi

E . agiris k, k,,,;; novamexicanus ' 5 ,41'3'

<D. sici l is

- D. sanguineus

! 8 9 D. shoshone

Fig. 10. A second-order (community-type) plexus and species-association plexus for the same copepod samples (Whittaker & Fairbanks, 1958, figs. 3 and 5) . On the left the six community types bounded on the diagonal of Table I are arranged by percentage similarities (numbers on connecting lines) of composite samples. For each group are entered a characteristic species, mean total salt content of the water (parts per million, the first number), and average micro- crustacean sunlmer standing crop (individuals per litre, the second number). On the right, copepod species are arranged by percentage co-occurrence values (numbers on connecting lines) ; lengths of the solid lines are scaled for these values. The heavy line connecting Eucyclops agi7is and Cyclops bicuspidatus is a centre from which the remaining species radiate on three environmental axes-oligotrophy (Diaptomus ashlandi and Epischura nevadensis), salinity (C. oernalis, D. sicilis, and D. novamexicanus), and seasonal impermanence ( D . sanguineus and D. shoshone).

It is a special value of a plexus, however, that it can sometimes be constructed and interpreted in circumstances which make the direct approach impossible. In favourable cases the plexus may arrange community types in such a way as to render apparent the

Gradient analysis of vegetation 241

design in relation to environments of the pattern of communities from which the samples were taken, Experience suggests, however, cautionary observations. The fallibility of similarity measurements, particularly when bimodal distributions are involved, implies the fallibility of plexuses constructed from them. The limitations of a two-dimensional surface imply that representation of complex relationships is often very difficult. Different (though related) plexuses may result from application of different similarity measurements to the same samples, and from application of the same similarity measurement to different strata or fractions of the same samples, as in the studies of Beah & Cope (1964) and Barkman (1965). Both the limitations of the approach and the real interest of its results in favourable circumstances should be recognized in balance with each other.

The procedures of sample similarity computation, matrix compilation and plexus construction can be turned through 90 degrees. Similarities of species distributions are then measured and values from the resulting matrix used to construct a species plexus. A number of authors have published plexuses for species association (Vries, 1953; Vries et al. 1954; McIntosh, 1957, 1962; Omura & Hosokawa, 1959; Welch, 1960; Agnew, 1961 ; Looman, 1963; Ramsay & De Leeuw, 1964; Groenewoud, 1965b; Beak, 1965; Yarranton, 1966).

When both community and species plexuses are formed from vertical and horizontal analyses of the same community table, they may produce figures which are convergent in their indication of environmental relations. Fig. 10 illustrates a relatively simple case in which the two plexuses can be effectively interpreted in relation to each other and three environmental axes. A more complex treatment was carried out by Hegg (1965) using an association measurement of Dice (1945) to arrange species in a plexus and in ecological groups characterizing lower and higher level community types.

(b) Conclusion (I) Sample similarity values from a matrix may be used to construct a plexus, a

reticulate diagram in which the lines connecting samples are, so far as possible, proportional to ecological distances between these samples. Plexuses can be constructed also on the basis of similarity values for community types (as represented in composite samples) and for species (from measurements of relative distributional similarity of species, Fig. 10).

(2) Various limitations effect these techniques, but in favourable circumstances they produce arrangements in which (a) ecological distances between samples, or types, or species, are represented in the spacing of these in the plexus, (b) the design of the plexus as a whole represents a compositional hyperspace, and (c) known environ- mental relations of the samples permit effective interpretation of that hyperspace in terms of environmental gradients.

(4) Early Wisconsin gradient analysis (a) Discussion The first major studies of the Wisconsin school analyzed upland forests from the

southern (Curtis & McIntosh, 1951) and northern (Brown & Curtis, 1952) parts of 16 Biol. Rev. 42

242 R. H. WHITTAKER the state. Samples were taken from all suitable forest tracts available, so far as possible, by the random-pairs technique of Cottam & Curtis (1949, 1956; Grieg-Smith, 1964), a technique suited to obtaining data from extensive forest stands on level terrain, in contrast to the concentrated, o. I-hectare samples used by myself on mountain topography. The Wisconsin authors found that the samples did not simply fit into a number of distinct community-types, and they sought other means of analysis.

From the ninety-five southern samples, eighty were arranged by their ‘leading dominants ’-the tree species having the highest value of the Wisconsin synthetic importance value (the sum of relative frequency, relative density, and relative basal area). Four dominance types, each characterized by dominance of a single tree species, resulted. Importance values of species were averaged for the samples in each of these dominance types, to produce four composite samples. The samples from the northern part of the state were similarly classified into eleven dominance types. Different ways of arranging the four or eleven composite samples in tables were tried. In only one arrangement (and its mirror image) would importance values decrease progressively through the columns away from the peak values for the species. By study of the tables, weights were assigned to species, ranging from I for species most strongly concentrated at one side of the table to 10 for those most strongly concentrated at the other side. Weighted averages were computed, and the samples were regrouped by ranges of values of the weighted averages into steps of a composite transect.

These techniques of ordination are very close to those I have used (Whittaker, 1951, 1956, 1960), except that trial arrangements of dominance types (‘method of leading dominants ’) replace field transects and topographic composite transects as the preliminary ordination from which species weights are derived. Success of the trial arrangement of dominance types depends both on unimodality of the populations of major species and on preponderance of the effects of one complex-gradient over those of others which would introduce multidimensional relations among the dominance types.

The tables for trial arrangement of dominance types and the composite transects by weighted averages both supported the ideas of vegetation continuity and species individuality (Curtis & McIntosh, 1951; Brown & Curtis, 1952; Curtis, 1955; McIntosh, 1958). Distributions of species populations were observed to be of binomial form (Brown & Curtis, 1952; McIntosh, 1958; Maycock & Curtis, 1960) as suggested by Gause (1930) and Whittaker (1951,1952, 1956). Characteristics of soils were shown to vary, apparently continuously, along the gradient (Brown & Curtis, 1952; Maycock & Curtis, 1960). Distributions of understory plant species (Gilbert & Curtis, 1953) of mosses and lichens on bark (Hale, 1955; Culberson, 1955), of birds (Bond, 1957) and of soil microfungi (Tresner, Backus & Curtis, 1954; Christensen, Whittingham & Novak, 1962) were studied, and these were found to form continua paralleling that of the forest trees. A prairie continuum was investigated by Curtis (1955; Orpurt & Curtis, 1957; cf. Dix & Butler, 1960; Looman, 1963; Knight, 1965), usingspecies weights derived from field observation of distributional relations of species to topo- graphy. Dix (1959 see also Horikawa & Itow, 1958; Itow, 1963), used weighted averages for indication of intensity of grazing effects and as expressions of ecological

Gradient analysis of vegetation 243 distance along a grazing ecocline, an approach related to that of Dyksterhuis (1948). Some of the early Wisconsin studies used measurements of similarity of species distribution (Gilbert & Curtis, 1953; Hale, 1955; Culberson, 1955) and sample similarity (Culberson, 1955) in addition to the weighted-average ordination. Additional applications of trial arrangement of dominance types and ordination by weighted averages include Habeck (1959a, b), Christensen, Clausen & Curtis (1959), Maycock & Curtis (1960), Bray (1960), Lindsey, Petty, Sterling & van Asdall (1961), Anderson (1963), Larsen (1965), and Buell, Langford, Davidson & Ohmann (1966).

To the extent that the characteristics of the vegetation gradient emerge from the trial arrangement of dominance types, rather than from field observation, the arrange- ment of samples in early studies of the Wisconsin school is indirect ordination. Some studies, however, are in fact direct ordination (e.g. Curtis, 1955 ; Maycock, 1963). Although details of techniques differ, my own work and early work of the Wisconsin school are closely parallel studies in gradient analysis, with similar methods leading to similar results on the gradient relations of environments, species populations and communities.

(b) Conclusion (I) In early work of the school of Wisconsin, samples were ordinated by grouping

the samples into dominance types and arranging composite samples for these domin- ance types into ecological series in which species importance values decline smoothly away from peak values. Weights are applied to species according to their relative positions in the ecological series and used to compute weighted averages and arrange samples in a composite transect. (2) The Wisconsin transects gave results convergent with those from my work in

direct gradient analysis, including (a) vegetational continuity, (b) binomial distribu- tions of species populations, and (c ) species individuality. Studies of gradient relations of environments and soils, trees and subordinate plants, animals, and saprobes, in upland forest continua of Wisconsin provided also an effective illustration of the concept of ecocline.

( 5 ) Wisconsin comparative ordination (a) Discussion With the publication of Bray & Curtis (1957), the Wisconsin school turned in a

new direction. Clausen (1957~) and Bond (1957) calculated percentage similarities (of forest herb and forest bird populations) for all samples in their sets compared with one another, chose least similar samples as end points of a gradient, and arranged other samples in sequence along the gradient by their relative similarities to the two end points. This approach, of indirect ordination by sample comparison, was elabo- rated by Bray & Curtis (1957) into a system by which several axes of ordination could be extracted from the sample comparisons. Comparative ordination has two significant advantages. First, it makes possible the ordination of samples without prior assump- tions about environmental gradients; it will even in some cases lead to the recognition of gradient relationships which were not observed in the field. Second, it provides means of arranging a set of samples in relation to two, three, or more axes of variation,

16-2

244 R. H. WHITTAKER multidirectional treatment that may be possible with weighted averages (Ellenberg, 1950, 1952; Waring & Major, 1964) but is usually difficult. The technique is basically simple, but its details are less so; the essential design of the technique may be outlined first.

(a) Samples are taken from an area of vegetation, or range of communities, that is to be studied. (b) Characteristics of the samples which seem most useful for ordination are chosen. (c) Using these characteristics, sample similarity values are computed for every sample compared with every other sample and the similarity values are compiled in a matrix. (d) For each sample the total of its similarity values with all other samples is obtained. That sample which has the lowest similarity total is chosen as one end point of the first axis of ordination. That sample which has the lowest similarity value with the first sample chosen becomes the opposite end point of that axis. (e ) All samples of the set are ordinated along the first axis on the basis of their relative similarities to the two end-point samples. cf) Samples in the intermediate range of this axis are observed and from them the pair of samples with lowest similarity to each other and the first end-point samples are chosen for end-point samples of a second axis. (g) All samples are subjected to a second ordination in relation to this axis, on the basis of their similarity values with the new pair of end-point samples. (h) SampIes in the intermediate ranges of both the first and second axes are observed, and from them the pair of samples with lowest similarity to each other and the second end-point samples are chosen for end points of the third axis. (i) All samples are subjected to a third ordination on the basis of their similarities with the third pair of end-point samples.

The Wisconsin school has its own choice of techniques for the various stages of the method. (a) Forest samples are taken by random pairs or other plotless techniques. (b ) The characteristics of forest samples chosen for ordination have included densities, basal areas and frequencies of larger trees, of saplings and of shrubs, and frequencies of herbs. (c) The Wisconsin ‘index of similarity’ is, as discussed above, a variant on percentage similarity in which importance values are converted to percentages first horizontally, then vertically. (e ) A number of means of computing relative positions of samples between two end-point samples have been tried. Clausen (1957a, b ; cf. Bond, 1957) and Maycock & Curtis (1960) used a simple algebraic expression of position, Bray & Curtis (1957) a geometric treatment of sample similarities. Beals (1960) subtracted similarities from a threshold of 85% and computed position along the axis (x) by a relation based on the Pythagorean theorem, x =(L2+D:-Di) /zL, in which D, and D, are distances of a given sample from the end-point samples and L is the distance of the end-point samples from each other. Distance of the sample from this axis (hence along other axes of variation) may be estimated from e = , / (D;-X~) . Loucks (1962) used an angular transformation of percentage similarity values and computed positions by the formula of Beals (1960).

Some other variations may be mentioned. Bray (1956) and Swindale & Curtis (1957) used indirect ordination of species to derive species weights for weighted-average ordina- tion of samples. Curtis (1959) brought twenty-eight community types of Wisconsin into an ordination by a matrix and plexus treatment, computing community-type

Gradient analysis of vegetation 245 similarities from presence values for ground-level species. Patterns of environmental characteristics, community characteristics and species populations were plotted in rela- tion to the two main axes (temperature and moisture) of the three-dimensional plexus. Curtis’s book (1959), culminating his work, is a masterly treatment of the vegetation of a state, in which ordination is a means to the purpose of organizing and presenting a wealth of information on Wisconsin plant communities.

Maycock & Curtis (1960) used weighted averages (with weights based on field observations of species along the moisture gradient) to select the end-point samples for the first ordination. The second and third axes were derived in the usual manner, by choice of stands of lowest similarity among those of intermediate positions on the axis or axes already analyzed. The approach thus combines direct ordination for an ob- served principal axis of vegetational variation with indirect ordination for secondary axes of less evident meaning.

Loucks ( I 962) compared a three-dimensional indirect ordination by sample similarity with an ordination by environmental scalars-synthetic scales derived by combining values for several environmental factors. Use of three scalars (for moisture conditions, soil nutrients and local climate) produced an ordination related to that from sample comparison, although the axes of the two ordinations were obliquely related to one another. Monk (1965) arranged samples by percentage similarities of their values for five soil variables, thus indirectly ordinating samples by environmental characteristics. Knight (1965) showed occurrence of marked trends in representation of different plant forms along the moisture gradient in a prairie. Samples could be ordinated by representation of plant forms and the resulting ordination was strongly related to that by representation of species. Gittins (1965~2, b ; see also Beals, 1965) has shown that Bray & Curtis’s system may be used in inverse form, ordinating species by their distributional association in samples, with results closely related to those of ordination by sample comparison. Other studies applying indirect comparative ordination following Bray & Curtis (1957) include Ayyad & Dix (1964), McIntosh & Hurley (1964), Gittins (19654 c), White (1965), Larsen (1965), and Swan & Dix

Two limitations of the Wisconsin method may be observed. First, in some circum- stances the comparison with end-point samples is a less sensitive basis of ordination than weighted averages. Comparison with end points uses distributional information less efficiently, and significance of the similarity values may be reduced by unrecog- nized bimodalities of species populations. It is probable that, when a primary gradient is recognized in the field, effectiveness of the ordination may be increased by Maycock & Curtis’s (1960) approach of direct, weighted-average ordination along this primary gradient, combined with indirect, comparative ordination along second and third gradients.

Second, the orientation of axes is determined by extreme samples (Austin & Orloci, 1966). The technique may give the impression that its arithmetic produces a more objective choice of primary axis than an observed gradient accepted for direct ordina- tion. The objectivity is illusory, for the particular orientations of axes are determined by accidents of sample selection in the field. The effect of sample choice on the choice

(1966).

246 R. H. WHITTAKER of axis is not, however, as serious an objection as it might seem. In many cases the technique will select first end-point samples which reasonably indicate the major axis of vegetational variation. The fact that this principal axis may be obliquely related to environmental gradients as ordinarily accepted is not fundamentally objectionable. It is also possible to design techniques for determination of the first axis by the long- axis of variation in the full sample set, rather than by extreme stands (Orloci, 1966).

Beyond the question of oblique axes, indirect ordination will in some cases arrange samples along axes that do not correspond to environmental gradients at all. I n the ordination by Beak & Cottam (1960) the central part of the community pattern was occupied by samples including successional species of Populus and Betula. Samples representing a moisture gradient were curved around these, with the end points of the gradient, the wettest and driest samples, close together on one side of the pattern. The arrangement has ecological meaning. It expresses both the compositional relation- ship of the samples dominated by widespread successional species with all other samples of the pattern, and the sharing of bimodal species by samples from the ex- tremes of the moisture gradient (cf. Maycock, 1963)-extremes which may be alike in some qualities of environment other than position along the ‘moisture gradient’. A Wisconsin ordination is a constructed expression of similarities in species composi- tion of samples; correspondence of its axes to environmental gradients may be sought but not assumed. If, however, ordination itself is the objective, then a position of relativism may be taken on oblique axes and patterns like that of Beals & Cottam (1960). Any compositional axes will serve for ordination and express compositional relationships of samples, and there is no need to require of these axes any particular direction or relation to environmental gradients recognized by ecologists.

For visual representation of results the Wisconsin school has developed charts and models in which sample positions, magnitudes of environmental factors, species importance values and population centres, and community types and trends are shown in relation to the three axes of ordination. Fig. 11 (Ayyad & Dix, 1964; cf. Bray & Curtis, 1957, fig. 7) illustrates the scatter of samples in a three-dimensional hyperspace, with the axes taken in pairs for two-dimensional representation. Sizes of the circles at sample points indicate importance values for the two species. The figures thus represent also the three-dimensional equivalent of the binomial curve and solid, an ‘atmospheric distribution’ (Bray & Curtis, 1957) in which importance values decline in all directions away from the population centre for the species. Fig. 12

(Bray & Curtis, 1957) shows centres of species populations in relation to the three axes of sample ordination, as an alternative to a species-association plexus.

(b) Conclusion ( I ) Like most other indirect techniques, Wisconsin comparative ordination is a

system of extracting directions of variation from similarity measurements. Values of percentage similarity of samples are compiled into a matrix. The sample which has lowest total similarity with other samples becomes the first end-point sample, and the sample with lowest similarity to it becomes the second end point. Usually three pairs of end-point samples are chosen to define three axes of variation. The samples are

Gradient analysis of vegetation 247 ordinated in relation to these axes by their similarities with the end-point samples. A number of other techniques have been used.

(2) The axes are compositional gradients, and similarity values are used as ecolo- gical distances to locate samples along these axes. The axes should also represent ecoclines of changing community characteristics in relation to environmental complex- gradients (which may include successional or disturbance gradients). Relations of axes to environments in the field will not always, however, be intelligible ones. The system of axes defines a compositional hyperspace which should represent (in a transformed version of different proportions) the pattern of environments and communities in the landscape from which the samples were taken. Relations of environmental factors, species populations and community characteristics to one another in the hyperspaces have been studied by various authors of the school.

Z 2

Fig. I I . Sample ordination and patterns of species importance in a three-dimensional hyper- space (Ayyad & Dix, 1964, fig. 2). Centres of circles locate samples in relation to they (ordinate) and x (abscissa) axes on the left, the z and x axes in the middle, and the z and y axes on the right. Sizes of circles represent relative density values in these samples for two species, Agro- pyron dasystachum above, and Koeleria rristata below.

(3) Theoretical results are in accord with those of direct gradient analysis and the earlier Wisconsin research. The study of Bray & Curtis (1957) stated the concepts of compositional axis and atmospheric distribution (n-dimensional binomial solid) in application to species distributions. The strong point of the Wisconsin school's later work, however, is its versatile exploration of techniques of ordination.

R. H. WHITTAKER 248

(a) Discussion (6) Factor analysis

The reader may have noted the resemblance of Wisconsin comparative ordination to the statistical techniques known as ‘factor analysis ’. Wisconsin ordination is, in fact, an approximation to factor analysis for ecological purposes, similar in design and objectives but based on simpler computations. Formal factor analysis (or principal component analysis) can also be applied to community samples, however, as shown by Goodall (1954~) and Dagnelie (1960,1962b; see also Williamson, 1961 b ; Reyment, 1963 ; Colebrook, 1964; Groenewoud, 1965 a ; and Orloci, 1966). Ecologists interested in such applications may take advantage of the effective review by Greig-Smith (1964) and a discussion of notable lucidity by Dagnelie (1962b), to which the present account owes much. The computations are discussed by Thurstone (1947) and in other texts.

Fig. 12. An ordination of centres of species distribution in a three-dimensional hyperspace (Bray & Curtis, 1957, fig. 8). Tree species indicated by genus and species initials: Acer saccharum, Carya cord$mis, C . ovata, Fraxinus americana, &hns cinerea, J. nigra, Ostrya virginiana, Populus grandidentata, Prunus serotina, QWW alba, Q. borealis, Q . macrocarpa, Q. velutina, Tilio americana, Ulrnus americana, U. rubra.

Measurements of distributional similarities of species in samples are first summarized in a matrix. Coefficients of correlation (comparing species distributions by importance values in samples) or coefficients of point correlation (comparing species distributions only by occurrence in samples) have been preferred to other similarity measurements for factor analysis. Values for distributional ‘correlation ’ of each species with itself must be entered on the diagonal of the full matrix. The most appropriate of such values for factor analysis are estimated or computed ‘communalities’. The matrix may be analyzed to extract from it underlying, initially unknown, ‘factors ’ which, acting

Gradient analysis of vegetation 249 together in different combinations with different efficacies, determine the co-occurrence of species in samples.

Effects of ‘factors’ on species importance values may be expressed as

zji = ail F,, + aj2 F2, + ai3 F3, . . . + a,, Fmi + a, S,,.

In this zii is the importance value of species j in sample i, Fl,, F,,, . . ., Fm, are the com- mon ‘factors’ affecting more than one species and determining their importance values in samples, and Sji is a special factor for that sample and species only. The coefficients a,,, aj2, ..., aim are ‘loadings’. These express the correlations of the distribution of species j with the extracted common factors; but the loadings are interpreted as expressing the ‘efficacy’ of the factors in determing the species distribution, or the ‘responsiveness’ of the species’ distribution to the factors. The analogy with an equation of multiple regression will be noted. The technique of factor analysis may in fact be conceived as solving first for coefficients (i.e. loadings) and second for values of unknown independent variables (i.e. extracted factors) when only the measurements of dependent variables (importance values) in samples are known. The solution must be obtained from analysis of the matrix of coefficients of correlation of the dependent variables with one another.

These coefficients of correlation may be represented by

yjk = ajlakl+aj2ak2+aj3ak3... +ajmakm.

In this, rjk is a coefficient of correlation for distributional similarity of species j and I t . The magnitude of rjk is taken to express the degree to which species j and K are alike in their distributional ‘response’ to combined effects of the factors F,, F,, ..., F,, which correspond to the successive terms of the equation. Since values of rjk, etc., are known, the matrix may be solved to obtain the loadings. An infinite number of solutions are possible. Factor analysis computations are designed to obtain a solution which is optimal in the sense of a maximum value for each successive column of loadings (for different species on a given factor, see bottom row of Table 2), hence maximum significance of a given factor in determining differences in species repre- sentation in samples. When the first factor has been extracted, a new matrix of residual coefficients of correlation, expressing the remaining effects of all other factors on species distribution and correlation, is computed. This matrix may be solved for a second factor, and a residual matrix may again be obtained and solved for a third factor, and so on until species correlations are judged sufficiently accounted for by the factors extracted.

Matrix analysis to this point yields loading values (Table 2). The right-hand column of the table (hq) gives the sums of the squares of the loadings for each species. These values (communalities) consequently express the relative ‘ responsiveness ’ of species to the full set of factors extracted. The loadings may serve as co-ordinate values for ordinating species in a first (loading) hyperspace for which the extracted factors are axes. Such an ordination is illustrated in Fig. 13. Distance of a species’ point from the origin on a given axis (its loading value for that factor) indicates the species’ respon- siveness to that factor, The shortest distance between the point and the origin is h,.

R. H. WHITTAKER the square root of the communality. Species near the origin are consequently those least correlated with, or least responsive to, the factors extracted.

The figure illustrates further steps of Dagnelie's (1960, 1962 b) interpretive treat- ment. The circles include species known from field observation to have their popula- tions centred in certain kinds of environments-dry poor soils (C), moist rich soils (A), and wetter soils (B). The species included are thus ecological groups possessing indicator value; they are also diagnostic-species groupings for the school of Braun- Blanquet and may be used to characterize community types. Circle (D) includes species of low communality which occur through the sample set with weak correlation with the factors ; these are 'companions' in the terminology of phytosociology. Axes of the ordination may be rotated together, or separately rotated into oblique relation to each other, and new loading values may be calculated. The dashed lines in Fig. 13 are axes rotated to run through the three ecological groups; the new factor axes may then be identified with complex-gradients of soil characteristics.

Table 2. Matrix of loadings on three factors for fourteen species of French beech forests (shortened from table 34, Dagnelie, 1960, with a; summed for these species only)

Loadings

Fagus silvatica (h) Sorbus aucuparia (h) Anemone nemorosa Milium effusum Festuca silvatica Luzula nemorosa Oxalis acetosella Juncus sp. Carex remotn Vaccinium myrtillus Deschainpsia jlextmsa Dicranuni scoparium Leucobryuni glaucum Hypnum cupressiforrne

0'47 0.28 0.48 0.40 0.64 0.51

044

0.24 - 0.65 -0.57 -0.52 - 0.48 - 0'29

0'1 I

012 0.19

- 0.43 0.14

-0.37 0.33

0.46 0.28 0'32 0.38

-0.18 - 0.24 - 0.36

-0.10

0.37 - 0.38 -0.17 0.14 0.15 0.30 0'43 0.27 0'22

-0.19 -0.18 0.27 0.17 0.43

0'37 0.26

0'44 019 0.56 0.46 0.38 0.29 0.19 056 0.50 0.37 0.32 0.39

Car" 2.96 I .26 1.10

Given loadings, values may be computed for the magnitudes of the extracted factor which, acting through the responses of the species, determine composition of each of the samples. Dagnelie (1960, pp. 140-6) describes a simplified approach to this compu- tation. When these factor values have been computed, the position of each sample in relation to the factors is defined by them; and the samples may be ordinated in a second (factor-value) hyperspace (Fig. 14). As further steps, Dagnelie (1960, 1962b) has plotted species-importance values and forest-production values in relation to the factor-value hyperspace of Fig. 14, thus relating the extracted factors as presumed complex-gradients with gradients of species populations and community characteristics.

Techniques of factor analysis so far described are ' R' techniques, treating vegeta- tion samples as individuals and species occurrences or importance values as attributes or tests. Like other techniques of indirect ordination, those of factor analysis can be

Gradient analysis of vegetation 25 1

turned through 90 degrees to treat species as ‘individuals’ and the samples in which they occur as ‘attributes’ in ‘Q’ techniques (Dagnelie, 1960). Dagnelie (1960, 1962b) has also applied factor analysis to nine attributes of environment of his beech samples. The matrix of correlation is in this case one of coefficients of correlation between factor gradients or other characteristics of sample environments ; the extracted factors are common factors of environment which may be assumed to underlie the particular factor gradients.

-0.5 t ‘2

Fig. 13. An ordination of species (numbered points) in a loading hyperspace, from factor analysis of French beech forests (Dagnelie, 19623, fig. 7). Circles enclose four ecological groups of species related to environmental factors and community types; dashed lines are axes rotated so that extracted factors will more nearly represent the environmental determinants of these groups.

As is the case with Wisconsin comparative ordination, extracted factors before rotation are as likely as not to be oblique in relation to environmental complex- gradients that ecologists recognize. Some extracted factors prove very difficult to interpret ecologically. It is probable that some of these are environmental effects which an ecologist is unable to recognize and that others are false factors resulting from the arithmetic of species correlations. Factor analysis is indeed not a simply ‘objective’ technique from the calculations of which will emerge the ‘true’ factors affecting a pattern of vegetation. The technique is demanding and it is unlikely to be profitable unless ecological skill, understanding and thoughtfulness are invested along with labour and computer time.

The most serious problems affecting factor analysis may result from the nature of compositional gradients. In psychological application, it may be assumed that

R. H. WHITTAKER attributes (test scores) bear a relation to one another and the extracted factor that, if not strictly linear, is at least monotonic or unidirectional. The relation of environmental factor gradients to one another and an extracted common gradient is of analogous design, but the relationships of populations in a compositional gradient are not. Variation of a given attribute (species importance value) along the compositional gradient is normally curvilinear (the slope of the binomial curve), but will often also be bidirectional, sloping in opposite directions on the two sides of the mode. Slope of the importance value may reverse itself more than once along the gradient when the population distribution is bimodal.

If the samples are drawn from a narrow range of a compositional gradient, the species may form three distributional groups of about equal size. Species of two groups have their modes near or beyond the opposite limits of the range of the gradient sampled, and the importance values for species of these two groups decline in opposite directions along the gradient. Positive correlations within these groups and negative correlations between them are responsible for the extracted factor, in relation to which they have high loading values of opposite signs, and for the effectiveness of the ordination.

As samples are taken from a larger range of a gradient, a larger number of species will have their modes between the extremes of the gradient as sampled. The propor- tion of species with unidirectional slopes decreases in the total set of species, while the proportion with bidirectional slopes increases. Effectiveness of the unidirectional species for the factor analysis is decreased, especially within the central range of the gradient. Within this central range, positive correlations may be expected among species whose modes are close together and whose importance values decline together in both directions away from their modes (cf. Groenewoud, 1965~) . The implication of these correlations for factor analysis is unestablished, but unlikely to be favourable. It consequently seems that factor analysis is best suited for intensive studies of sample sets from narrower ranges of environments (Greig-Smith, I 964).

(b) Conclusion ( I ) Mathematically more advanced techniques of factor analysis can be applied to

vegetation samples, with results similar to those of Wisconsin comparative ordination. In this approach: (a) A matrix of distributional similarities of species is compiled. (b) The matrix is solved to yield one or more extracted ‘factors’ and ‘loadings’ of species expressing the degree to which their distributions are correlated with those factors. The loadings provide an ordination of species in a hyperspace of which the factors are axes (Fig. 13). (c) Factor values are computed for the samples. The factor values for a sample are estimates of magnitudes of the different extracted factors which (acting through the different loading values) determine species composition of the sample. Factor values permit ordination of samples in a hyperspace having the same extracted factors as axes, but different scales of magnitudes (factor values rather than loadings) along these axes (Fig. 14). (d)Values for environmental factor gradients, species populations and community characteristics for the samples can be plotted in the factor-value hyperspace and thus related to one another and the extracted factors.

Gradient analysis of vegetation 253 (e ) Groups of species which are close together in the loading hyperspace, and groups of samples which are close together in the factor-value hyperspace, also provide a basis for vegetational classification if this is desired.

(2) These techniques: (a) are often laborious in relation to their research rewards; (b) produce extracted factors which require ecological understanding from other sources, and often rotation in the hyperspace, for their interpretation; (c) can probably in some cases produce extracted factors which are ecologically unidentifiable or even meaningless in relation to environmental gradients ; ( d ) are subject to limitations resulting from the nature of species distributions and the limitations of correlations as means of expressing distributional relations of species; and (e ) are increasingly affected by these limitations as the range of environments and communities represented in the sample set increases. The techniques consequently seem most appropriate for sample sets from narrower ranges of environment on which an investigator is willing to make heavy investment of effort and interpretation.

old c""' 0

classification 1 2

- 2 t 0

I O 0

+ 2 0

1" 0 0

Fig. 14. An ordination of samples in a factor-value hyperspace, from factor analysis of French beech forests (Dagnelie, 1962, fig. 9). Samples have been classified into community types by a conventional classification by undergrowth, left, and by representation of the ecological groups of Fig. 13, right, as indicated by the symbols at sample points. Oldclassification: 0, beech forest with Festuca silvatica; -, beech forest with Luzula nemorosa (fresh variants); + , beech forest with L. nemorosa (dry variant); 0, intermediate beech forest; I, beech forest with Vaccinium myrtillus. New classification by representation of ecological groups (see Fig. 13): 0, groups A and B ; -, groups B ; + , groups B and C; I group C.

(3) The extracted factors may be regarded as compositional gradients. Factor analysis applied to natural communities thus converges with other techniques of indirect gradient analysis in meaning. An indirect gradient analysis should make possible : (a) a treatment causing the emergence from sample data of axes representing directions of variation in community composition, (b) use of these axes to ordinate samples and species in a hyperspace, (c) identification of environmental complex- gradients corresponding to these axes, in some cases, (d ) observation of variations in environmental factors, species importance values and community characteristics through the hyperspace, and (e ) comprehension, through the ordination and these observations, of the pattern of variation of the vegetation sampled.

254 R. H. WHITTAKER

IV. CONCLUSION AND SUMMARY

(I) Choice of techniques Within gradient analysis as a method, two approaches may be distinguished-

direct analysis in relation to an environmental gradient accepted as given and indirect analysis in which axes of variation are obtained from similarity measurements. Within each of these approaches a variety of techniques may be applied to different problems and purposes ; choice among these techniques cannot be categorically stated. Rewards are often greatest, in fact, for the ecologist who experiments with more than one technique for clarifying the relationships in his sample set.

Advantages of direct and indirect techniques are evenly balanced in application to different circumstances. Direct gradient analysis has marked advantage in clarity and interpretability of its results, when conditions are favourable for the direct approach. In some circumstances, however, it is better not to accept environmental gradients as given, as a basis of ordination. Matrix and plexus techniques and Wisconsin compara- tive ordination can be applied to research circumstances and kinds of sample sets for which direct gradient analysis is inappropriate or ineffective. Direct gradient analysis is usually the approach of preference when environmental complex-gradients are recognized and samples can be taken to represent the gradients adequately in the sample set; indirect gradient analysis is the approach of preference in other circumstances.

As regards techniques of more, or less, mathematical complexity, varied purposes must be distinguished-the effort to understand a pattern of vegetation by the most direct and effective means, the testing of hypotheses on vegetation structure, and the exploratory application of mathematically advanced techniques to see what they may accomplish. All are well justified, but the first is the most common. Mathematical treatment has often the great merit of revealing relationships which are inherent in study material but otherwise unrecognizable. I t is the case, however, and no fault of mathematically inclined ecologists, that the character of species distributions and compositional gradients imply diminishing returns from complex mathematical treat- ment of vegetation. Advances in theory of vegetation structure and classification have come predominantly from simpler techniques, from the results of which the basis of complex and indirect techniques may be understood. This is by no means to oppose mathematical treatment, but to balance against its interest and value the fact that for some kinds of vegetation research simpler and more direct methods, carried out with skill and perceptiveness, are more effective.

(2) Concepts A natural community and its environment are linked together by cyclic and inter-

determinant processes into an ecosystem. A landscape may be conceived as a pattern of ecosystems related to one another along environmental gradients. One may choose to distinguish, as aspects of the landscape pattern, an environmental pattern and a community pattern, which vary in parallel in the landscape space.

Along a single environmental gradient chosen for study, environments and com-

Gradient analysis of vegetation 25 5 munities change together, forming an ecosystemic gradient or ecocline. The environ- mental part of the ecocline, comprising many gradients of particular environmental factors, is a complex-gradient. The parallel gradient of communities is a coenocline, and the coenocline includes various trends or gradients of particular characteristics of communities. Species populations are generally distributed along complex-gradients in the form of binomial curves, with their densities tapering on each side of the popula- tion optima. Form of the curve and location of the optimum express in part the species’ genecology-its adaptive centre and range of genetic differentiation as ex- pressed in population distribution. Species evolve toward adaptive differences, as indicated by different locations of their population optima, different niches within communities, and the overlap of population distributions made possible by different niches. Because of their scattered centres and distributional overlap, the species populations along a complex-gradient form a population continuum, a compositional gradient. A compositional gradient, and the coenocline of which it is one expression, may be divided into segments, and the resulting sequence of community types is an ecological series.

More than one ecocline may be recognized in a landscape pattern as directions of variation of its environments and communities. Two or more complex-gradients as axes define an environmental hyperspace, which may represent in abstract form much of the environmental variation in the landscape. Two or more compositional gradients as axes define a compositional hyperspace representing much of the variation in vegetation in the landscape. In a hyperspace the binomial distributions of species become binomial solids or atmospheric distributions. Groups of species whose population centres are close together along a gradient or in the compositional hyper- space are ecological groups. Relative similarities of vegetation samples express their ecological distances-their degrees of separation from one another along compositional gradients or in compositional hyperspaces.

Representation of ecological groups and measurements of sample similarity provide means by which samples can be located in the compositional hyperspace. The process of locating or arranging samples, or species, or community types in relation to one another along gradients or in a hyperspace is ordination. An ordination of community types in relation to more than one gradient, a multidimensional ecological series, forms a community-type pattern. The community types of the pattern are to be understood as abstractions from the landscape pattern, interconnected by those gradients of environments, species populations and community characteristics which gradient analysis provides the means of studying in relation to one another.

(3) Perspective The four preceding paragraphs state a system of interlinked concepts through which

the method and the relations of its various techniques to one another may be under- stood, and in which the principal theoretical contribution of gradient analysis is embodied, This theoretical advance is the replacement of the conception of vegetation as consisting of the units into which ecologists have classified that vegetation, units often of unclear relation to one another, by the conception of vegetation as a pattern of

R. H. WHITTAKER intergrading communities, a complex population continuum. Research techniques based on gradient relations and a multidimensional pattern, co-ordinate system, or hyperspace are appropriate to this conception.

Scientific knowledge may be viewed as a structure of relationships, and relationships of relationships, among unknown entities (Hutchinson, 1953). It is desired that the relationships should be, so far as possible, quantitatively expressed, experimentally established, tightly and coherently interrelated in deductive systems, and causal or predictive from lower levels of organization to higher. Results of gradient analysis cannot yet claim experimental verification, exact mathematical interlinkage, or predic- tive value except in a limited sense. Further understanding of the basis, in genetics and population dynamics, of observed patterns of species distribution must await further, more difficult research. Vegetation analysis is not one of the exact sciences; character, complexity, and relative looseness of the relationships studied imply that appropriate methods and feasible achievements differ widely from those of physical sciences (Whittaker, 1957, 1962; Slobodkin, 1965). It may be a fair measure of the contribution of gradient analysis that it has provided, despite these limitations, a clearer, deeper, and more coherent understanding of vegetation relationships, together with quantitative techniques appropriate to these relationships and a wide range of ecological problems.

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