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Measuring compositional change along gradients

Mark V. Wilson I & C. L. Mohler 2,* 1 Environmental Studies Program, University of California, Santa Barbara, CA 93106, U.S.A. 2 Ecology and Systematics, Cornell University, Ithaca, NY 14853, U.S.A.

Keywords: Beta diversity, Gradient analysis, Multivariate methods, Niche, Ordination, Siskiyou Mts., Species distributions, Species turnover, Succession

Abstract

A new procedure for measuring compositional change along gradients is proposed. Given a matrix of species-by-samples and an initial ordering of samples on an axis, the 'gradient rescaling' method calculates l) gradient length (beta diversity), 2) rates of species turnover as a function of position on the gradient, and 3) an ecologically meaningful spacing of samples along the gradient. A new unit of beta diversity, the gleason, is proposed. Gradient rescaling is evaluated with both simulated and field data and is shown to perform well under many ecological conditions. Applications to the study of succession, phenology, and niche relations are briefly discussed.

Introduction

The observation that community composition varies along environmental gradients is basic to much work in community ecology. Gradients not only form the context for many field studies (see Whittaker 1967, 1972) but figure heavily in much theoretical work on niche relations (Hutchinson, 1958; McNaughton & Wolf, 1970; Whittaker et al., 1973). Despite the large role of compositional gra- dients, or coenoclines as they are sometimes called (Whittaker, 1967), statistical methods for the anal- ysis of continuous compositional variation remain rather primitive.

We address three related problems. First, one

* The authors thank E. W. Beals, R. Furnas, P. L. Marks, R. K. Peet, O. D. Sholes and the late R. H. Whittaker for helpful comments. This work was supported in part by a National Science Foundation grant to Robert H. Whittaker of the Section of Ecology and Systematics at Cornell University, and in part by Mclntire-Stennis Grant No. 183 7551 and a grant from the National Park Service, l~oth to Peter L. Marks of the Section of Ecology and Systematics at Cornell University.

Vegetatio 54, 129-141 (1983). Dr W. Junk Publishers, The Hague. Printed in the Netherlands.

may want to know the biological length of some coenocline, which is to say, the total amount of compositional change associated with the gradient. Such a measure is particularly useful when compar- ing two or more gradients (Whittaker, 1960; Peet, 1978). For example, if Janzen (1976) is correct in believing that mountain passes of given elevation are biologically 'higher' in the tropics, then the amount of compositional change from sea level to the pass should be greatest at tropical latitudes and decline toward the poles. Whittaker (1960, 1965, 1972) refers to compositional differentiation along gradients as beta diversity, in contrast to the alpha diversity or richness of a single sample. He and subsequent authors have measured beta diversity in units of half-changes (HC), analogous to the half- life of radioactive elements. Roughly speaking, a coenocline of l HC has endpoints which are 50% similar, a coenocline of 2 HC's has endpoints which are each 50% similar to the midpoint, etc. However, measurement of coenocline length in half-changes is complicated by apparently random composition- al variation among replicate samples, henceforth referred to as 'noise', by the non-linear relation

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between similarity measures and sample separation along the gradient (Gauch, 1973), and by the pres- ence of other, unmeasured gradients. To our knowledge, previous workers have not attempted to test the accuracy of Whittaker's procedure for determining beta diversity in half-changes, and al- ternative procedures are few.

Second, one may want to compare rates of spe- cies turnover at various points on a gradient, for example, to define objectively ecotones between community types (Whittaker, 1960; Beals, 1969) or to locate periods of rapid change during a succes- sional or phenological sequence. Turnover rate may be expressed as beta diversity per unit gradient as measured over an infinitesimal length of gra- dient. (Conversely, beta diversity is turnover rate integrated over the gradient (Bratton, 1975).) Whit- taker, Beals & Bratton all estimated compositional turnover using the measures percentage similarity (PS) or percentage difference (PD= 100-PS) (Goodall, 1978). This approach may lead to prob- lems because: l) random variation in composition contributes to PD; 2) non-monotonic change in species abundance may cause PD to underestimate compositional change; and 3) spacing of samples along the gradient reflects the investigator's percep- tion of the environment rather than that of the species which compose the coenocline.

Third, one may want a natural, species-defined spacing of samples. In general, any ordination pro- duces a species-based but undefined spacing; in this paper we consider in particular the spacing defined explicitly by a constant species turnover rate along the gradient. When turnover rate is constant the separation between samples is an expression of sample distinctiveness, that can be used in weight- ing niche metrics (Colwell & Futuyma, 1971; Pie- lou, 1972). Furthermore, if habitat axes are rescaled in beta diversity units, breadth and overlap meas- ures for different systems can be compared and multi-dimensional niche metrics can be computed directly rather than by the addition or multiplica- tion of several one-dimensional measures (cf. Le- vins, 1968; Pianka, 1973; Cody, 1974).

We believe the technique presented here, gra- dient rescaling, resolves the problems of measuring compositional change, rates of species turnover, and axis scale. The gradient rescaling approach is treated in the following sequence. First, we give the rationale of using species turnover as a basis for

gradient analysis and develop the computational procedures for measuring beta diversity in half- changes. Second, we describe our evaluation of the technique using simulated data sets. Third, we illus- trate the use of the technique with field data. Final- ly, we discuss and compare methods and mention some applications of our approach to the study of compositional change along gradients.

Gradient rescaling

Rationale

The relationship between beta diversity, rate of species turnover, and gradient scale may be ex- pressed as:

b

fl = f g (x )dx (1) fl

where/3 is the beta diversity of the compositional gradient between points a and b, R is the species turnover rate and x are values along X, the initial scaling of the gradient. Field measurements of en- vironmental factors along the gradient X are, of course, in a scale chosen by the investigator (e.g., elevation in meters or acidity in pH units). The choice of scale must be arbitrary, reflecting the investigator's notion of what is easy, traditional or interesting to record. Since the investigator's scale of measurement does not necessarily reflect how organisms respond to changes in environmental conditions, several authors have proposed rescaling gradient axes using species distributions as an 'ecoassay' (e.g. Colwell & Futuyma, 1971) of the environmental gradient. Gauch et al. (1974) and Ihm & Groenewoud (1975) proposed scaling axes such that species distributions most closely ap- proximate Gaussian curves. Austin (1976) dis- counted the generality of Gaussian curves in nature and used the more flexible beta distribution of sta- tistics in an algorithm similar to that of Gauch et al. (1974). In contrast to these curve-fitting ap- proaches, Hill & Gauch (1980) scaled reciprocal averaging ordination axes so that, on the average, the species in each sample have distribution curves of unit standard deviation.

Our approach is based on the view that composi- tional turnover is the essence of community gra- dients, and that environmental change is ecological-

ly significant primarily to the extent that it influen- ces the relative abundances of species. Accordingly, we scale community gradients so that the rate of compositional change is constant throughout. In this scaling, each portion of the gradient is ecologi- cally equivalent to all other portions. This scaling allows the biological length of the compositional gradient and the rate of species turnover along physically-defined gradients (e.g., elevation), to be accurately calculated. Our approach starts with the relationship between gradient scale, rate of turn- over and beta diversity shown in equation (1). There are three phases. We first find a new scaling, X', such that R(x')= k, for all x', where k is a constant. That is, we find a new set of relative sample positions such that the rate of species turn- over along the gradient is constant. Second, we calculate the compositional length or beta diversity of the gradient as:

/3= k (b -a)

where b - a is the distance between endpoints in terms of the scaling X'. The value of/3 is independ- ent of the initial scaling. Finally, R*(x), the rate of compositional change along the original gradient, is computed by using the original scaling to trans- form R(x'). Details of this procedure are presented below.

Early in the development of gradient rescaling we devised a measure of beta diversity which is an alternative to the half-change. This we name the gleason, in honor of H. A. Gleason, who first point- ed out the importance of continuous compositional change along environmental gradients (Gleason, 1926). One gleason (G) is the amount of composi- tional turnover which would occur if all changes were concentrated into a single species whose abundance changed 100%. For example, a 1 G coenocline might consist of three species, two of which change from 50% to 0% relative abundance as the third species changes from 0% to 100% rela- tive abundance. There is no general function which can interconvert gleasons and half-changes. In the following sections we develop computational procedures for both measures.

Computations

We start with field observations of species abun-

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dances within a sequence of samples, X, along an ecological gradient. This gradient can be either a direct environmental gradient or an indirect, com- positional gradient derived by ordination. We con- sider two measures of species turnover or composi- tional change: half-changes and gleasons. Compu- tations for both measures are performed by a FORTRAN program, GRADBETA, which is available with documentation from Cornell Ecol- ogy Programs, Ecology and Systematics, Cornell University, Ithaca, NY 14853 (Wilson & Mohler, 1980).

For either gleasons or half-changes species im- portance values should be standardized so that sample totals are equal. After this standardization, measures of compositional change are measures of proportional turnover. In this way, the influence of trends in total biomass are minimized in the inter- pretation of compositional turnover.

Field data always include variation due to sample error or the effects of chance and uninvestigated ecological factors. Such noise causes the observed rate of compositional change to be higher than the change attributable to species distributions along recognized gradients. Noise is often ignored in eco- logial analyses, but this practice may lead to misin- terpretation of ecological patterns.

A convenient index of noise is internal associa- tion (IA), the expected similarity of replicate sam- ples. IA decreases as the average noise level of data increases. Any index of compositional similarity can be used to estimate IA; our technique uses percentage similarity (Whittaker, 1975; Goodall, 1978). Replicate samples are equivalent to samples whose separation along the gradient is zero. There- fore, in a plot of sample percentage similarity of each pair of samples versus the gradient separation of each pair, an estimate of IA may be obtained by extrapolating percentage similarity to zero separa- tion (Whittaker, 1960). Whittaker accomplished this by fitting a straight line to the central portion of a plot of the logarithm of percentage similarity versus gradient separation. We make two refine- ments. First, extrapolation is accomplished by line- ar least-squares regression instead of by a hand-fit- ted line. Second, instead of an arbitrary spacing between pairs of samples, we use the ecologically more meaningful gradient distance derived by the gleasons or half-changes procedure.

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Half-changes Whittaker (1960) defined the half-change as the

ecological distance between two samples with sim- ilarity of species composition 50% that of two repli- cate samples. The Whittaker method for measuring the beta diversity of an entire gradient is based on the posited relationship:

HC(a,b) = [log(IA) - log(PS(a, b))]

log2

where PS(a,b) is the percentage similarity of sam- ples a and b, IA is the expected similarity of repli- cate samples, and HC(a,b) is the beta diversity of the gradient segment from a to b, in half-changes.

This half-change formula implies two basic as- sumptions. The first assumption is that sample sim- ilarity is negatively exponentially related to ecolog- ical distance as measured in half-changes, so that as ecological distance increases the rate at which sam- ple similarity decreases is constant (cf. Gauch, 1973). Whittaker (1960, 1972) presents data sets that support this assumption. The calculations of half-changes in our technique are also based on this exponential relationship. The second assumption is that samples are evenly spaced along the ecological gradient. This assumption, which is not met by most field data, is avoided by our technique.

In outline our method is as follows. First, an index of compositional turnover rate is computed for each sample point on the gradient and the among-sample variance in this index is determined. The turnover index used in the procedure is basical-

HC(a,b) ly the average of- for all samples b in

Ax(a,b) some vicinity of a, where HC(a,b) is the number of half-changes between samples a and b, and Ax(a,b) is the gradient distance between a and b. The ratio

HC(a,b) is the slope of a vector in a 2-space

Ax(a,b) whose axes are gradient distance and composition- al distance (in half-changes). The appropriate aver- age for the ratios is thus the slope of the vector sum. Second, using a path-of-steepest-descent algorithm (Berington, 1969), the sample positions are adjust- ed iteratively until the variance in turnover rate has reached a local minimum. This produces the new scaling, X'. Third, the final turnover rate and new sample positions are used to compute a set of

corrected turnover rates for the original scaling and a set of par~al beta diversity values corresponding to the intervals around each sample. Total beta diversity is then computed as the sum of the partial beta diversities. See Wilson & Mohler (1980) for details of the numerical application...