Definitions of diversity. Measuring species diversity. Choosing an index and interpreting diversity measures. Sampling problems. Structural diversity. Applications of diversity measures. Summary.

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Ecological Diversity and its Measurement

THE 'characteristic' defined by Yule1 and the 'index of diversity' defined by Fisher2 are two measures of the degree of concentration or diversity achieved when the individuals of a population are classified into groups. Both are defined as statistics to be calculated from sample data and not in terms of population constants. The index of diversity has so far been used chiefly with the logarithmic distribution. It cannot be used everywhere, as it does not always give values which are independent of sample size ; it cannot do so, for example, when applied to an infinite population of individuals classified into a finite number of groups. Williams3 has pointed out a relationship between the characteristic and the index of diversity when both are applied to a logarithmic distribution. The present purpose is to define and examine a measure of concentration in terms of population constants.

Measurement of diversity

INTRODUCTION A renewed interest in objeetive and quantitative approaehes to the elassifieation of plant communities has led, within the past decade, to an extensive exalllination of systematic theory and technique. This examination, ineluding the work of Sorenson (1948), Motyka et al. (1950), Curtis & McIntosh (1951), Brown & Curtis (1952), Ramensky (1952), Whittaker (1954, 1956), Goodall (1953a, 1954b)? deVries (1953), Guinoehet (1954, 1955), Webb (1954), Eughes (1954) and Poore (1956) has acconlpanied theoretie studies in taxonomy [Fisher (1936), Womble (1951), Clifford & Binet (1954), Gregg (1954)] and in statisties (Isaaeson 1954). It is a Gonclusion of many of these studies that nature of unit variation is a naajor problenl in systematies, and that whether this variation is diserete, continuous, or in some other form, there is a need for appliGation of (uantitative and statistical methods. In eeologic elassifieation, an inereased use of ordinate systellls, sr hiGh has been stimulated by the developnlent of more effieient sampling teehniques and the collection of stand data on a large seale, has prompted the proposal of the term \"ordination\" ( Goodall 1953b ) . Goodall (1954a) has defined ordination as \"an arrangenlent of units in a unior multi-dinlensional order\" as synonylllous with \"Ordnung,\" (Ramensky

An Ordination of the Upland Forest Communities of Southern Wisconsin

Species richness is a fundamental measurement of community and regional diversity, and it underlies many ecological models and conservation strategies. In spite of its importance, ecologists have not always appreciated the effects of abundance and sampling effort on richness measures and comparisons. We survey a series of common pitfalls in quantifying and comparing taxon richness. These pitfalls can be largely avoided by using accumulation and rarefaction curves, which may be based on either individuals or samples. These taxon sampling curves contain the basic information for valid richness comparisons, including category‐subcategory ratios (species-to-genus and species-toindividual ratios). Rarefaction methods ‐ both sample-based and individual-based ‐ allow for meaningful standardization and comparison of datasets. Standardizing data sets by area or sampling effort may produce very different results compared to standardizing by number of individuals collected, and it is not always clear which measure of diversity is more appropriate. Asymptotic richness estimators provide lower-bound estimates for taxon-rich groups such as tropical arthropods, in which observed richness rarely reaches an asymptote, despite intensive sampling. Recent examples of diversity studies of tropical trees, stream invertebrates, and herbaceous plants emphasize the importance of carefully quantifying species richness using taxon sampling curves.

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Quantifying biodiversity: procedures and pitfalls in the measurement and comparison of species richness

Evolution and measurement of species diversity

Part 1. It is shown that in a large collection of Lepidoptera captured in Malaya the frequency of the number of species represented by different numbers of individuals fitted somewhat closely to a hyperbola type of curve, so long as only the rarer species were considered. The data for the commoner species was not so strictly `randomized', but the whole series could be closely fitted by a series of the logarithmic type as described by Fisher in Part 3. Other data for random collections of insects in the field were also shown to fit fairly well to this series. Part 2. Extensive data on the capture of about 1500 Macrolepidoptera of about 240 species in a light-trap at Harpenden is analysed in relation to Fisher's mathematical theory and is shown to fit extremely closely to the calculations. The calculations are applied first to the frequency of occurrence of species represented by different numbers of individuals--and secondly to the number of species in samples of different sizes from the same population. The parameter ` alpha ', which it is suggested should be called the `index of diversity', is shown to have a regular seasonal change in the case of the Macrolepidoptera in the trap. In addition, samples from two traps which overlooked somewhat different vegetation are shown to have ` alpha ' values which are significantly different. It is shown that, provided the samples are not small, ` alpha ' is the increase in the number of species obtained by increasing the size of a sample by e (2.718). A diagram is given (Fig. 8) from which any one of the values, total number of species, total number of individuals and index of diversity (alpha), can be obtained approximately if the other two are known. The standard error of alpha is also indicated on the same diagram. Part 3. A theoretical distribution is developed which appears to be suitable for the frequencies with which different species occur in a random collection, in the common case in which many species are so rare that their chance of inclusion is small. The relationships of the new distribution with the negative binomial and the Poisson series are established. Numerical processes are exhibited for fitting the series to observations containing given numbers of species and individuals, and for estimating the parameter alpha representing the richness in species of the material sampled; secondly, for calculating the standard error of alpha, and thirdly, for testing whether the series exhibits a significant deviation from the limiting form used. Special tables are presented for facilitating these calculations.

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The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population

Patterns in the Balance of Nature

Patterns in the Balance of Nature and Related Problems in Quantitative Ecology.

SUMMARY
Evidence is brought forward that in comparing the numbers of insects caught under varying conditions, with particular reference to captures in a light trap, more consistent results are obtained if the geometric means are compared than by the use of arithmetic means. This is most conveniently done by summing the logarithms of the numbers instead of the numbers themselves. If any of the numbers in the series is zero it has been found practical to add one unit to all the captures in the series and so deal with log (n+1) instead of log n.

The use of the logarithms prevents the swamping of the results in a series of observations by very high numbers on a single night. It also gives a more normal distribution of departures from a mean. As a result of the latter it is possible to apply the statistical formulae for standard deviation etc. which are not applicable to the skew curve obtained by the use of the departures of the numbers themselves from an arithmetic mean.

C. B. Williams

Papers

The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population

Patterns in the Balance of Nature

Patterns in the Balance of Nature and Related Problems in Quantitative Ecology.

Patterns in the Balance of Nature and Related Problems in Quantitative Ecology.

The use of logarithms in the interpretation of certain entomological problems