From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities . MjM Software Design, Gleneden Beach, Or

From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities . MjM Software Design, Gleneden Beach, Or

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## From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities . MjM Software Design, Gleneden Beach, Or

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**CHAPTER 4**Species Diversity Tables, Figures, and Equations From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities.MjM Software Design, Gleneden Beach, Oregon http://www.pcord.com**Alpha diversity: diversity in individual sample units**• Beta diversity: amount of compositional variation in a sample (a collection of sample units) • Gamma diversity: overall diversity in a collection of sample units, often "landscape-level" diversity**Proportionate diversity measures**For an observed abundance xi, (numbers, biomass, cover, etc.) of species i in a sample unit, let pi = proportion of individuals belonging to species i: • a = constant that can be assigned and alters the property of the measure • S = number of species • Da = diversity measure based on the constant a. The units are "effective number of species" Can think of a as the weight given to dominance of a species.**Figure 4.1. Influence of equitability on Hill's (1973a)**generalized diversity index. Diversity is shown as a function of the parameter a for two cases: a sample unit with strong inequitability in abundance and a sample unit with perfect equitability in abundance (all species present have equal abundance; see Table 4.1).**D0 = species richness**When a = 0, Da is simply species richness.**D2 and Simpson's index**Simpson’s (1949) original index (1/D2) is a measure of dominance rather than diversity The complement of Simpson's index of dominance is and is a measure of diversity. It is the likelihood that two randomly chosen individuals will be different species.**D1 and Shannon-Wiener index**If a = 1 then D1 is a nonsense equation because the exponent is 1/0. But if we use limits to define D1 as a approaches 1 then**The logarithmic form of D1 is the Shannon-Wiener index**(H’), which measures the “information content” of a sample unit: The units for D1 are "number of species of equal abundance“ The units for H' are the log of the number of species of equal abundance.**Box 4.1. How is information related to uncertainty?**For plot 1 For plot 2**Table 4.2. Some measures of beta diversity. See Wilson and**Mohler (1983) and Wilson and Shmida (1984) for other published methods. “DCA” is detrended correspondence analysis. A direct gradient refers to sample units taken along an explicitly measured environmental or temporal gradient. Indirect gradients are gradients in species composition along presumed environmental gradients**Figure 4.2. Example of rate of change, R, measured as**proportional dissimilarity in species composition at different sampling positions along an environmental gradient. Peaks represent relatively abrupt change in species composition. This data set is a series of vegetation plots over a low mountain range. In more homogeneous vegetation, the curve and peaks would be lower.**The amount of change, b, is the integral of the rate of**change: where a and b refer to the ends of an ecological gradient x.**Figure 4.3. Hypothetical decline in similarity in species**composition as a function of separation of sample units along an environmental gradient, measured in half changes. Sample units one half change apart have a similarity of 50%.**Wilson and Mohler (1983) introduced "gleasons" as a unit of**species change. This measures the steepness of species response curves. It is the sum of the slopes of individual species at each point along the gradient. where Y is the abundance of species i at position x along the gradient. This can be integrated into an estimate of beta diversity along a whole gradient with where PS(a,b) is the percentage similarity of sample units a and b and IA is the expected similarity of replicate samples (the similarity intercept on Fig. 4.3).**Minchin measured beta diversity using the mean range of the**species’ physiological response function: where ri is the range of species i along the gradient, L is the length of the gradient, and r and L are measured in the same units.**Oksanen and Tonteri (1995) proposed the following measure of**total gradient length: where A is the absolute compositional turnover (rate of change) of the community between points a and b on gradient x.**This semi-log plot is the basis for Whittaker's (1960)**method of calculating the number of half changes along the gradient segment from a to b, HC(a,b): • where • PS(a,b) is the percentage similarity of sample units a and b • IA is the expected similarity of replicate samples (the y intercept on the figure just described).**Beta turnover**measures the amount of change as the "number of communities." where g = the number of species gained, l = the number of species lost = the average species richness in the sample units:**The simplest descriptor of beta diversity and one that can**be applied to any community sample, is where is the landscape-level diversity and is the average diversity in a sample unit. Whittaker (1972) stated that a generally appropriate measure of this is where w is the beta diversity, Sc is the number of species in the composite sample (the number of species in the whole data set), and S is the average species richness in the sample units.**As a rule of thumb:**• w < 1 is low • 1 < w< 5 is medium • w > 5 is high**Half changes corresponding to the average dissimilarity**among sample units: This can be rewritten as Figure 4.4. Conversion of average dissimilarity, measured with a proportion coefficient, to beta diversity measured in half changes (D).**First-order jackknife estimator**(Heltshe & Forrester 1983, Palmer 1990) • where • S = the observed number of species, • r1 = the number of species occurring in only one sample unit, and • n = the number of sample units.**The second-order jackknife estimator (Burnham & Overton**1979; Palmer 1991) is: where r2 = the number of species occurring in exactly two sample units.**Evenness**An easy-to-use measure (Pielou 1966, 1969) is "Pielou's J" • where • H' is the Shannon-Wiener diversity measure • S is the average species richness. • If there is perfect equitability then log(S) = H' and J = 1.**Hayek and Buzas (1997) partitioned H' into richness and**evenness components based on the equation • where • E = eH'/S • e is the base of the natural logarithms.**Figure 4.5. Species-area curve (heavy line) used to assess**sample adequacy, based on repeated subsampling of a fixed sample (in this case containing 92 sample units and 122 species). Dotted lines represent 1 standard deviation. The distance curve (light line) describes the average Sørensen distance between the subsamples and the whole sample, as a function of subsample size.**Species – Area equations**• Arrhenius (1921): • where • S is the number of species, • A is the area of the sample, and • c and b are fitted coefficients. • In log form:**Table 4.3. Species diversity of epiphytic macrolichens in**the southeastern United States. Alpha, beta, and gamma diversity are defined in the text (table from McCune et al. 1997).