# Quantifying non-spatial species diversity

Species diversity, a combination of species richness and relative abundance (Newton, 2007), is not the only aspect of biodiversity, but a rather important and the most commonly considered one (Kimmins, 2004). In the past, most importance has been assigned to species diversity and there is a wide range of approaches to quantifying this aspect of diversity.

In the light of large-scale forest destruction and associated species loss, which may dramatically increase as a result of climate change and of the growth of the human population, the monitoring of species diversity assumes quite some importance. Forest management and conservation have a particular influence on and responsibility for species diversity. In this context, the dynamics of forest succession and the effect of disturbances are crucial aspects of diversity research.

Species diversity is usually defined in terms of species richness and abundance. Species richness is simply the number of species whilst abundance is a density measure, i.e. the number of organisms of a certain species per space unit. Abundance can be expressed in absolute and in relative terms (Gaston and Spicer, 2004).

Species richness is often interpreted as a surrogate measure for other kinds of biodiversity: More species usually lead to greater genetic variation, i.e. there is a greater diversity of genes in the population. This implies greater ecological variation and a better exploitation of niches and habitats (Magurran, 2004; Gaston and Spicer, 2004; Krebs, 1999}.

A species diversity index is a mathematical expression of species diversity in a community and provides important information on the occurrence and distribution of species in a community (Krebs, 1999). Popular examples of typical species diversity measures include the Shannon and Simpson indices. Both indices take the relative abundance of different species into account rather than simply expressing species richness. The Shannon index (see Eq. (1); Shannon and Weaver, 1949) is an information theory index and was originally proposed to quantify the entropy, i.e. the uncertainty of information in strings of text (Krebs, 1999). It measured the uncertainty of the next letter in a coded message or the next species to be found in a community. A monospecies forest would have no uncertainty and H‘ = 0. The Shannon index is affected by both the number of species and their equitability or evenness. By contrast, the Simpson index (see Eq. (2); Simpson, 1949) is a dominance or concentration measure. The evenness forms are often used as standardisation to allow for comparisons between different monitoring sites (Pretzsch, 2009) and is constrained between 0 and 1.

(1) $E_{H’} = \frac{H’}{\log\; s} \hspace{0.2cm} \text{with} \hspace{0.2cm} H’ = -\sum_{i=1}^s p_i \log(p_i)$

pi is the proportion of individuals found in the ith species and s is the number of species. There are different variants of the Simpson index and this is the version suggested by \(Magurran, 2004, p. 116). It uses the reciprocal as opposed to the complement form for calculating the evenness measure. Whilst the Shannon measure emphasizes the species richness component of diversity the Simpson index is weighted by abundances of the commonest species. The Shannon and Simpson measures are among the most meaningful and robust diversity measures available (Krebs, 1999; Magurran, 2004).

Both species diversity measures can be calculated based on tree number proportions and basal area proportions. Basal area, g (measured in m2), is the cross-sectional surface area of tree stems at 1.3 m above ground level. In the case of basal area proportions the conspecific tree sizes are taken into account and not only their numbers.

The Shannon index is easy to compute in R. In lines 1 and 3 the total species specific number of trees and basal areas are determined. The Shannon index is calculated in lines 4f. and 6f. for tree number and basal area proportions, respectively. The corresponding evenness measures are calculated in lines 8 and 9. We used the natural logarithm here and it is also common to apply the binary logarithm.

sph <- tapply(myData$treeno, myData$species, length)
myData$ba <- pi * (myData$dbh / 200)^2
ba <- tapply(myData$ba, myData$species, sum)
(ShannonStems <- sum(stems.species / sum(stems.species) *
(-log(stems.species / sum(stems.species))))
(ShannonBasalArea <- sum(basalArea.species /
sum(basalArea.species) * (-log(basalArea.species /
sum(basalArea.species))))
ShannonStems / log(numberSpecies)
ShannonBasalArea / log(numberSpecies)

The index by Simpson (1949) gives a probability of any two individuals drawn at random from an infinitely large population belonging to different species. Simpson suggested that this probability was inversely related to diversity.

(2) $E_{D} = \frac{D}{s} \hspace{0.2cm} \text{with} \hspace{0.2cm} D = \frac{1}{\sum_{i=1}^s p_i^2}$

The R code for the Simpson index follows largely the same structure as the one above for the Shannon index. Also here the index is calculated both for tree number and basal area proportions.

(SimpsonStems <- 1 / sum((stems.species /
sum(stems.species))^2))
(SimpsonBasalArea <- 1 / sum((basalArea.species /
sum(basalArea.species))^2))
SimpsonStems / numberSpecies
SimpsonBasalArea / numberSpecies

Other species diversity indices are described in detail in Krebs (1999), Magurran (2004) and in Staudhammer and LeMay (2001).

Literature

Gaston, K. J. and Spicer, J. I., 2004. Biodiversity. An introduction. Blackwell Publishing. Oxford, 191p.

Kimmins, J. P., 2004. Forest ecology – a foundation for sustainable management. 3rd edition. Pearson Education Prentice Hall. Upper Saddle River, NJ. 700p.

Krebs, C. J., 1999. Ecological methodology. 2nd edition. Addison Wesley Longman. New York, 620p.

Magurran, A. E., 2004. Measuring biological diversity. Blackwell Publishing. Oxford, 256p.

Newton, A. C., 2009. Forest ecology and conservation. A handbook of techniques. Oxford University Press. Oxford, 454p.

Pretzsch, H., 2009. Forest dynamics, growth and yield. From measurement to model. Springer, Heidelberg, 664p.

Shannon, C. E. and Weaver, W., 1949. The mathematical theory of communication. University of Illinois Press. Urbana, 35p.

Staudhammer, C. L. and LeMay, V. M., 2001. Introduction and evaluation of possible indices of stand structural diversity. Canadian Journal of Forest Research 31: 1105-1115.

Simpson, E. H., 1949. Measurement of diversity. Nature 163: 688.

## By Arne Pommerening

My background is in forest science with a PhD in forest biometrics (from Göttingen University (Germany) and a Habilitation in forest biometrics (from BOKU University Vienna (Austria). For eleven years I have been working in the fields of quantitative forest management and quantitative ecology at Bangor University (North Wales, UK) before working for a short while in Switzerland. Since 2014 I work as a Professor in Mathematical Statistics Applied to Forest Science at the Swedish University of Agricultural Sciences (SLU in Umeå and my research areas include woodland structure analysis and modelling, spatio-temporal dynamics of plant point patterns, individual-based modelling with a focus on plant interactions, plant growth analysis, methods of quantifying and monitoring biodiversity and the analysis of human behaviour of selecting trees. Much of my research is computer-based using simulation experiments and my research is strongly interdisciplinary and international.