Concepts of diversity

1. Concepts of diversity Peter Shaw USR

2. At the moment you probably have a good idea of what ecological diversity is. By the end of the lecture you will know more but probably feel more confused! This is because diversity is not a unitary concept but has several strands; different emphases give rise to different answers. (And as an analytical tool for actual practical decisions, I think that diversity generates more heat than light.) • Diversity is a deceptive concept. Magurran (1991) likens the concept of diversity to an optical illusion; the more it is examined, the less clear it becomes. • What I want to cover: • Types of diversity • Spercies abundance curves • Diversity indices • The usefulness thereof

3. Ecologists recognise that the diversity of an ecological system has 2 facets: • species number (=richness) • evenness of distribution Both these systems have 30 individuals and 3 species. Which is more diverse? Sp. A 10 Sp. B 10 Sp. C 10 Sp. A 1 Sp. B 1 Sp. C 28 A 10 B 2 C 3 D 1 E 1 F 1 And how about this?

4. Figure 2.1 The difference between species richness and diversity. The diagram shows two samples, both containing the same number of species and individuals (16 individuals from 4 species), but with a clearly different balance of species composition. Habitat 1 (low diversity) Habitat 2 (high diversity)

5. Types of diversity The word has multiple meanings, multiple facets. There is a primary distinction into 3 levels, corresponding to different measurement scales: α, ß and γ Alpha diversity is the diversity of organisms within a selected habitat or sample, and is quantified by indices and by rank abundance models. Beta diversity is an index of the rate of increase of alpha diversity as new habitats are sampled, so is a measure of the turnover of species along a spatial gradient. Finally, gamma diversity is the full diversity (species richness) of an entire sampled landscape or gradient.

6. More on α, ß and γ diversity Scale: These definitions of alpha, beta and gamma diversity are scale dependent, so that a patch size that a mammal ecologist would consider to be one habitat (measuring alpha diversity) would be a mosaic of micro-habitats which a microbial ecologist might count as containing gamma diversity. Scalar/vector: α and γ diversity are scalar quantities, ie may be represented by a single number. ß diversity is a vector: it must have a directional component as well as a magnitude.

7. Species richness This is what people often (not always) mean when they talk of biodiversity. The value of species richness is, in principle, easy: you list all the species in the habitat and count them. IMHO this is as good, if not better, than any of the more complex indices as a tool to assess the ‘value’ of a habitat. BUT this has flaws: 1: This index is usually criticised as being too sensitive to the occurrence of rare species. 2: Are you sure of your list? The more samples you take the more species you find – this links to rank abundance curves (coming next..). And that’s assuming your taxonomic basis is solid. There are two unrelated tools available to deal with the problem that you can’t take infinite samples: Bootstrapping and Rarefaction.

8. Bootstrapping Here you (or rather your PC) constructs a curve showing how the rate of species appearance declines with increasing sample number. A function is then fitted and its asymptote estimated. In effect this tells you how many species you WOULD have found if you had taken infinite samples – the estimated species richness of the site. This comes with one health warning: GIGO! Number spp Estimated asymptote Number samples

9. Here you want to compare species richness in two populations from which you took unequal numbers of observations For example you might use this to estimate how many spp would have been found if you sampled 1000 organisms each time. You take Q samples, totalling N organisms from Q species (species 1 has S1 individuals, Sp2 has S2 etc). You want to estimate how many spp would have been found in a sample of R organisms: Rarefaction N R Where Expected number of species in sample of K organisms N-Si R N! (N-R!)*R! i=Q Σ 1- Number of ways of picking N items from a list of R when order does not matter N R i=1

10. Rank Abundance curves Since the idea of diversity indices is to encapsulate the relative abundances of different species, a logical extension of this is simply to examine the relative abundances visually! The standard tool here is a rank abundance curve, where the Y axis is log-transformed density and the X axis is rank order of density (1st, 2nd, 3rd, …87th)

11. The actual pattern described by these lines can be shoe-horned into different mathematical models, each making different assumptions about how the community is arranged. The actual patterns seen in the field range over most of the available models and never quite fit any of them! Broken stick: a space/resource is broken into random segments, each length corresponding to a species’ resource allocation (forest birds). Log-normal (forest plants) Geometric: each successive sp occupies a constant proportion of the remaining space/resource (sub-alpine plants)

12. Log series alpha

13. Diversity indices • Today I only intend to cover 2 indices (my favourites!). These are: The Simpson index, The Shannon index • Be aware that there are many more. Hill (1973) noted that one function could generate infinitely many diversity indices, each credible. Here was have a total of N individuals from S species: we define the proportion of the ith species as pi. When n = 0 this is S – species richness. N=1 turns out to be the Shannon index (tho the maths are rather tricky..). N=2 gives us the Simpson index. N=3,.. are unused but perfectly good indices. Higher values of N downweight rare species. 1/(1-n) i=S Σ Pin Calculate: i=1

14. The Simpson Index D • This is an intuitively simple, appealing index. It is the probability that two consecutive samples drawn from the same population will be different species. • It involves sampling individuals from a population one at a time (like pulling balls out of a hat). • What is the probability of sampling the same species twice in 2 consecutive samples? Call this p. • If there is only 1 species, p = 1.0 • If all samples find different species, p = 0.0

15. The probability of sampling species i = pi. • Hence p(sampling species 1 twice) = pi * pi. • Hence p(sampling any species twice) • = p(sp1) + p(sp2)… +p(spN) • Hence the simplest version of this index = i pi * pi • This has the counter-intuitive property that 0 = infinite diversity, 1 = no diversity • Hence the usual formulation is: • Simpson’s diversity D = 1 - i pi * pi

16. Applying the Simpson index to the communities listed previously: Sp. A 10 p = 1/3 Sp. B 10 p=1/3 Sp. C 10 p = 1/3 Sp. A 1 p=1/30 Sp. B 1 p = 1/30 Sp. C 28 p = 28/30 D = 1-(1/9 + 1/9 + 1/9) = 0.667 D = 1-(1/900 + 1/900 + 784/900) = 14/900 = 0.0156 Applied to the demo communities given previously, this tells what was already obvious: the first community had a higher diversity (measured as evenness of species distribution)

17. Figure 2.3: The spreadsheet showing the calculation of Simpson’s index for two observations within the Wimbledon common dataset. (This sheet shows the cell entries used. Circled cells should be copied and pasted up to their greyed-out boundaries). =E4*E4 =F4*F4

18. The Shannon (or Shannon-Wiener) index • Very often mis-named as the Shannon-Weaver index, this widely used index comes out of some very obscure mathematics in information theory. It dates back to work by Claude Shannon in the Bell telephone company labs in the 1940s. • It is the most commonly used, most mis-used and least understood (!) diversity index.

19. To calculate: • H = -i[pi*log(pi)] • Note that you have the choice of logarithm base. Really you should use base 2,when H defines the information content of the dataset in bits. • To do this use • log2(x) = log10(x) / log10(2)

20. An oddity is that the index varies with number of species, as well as their evenness. • The maximum possible score for a community with N species is log(N) – this occurs when all species are equally frequent. • Because of this (and log-base problems, and the difficulty of working out what the score actually means) I prefer to convert H into an evenness index; H as a proportion of what it could be if all species were equally frequent. This is E, the evenness. • E = H / log(N). • E = 1 implies total evenness, E  0 implies the opposite. • This index is independent of log base and of species number.

21. Applying the Shannon index to the communities listed previously: Sp. A 10 p = 1/3 Sp. B 10 p=1/3 Sp. C 10 p = 1/3 Sp. A 1 p=1/30 Sp. B 1 p = 1/30 Sp. C 28 p = 28/30 H10 = -3*(log10(1/3)*1/3) = -3*(log10(1/3)*1/3) = 0.477 E = H / log(3) = 1, ie this community is as even as it could be H10 = 1/30*log10(1/30) + 1/30*log10(1/30) + 28/30*log10(28/30) = 0.126 E = H / log(3) = 0.265

22. Figure 2.5: The spreadsheet showing the calculation of Shannon’s index for two observations within the Wimbledon common dataset. This sheet shows the cell entries used. Circled cells should be copied and pasted up to their greyed-out boundaries. Note that the value 8 used in the equitability calculation is the number of species in the dataset. This could also be obtained by counting the number of names in column B.

23. Other diversity indices The Berger-Parker Index is the simplest and most easily understood diversity index, since it only calculates the proportion of the commonest species in a sample: d = Nmax/N The Brillouin index It is calculated as: HB = ln(N!) - Σln(ni!) N Where HB = the Brillouin index, N = total number of individuals in the sample, ni = number of individual of species i, N! means the factorial of N = 1 * 2 * 3 * 4... * N, ln(x) = natural logarithm of x (or logarithm base e)

24. Measures of Beta diversity I have said a lot about alpha diversity – since this is easy! Beta diversity is more messy since alpha diversity is mixed with spatial information. This is handled in several ways which can be squeezed under the heading of Ordination: this includes direct gradient analysis (plotting species along a pre-measured axis), mosaic diagrams and indirect ordinations (later in the course). One particular form of ordination, Bray-Curtis ordination, was invented to handle beta diversity along perceived spatial gradients.

25. Figure 4.2. Vegetation zonation away from a stream edge in a floodplain forest (redrawn from Hughes & Cass 1997). Onoclea zone Open water Typha swamp Closed tree canopy Hardwood canopy Shrub zone 0 25 50 75 Distance from stream edge, m

26. 4000 Alpine dwarf scrub Figure 4.3 A mosaic diagram, in this case showing the distribuition of vegetation types in relation to elevation and moisture in the Sequoia national park. This is an example of a direct ordination, laying out communities in relation to two well-understood axes of variation. Redrawn from Vankat (1982) with kind permission of the California botanical society. Subalpine conifer Wet meadows 3000 Elevation, m Lodgepole pine Red fir Jeffrey pine pinyon 2000 Mixed conifers Ponserosa pine Montane hardwood Montane hardwood 1000 Chaparral hardwoods Valley-foothill hardwood Annual grassland Moist….…………………Dry

27. Does diversity matter? I want to avoid any attempt at analysing the ethical question ‘Does Biodiversity Matter?’ and remain focussed on assessing whether there are any sound ecological reasons for using diversity measures beyond giving you one more statistic in your thesis :=) The diversity – stability hypothesis. (questionable, probably a long-lived myth) The diversity – functionality hypothesis: diverse ecosystems function ore effectively as measured by rates of energy capture, nutrient cycling etc. This comes out of John Lawton’s Ecotrons, mini-ecosystems where species assemblage is controlled. Diversity driving conservation decisions: Nice idea but in practice diversity indices tend to be lowest for really important habitats like lowland heaths or the mountain gorilla habitat!!