CONCORDANCIAS LEYES AÑOS ANTERIORES

The relation between species abundance and degree of aggregation is still debated as an important issue of ecological theory (He, Legendre et al.,1997; Plotkin, Potts et al., 2000; Condit et al., 2000; Morlon et al., 2008). Moreover, as pointed out in

Morlon et al., 2008, the correlation between these two quantities strongly depends on how they are measured. For example, for Pasoh forest, in He, Legendre et al.,

1997 the proposed Donnelly clumping index based on nearest-neighbour distance shows a slightly positive correlation between abundance and aggregation. By con- trast, the relative neighbourhood density Ω0−10 (Condit et al., 2000; Harte, Conlisk et al.,2005;Ostling et al.,2000) and the Cramer-von Mises-type k statistic (Plotkin, Potts et al., 2000) are negatively correlated to abundance.

To investigate if mTp and Knuth aggregation parameters are similarly correlated with species abundance or not, we compute on one hand the correlation between the abundance and the following mT p quantities: the mean number of parents

ρX· |W|, the mean clump radius σX·

q

x 0 200 400 600 800 1000 y 0 100 200 300 400 500 Generated species x 0 200 400 600 800 1000 y 0 100 200 300 400 500 Real species x y 0 200 400 600 800 0 100 200 300 400 500 1000 x 0 200 400 600 800 1000 0 100 200 300 400 500

REAL SPECIES mTp SPECIES

Quassia Amara 1000 x 0 200 400 600 800 1000 y 0 100 200 300 400 500 Generated species x 0 200 400 600 800 1000 y 0 100 200 300 400 500 Real species x y 0 200 400 600 800 1000 0 100 200 300 400 500 x 0 200 400 600 800 1000 0 100 200 300 400 500 Posoqueria Latifolia 1000 x 0 200 400 600 800 1000 y 0 100 200 300 400 500 Generated species x 0 200 400 600 800 1000 y 0 100 200 300 400 500 Real species x y 0 200 400 600 800 1000 0 100 200 300 400 500 x 0 200 400 600 800 1000 0 100 200 300 400 500 Soroacea Affinis 1000

Figure 2.10: Plot of three species distribution from the BCI surveyed area (left column) and plot of the same species distribution generated according to the mT p with parameters fitted from the real data (right column). Species are selected to display the three cases : i)(Quassia amara, top) a(mT p) ≫ a(real), meaning that the Knuth’s method detects a finer structure for the real species compared to the generated one; ii) (Posoqueria latifolia, middle) a(mT p) ≈ a(real) Knuth’s method recognises as similar the two spatial structures and therefore the hypothesis that the species is distributed according to a mT p is not rejected; iii) (Soroacea affinis, bottom) a(mT p) ≪ a(real), the Knuth’s methods detects a finer structure for the generated species with respect to the real one.

2.4. Application to the BCI ecological database

Figure 2.11: Frequency histograms of anisotropy index for the real BCI species and the ones generated by a modified Thomas process with parameters fitted by data.

µX and the relative neighbourhood density Ω0−10 (see Morlon et al., 2008) and on

the other hand the correlation of species’ abundance with Knuth optimal bin area and with index of anisotropy (see figure 2.12). These latter, as we have seen, give us information on how structured is the data density function and how far it is from a uniform or isotropic one.

Log10(abundance) Log2(number of clumps)

(b)

Log

10

( )

Mean number of clumps:

Log

10

(

)

Log10(abundance)

Mean clump radius:

Log10(abundance) Log2(number of clumps)

Log

10

( )

Relative neighborhood density: Ω0-10

Log 10 (Ω 0-10 ) Log10(abundance)

Abundance per clump:

Log10(abundance) Log2(number of clumps)

Log

10

(

)

Knuth index of anisotropy:

Log

10

(

)

Log10(abundance)

Knuth optimal bin size:

Figure 2.12: Correlation between mTp and Knuth’s parameters and the abundance of a species.

From the determination coefficients, only the relative neighbourhood density shows a negative correlation with abundance, while the other mTp parameters result to be slightly positive correlated with it. This is in accordance with the literature (Morlon et al., 2008). Knuth optimal bin area and index of anisotropy are, instead, insignif- icantly correlated with the abundance with respect to the determination coefficient (bottom panels offigure 2.12). This is quite reasonable because Knuth optimal grid depends only on data distribution, and not on their abundance.

3

Diversity and Similarity Indexes

Estimating biodiversity of forests is a central issue in modern conservation ecology. Both from the theoretical and field application point of view it represents a daunting challenge.

The very same word biodiversity may, and actually does, assume many different meanings and refer to a vast number of notions depending on the subject under study, so that different additional terms have been introduced to reflect the multiple aspects of this important concept (Colwell, 2009).

Following Whittaker1 _{(Whittaker, 1960), we distinguish between alpha, beta and}

gamma-diversity in relation to the scale of investigation. In particular, we use the

term alpha-diversity when biodiversity is measured at the scale of a single sample, while beta-diversity or turnover2 _{refers to the change in species composition between}

samples. Finally, we talk about gamma-diversity when describing the diversity of an assemblage of samples.

In the first part of this chapter we will explore in details some of these notions and we will present the most important indexes introduced in literature to measure ecological diversity in community composition. We will then study how to insert them in the context of point processes’ theory. This will be useful in the next chapter, where we will focus on the more specific problem of describing the decay of similarity between two regions of a landscape as a function of the distance between them.

1_{In Whittaker, 1972the author proposed a new terminology to refer to different diversities in}

space (Magurran, 2013). Indeed, he distinguished between seven spatial scales: within sample (point diversity), between samples within a defined habitat (pattern diversity), within a defined habitat (alpha-diversity), between habitats within a defined landscape (beta-diversity), within a defined landscape (gamma-diversity), between landscapes within a defined biogeographic province (delta-diversity) and within a defined biogeographic province (epsilon-diversity). Here we stick with the original definition of 1960.

3.1

The concepts of diversity and similarity

The first and most intuitive indicator of diversity is the species richness, term coined by McIntosh to indicate the total number of species S of a community (Magurran,

2013; McIntosh, 1967). This is the oldest mathematical descriptor, and yet the poorest, since it does not give any information of the actual composition of the community and it clearly cannot be used for comparisons between ecosystems, if not in a trivial way.

However, despite its simplicity, estimating species richness from small samples to bigger areas is not trivial at all. Indeed, lots of estimators have been introduced to tackle the problem of inferring the number of missing species which we do not see in the surveyed area, but which are present in the ecological community under study (seeChapter 5). The major problem is that usually the species estimates depend on the sampling effortGaston, 1996. In literature, there have also been proposed other simple diversity indexes based on species richness which, apart from the observed number of species S, they also take into account the observed number of individuals

N in order to reduce the dependence on the sample scale (Magurran, 2013). Two

examples are the Margalef’s diversity index (S −1)/ln N and (Clifford and Stephen- son, 1975) and Menhinick’s index S/√N. Anyway, even these latter lack in giving

information about the composition of the sampling unit under study.

Complementary to the notion of species richness, are the ones of evenness (Heip et al., 1998; Pielou, 1969, 1975; Smith and Wilson, 1996; Gray, 2000) and domi-

nance or concentration (Magurran, 2013; Hill, 1973; Jost, 2010b). In particular, a

community has high evenness if all species are equally abundant, whereas it has high dominance if there is one or few species with population much larger than all the oth- ers. Both these concepts are therefore strictly connected to the species-abundance distribution of the community, which gives information on how similar species are in their abundances and it is thus related to Preston’s concepts of commonness and rarity of species (Preston, 1948).

Any index of measure which takes into account both species richness and evenness is called an heterogeneity measure (Magurran,2013;Good,1953;Gray,2000). They can be distinguished in two classes, depending on whether they make assumptions on the species-abundance distribution of the community (parametric indexes) or not (non-parametric indexes). Examples of parametric indexes are the α parameter of the log-series distribution (see Chapter 5) and the λ parameter of the log-normal (Taylor, 1978), this latter defined as the ratio between the observed number of species and the standard deviation of the distribution. Log-series and log-normal distributions have both been widely used in literature to describe the abundances of species (Slik et al., 2015; Ter Steege, Sabatier et al.,2017;White et al., 2012; Aza- ele, Cornell et al.,2012;Magurran and Henderson, 2003;Preston, 1948). Simpson’s index D (Simpson, 1949) and Shannon’s information index (Shannon and Weaver,

1949), are instead examples of non-parametric indexes of diversity (see section be- low).

In what follows we will describe some of the most important diversity indexes which have been introduced in literature to describe the alpha-diversity of a community.