Decomposition of species richness and its phylogenetic analogues into withinand between-group (alpha and beta) components is widely used (Whittaker 1972; Faith et al. 2009). However, these take no notice of abundance differences between sites. Conservationists using these measures cannot distinguish a site whose species are equally abundant from a site with the same species but with a highly skewed abundance distribution whose most phylogenetically distinctive species are rare. The former site would be a better bet for conservation. These considerations, and others, motivate the development of decomposition theory for abundance-based phylogenetic diversity measures. The decomposition also leads to abundance-sensitive measures of phylogenetic similarity and complementarity.

When there are N assemblages, the phylogenetic Hill numbers q D (T ) (Eqs. 4a

and 4b) and phylogenetic diversity qPD(T) (Eqs. 5a and 5b) of the pooled assemblage can be multiplicatively decomposed into independent alpha and beta components (Chiu et al. 2014). We briefly describe the decomposition of the measure q D (T ) here for the ultrametric case, and only summarize the decomposition of the measure qPD(T). The extension to the non-ultrametric case for both measures is obtained by simply replacing all T in the formulas with the mean branch length T of the pooled assemblage.

To begin the partitioning, a pooled tree is constructed for the N assemblages. Assume that there are S species in the present-day assemblage (i.e., there are S tip nodes). For any tip node i, let zik denote any measure of species importance of the ith species in the kth assemblage, i = 1, 2, …, S, k = 1, 2, …, N. The measure zik is referred to as “abundance” for simplicity, although it can be absolute abundances,

relative abundances, incidence, biomasses, cover areas or any other importance

S

measure. Define z+ k = åzik

i =1 (i.e., the “+” sign in z+k denotes a sum over the tip

N

nodes only) as the current size of the kth assemblage. Let z++ = åz+ k be the total

abundance in the present-day pooled assemblage.

Now consider the phylogenetic tree in the time interval [−T, 0], and in the pooled assemblage define BT and Li as in section “Phylogenetic Hill numbers and related measures”. We extend the definition of zik to include all nodes and their corresponding branches by defining zik for all i∈BT as the total abundances descended from branch i. (Here the index i can correspond to both tip-node and internal node; if i is a tip-node, then zik represents data of the current assemblage as defined in the preceding paragraph.) As shown in Fig. 2 of Chiu et al. (2014), the diversity for each individual assemblage can be computed from the pooled tree structure, and only the node abundances vary with assemblages.

In the pooled assemblage, the node abundance for branch i (i∈BT) is zi + = åzik

k =1

with branch relative abundance zi+/z++, so the phylogenetic gamma diversity of order

q can be calculated from Eq. (4a) as

The limit when q approaches unity exists and is equal to

The gamma diversity is the effective number of equally abundant and equally distinct lineages all with branch lengths T in the pooled assemblage.

Chiu et al. (2014) derived the following phylogenetic alpha diversity for q ≥ 0 and q ≠ 1:

The alpha diversity is interpreted as the effective number of equally abundant and equally distinct lineages all with branch lengths T in an individual assemblage. When normalized measures of species importance (like relative abundance or relative biomass) are used to quantify species importance, we have z++ = N in Eqs. (8a) and (8b). The alpha formula then reduces to a generalized mean of the local diversities with the following property: if all assemblages have the same diversity X, the alpha diversity is also X (Jost 2007). For non-normalized measures of species importance, like absolute abundance or biomass, this property does not hold. This is because when species absolute abundances are compared, for example, a threespecies assemblage with absolute abundances {2, 5, 8} will not be treated as identical as another three-species assemblage with absolute abundances {200, 500, 800}. However, these two assemblages are treated as identical when only relative abundances are compared.

Chiu et al. (2014) proved that the phylogenetic gamma Hill number (Eqs. 7a and 7b) is always greater than or equal to the phylogenetic alpha Hill number (Eqs. 8a and 8b) for all q ≥ 0 regardless of species abundances and tree structures. Based on a multiplicative partitioning, the phylogenetic beta diversity is the ratio of gamma diversity to alpha diversity:

When the N assemblages are identical in species identities and species abun-

dances, then q D (T ) = 1 for any T. When the N assemblages are completely phylo-

genetically distinct (no shared lineages), then q D (T ) = N , no matter what the

b

diversities or tree shapes of the assemblages. The measure q D (T ) thus quantifies

the effective number of completely phylogenetically distinct assemblages in the interval [−T, 0]. As proved by Chiu et al. (2014), the phylogenetic beta diversity

q (T ) is always between unity and N for any given alpha value, implying alpha

and beta components are unrelated (or independent) for both measures, q D (T ) and qPD(T); see Chao et al. (2012) for a rigorous discussion of un-relatedness and independence of two measures. When all lineages in the pooled assemblage are completely distinct (no lineages shared) in the interval [−T, 0], the phylogenetic alpha, beta and gamma Hill numbers reduce to those based on ordinary Hill numbers. This includes the limiting case in which T tends to zero, so that phylogeny is ignored.

Parallel decomposition can be made for the phylogenetic diversity qPD(T), and

we summarize the following relations: q PD (T ) = q D (T )´T and

q PD (T ) = q D (T )´T.

Under a multiplicative partitioning scheme, we have

q PD

(T ) = q PD (T ) / q PD (T ) = q D

(T ) , i.e., the beta components from partitioning the phylogenetic Hill numbers q D (T ) and phylogenetic diversity qPD(T) are identical, implying the interpretation and the corresponding similarity or differentiation measures (in the next section) are also identical. Thus, it is sufficient to focus only on the measure q D (T ) , which will be referred to as the phylogenetic beta diversity or beta component for simplicity.

For each of the two measures, q D (T ) and qPD(T), alpha and gamma diversities obey the replication principle. Then the beta diversity formed by taking their ratio is replication-invariant (Chiu et al. 2014). That is, when assemblages are replicated, the beta diversity does not change. Therefore, when we pool equally-distinct subtrees, such as pooling equally-ancient subfamilies, the beta diversity is unchanged by pooling the subfamilies if all subfamilies show the same beta diversity (“consistency in aggregation”).

We now give the phylogenetic beta diversities for the special cases of q = 0, 1 and 2.

(a) When q = 0, we have 0 D (T ) = L (T ) / L (T ) , where L (T) denotes the total

b g a γ

branch length of the pooled tree (the gamma component of Faith's PD) and

Lα(T) denotes the average length of individual trees (the alpha component of

Faith's PD).

(b) When q = 1, the phylogenetic beta diversity of order 1 is

where HP,γ and HP,α denote respectively the gamma and alpha phylogenetic entropy. When the species importance measure zik represents the ith species relative abundance in the kth current-time assemblage, then z+ k = 1, z++ = N, z+ k / z++ = 1 / N. In this special case, we have 1D (T ) = exp é(HP,gHP,a) / T ù . Thus an additive

decomposition for phylogenetic entropy HP holds (Pavoine et al. 2009; Mouchet

and Mouillot 2011), as for ordinary Shannon entropy (Jost 2007).

(c) When q = 2, the phylogenetic beta diversity can be expressed as

In the special case of z+ k = 1, z++ = N , this phylogenetic beta diversity of order 2 can be linked to quadratic entropy as

where Qγ and Qα denote respectively the gamma and alpha quadratic entropy. The above formula is also applicable to non-ultrametric trees by replacing all T with T , the mean branch length in the pooled assemblage; see Chiu et al. (2014, Appendix

C) for a proof.

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