Hedging our bets: the expected contribution of species to future phylogenetic diversity

Within a given taxonomic group, individual biological species are generally considered to be of equal or near-equal biodiversity value. So, for instance, areas with a greater number of species are more valuable than those with fewer (Myers et al. 2000). When wild species are ranked by value, this is usually based on their threat of extinction (see, e.g. SARA 2002). However, as pointed out by Cousins (1991), species are discovered and identified because they are different from other species, which suggests that they may differ in value. In the context of conservation, Avise (2005) has highlighted five different currencies for valuing species: rarity, distribution, ecology, charisma, and phylogeny. Here, we consider the value of a species based on its position in a phylogeny. A phylogeny is the directional, acyclic graph depicting relationships between leaves (species), which we define formally in the next section. A phylogeny generally has a root (which assigns direction) and edge weights that can represent unique feature diversity (e.g. as measured by evolutionary time or genetic distance). Species can be defined by the features they possess, and one measure of their worth is their expected contribution of unique features. In this way, we can use a phylogeny to assign a measure of evolutionary value to a species based on its expected contribution of unique features. Because of the highly imbalanced shape of the Tree of Life, some species in a phylogeny will have far fewer close relatives than others in that phylogeny (Mooers and Heard 1997), and these more distantly-related species will be expected to contribute more unique features (Faith 1992).

Phylogenetic measures of conservation value have a long pedigree (see, e.g. Alschul and Lipman 1990; May 1990) and have begun to be explored in some detail (Haake et al. 2005; Hartmann and Steel 2007; Pavoine et al. 2005a, 2005b; Redding and Mooers 2006). So, for example, Pavoine and colleagues presented one new measure of originality, a set of sampling weights such that the expected pairwise distance on the tree is maximized. Haake and colleagues extended the ‘Shapley value’ (Shapley 1953) from co-operative game theory to the conservation setting to calculate the average distance of a focal species to all possible subsets of taxa. For both measures, more original species are those expected to contribute more to the resulting sets. Yet another measure that uniquely apportions the tree to its tips (Isaac et al. 2007) and which is the focus of a new international conservation initiative (the EDGE initiative, Zoological Society of London) scales almost perfectly with the Shapley value (unpublished results).

One question with these measures concerns the sets that individual species are asked to complement. For instance, given known extinction probabilities for species, some future sets are much more likely than others and so some species will be more valuable because their close relatives are less likely to be included in future sets. Here we formalize this idea to extend the Shapley value of a species to include pre-assigned extinction probabilities. We then compare our measure with the original Shapley value using the prosimian primates as a test case.

Definition Let $\mathcal{T}$ be a rooted or unrooted phylogenetic tree with leaf set $X$, together with an assignment of positive lengths to the edges (branches) of $\mathcal{T}$. We let $l(e)$ denote the length of edge $e$, and let $E(\mathcal{T})$ denote the set of edges of $\mathcal{T}$. For a subset $S$ of $X$, let $PD(S)$ denote the phylogenetic diversity of $S$ defined as follows. If $\mathcal{T}$ is unrooted then $PD(S)$ is the sum of the lengths of the edges (branches) of $\mathcal{T}$ in the minimal subtree that connects $S$. If $\mathcal{T}$ is rooted, then $PD(S)$ is the sum of the lengths of the edges of $\mathcal{T}$ in the minimal subtree of $T$ that connects $S$ and the root of the tree. Figure 1 illustrates these concepts, and includes values at the tips that we will use in the next section. Note that although the branch lengths in this example are clock-like, this assumption is not required in any of the results we describe.

For a leaf $i\in X$, and a subset $S\subseteq X-\{i\}$ let

The quantity $\Delta_{PD}(S,i)$ measures how much phylogenetic diversity $i$ contributes to the tree that one obtains from $\mathcal{T}$ once species not in $S$ have been pruned out (for example if they go extinct). Alternatively, $\Delta_{PD}(S,i)$ is the marginal increase in phylogenetic diversity of $S$ if $i$ is added.

Now, suppose that each species has an associated extinction probability $P(ext)$ (which may vary from species to species) — for example, this may be the probability that the species is extinct in (say) 100 years from now (either globally, or in some specified community). We will denote this $P(ext)$ value for species $j$ by $\epsilon_{j}$. In this paper we consider the simplest model which assumes that the extinction of each species in $X$ comprise independent events. Given $i\in X$, let $\mathcal{S}_{i}$ denote the random subset of species in $X-\{i\}$ which survive (i.e. do not go extinct).

A related but different index, based on the Shapley value in co-operative game theory, has recently been described by Haake et al. (2005). This index, denoted here as $\psi^{\rm sh}$ can be defined (for unrooted trees) as follows: For $i\in X$,

This index has certain appealing properties. In particular, $\sum_{i\in X}\psi^{\rm sh}_{i}=PD(X)$, and there is a simple formula for quickly computing $\psi^{\rm sh}_{i}$. The index $\psi^{\rm sh}$ also has a stochastic interpretation, but this is not based on extinction or survival of species, rather on the expected contribution to $PD$ of each species under all possible orderings of the total set of species (for details see Haake et al. 2005). The index $\psi^{\rm sh}$ allocates existing $PD$ ‘fairly’ amongst the species, whereas $\psi$ quantifies the expected contribution of each species to future $PD$.

Computing the HED index directly via (1) could be problematic as it requires summation over all the subsets of $X-\{i\}$ and this grows exponentially with $|X|$. However we now show that the index can be readily and quickly computed, both for rooted and unrooted trees. This polynomial-time algorithm for computing $\psi$ thus complements (but is quite different to) the polynomial-time algorithm described by Haake et al. (2005) for computing $\psi^{\rm sh}$.

We compared the HED index with the Shapley (Haake et al. 2005) values for the Prosimians (Mammalia: Primata), a group of approximately 50 species with a broad range of extinction probabilities. This group includes the Aye-Aye, the lemurs, the lorises and galagos. We made use of a recent dated Supertree of the order Primates (Vos and Mooers 2004; Vos 2006), see Fig. 2, and Red List risk designations from the IUCN (www.iucnredlist.org, accessed February 2006). Following Isaac et al. (2007) and Redding and Mooers (2006), we first converted the five categories of risk (CR, EN, VU, NT, and LC) to probabilities of extinction. Under the IUCN criteria, the species in the VU category are given a $P(ext)$=0.1 over the next 100 years. We gave the lowest and highest threat categories very conservative probabilities of extinction over the next 100 years of 0.001 and 0.9 respectively, leaving $P(ext)=0.5$ for EN, and $P(ext)=0.01$ for NT: this scale is very similar to that calculated from real population viability analyses for birds (Redding and Mooers 2006). We are primarily interested in how the ranking of species changes using different approaches.

The bivariate correlation between the metrics is high (0.94). Both measures chose the Aye-Aye ($Daubentonia$ $madagascarensis$) as the most important species, followed by $Perodicticus$ $potto$. Interestingly, the three most highly ranked species under current conservation policy (the critically endangered lemurs $Propithecus$ $tattersalli$, $Hapalemur$ $simus$, $H.$ $aureus$) are nested well up in the tree (Figure 2) such that none of them were chosen in the top ten for either SV or HED. If we compare the rest of the rankings for these two metrics, the largest single difference is for the two $Arctocebus$ species: they rank twelfth under SV (being relatively isolated on the tree), but only twenty-sixth under HED: because neither is severely threatened, the chances are good that their common path will persist.

Both measures are very heavily influenced by the pendant edge ($PE$) length of the focal species (with correlations $PE$ vs. SV=0.94, and $PE$ vs. HED=0.98). $PE_{i}$ is always part of the marginal increase to $S$, while interior nodes are most likely represented with high probability, especially for larger and more balanced trees. $PE$ is, however, a poor predictor of HED for $Propithecus$, for $H.$ $griseus$, and for $Nycticebus$ $couang$ (Figure 2). The first two groups contain the three most endangered species, increasing the value of close relatives. $Nycticebus$ is an isolated genus, and $N.couang$’s sister species is listed as vulnerable ( $P(ext)=0.1$). Likewise for $Propithecusverreauxi$ - although it has close relatives and so a short PE, these relatives are at high risk of extinction, which increases its value; this is what we saw with species $D$ in figure 1.

This research was supported by a grant from the Marsden Fund, New Zealand to MS and AOM. AOM received additional support from NSERC Canada and the Institute for Advanced Study, Berlin. We thank Dave Redding, Klaas Hartmann, and Andy Purvis for discussion.