An Example

We apply the phylogenetic diversity measures and similarity (or differentiation) measures considered in this chapter to a real conservation biology case discussed by Pavoine et al. (2009), a heavily-fished assemblage of 52 rockfish species of the genus Sebastes collected for 20 years over three decades (1980–1986, 1993–1994, 1996, 1998–2007) from the Southern California Bight, USA. The phylogenetic tree for these 52 species was obtained from Hyde and Vetter (2007); see Fig. 3a. The age of the root for these species is around 7.9 million years (Myr).

We separate the data into three decades: 1980s, 1990s and 2000s, which will be referred to as Assemblages (and Decades) I, II and III respectively. Within each decade's assemblage, species abundances are pooled. The species relative abundances for the three assemblages are shown in Fig. 3a. There were 48, 44 and 39 species in Decades I, II and III, respectively. (Note that each data point here is a mean of many years' observations.) A sub-tree containing only the six dominant species (those with relative abundance >8 % in at least one assemblage) is shown in Fig. 3b. All six species are shared in the three assemblages and four of them have been in isolated lineages for 6 Myr.

As suggested in section “Phylogenetic Hill numbers and related measures”, we present for each assemblage two types of profiles. In Fig. 4a, we plot the measure q D (T ) as a function of order q, 0 ≤ q ≤ 3, for two selected values of temporal perspectives: T = 0 (phylogeny is ignored) and T = 7.9 Myr (whole phylogenetic tree in Fig. 3a is considered). In Fig. 4b, we plot q D (T ) and qPD(T) as functions of T separately for q = 0, 1, and 2 for 0 ≤ T ≤ 10.

Based on our phylogenetic diversity measures, all profiles in Fig. 4 reveal that the diversity in the most recent decade (Decade III) is the lowest among the three decades in the rockfish assemblage. This implies an appreciable loss of species (as shown in the first type of profile for T = 0), loss of lineages (as shown in the second type of profile based on the measure q D (T ) ), and loss of evolutionary history (as

Fig. 3 (a) The phylogenetic tree of 52 rockfish species of the genus Sebastes (Hyde and Vetter 2007) and the species relative abundances in three assemblages: 1980s (Decade I), 1990s (Decade II) and 2000s (Decade III). The age of the root is T = 7.9 Myr. (b) A sub-tree contains only the dominant species (those with relative abundance >8 % in at least one assemblage), and these species are marked in figure (a). All six species are shared by the three assemblages and four of them diverged around 6 Myr ago (i.e., they have been in isolated lineages for 6 Myr) (See Pavoine et al. (2009) for details)

shown in the second type of profiles based on the measure qPD(T)) over the three decades.

When species/lineage abundances are discounted (q = 0 in the left panels of Fig. 4b), both lineage richness (based on the measure 0 D (T ) ) and total branch lengths (based on the measure 0PD(T), i.e., Faith's PD) exhibit the expected ordering: Decade I> Decade II > Decade III. When species/lineage abundances are counted (i.e. q = 1 and 2 in Fig. 4b), the profiles for Decades I and II cross because the assemblage of Decade II has more even abundant species than that of Decade I (see the first type of profiles for T = 0 and Fig. 3a, b). Note that if the time-depth is greater than 6 Myr (including the age of the root), then all the abundance-sensitive phylogenetic measures for the three assemblages are very close because most of the dominant species began to diverge around 6 Myr (Fig. 3b). This also explains the closeness of the three profiles in the first type of profile for T = 7.9 Myr (the right panel in Fig. 4a).

Fig. 4 (a) The first type of diversity profile plots q D (T ) as a function of order q, 0 ≤ q ≤ 3, for two selected values of temporal perspectives: T= 0 (non-phylogenetic case) and T = 7.9 Myr (the age of the root of the phylogenetic tree in Fig. 3a). (b) The second type of diversity profile plotsq D (T ) (phylogenetic Hill number) and qPD(T) (phylogenetic diversity) as functions of T, 0 ≤ T ≤ 10, separately for q = 0, 1 and 2

To illustrate the phylogenetic differentiation among assemblages, we focus on measuring the phylogenetic differentiation between any two decades for three pairs (i.e. Decades I vs. II, Decades I vs. III and Decades II vs. III). To see how the phylogenetic differentiation measures vary with the time perspective q and with the order T, we show two types of profiles for each of the two differentiation measures 1CqN (T ) and 1-UqN (T ) in Figs. 5 and 6. In Fig. 5a, we present the first type of profile that plots the measure 1CqN (T ) as a function of q where q is in the range [0, 3] for two time perspectives: T = 0 (non-phylogenetic case) and T = 7.9 Myr (the age of the root node). In Fig. 5b, the same type of differentiation profile is shown for the other measure 1-UqN (T ) . Then in Fig. 6a, b, we present the second type of

Fig. 5 (a) Differentiation profiles of the measure 1CqN (T ) and (b) of the measure 1-UqN (T )

as a function of order q, 0 ≤ q ≤ 3, for two specific time perspectives: T = 0 (left panels, corresponding to non-phylogenetic differentiation profiles), and T = 7.9 Myr (right panels, corresponding to the profiles for the age of the root node of the pooled phylogenetic tree in Fig. 3a) for three pairs of assemblages (I vs. II, I vs. III, and II vs. III)

profile that shows the two measures as a function of temporal perspective T, 0 ≤ T ≤ 10, for q = 0, 1 and 2 separately.

Based on the two phylogenetic differentiation measures, all profiles in Figs. 5 and 6 show consistent patterns. When species/lineages abundances are discounted (q = 0), the differences among the differentiation measures of the three pairs of assemblages are not appreciable, as shown in the two left panels in Fig. 6 and in the initial point in each of profiles in Fig. 5. When species/lineages abundances are counted (q > 0), the compositional differentiation between Decades I vs. II is generally close to that between Decades I vs. III, and the differentiation between two recent decades (Decades II vs. III) is much lower than any of the other two pairs. This implies that the composition of species/lineage abundances has changed after 1990. Examining the relative abundances for those dominant species listed in Fig. 3b, we see that the most abundant species S. paucispinis (23 %) in Decade I became less abundant in both Decade II (9 %) and Decade III (11 %); the second most abundant species S. mystinus (11 %) in Decade I became quite rare in both Decade II (4 %) and Decade III (5 %). Also, the species S. miniatus in Decade I was rare, but it became the most dominant species in both Decade II (12 %) and Decade III (25 %). These compositional changes for dominant species help explain the above findings.

As the time perspective T becomes large, more dominant shared lineages are added to the two assemblages, implying the differentiation between any two assemblages should exhibit a non-increasing trend as T is increased. Our two differentiation measures for q > 0 in Fig. 6 show the expected decreasing trend, and the decline rates differ for q = 1 and q = 2. Based on Fig. 3b, we see that most of the dominant and isolated species began to diverge around 6 Myr ago. Thus, the two differentiation profiles for q = 1 and 2 start to decrease sharply around 6 Myr especially for order q = 2. Since the node abundances near roots (where the differentiation values are near zero) are relatively high and dominant in the whole tree, all values of the phylogenetic differentiation measures for T = 7.9 Myr (the first type of profile for T = 7.9 Myr in the right panel of Fig. 5) are substantially lower than their corresponding non-phylogenetic differentiation measure by comparing two figures (T = 0 and T = 7.9 Myr) in each row of Fig. 5. The two types of profiles (in Fig. 5a, b, and 6a, b) demonstrate that the two differentiation measures 1CqN (T ) and 1-UqN (T ) can incorporate the differences in both tree structure and lineage abundances.

Fig. 6 (a) Differentiation profi of the measure 1CqN (T ) and (b) of the measure 1-UqN (T ) ,

as a function of the time perspective (or time-depth) T, 0 ≤ T ≤ 10, for q = 0 (left panel), q = 1 (middle

panel), and q = 2 (right panel) for three pairs of assemblages. All measures are computed for the interval [−T, 0], where T varies from 0 to 10

In summary, our phylogenetic diversity measures have shown an appreciable loss of species, lineage and evolutionary history in rockfish assemblage over time due to fishing pressure, and our phylogenetic differentiation measures show a pronounced change of species/lineages composition after 1990.