Properties of biodiversity indices that model future extinction risk

TL;DR

This work provides a simple and explicit description of the mean and variance of the HED(GE) indices, and derives various equalities for feature diversity, and an inequality if species extinction rates are correlated with feature types.

Abstract

The loss of biodiversity due to the likely widespread extinction of species in the near future is a focus of current concern in conservation biology. One approach to measure the impact of this extinction is based on the predicted loss of phylogenetic diversity. These predictions have become a focus of the Zoological Society of London's 'EDGE2' program for quantifying biodiversity loss and involves considering the HED (heightened evolutionary distinctiveness) and HEDGE (heightened evolutionary distinctiveness and globally endangered) indices. Here, we show how to generalise the HED(GE) indices by expanding their application to more general settings (to phylogenetic networks, to feature diversity on discrete traits, and to arbitrary biodiversity measures). We provide a simple and explicit description of the mean and variance of such measures, and illustrate our results by an application to the phylogeny of all 27 extant Crocodilians. We also derive various equalities for feature diversity, and an inequality if species extinction rates are correlated with feature types.

Figures (5)

• Figure 1: A rooted phylogenetic tree $T$ (left) and a rooted phylogenetic network $N$ (right) with leaf set $X = \{1,\ldots,5\}$ and on-negative edge lengths ($\ell_i \geq 0$ for each $i$). The edges are directed downwards.
• Figure 2: The nesting relationship between the different diversity measures described in this paper. We show later that two of these diversity classes (feature diversity and PD on networks) are equivalent as sets, despite their quite different definitions. However, all other nestings are strict.
• Figure 3: The expected values ($\psi_x$ and $\psi'_x)$ and standard deviations ($\sqrt{\Phi_x}$ and $\sqrt{\Phi'_x}$) of $\Delta_\varphi(S_x,x)$ and $\Delta'_\varphi(S_x,x)$, respectively, for the Crocodilian data-set. Both statistics vary widely across the species. See text for further details.
• Figure 4: The distribution of $\Delta'_\varphi(S_x,x)$ for the two alligator species at the top of Fig. \ref{['fig:croc']} ($x_1=$Alligator mississippiensis and $x_2=$Alligator sinensis), together with their associated means and standard deviations. The $x$-axis is in units of millions of years.
• Figure 5: A rooted phylogenetic network $N_\mathbb F$ realising the feature distribution $\alpha_1 = \{f_1, f_2\}$, $\alpha_2 = \{f_2, f_3\}$, and $\alpha_3 = \{f_1, f_3\}$.

Theorems & Definitions (8)

• Proposition 2.1
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 5.1
• proof