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.
Following Daniel P. Faith's seminal paper in 1992 [6], phylogenetic diversity (PD) has become an increasingly prominent measure of biodiversity. Defined on a weighted phylogenetic tree, PD sums the branch lengths connecting a given subset of species, aiming to capture biodiversity more completely than species richness alone.
Preserving PD and, more broadly, the ‘Tree of Life’ has become a central [5,23], though sometimes contested, goal in conservation biology [9,22]. Moreover, the concept of PD has inspired an extensive framework of PD-based metrics, including those that integrate it with extinction risk, primarily via the “Edge of Existence” programme [8,11], see also [18]. However, subsets of species that maximise phylogenetic diversity may not necessarily maximise other measures of biodiversity, such as feature [12] or functional diversity [14].
In this paper, we generalise concepts that were introduced and studied in [20] and [24] and which found application in the Zoological Society of London’s EDGE2 approach to quantifying biodiversity risk [8]. Our extensions are in two directions. First, we move beyond PD on trees to more general diversity measures for assigning a score to any subset of species. Some of these measures (e.g. feature diversity) share the properties that PD has of being submodular and monotone. However, these two properties could be overly restrictive.
For example, submodularity implies that the biodiversity score of a set S of species (denoted φ(S)) is less than or equal to the sum of the scores of the individual species in S (formally, φ(S) ≤ x∈S φ(x)). However, if the biodiversity score of S incorporates positive ecological interactions (whereby ’the whole is more than the sum of the parts’) then this submodularity condition may well be inappropriate.
Similarly, monotonicity implies that the biodiversity score assigned by φ to any subset S of species never decreases if another species x is added to S. However, if φ is a measure of future biodiversity, and species x is likely to cause the extinction of species in S, then monotonicity could also be problematic.
A second extension of the EDGE2 approach is to establish that the same type of product expression that arises for the EDGE2 score (an expected value of a random variable) can be applied to any biodiversity measure. We also provide an exact formula for its variance and, in the case of feature diversity (including PD), an explicit method for computing this variance. We apply our results to study the two EDGE2 indices -a measure of phylogenetic irreplaceability and a measure of the expected gain following conservation (both based on PD) -alongside their corresponding standard deviations for a data set of extant Crocodilians and briefly discuss our findings.
We begin by introducing some notation and recalling some earlier definitions.
1.1. Definitions. Let X be any finite set whose elements we refer to as ‘species’.
For each x ∈ X, each subset S x ⊆ X \ {x}, each choice I x ∈ {∅, {x}}, and any function φ : 2 X → R, let ∆ φ (S x , x) := φ(S x ∪ {x}) -φ(S x ) and ∆ ′ φ (S x , x) := φ(S x ∪ {x}) -φ(S x ∪ I x ). When φ is a biodiversity measure, ∆ φ (S x , x) can be interpreted as the marginal gain in the diversity of S x obtained by actively conserving species x. In contrast, ∆ ′ φ (S x , x) measures the change in diversity of S x relative to a scenario in which x is actively conserved and a scenario in which x is not actively conserved and, therefore, may either go extinct (I x = ∅) or remain extant (I x = {x}).
Now consider these two indices when S x and I x are random variables. Specifically, suppose that each species x ∈ X independently becomes extinct with probability ϵ x (or not, with probability 1-ϵ x ). This model has been referred to as the (generalised) field-of-bullets model of extinction, dating back to Raup [17]. We refer to it here as the g-FOB model. Thus, I x = {x} with probability 1 -ϵ x , and I x = ∅ with probability ϵ x , and each other species x ′ ∈ X \ {x} is (independently) present in the surviving set S x with probability 1 -ϵ x ′ .
For x ∈ X, let:
]. Note that ψ x generalises the heightened evolutionary distinctiveness (HED) score of species x, whereas ψ ′
x generalises its heightened evolutionary distinctiveness and globally endangered (HEDGE) score. Both HED and HEDGE were originally defined in the context of PD by [20] and correspond to the ED2 (irreplaceability) and EDGE2 (expected gain) metrics presented in [8].
In addition, we also consider the variance of these measures. Let
When φ is some biodiversity measure applicable to any collection of species, these two indices have a clear meaning. For example, ψ ′
x is the expected additional biodiversity that results from protecting species x (i.e. setting the extinction risk of x to zero rather than leaving it at ϵ x ) under a g-FOB model, whereas ψ x is the expected additional biodiversity (under the g-FOB model) that results from protecting s
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