The expected loss of feature diversity (versus phylogenetic diversity) following rapid extinction at the present

The current rapid extinction of species leads not only to their loss but also the disappearance of the unique features they harbour, which have evolved along the branches of the underlying evolutionary tree. One proxy for estimating the feature diversity (FD) of a set S of species at the tips of a tree is 'phylogenetic diversity' (PD): the sum of the branch lengths of the subtree connecting the species in S. For a phylogenetic tree that evolves under a standard birth-death process, and which is then subject to a sudden extinction event at the present (the simple 'field of bullets' model with a survival probability of s per species) the proportion of the original PD that is retained after extinction at the present is known to converge quickly to a particular concave function [Formula: see text] as t grows. To investigate how the loss of FD mirrors the loss of PD for a birth-death tree, we model FD by assuming that distinct discrete features arise randomly and independently along the branches of the tree at rate r and are lost at a constant rate [Formula: see text]. We derive an exact mathematical expression for the ratio [Formula: see text] of the two expected feature diversities (prior to and following an extinction event at the present) as t becomes large. We find that although [Formula: see text] has a similar behaviour to [Formula: see text] (and coincides with it for [Formula: see text]), when [Formula: see text], [Formula: see text] is described by a function that is different from [Formula: see text]. We also derive an exact expression for the expected number of features that are present in precisely one extant species. Our paper begins by establishing some generic properties of FD in a more general (non-phylogenetic) setting and applies this to fixed trees, before considering the setting of random (birth-death) trees.