The biological meaning of D

Since Yule (1925) the default model of a diversifying clade has been a memoryless birth process: every lineage speciates at the same instantaneous rate, independent of how long it has existed. Nee, May & Harvey (1994) built the standard likelihood of a reconstructed tree on exactly this assumption. Yet real phylogenies persistently violate it in two ways a constant-rate process cannot produce: diversification slows down toward the present (Pybus & Harvey's γ<0), and clades harbour long, isolated terminal branches — living fossils — lineages that have persisted for tens of millions of years without splitting. Two repairs dominate: making the rate depend on standing diversity (diversity-dependence), and making it depend on lineage age — protracted speciation, where speciation takes time to complete (Rosindell et al. 2010; Etienne & Rosindell 2012), and explicit age-dependent birth–death models (Hagen et al. 2018). The covariate D operationalises this second repair at the level of a single lineage.

Let t_s be the last time lineage s speciated and I_s=t−t_s its isolation time — the length of its current pendant edge. With N lineages alive and mean isolation M:

So D is simply the centered pendant edge: how much longer (D>0) or shorter (D<0) a lineage has gone without speciating than its living peers. Because Σ_sD_s=0 by construction, D carries no information about the clade as a whole — only about the relative position of a lineage within it.

This ties D to two familiar measures. For a present-day lineage, I_s is its terminal branch length, so D at the tips is the centered terminal branch: the raw material of evolutionary distinctiveness (Isaac et al. 2007 — long pendant = distinct, "on the EDGE") and, inversely, of the tip diversification rate DR (Jetz et al. 2012 — short pendant = fast recent diversification). D>0 is thus a relatively isolated, relictual lineage; D<0 a recently-radiated one. Evaluated at every branching event, D makes this instantaneous and relative.

• D is right-skewed (pooled skew 2.3): the great majority of lineages have near-average or short pendants, punctuated by a thin tail of extreme isolates. Only 22.5% of events involve a lineage older than its clade's mean — clades are dominated by the recently diversified.
• D is decoupled from richness. Its correlation with N is ≈0 (within-tree median |r|=0.19) and its spread is flat across N-quintiles. How isolated a lineage is says nothing about how species-rich its clade is: relicts occur at every diversity level.
• Extreme isolation is ubiquitous. In many clades a single lineage carries a pendant edge spanning almost the entire history of the group (Table 1) — the living-fossil signature, and the atlas shows it is the rule, not the exception.
• Splitting favours the young. The lineage that speciates has, on average, a shorter-than-mean pendant (D̄<0). Descriptively, speciation falls on recently-diversified lineages more than chance — the direction of an age effect — though reconstruction's "pull of the present" confounds it, so it is a pattern to be tested by the model, not read off directly.

Table 1. The most isolated lineage in a sample of atlas clades: its pendant edge as a fraction of crown age. A ratio near 1 means a lineage isolated for essentially the clade's entire history.

Placed in the rate, λ_s = f(⋯ + β_DD_s), the coefficient β_D tests age-dependent speciation directly, and its sign is interpretable:

Because Σ_sD_s=0, β_D acts only on which lineage splits, never on the clade's overall rate: a pure test of internal, age-structured heterogeneity, cleanly separated from diversity-dependence (carried by N). This is why D and N are the two orthogonal covariates — one asks when the clade diversifies, the other which lineages within it do.

D is not an abstract statistical residual: it is relative phylogenetic isolation — the quantity conservation biology calls evolutionary distinctiveness and macroevolution reads as tip rate, made instantaneous and mean-centered. The atlas shows it is right-skewed, richness-independent, and dominated by a tail of ancient relicts — a covariate biologically legible on its own and, through β_D, one that turns the model into a direct test of whether speciation is age-structured. With N it forms a two-covariate model whose axes answer the two questions a phylogeny can pose: the tempo of diversification, and the age structure of who diversifies.