Table 3.

Details of the six macroevolutionary models that were simulated. X₀ = ancestral value of trait X; X_i = value of trait X in species i; λ_i = instantaneous speciation rate in species i; λ₀, λ₁ = instantaneous speciation rate for species in the 0 state or 1 state, respectively; r_ab = per-lineage rate of transition of X from state a to state b; t_i = time since species i was last involved in a speciation event. Models 1–5 were simulated using MeSA (www.agapow.net/software/mesa), model 6 with PhyloGen (tree.bio.ed.ac.uk/software/phylogen/).

model	details
1. punctuationally evolving key trait [60,61]	X0 = 100; Xi changes (in both daughters) only at speciation events; changes are drawn from a normal distribution, μ = 0, σ = 50. If Xi becomes negative it is set to 0. λi = 0.001 + Xi/100 000
2. gradually evolving key trait [60,61]	X0 = 100; X changes continuously by Brownian motion with μ = 0, σ = 5 per time unit; Xi and hence λi were assessed every 0.1 time units and at every speciation event. Negative Xi were truncated to 0. λi = 0.0001 + Xi /10 000
3. binary key trait	X0 = 0, λ0 = 1, λ1 = 10. r01 = r10 = 0.05. States and rates were assessed every time unit and at every speciation event
4. fast-evolving binary key trait	X0 = 0, λ0 = 1, λ1 = 10. r01 = r10 = 10. States and rates were assessed every time unit and at every speciation event
5. patency [61]	λi = max(5–15ti, 0.6); ages and rates were assessed every 0.001 time units and at every speciation event
6. spatial model	Initial λ = 1. Ancestral species placed on an infinite square grid (i.e. each cell is adjacent to four others). Species occupy only one cell; they are selected at random to speciate but can do so only if they are adjacent to at least one empty cell