We construct a model for the evolution of continuous traits of species consisting of a quasi-birth-and-death process (QBD) {(X(t), φ(t)) : t ≥ 0} [1, 2] evolving within a phylogenetic tree [3].

We assume that the values of the continuous trait lie in the interval [a, b]. We partition this interval into subintervals labelled 0, . . . , N. Then the level X(t) = n of the QBD corresponds to the nth partition. Further, the phase φ(t) = i of the QBD models the direction and strength of selection on the trait.

We build on the ideas in [3] to derive methodology for the computation of the likelihood of observing the phylogenetic species tree given its topology and trait values observed at its tips.