Trait Evolution on Phylogenetic Networks 3

14 Species may evolve on a reticulate network due to hybridization or other gene flow rather than 15 on a strictly bifurcating tree, but comparative methods to deal with trait evolution on a network 16 are lacking. We create such a method, which uses a Brownian motion model. Our method seeks 17 to separately or jointly detect a bias in trait value coming from hybridization (í µí»½) and a burst of 18 variation at the time of hybridization (í µí±£ !) associated with the hybridization event, as well as 19 traditional Brownian motion parameters of ancestral state (í µí¼‡) and rate of evolution (í µí¼Ž !) of 20 Brownian motion, as well as measurement error of the tips (SE). We test the method with 21 extensive simulations. We also apply the model to two empirical examples, cichlid body size and 22 Nicotiana drought tolerance, and find substantial measurement error and a hint that hybrids have 23 greater drought tolerance in the latter case. The new methods are available in CRAN R package 24 BMhyd.