Estimating primate divergence times by using conditioned birth-and-death processes

The fossil record provides a lower bound on the primate divergence time of 54.8 million years ago, but does not provide an explicit estimate for the divergence time itself. We show how the pattern of diversification through the Cenozoic can be combined with a model for speciation to give a distribution for the age of the primates. The primate fossil record, the number of extant primate species, and information about the structure of the primate phylogenetic tree are combined to provide an estimate for the joint distribution of the primate and anthropoid divergence times. To take this information into account, we derive the structure of the birth-and-death process conditioned to have a subtree originate at a particular point in time. This process has a size-biased law and has an immortal line running from the root of the tree to the root of the subtree, with species on the spine having modified offspring and length distributions. We conclude that it is not possible, with this model, to rule out a Cretaceous origin for the primates.     

On the use of grammars in vegetation analysis

Grammars provide a means of generating large amounts of variability from relatively small resources, and seem especially relevant when the variation to be modelled is discontinuous since they describe the organisation symbols from some alphabet. In order to use grammars, it is desirable that they be inferred from observational data, and we present three examples of such inference. Two involve temporal successions, while the third is based on spatial transects of aquatic vegetation in wetlands in Victoria, Australia. The temporal successions were adequately represented by simpler regular grammars, suggesting that the processes involved may be modelled in a relatively simple manner. The spatial data requires a context-free grammar, reflecting the greater complexity of the spatial series, the uniqueness of each site, and the difficulties of modelling parallel systems with serial models. Examining the grammars further leads to some suggestions concerning the features of them which are important for ecological interpretation. It further leads to a novel view of the nature of groups in classification.