Component-compatibility in historical biogeography

-The problems of reconstructing historical relationships for areas of endemism from distributional data for groups of taxa and the cladistic relationships among the members of those groups can be solved by applying the two principles of parsimony and mutual inclusion or exclusion (compatibility) of components. Components can be extracted from a data matrix by means of transcription into partial monothetic sets. The data matrix thus derived represents the distribution over areas for the monophyletic groups in one or more cladograms. It is derived from two different matrices by boolean multiplication. The first matrix gives the binary representation of distributions of taxa over areas of endemism; the second describes the cladogram for the same taxa, in terms of character states converted into binary form by additive binary coding. The derived data matrix can be used in historical biogeography to represent the given phyletic data ( Assumption 0 here newly defined), and can be amended to reflect Assumptions 1 or 2 to accomodate the problems of wide-spread taxa and missing areas. Areacladograms are determined from the derived matrix by searching for the largest sets of mutually compatible components. Area-cladograms are evaluated in terms of support (vicariance) and contradiction (ad hoc interpretations such as dispersal and extinction). Area-cladograms that best fit the data matrix regarding the balance between support and contradiction are selected as the best possible recontructions of relationships among the areas of endemism. The procedure is illustrated by the example of the poeciliid fish genera Heterandria and Xiphophorus, and several other standard examples.