Application of fundamental equations to species-area theory.

Background: Species-area relationship (SAR), endemics-area relationship (EAR) and overlap-area relationship (OAR) are three important concepts in biodiversity study. The application of fundamental equations linking the SAR, EAR and OAR, can enrich the axiomatic framework of the species-area theory and deepen our understanding of the mechanisms of community assembly.

Results: Two fundamental equations are derived and extended to power law model and random replacement model of species-area distribution. Several important parameters, including the overlap index and extinction rate, are defined and expressed to enrich the species-area theory. For power law model, both EAR and OAR have three parameters, with one more parameter of the total area than SAR does. The EAR equation is a monotonically increasing function for parameter c and z, and a monotonically decreasing function for parameter A. The extinction rate, with two parameters, is a monotonically increasing function for parameter z, and a monotonically decreasing function for parameter A. The overlap index is a monotonically increasing function for parameter A, and a monotonically decreasing function for parameter z, independent of parameter c.

Conclusions: The general formats of SAR, EAR, OAR, overlap index, overlap rate, sampling rate and extinction rate, are derived and extended to power law model and random replacement model as the axiomatic framework of species-area theory. In addition, if the total area is underestimated, the extinction rate will be overestimated.