Species Area Relationship (SAR): Pattern Description with Geometrical Approach
Several formulations are describing the pattern of species-area relationship, log-log linear, semi-log linear, among others. These patterns mainly explain the species-area relationship for large areas, and for the small area, they provide significant differences from real data. We consider the geometric position of individuals of species, and base on that, we find the probability of observing at least one individual of the species. We apply a translation of the well-studied problem of mixed salt-water in a tank to describe the formula of SAR. For a rectangular sample area the species-area relationship follows the pattern, with some simplification, $S=c|A^β+a|^z$, where $S$ is the number of species in the area of size $A$ and $a,c,z$, and $β$ are constants with $z<1$ and $β\leq1$. We also show how the constant $z$ relates to some macroecological patterns, namely spatial aggregation, percentage of area coverage, and the core-satellite model. We exemplify our method using data on tropical tree species from a 50ha plot in Barro Colorado Island (BCI), Panama, using all individuals.