The Variations of the Numbers of Species Recorded 1-, 2-, … x-times (Singletons, Doubletons, … x-tons) with Increasing Sampling-size : An Analytical Approach Using Taylor Expansion

The process of species accumulation, during progressive sampling, results in the regular, monotonic increase of the number of recorded species with sampling size. On the contrary, the numbers f1(N), f2(N), f3(N), …, fx(N) of those species recorded 1-, 2-, 3-, …, x-times at sampling-size N all show non-monotonic variations with N. The major characteristic elements of this non-monotonic variations (namely: the maximum reached at ∂fx (N)/∂N = 0 and the inflexion point at ∂2fx (N)/∂N2 = 0) provide interesting cues regarding the degree of advancement of sampling completeness. Such cues yet remain undetectable however along the regular, monotonic increase of the species accumulation curve itself. Although usually unrecorded, the variations of the fx(N) may yet be computed and, accordingly, the associated cues above thereby made available in practice. This computation involves the Taylor expansion of the fx(N), making use of recently derived mathematical properties of the species accumulation process. For common practice, focus is placed upon the variations of the fx(N) of lower-orders (i.e. f1(N), f2(N), f3(N), f4(N)), which is sufficient to disclose information of particular relevance in assessing the progress of sampling towards completeness.