Extrapolation of the Species Accumulation Curve Associated to “Chao” Estimator of the Number of Unrecorded Species: A Mathematically Consistent Derivation

Abstract

Incomplete samplings are doomed to become common practice for many inventories of biodiversity, thereby inviting to extrapolate what the rate of accumulation of newly recorded species would be if sampling was to be continued any further. For this purpose, a new derivation is provided for the extrapolation of the Species Accumulation Curve associated to the “Chao” estimator of the number of unrecorded species. This new derivation strictly complies with the general mathematical relationship constraining the shape of any expression of the Species Accumulation Curve, while the extrapolation previously proposed by Chao & Chiu [1] does not. The mathematically relevant formulation for the extrapolation R(N) of the Species Accumulation Curve associated to “Chao” estimator is thus: R(N) = R(N0) + [f12/(2 f2)](1 – exp[– (2f2/f1/N0 ). (N – N0)]), with N0 as the actual sample size, f1 and f2 as the numbers of species actually recorded once and twice and R(N) as the extrapolated number of species expected to be recorded as a function of sample size N (N> N0). Accounting for the constraining relationship mentioned above is also essential in another respect: it allows to extrapolate separately the numbers of species expected to be recorded 0-, 1-, 2-, >2- times, thereby permitting to analyse rationally the process of species accumulation during continuously growing sampling. At last, the preferred range of applicability of both “Chao” estimator and the associated extrapolation of the Species Accumulation Curve estimator is discussed, by comparison with the alternative type estimator Jackknife-2.