Detecting target species: with how many samples?

The detection of target species is of paramount importance in ecological studies, with implications for environmental management and natural resource conservation planning. This is usually done by sampling the area: the species is detected if the presence of at least one individual is detected in the samples. Green & Young (Green & Young 1993 Sampling to detectrare species. Ecol. Appl. 3, 351-356. (doi:10.2307/1941837) introduce two models to determine the minimum number of samples n to ensure that the probability of failing to detect the species from them, if the species is actually present in the area, does not exceed a fixed threshold: based on the Poisson and the Negative Binomial distributions. We generalize them to two scenarios, one considering the area size N to be finite, and the other allowing detectability errors, with probability δ. The results in Green & Young are recovered by taking N → ∞ and δ = 0. Not taking into consideration the finite size of the area, if known, leads to an overestimation of n, which is vital to avoid if sampling is expensive or difficult, while assuming that there are no detectability errors, if they really exist, produces an undesirable bias. Our approximation manages to skirt both problems, for the Poisson and the Negative Binomial.