Detecting species at low densities: a new theoretical framework and an empirical test on an invasive zooplankton

Species often occur at low abundance and, as a result, may evade detection during sampling. Here, we develop a general theoretical model of how the probability of detecting a species in a discrete sample varies with its mean density in that location based on sampling statistics. We show that detection probability at low densities can be approximated with a simple one parameter saturating function: The probability of detection ( P ) should increase with the mean number of individuals in a discrete sample ( N ) as P  ~ αN/(1 + αN). We further show how the parameter α will be affected by species spatial aggregation, within‐sample observability, and gear catchability. We use the case of the invasive zooplankter, spiny water flea ( Bythotrephes longimanus ), to demonstrate that this theoretical model fits the empirical pattern of detectability. In Lake Mendota, WI, Bythotrephes went undetected despite rigorous long‐term zooplankton monitoring (possibly as long as 14 yr with 15 sampling dates/yr). Using our modeling framework, we found that the likelihood of detecting Bythotrephes was low unless peak annual densities exceeded 0.1 individuals/m 3 over multiple years. Despite using models based on Bythotrephes ecology to identify ways to increase detectability, detection probabilities remained low at low densities, such that cases of missed detection such as Lake Mendota are possible despite rigorous monitoring effort. As such, our theoretical model provides a simple rule of thumb for estimating sampling effort required to detect rare species when densities are expected to be low: The minimum samples required ( S ) to reliably detect a species can be approximated as S  ~   (1 +  N )/ N .