Successful eradication of invasives is facilitated by early detection and prompt onset of control. However, realizing or verifying that a colonization has occurred is difficult for cryptic species especially at low population densities. Responding to the capture or unconfirmed sighting of a cryptic invasive species, and the associated effort to determine if it indicates an incipient (small, localized) population or merely a lone colonizer, is costly and cannot continue indefinitely. However, insufficient detection effort risks erroneously concluding the species is not present, allowing the population to increase in size and expand its range. Evidence for an incipient population requires detection of ≥1 individual; its absence, on the other hand, must be inferred probabilistically. We use an actual rapid response incident and species-specific detection estimates tied to a known density to calculate the amount of effort (with non-sequential detections) necessary to assert, with a pre-defined confidence, that invasive brown treesnakes are absent from the search area under a wide range of hypothetical population densities. We illustrate that the amount of effort necessary to declare that a species is absent is substantial and increases with decreased individual detection probability, decreased density, and increased level of desired confidence about its absence. Such survey investment would be justified where the cost savings due to early detection are large. Our Poisson-based model application will allow managers to make informed decisions about how long to continue detection efforts, should no additional detections occur, and suggests that effort to do so is significantly higher than previously thought. While our model application informs how long to search to infer absence of an incipient population of brown treesnakes, the approach is sufficiently general to apply to other invasive species if density-dependent detection estimates are known or reliable surrogate estimates are available.