The problems of controlling exotic species have been acknowledged as serious threats to an indigenous ecosystem as well as to society. In response to these threats, various management programs of exotic species have been proposed for supporting eradication in many regions. Although eradication is regarded as the first-best solution, such attempts have been unsuccessful in reality due to mainly two factors: (1) stock-dependent catchability, and (2) uncertainties. This article demonstrates when to aim at eradication through addressing an optimal adaptive management strategy in the framework of a bio-economic model with the aforementioned factors. The study sets out that the sensitivity of catchability in response to a change in the existing stock determines whether or not aiming at eradication is justified. The results also show that process uncertainty associated with stock growth significantly affects the timing of removal actions for eradication, and an increase in the degree of uncertainty could help achieve eradication in a cost-effective manner if we optimally adapt our removal actions to the uncertainty.