A recent study has inferred that the red fox (Vulpes vulpes) is now widespread in Tasmania as of 2010, based on the extraction of fox DNA from predator scats. Heuristically, this inference appears at first glance to be at odds with the lack of recent confirmed discoveries of either road-killed foxes--the last of which occurred in 2006, or hunter killed foxes--the most recent in 2001. This paper demonstrates a method to codify this heuristic analysis and produce inferences consistent with assumptions and data. It does this by formalising the analysis in a transparent and repeatable manner to make inference on the past, present and future distribution of an invasive species. It utilizes Approximate Bayesian Computation to make inferences. Importantly, the method is able to inform management of invasive species within realistic time frames, and can be applied widely. We illustrate the technique using the Tasmanian fox data. Based on the pattern of carcass discoveries of foxes in Tasmania, we infer that the population of foxes in Tasmania is most likely extinct, or restricted in distribution and demographically weak as of 2013. It is possible, though unlikely, that that population is widespread and/or demographically robust. This inference is largely at odds with the inference from the predator scat survey data. Our results suggest the chances of successfully eradicating the introduced red fox population in Tasmania may be significantly higher than previously thought.