Summary1. Invasive vertebrate species are a world-wide threat to biodiversity and agricultural production. The presence of foxes, one of the most damaging invasive vertebrates introduced to Australia, has now been confirmed in the island state of Tasmania, placing at risk many species of native vertebrates and substantial agricultural industry. 2. Effective eradication of such a rare but elusive carnivore requires robust strategies informed by novel but systematic detection. 3. We combine DNA detection approaches for trace samples with systematic stratified and opportunistic surveys of carnivore scats to estimate the current distribution of foxes in Tasmania. We use that DNA evidence and other hard evidence provided by carcasses and other material to build a predictive model of fox habitat suitability for all of Tasmania. 4. We demonstrate that this destructive species is widespread in northern and eastern Tasmania but has not yet reached the limits of its range. The widespread nature of this distribution reveals that targeting fox activity hotspots only for eradication is unlikely to be successful and that a strategic and statewide approach is required. Our habitat suitability model can provide a basis for prioritizing areas for fox management. 5. Synthesis and applications. Our approach highlights the importance of early and preemptive surveys of recently established, and therefore rare, invasive species and the necessity of providing a sound and defensible approach to determining the distribution of the invasive species. This approach provides a template for the systematic detection of rare cryptic carnivores.

Summary1. Polymerase chain reaction (PCR) diagnostic tests are increasingly applied to the identification of wildlife. Yet rigorous verification is rare and the estimation of test accuracy (the probability that true positive and true negative samples are correctly identified -test sensitivity and specificity, respectively), particularly in combination with sequencing, is uncommon. This is important because PCR-based tests are prone to contamination in sampling and the laboratory. 2. Here, we use an experimental case-control approach to estimate the sensitivity and specificity of a sequential PCR-based wildlife detection test used to identify incursions of red foxes into Tasmania from predator faeces (scats). 3. Our results show that the sensitivity of the fox test is high ( $ 94%) for the PCR-based test on its own, but this decreases to $ 84% when combined with the DNA sequencing step. In contrast, the specificity increases from $ 96% in the PCR-only test to~99Á6% after inclusion of the DNA sequencing step. 4. The intense public scrutiny of the fox eradication programme in Tasmania has undoubtedly influenced the application of a sequential PCR test that maximizes specificity at the expense of sensitivity and so increases the risk that scats containing fox DNA would not be detected. This could lead to the establishment of foxes in Tasmania as a consequence. 5. Synthesis and applications. Importantly, the estimation of the sensitivity and specificity of sequential tests enables decisions about the risk associated with mistaken identification (i.e. false negatives vs. false positives) to be quantified for decision-makers. The cost of false-negative errors should be balanced against the costs of false-positive errors, which could include the expenditure incurred in the application of unnecessary management actions were foxes not in fact present. Understanding the risks and costs associated with both false-negative and false-positive errors is therefore a key component to the decision-making process for the management of the Tasmanian fox incursion.

The ability to detect the incursion of an invasive species or destroy the last individuals during an eradication program are some of the most difficult aspects of invasive species management. The presence of foxes in Tasmania is a contentious issue with recent structured monitoring efforts, involving collection of carnivore scats and testing for fox DNA, failing to detect any evidence of foxes. Understanding the likelihood that monitoring efforts would detect fox presence, given at least one is present, is therefore critical for understanding the role of scat monitoring for informing the response to an incursion. We undertook trials to estimate the probability of fox scat detection through monitoring by scat‐detector dogs and person searches and used this information to critically evaluate the power of scat monitoring efforts for detecting foxes in the Tasmanian landscape. The probability of detecting a single scat present in a 1‐km2 survey unit was highest for scat‐detector dogs searches (0.053) compared with person searches (x¯≅0.015) for each 10 km of search effort. Simulation of the power of recent scat monitoring efforts undertaken in Tasmania from 2011 to 2015 suggested that single foxes would have to be present in at least 20 different locations or fox breeding groups present in at least six different locations, in order to be detected with a high level of confidence (>0.80). We have shown that highly structured detection trials can provide managers with the quantitative tools needed to make judgments about the power of large‐scale scat monitoring programs. Results suggest that a fox population, if present in Tasmania, could remain undetected by a large‐scale, structured scat monitoring program. Therefore, it is likely that other forms of surveillance, in conjunction with scat monitoring, will be necessary to demonstrate that foxes are absent from Tasmania with high confidence.

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