Wildlife collision data are ubiquitous, though challenging for making ecological inference due to typically irreducible uncertainty relating to the sampling process. We illustrate a new approach that is useful for generating inference from predator data arising from wildlife collisions. By simply conditioning on a second prey species sampled via the same collision process, and by using a biologically realistic numerical response functions, we can produce a coherent numerical response

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... p between predator and prey. This relationship can then be used to make inference on the population size of the predator species, including the probability of extinction. The statistical conditioning enables us to account for unmeasured variation in factors influencing the runway strike incidence for individual airports and to enable valid comparisons. A practical application of the approach for testing hypotheses about the distribution and abundance of a predator species is illustrated using the hypothesized red fox incursion into Tasmania, Australia. We estimate that conditional on the numerical response between fox and lagomorph runway strikes on mainland Australia, the predictive probability of observing no runway strikes of foxes in Tasmania after observing 15 lagomorph strikes is 0.001. We conclude there is enough evidence to safely reject the null hypothesis that there is a widespread red fox population in Tasmania at a population density consistent with prey availability. The method is novel and has potential wider application. PeerJ reviewing PDF | ABSTRACT 10 Wildlife collision data are ubiquitous, though challenging for making ecological inference due to typically irreducible uncertainty relating to the sampling process. We illustrate a new approach that is useful for generating inference from predator data arising from wildlife collisions. By simply conditioning on a second prey species sampled via the same collision process, and by using a biologically realistic numerical response functions, we can produce a coherent numerical response relationship between predator and prey. This relationship can then be used to make inference on the population size of the predator species, including the probability of extinction. The statistical conditioning enables us to account for unmeasured variation in factors influencing the runway strike incidence for individual airports and to enable valid comparisons. A practical application of the approach for testing hypotheses about the distribution and abundance of a predator species is illustrated using the hypothesized red fox incursion into Tasmania, Australia. We estimate that conditional on the numerical response between fox and lagomorph runway strikes on mainland Australia, the predictive probability of observing no runway strikes of foxes in Tasmania after observing 15 lagomorph strikes is 0.001. We conclude there is enough evidence to safely reject the null hypothesis that there is a widespread red fox population in Tasmania at a population density consistent with prey availability. The method is novel and has potential wider application. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 29 trends in wildlife abundance. Doubts, however, remain on the appropriate methods for analysing such data, 30 and whether useful information can be retrieved. Indeed, a key challenge when making inference from 31 wildlife collision data is that it is essentially presence-only data. As such, we typically cannot resolve 32 the incidence rate over the temporal and areal extent of the study from the raw data alone as we do not 33 have information about the vehicle movement (traffic) factors (e.g. speed, volume) and the abundance and 34 behaviour patterns of the wildlife that leads to their interactions/collisions with vehicles. These factors 35 clearly matter. For example, Finder et al. (1999) give an example of landscape factors influencing the 36 collision rate of vehicles with wildlife, D'Amico et al. (2015) show that higher abundance leads to a 37 higher collision rate, and Hobday and Minstrell (2008) show that vehicle speed influences the probability 38 of a vehicle-wildlife collision. 39 Despite its limitations, wildlife collision data are sometimes the only source of information that a 40 species is present in an area of interest (e.g. Boles et al., 1994; Lubis, 2005) . Such data may be used as an 41 alternative method of resighting to make inference on marked predator populations (e.g. McClintock et al., 42 2015). Conversely, if sampling effort can be quantified, it should be possible to use a lack of wildlife 43 collision data for a particular species to make inference on the probability of species presence/absence, 44 and this is the key motivation of the new analysis approach that follows. 45 PeerJ reviewing PDF | Manuscript to be reviewed We note that where a numerical response function between a predator and its prey is known to 46 exist, one can evaluate the expectation of the abundance of the predator species conditional on the 47 113 inferring the expected number of wildlife collisions, though these data are typically not available. Indeed, 114 for our case study, compiling aircraft movement data and attributes of Australian airfields is an onerous 115 task. There are c.1,650 airfields covered by air services Australia, and even if traffic movement data were 116 collected for all airfields there is still uncertainty about the resident population of species of interest and 117 their access to the airport runways (perimeter fencing integrity etc.). Thus collection of more detailed 118 information may not resolve the uncertainty around interpretation of the raw fox runway strike counts, 119 and for wildlife collision data more generally. 120 To address these issues we present an alternative method of analysis. European hares (Lepus eu-121 ropaeus Pallas 1778) and European rabbits (Oryctolagus cuniculus Linnaeus 1758), collectively termed 122 lagomorphs, are consistently a significant component of fox diet (Saunders et al., 1995), and there is 123 expected to be a numerical response relationship between the two (Pech et al., 1992). The distribution of 124 foxes on mainland Australian is strongly influenced by the distribution and abundance of lagomorphs 125 where they occur concurrently over the vast majority of their ranges. The numerical response of foxes to 126 rabbits is strongly non-linear and largely concave (Pech et al., 1992) . 127 We propose using the runway strike rate of lagomorphs by aircraft as a proxy for both the hazard 128 rate posed by aircraft movements as well as reflecting the productivity of the airfield environment and 129 available access of the runway for small mammals (in this case foxes and lagomorphs). We aggregate the 130 strike data for each state in Australia and analyse the number of fox runway strikes conditional on the 131 number of lagomorph runway strikes. A degree of aggregation is necessary to overcome computational 132 problems that would arise from the sparseness of these data. Other spatial resolutions would be possible, 133 though the state-based approach had the advantage of the data already being classified by state. It is 134 possible that the distribution of foxes and rabbits is not constant across a state. Indeed, in the Northern 135 Territory the northernmost limit for rabbits and foxes is around Tennant Creek. The variable runway strike 136 3/8 PeerJ reviewing PDF | Manuscript to be reviewed 205 in Tasmania. Note, however, that both Caley and Barry (2014) and Caley et al. (2015) infer that extinction 206 is the most likely outcome. Our results are consistent with the rarity of road killed foxes in Tasmania 207 in comparison to mainland Australia where they numerous -there have only been three fox carcasses 208 found on Tasmanian roads and one produced by a hunter that were considered credible by Tasmanian 209 authorities. We note, however, that the provenance of these carcasses is the subject of a polemical debate, 210 and the deliberate hoaxing for some, or all of these carcasses cannot be definitively ruled out. 211 Looking to the future, if no further credible evidence of free-living foxes is found in Tasmania, 212 then exactly how widespread and abundant foxes have been in Tasmania will undoubtedly be subject to 213 ongoing debate. Excluding the current study, there is possibly considerable irreducible uncertainty in the 214 provenance of the data used to date (as, for example, argued by Marks et al., 2014). We note another 215 novel, independent observational process arising from the predation of foxes by wedge-tailed eagles 216 (Aquila audax, Latham 1802) that should provide additional, independent inference. Like their northern 217 hemisphere counterpart the golden eagle (Aquila chrysaetos, Linnaeus 1758), wedge-tailed eagles are 218 known to effectively prey on red foxes, and red fox remains are consistently found in wedge-tailed eagle 219 diets (at non-trivial percentages) in a wide range of habitats wherever foxes are present (e.g. Olsen et al., 220 2010; Sharp et al., 2002; Parker et al., 2007; Brooker and Ridpath, 1980; Glen et al., 2016). Their range 221 6/8 PeerJ reviewing PDF | Manuscript to be reviewed encompasses all of the parts of Tasmania of interest. The nest locations of the wedge-tailed eagle pairs 222 could be carefully searched for fox remains. Making inference from such data, particularly if no fox 223 remains are found, will need to be conditional on an appropriate eagle-fox detection model. Factors such 224 as the territory size of eagle pairs will set the spatial resolution of the resulting inference. 225 Traditional methods of inferring extinction have focussed on making inference from the sighting 226 record (e.g. Solow, 1993Solow, , 2016, where prior positive observations are used to estimate the sighting 227 rate/probability given species presence, and the probability of extinction is estimated accordingly given 228 the time or then number of surveys since the last sighting (see Caley and Barry (2014) for a recent Bayesian 229 implementation). The method illustrated here can make inference on the probability of extinction without 230 the need for prior sightings from the area of interest, provided there are data on a functionally linked 231 species available. Our approach is a logical extension of the empirical findings studies such as Barrientos 232 and Bolonio (2009), who showed that the presence of rabbits adjacent to roads increases the rate of 233 road-kill of the European polecat (Mustela putorius, Linnaeus 1758). Calibrating the numerical response 234 function is the extension that enables inference on the abundance of the predator of interest. Ideally 235 the choice for the form of the numerical response function would be informed by previous studies in 236 conjunction with the data. The Type III Holling numerical response used here is a more general form 237 than, for example, the simpler Type II Holling numerical response. In our example the inference from the 238 simpler model is near identical. 239 Of course, prior sightings are not required for estimating detection probabilities if the detection 240 power of the search effort is known independently. Quantifying detection probabilities becomes difficult, 241 however, when search effort varies in space and time, along with the wildlife species of interest. This 242 necessitates computationally intensive methods that respect the complexity of the underlying population 243 and surveillance processes (Caley et al., 2015). The key feature of the approach we have illustrated here 244 is that by statistically conditioning on a second, biologically linked species, it essentially integrates over 245 the unknown factors underpinning the observation effort and hence detection probability. Reid (1995) 246 notes how natural the process of conditioning is as a tool in everyday statistics, and our example here 247 demonstrates how it can be used to extract useful inference from data, for which at first glance may appear 248 uninformative. 249 Finally, although we have illustrated our approach using wildlife collision data, it is applicable to any 250 observational process which samples both the predator species of interest and a prey species (or multiple 251 prey species), for which a numerical response relationship between the two is known to exist and can be 252 calibrated.