Tasmanian fox incursion.

There is understandably much concern about the possible biodiversity impacts of an established red fox population in Tasmania. Although there have been several attempted introductions of foxes into Tasmania over the previous 100 years, there has never been any conclusive evidence of an extant population [16]. However, sometime between 1998–2001 an unknown number of foxes were purportedly introduced into Tasmania and shortly thereafter, the first of a number of fox carcasses were discovered around the central north coast and midlands areas [15,17]. Subsequently, the scat-derived DNA evidence has been used to infer that the population was widespread in northern, eastern and southern Tasmania as of 2010 [15]. We note that the evidence underpinning this inference is contested [18].

Carcass discovery data.

Data on the discovery of fox carcasses were taken from publicly available data provided by the Invasive Species Branch of the Tasmanian Department of Primary Industries, Parks, Water and Environment (DPIPWE) (http://www.dpiw.tas.gov.au/). These carcasses are considered credible by authorities, comprising one hunter-killed fox in 2001 and three road-killed foxes (the last of which was discovered and reported in 2006) (Table 1). To avoid confusion we note that the relevant information is not limited to these four carcases. The failure to observe carcasses across extensive areas of Tasmania is also informative. It is this information (that no carcasses have been observed since 2006) that is the key observation underlying speculation about the status of the fox population in Tasmania.

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Table 1. Details of fox carcasses found in Tasmania.

https://doi.org/10.1371/journal.pone.0116631.t001

ABC computations.

ABC methods exploit the computational efficiency of modern simulation techniques by replacing the calculation of the likelihood with a comparison between the observed (D) and simulated (D’) data based on a model [D|θ] and a set of parameters θ that have prior distributions π(θ). The technical basis of the methods are explained in detail elsewhere, such as [12].

We construct a hierarchical model to describe the observed data. We create a simulation model of the foxes demographics and dispersal and incorporate uncertainties in our understanding of this using prior distributions on parameters. We then overlay a stochastic observation model to characterise the observation process, again reflecting uncertainties through prior distributions. The overall prior is π(θ).

The simplest ABC algorithm (ABC rejection sampler) in our case would then involve: (1) Sampling a set of model parameters θ* from the prior distributions π(θ) specified above; (2) Simulating a dataset D′ of road-killed and hunter-killed foxes using the hierarchical model; (3). Accepting the parameters θ* if a measure of the distance between the observed and simulated data is less than some arbitrary tolerance value ε(i.e. d(D,D′) ≤ ε where d is some distance function); (4) Repeating step one until a sufficient number of parameter combinations are accepted to characterise the posterior distributions of the parameters of interest. Whether or not the posterior distributions are correctly characterized depends on the tolerance between the observed data and accepted simulations. As ε→ 0, the correct posteriors are found with near certainty. The problem, of course, is that the probability of exactly reproducing the observed data may be so low that generating a sufficient number of samples for the posteriors may be computationally prohibitive. This is especially the case using rejection sampling which is inefficient due to repeatedly sampling areas of the parameter space that have a low probability of generating the observed data. Efforts to speed up the process generally involve the use of statistically sufficient summary statistics (if they exist) as a basis for estimating the discrepancy between D and D’, and the implementation of more efficient samplers. Toni et al. [12,19] provide more details.

Our measure of discrepancy was the sum over the observation period (conditional on the time of introduction) of the absolute difference between the number of observed and simulated carcasses, calculated separately for road kills and hunter kills. Approximate posterior distributions were generated using a an ABC sequential Monte Carlo (ABC SMC) sampler [12]— a particle filter. Starting with 1,000 particles (parameter combinations), the allowable discrepancy over the observation period between the simulated and observed road kill data was set to 2, 1, then finally zero. The allowable discrepancy for the hunter-killed data was set to 0 from the start. Hence for the final selection of accepted particles (and hence posterior distributions of parameters), the acceptance criteria was that the simulated road kills were exactly the same in number (n = 3) and timing (years 2003, 2005 and 2006) as was the simulated hunter kill (n = 1, year = 2001). No restriction was put on the location of simulated carcass discoveries, although this is clearly influenced by the value of model parameters. Thus there is additional information in the data that is not being used in the analysis. The reason for this is that the rejection rate of the sampler becomes too high if we incorporate these additional dimensions. Given this, the analysis is still well conditioned. The observed carcass discoveries are legitimate observation and the analysis conditions on this. While this data is aspatial, the interaction between the spatial observation and demographic processes means that a significant spatial component remains in the analysis.

Normal distributions were used as perturbation kernels for all parameters. We confirmed that the ABC SMC sampler was generating very similar posterior distributions to an ABC rejection sampler—our choice of the SMC sampler was to speed up computations.

Purpose & type.

In order to make inference about the likely demography, distribution and status of an invading fox population based on carcass discoveries, we require a spatio-temporal model containing the processes of interest. The purpose of the population spread model is to underpin inference on the spatio-temporal distribution of foxes in Tasmania, both now and into the future. To do this, the model needs to be simple enough that it is tractable computationally, and must marginalize to the process under question—are there foxes and where are they? In summary, the model must be “adequate” or “fit for purpose”.

We use a spatial model defined over a grid with each cell representing a 5 km × 5 km area which is deemed to be suitable or not to be colonized by a breeding fox population. This size cell is reasonable in terms of the typical home range size of a fox. Clearly this is a major abstraction of the fox population, but we argue it is adequate within the modelling framework used.

Assumptions include:

1. Introduced foxes are effective reproductive units from the first year of introduction (e.g. a least one male-female pair)
2. A female fox will always find a mate in each year (i.e. once occupied, a cell is capable of reproducing in each year).
3. The number of road kills in a cell is trivial in terms of cell population dynamics (i.e. has no effect on cell reproductive performance)
4. The hazard of being hit on a road is constant for all major roads.

Clearly some of these assumptions can be removed or relaxed (e.g. getting traffic volume data to underpin the hazard rate on roads). The ABC method requires that estimable parameters are assigned prior distributions. The details of these priors are given below.

Habitat suitability.

The prior distributions of habitat suitability are based on previous empirical studies conducted in mainland Australia summarised in [20]. The following land use classifications were deemed suitable to sustain a fox population—grazing of native pastures, forestry, plantations, modified pastures, cropping, horticulture, irrigated pastures and cropping, irrigated horticulture, intensive animal and plant production, rural residential, urban intensive uses and land in transition. Land use classifications deemed unsuitable were nature conservation (predominantly south-west Tasmania), managed resource protected areas, other minimal uses, and mining and waste. This produced a fox habitat suitability map with 48% of Tasmania deemed suitable for foxes, and a spatial distribution very similar to that of [17] and [15].

Locations of purported introductions.

Introductions were assumed to result in a breeding pair of foxes which are assumed to form a functional “occupied” cell. Based solely on the rumoured/alleged release locations of foxes [17], in each simulation we introduced foxes to cells near Longford, Oatlands, and St Helens. Clearly the distance from a major road that the introduced population first establishes could have a large effect of the timing of the observed road kills. To account for this, for any one simulation, first, the distance from a major road of the introduction was drawn from a uniform prior distribution on 0 to 25 km. An introduction cell with this setback from a major road was then selected with equal probability from the set of grid cells located within 30 km of the townships at the centre of the rumoured release locations. Note that we do not include the possibility of self- introduction at a major port, for which one case has been documented, and could have been occurring for many decades.

Timing of introductions.

The year of first introduction is rumoured to be 2001, although “accumulated evidence also indicates that such an act may have also occurred in 1999 and 2000” [17]. We backdate the rumoured date to include the possibility that foxes were introduced as early as 1995, and use a uniformly distributed prior on the period 1995–2001.

Dispersal.

Two dispersal kernels were used, one to reflect what could be considered “natural” dispersal behaviour within an established fox population, and the second “invasion” dispersal behaviour to accommodate the hypothesis (untested) that the invading low density (and persecuted) fox population could have different dispersal characteristics due to the lack of con-specifics. The natural spread kernel is based on a Weibull distribution fitted to the data of Coman et al. [21], who found that the great majority (70%) of red fox cubs were recaptured within 2 km of their original tagging location. For those that dispersed further, dispersal distances ranged up to 30 km with a mean of 11 km. The invasion kernel is circular, though with a flat density out to 30 km in all directions surrounding the occupied cell.

Survival & reproduction.

The cell survival parameter (q) is the probability that an occupied cell will retain a breeding fox population from year-to-year. A uninformative prior (Uniform [0,1]) was chosen, reflecting genuine uncertainty as to the effectiveness of the fox eradication program (e.g. no confirmed bait take by a fox as of end 2013), and the behaviour of the invading fox population, and the fact that past introductions have apparently failed [16]. The cell reproduction number (λ) which is the number of new occupied cells generated by occupied cell is similarly uncertain. Predation by Tasmanian devils (Sarcophilus harrisii, Boitard 1841) on fox cubs has been hypothesized anecdotally as a reason for failure of past introductions of foxes, though with the spread of devil facial tumour disease (DFTD) since about 1995 and associated major decline in devil abundance across much of northern and eastern Tasmania by 2006 [22], this hypothesized effect may be weakened considerably. A uninformative prior (Uniform [0,2]) was used for λ (i.e. somewhere between zero and the expected number of female foxes per litter assuming a 50:50 sex ratio). The net reproduction rate (NRR) for this simple system assuming survival before reproduction is (using the formula for the sum of a Geometric series): A NRR above unity is a necessary (though not sufficient) requirement for a population to establish.

Probability of road-kill.

Roads are a strong attractant for scavenging species such as foxes and they are at substantial risk of being involved in collisions with vehicles. For example, Snow et al. [23] estimated the annual survival rate for San Clemente island foxes (Urocyon littoralis, Baird 1857) was about 20% lower for individuals living near roads. We used the record of fox control on Philip Island [24] to help inform a prior on the probability that an occupied cell with a road passing through it will generate a road kill in any year. Over the 25-year period 1979/90–2004/05, there were 1,000 foxes recorded as being removed from Phillip Island, of which 35 were road kills. The fox population was thought to number at least 120–140 (removals of known cohort members) from 96/97 to 99/00. The island is about 100 km², which would be the equivalent of four 5 km × 5 km square cells, which could reasonably be assumed occupied in each of the years. So, the rate of road kill is 0.35 cell⁻¹ year⁻¹. The corresponding probability of at least one road kill will be about 0.30 cell⁻¹ year⁻¹. The population density of foxes in Tasmania is postulated as lower than that on Phillip Island, which also has a much higher density of roads, so the 0.30 was considered an upper bound. We were conservative on the low side, and used a Beta (10,90) distribution which has a mean of 10%, though considerable probability mass between 5% and 15%.

Probability of shot-kill.

The hunting “observational” process in Tasmania is substantial, with in excess of 4,000 deer hunting licenses issued annually by DPIPWE to hunt the fallow deer (Dama dama, Linnaeus 1758), whose range overlaps extensively with the habitat considered suitable for foxes, though we note a lack of overlap on the northern coastal fringe. Elsewhere hunting of small introduced game species (e.g. European rabbits Oryctolagus cuniculus Linnaeus 1758, European hares Lepus europaeus Pallas 1778) is a popular pastime across Tasmania (author’s personal observation). In addition, permits are issued for the control of native species that can cause browsing damage to agriculture, such as wallabies (Macropus spp.) and brushtail possums (Trichosurus vulpecula Kerr 1792). For the 2012–2013 season Wallaby licenses alone numbered 7,236 (DPIPWE – February 2013). Spotlight shooting, which is a recognized method to obtain fox carcasses, is a core method for much of this hunting. This effectively extends the hunting observational process to all of Tasmania other than the nature conservation estate. Despite the widespread nature of hunting in Tasmania, its effort is undoubtedly uneven in time and space. To allow for this, at the start of each simulation run, the probability of a cell being subject to potential hunting was randomly assigned using a 50% probability—that is hunting could occur on half the cells. This is an arbitrary choice on our part, and its main role is to accommodate for the possibility that a fox population is extant in a cell not subject to “observation” by shooting.

Generating a prior on the probability of shooting generating a shot fox from an occupied cell subject to hunting was difficult. Field et al. [25] estimated the probability of detecting foxes within a 1 km segment of a spotlight transect ranged between 6 and 18% depending on vegetation, but translating this to a typical annual probability of successfully shooting a fox within a 5 km × 5 km cell is difficult. We could again use the Phillip island data, although the intensity of spotlight searching effort was very high there—possibly much higher than would be typical for a hunted area in Tasmania. In the end we settled on a Beta (10,90) distribution (same as the probability of a road kill). This is possibly conservative on the low side, as Heydon & Reynolds [26] report a culling rate from Britain that would correspond to a much higher probability.

Ordering of events.

The ordering of events applied to occupied cells were survival, road or hunter kill, then reproduction.

Estimating time to next carcass discovery or extinction.

The trajectories for accepted simulations (see below) with an extant fox population as of the end of 2013 were projected stochastically out to 2022 using the parameters and state variables (e.g. spatial location of occupied cells as of 2013) unique to that simulation. These projections were used to estimate the time to the next fox carcass being detected, or the population going extinct, or either of these outcomes occurring.

Implicit prior on extinction probability and population size.

The choice of priors corresponded to an implicit prior probability of population extinction at the end of 2013 of 39.1%. Conditional on the population being extant, the number of occupied cells was variable (95% C.I. = 2–933) though typically large (median = 191).

Exploring prior sensitivity.

The sensitivity of the results to the choice of prior distributions may be of interest. Under the Bayesian paradigm if priors are formulated consistently based on one’s beliefs there should not necessarily be a requirement to present results arising from different priors. We note, however, that different people may have different prior beliefs, and hence a sensitivity analysis will assist their interpretation of the results. To facilitate this we conducted an analysis to investigate the sensitivity of our results to reasonable modifications of the prior distributions. The approach of rerunning the ABC computations with different priors is difficult to implement for our model due to the time needed to undertake computations. We can, however, use importance sampling to generate revised posterior distributions arising from alternative prior distributions. Briefly, using the modified prior distribution π′(θ) as the importance function we calculate the importance weights as π′(θ)/π(θ) and resample the posterior distribution with replacement using these weights to obtain the alternative posterior distribution[27].

The most informative prior parameters in our model relate to the observation process through road kills and hunting, hence it makes most sense to explore the sensitivity of the results to changes in prior distributions for these parameters. As the number of samples from our posterior distribution is reasonably small (n = 1,000), we are restricted somewhat in how far we can change the prior distribution before the importance weights become degenerate. We therefore explored increasing/decreasing the expected yearly probability that an occupied cell traversed by road generates a roadkill by ± 10% using the appropriate Beta distributions. Likewise, we explored the impact of altering the expected yearly probability that an occupied cell subject to hunting generates a hunter kill by ± 10%.

Model Fit.

An important requirement of any model is that its implicit properties are consistent with the observed data. To explore this we compared the observed data with the predictive distribution. Samples from the predictive distribution were generated by sampling parameters from the posterior and then using the simulation model to generate observations. We also simulated the prior predictive distribution by sampling directly from the priors. This allowed us to explore the influence of the priors.

Software & hardware.

All calculations was undertaken within the R computing environment [28] using the “raster” [29] and “simecol” [30] packages. Ten computer processor cores were utilised to run ten independent ABC SMC simulations with the random number generator for each processor seeded with a different starting value using the set.seed() function within R. Approximately 5,000 hours of processor time were required – considerable time savings would be expected if programmed more efficiently and/or in a faster computing environment.

The source R code is available on request from the corresponding author.