Efficiently locating conservation boundaries: Searching for the Tasmanian devil facial tumour disease front

Biological Conservation 142 (2009) 1333–1339

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Efficiently locating conservation boundaries: Searching for the Tasmanian devil facial tumour disease front Michael Bode a,*, Clare Hawkins b,c, Tracy Rout a, Brendan Wintle a a

Applied Environmental Decision Analysis Group, School of Botany, University of Melbourne, Parkville, Victoria, Australia Wildlife Management Branch, Tasmanian Department of Primary Industries and Water, Hobart, Tasmania, Australia c School of Zoology, University of Tasmania, Private Bag 5, Hobart, Tasmania 7001, Australia b

a r t i c l e

i n f o

Article history: Received 15 August 2008 Received in revised form 13 December 2008 Accepted 16 January 2009 Available online 16 March 2009 Keywords: Decision theory Spatial sampling Uncertainty

a b s t r a c t Conservation management actions and decisions are often defined by the location of ecological boundaries, for example, the present range of invasive or threatened species. The position of these boundaries can be cryptic, and managers must therefore directly sample sites, an expensive and time-consuming process. While accurate boundary location techniques have been considered by ecological theorists, the issue of cost-effective, or optimal boundary location has not. We propose a general framework for boundary location which incorporates both cost-efficiency and uncertainty. To illustrate its application, we use it to help locate an infectious disease front in the endangered Tasmanian devil population. The method ensures optimal spatial sampling by maximizing the expected information gained from each sample. When resources are limited, our method provides more accurate estimates of the boundary location than traditional sampling protocols. Using a formal decision theory sampling design encourages economically efficient actions, and provides defensible and transparent rationale for management actions. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction Ecologists have an abiding interest in biological transitions, and the boundaries that separate different biomes (Risser, 1995), ecological communities (Rahel and Hubert, 1991; Rundle et al., 1998; Connolly and Roughgarden, 2001), and species ranges (McWilliams, 1991; Kroel-Dulay et al., 2004). Conservation biologists are also interested in ecological boundaries: the spatial extent of diseases (Ron 2005; Peterson et al., 2004), threatened species (Berg et al., 1994; Dobson et al., 1997; Palma et al., 1999) and invasive species (Higgins et al., 1999; Guisan and Thuiller, 2005) represent important conservation boundaries, and identify the extent of conservation management requirements. While some ecological boundaries are readily and remotely identifiable (e.g., a forest–grassland transition), the location of more cryptic boundaries can be difficult to identify. This is particularly true in conservation situations, where the ranges of threatened or invasive species (a vital cue for conservation managers) contract or expand dynamically, and cannot be remotely sensed. In such situations, conservationists must use sampling techniques to actively search for boundaries. While techniques for precisely identifying the location of a multi-dimensional ecological bound-

* Corresponding author. Tel.: +61 3 8344 5422; fax: +61 3 9347 5460. E-mail addresses: [email protected] (M. Bode), [email protected]. gov.au (C. Hawkins), [email protected] (T. Rout), b.wintle@unimelb. edu.au (B. Wintle).

ary within a region or transect are highly developed in the ecological literature (Ludwig and Cornelius, 1987; Fortin, 1994; Fortin et al., 2000; Fagan et al., 2003), little attention has been paid to the question of efficiency. Current boundary detection methods require considerable amounts of data, yet conservation managers are often asked to locate multiple boundaries (e.g., the extent of a set of invasive species) over large areas, with limited budgets, in a short time. Conservation science currently lacks methods for efficiently locating ecological boundaries. In this paper we outline a general methodology that calculates an efficient strategy for locating ecological boundaries, given a constrained search budget. The method applies dynamic optimisation techniques within a Bayesian framework (Regan et al., 2006). We demonstrate the application of this method by focusing on a pressing Australian conservation problem: locating the position of a ‘‘disease front” in the Tasmanian devil (Sarcophilus harisii) population – an endangered Australian marsupial. 1.1. Case study: The Tasmanian devil facial tumour disease Devil facial tumour disease (DFTD) is a debilitating, infectious cancer (Pearse and Swift, 2006; Siddle et al., 2007) that has been found with dramatically increasing frequency in Tasmanian devils since the first documented case in 1996 (Hawkins et al., 2006). First identified in the north-east of Tasmania, DFTD quickly spread down the entire east coast, and is currently moving west across the island (Fig. 1). It is estimated that if current trends continue, no dis-

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individuals against the benefits of securing devils from a larger, more genetically diverse population. 2. Methods

Fig. 1. Verified cases of DFTD in Tasmania. This map represents the known information about the extent of the disease. Stars represent the locations of laboratory verified infections. The two black lines in the north-west of the island represent example transect locations, along which managers will send trapping expeditions to search for infected devils.

ease-free populations will exist on the island by 2013; without successful intervention, there is a strong possibility that the wild devil population will be extinct within the next 25–30 years (McCallum et al., 2007). Once very common, the species was first listed as a threatened species in 2006 under the Tasmanian Threatened Species Act (1995), and was uplisted to Endangered in 2008, under both this legislation and the IUCN Red List (Hoffman et al., 2008), as a direct result of the effects of DFTD. The nature of the disease (Siddle et al., 2007) suggests that effective management actions will be limited to the removal of diseased animals from infected devil populations, and the creation of ex situ insurance populations using individuals from uninfected regions (McCallum and Jones, 2006). Removal trials began in 2004 at one location (Jones et al., 2007), and the creation of multiple insurance populations (with an effective population size of 500 animals) began in 2005 (Jones et al., 2007; DPIW, 2006). The safest source of uninfected individuals for these insurance populations is in regions not yet infected by DFTD. To proceed, the spatial extent of the disease must therefore first be ascertained. Specifically, the boundary between uninfected and infected populations (the ‘‘disease front”) must be located. To safely source insurance animals from as large an area as possible, managers need to know precisely where this disease front is located. The more precise our understanding of the front location, the larger and more genetically varied the resultant insurance populations can be. Only a limited amount of resources have been made available to send out trapping expeditions to search for this conservation boundary. Using a ‘‘decision-theory” approach (Berger, 1985; Possingham et al., 2001), we can optimally position the trapping expeditions, and then use Bayes’ theorem to rationally synthesise the resulting sampling data with our existing information about the infection’s extent. Although the true position of the front will only ever be known probabilistically, this probability distribution is needed to weigh the risk of accidentally translocating infected

The ecological boundary model we employ is one-dimensional, but can be extended to two dimensions. The model domain is defined by a transect line that connects regions thought to be uninfected (the north-west coast), with the regions known to be diseased (Fig. 1). The trapping expeditions will take place somewhere along this transect. The model’s origin (x = 0) is located at the edge of DFTD’s known extent – this means that any locations in the negative domain (x < 0) are definitely infected. We have some understanding of the front location, and this limited information describes, with a prior belief distribution, the probability that the front is found at each location along the transect. Our approach considers the front location from a Bayesian perspective (McCarthy, 2007). The resources available to the trapping program are severely constrained, sufficient only for two trapping expeditions along each of five transects. By the end of the second expedition, our objective is to know the location of the DFTD front as precisely as possible, which is equivalent to minimising the standard deviation of our posterior belief distribution. We must achieve this objective through careful selection of the sites for the two trapping expeditions. 2.1. Prior belief distribution To make the most of the limited information available to conservation managers, our method begins with a prior belief distribution, which represents the manager’s a priori belief about the boundary location. Depending on the situation, this information can range from very precise, to completely uninformative. Our objective is to use the information gained from the trapping expeditions to refine this prior belief distribution. The point-data that represent the existing information on the extent of DFTD (Fig. 1) are a biased estimate of the DFTD front location. The front at present is likely to be substantially to the west of these points for two reasons. First, the information was collected some time in the past. Given the movement of the front to date, it is fair to assume that the disease has reached more westerly locations since these disease reports were verified. Second, diseased devils will not be discovered and diagnosed as soon as a local population is infected (i.e., at the exact time when the front moves through an area), because the prevalence of the disease will initially be quite low. Although late-stage diseased individuals would be easily visible to a casual observer from a distance of 20 m (Hawkins et al., 2006), the disease is generally only found in 20–30% of a population, even when well-established (C. Hawkins, unpublished data). DFTD identification requires the incidental discovery of symptomatic animals (typically by informed community members) followed by laboratory tests. This process can be considered a form of low-level sampling, and we therefore expect a substantial lag between the initial infection of a population, and discovery. We estimate that the average time between initial infection and identification is approximately 6 months, based on the observed prevalence of the infection at the time of initial identification. In the years since the initial infection, the apparent spread of DFTD across Tasmania has varied substantially through time, with an average westward speed of 17 km/year (McCallum et al., 2007). Assuming that the monthly speed of the front is sampled from a uniform distribution, U  [0, 2.84] (implying an expected front velocity of 17 km/year), we can construct an informative prior about the current location of the disease front by repeatedly choos-

M. Bode et al. / Biological Conservation 142 (2009) 1333–1339

ing 12 monthly movement distances, and constructing an expected distribution for the total annual movement. Following the central limit theorem, the resulting distribution will be normal, with a mean of 17 km (Fig. 2). This prior is a simple first approximation, and a spatially explicit, individual-based model is currently being developed by the Tasmanian Department of Primary Industries and Water’s Save the Tasmanian Devil Program. This will provide a much more accurate prior belief distribution (reflecting, for example, the heterogeneous underlying landscape through which the disease front moves). Alternatively, a less informative, or an uninformative prior could be used, if reliable information on the boundary location does not exist, or if there is concern about the potential influence that the prior might have on the optimal search decisions. 2.2. Using imperfect sampling data This prior belief distribution needs to be updated by the results of direct sampling along the transect. While this information will result in a more accurate understanding of the boundary’s location, it is not conclusive, as the information can only indicate whether the sampling is likely to have taken place behind, or in front of, the boundary. Additionally, the sampling methodology will rarely be perfectly accurate, and the possibility of false information should also augment the manager’s new posterior belief distribution. Both of these factors influence the search for the DFTD front. Despite the fact that they represent new empirical evidence, the data gathered during the DFTD trapping expeditions must be interpreted carefully before they are used to update our belief in the disease front location. A diagnosis of diseased individuals ensures that the sampling location is behind the front. However, failure to detect any diseased individuals is imperfect evidence that the front is yet to pass this location. While DFTD is considered sufficiently virulent to cause local extinction (McCallum et al., 2007), it is not particularly communicable (Pearse and Swift, 2006). If the disease front has just moved through an area (indicating that infected individuals are present), the incidence of DFTD will initially be quite low. Although incidence will increase as time passes, identifying the presence of DFTD in areas that have only been infected for a few years is likely to be much more difficult than identifying areas which have been infected for a long time. The increase in DFTD prevalence (the proportion of individuals in a population that are infected) through time can be estimated from data collected at two locations where the devil population was monitored before and during the infection process (C. Hawkins unpublished data). This data indicates that prevalence can be approximated by a linear function with a fixed maximum of 30%

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(qmax = 0.3), which is reached approximately 3 years after the front moves through a population. Because we are interested in distance to the front, rather than the time since the front passed a particular location, we transform time into distance using the expected velocity of the front (17 km/year). The expected prevalence of DFTD at a trapping location xT kilometres behind a front at location xF is therefore equal to:

qðxT Þ ¼ minfqmax ; mq mðxF  xT Þg;

ð1Þ

where xF is the location of the DFTD front, and mq ¼ 0:0059. The resources for each expedition allow for 5 days of trapping, which is not enough time to catch every individual in the local population. Based on analyses of previous trapping trips in similar habitat, we can estimate the number of unique individuals, n, that will be caught over the 5 day period. If DFTD prevalence is q(xT), then the probability of catching i infected animals (out of the n trapped) is defined by the binomial distribution:

Probðcatch i infected at xT Þ ¼ n Ci  ðqðxT ÞÞi  ð1  qðxT ÞÞni :

ð2Þ

(Note that there is no evidence to suggest that infected devils are more or less likely to be trapped.) The number of individuals it is possible to catch on a particular trapping expedition (n) will depend on the density of the local population. We will present results for two different levels of population density – in the low density example (e.g., the forests in south-west Tasmania), five nights of trapping (a single expedition) results in five individuals caught. In the high density example (e.g., the agricultural land in the north-west), 20 individuals are caught in the single trapping expedition. 2.3. Integrating trapping data with the prior belief distribution Given that the objective of the direct sampling is to reduce the standard deviation of our posterior belief distribution about the location of the boundary, we need to define how the outcomes of a trapping expedition will affect the prior belief distribution (Fig. 2). We do this using Bayes theorem (McCarthy, 2007). This theory calculates how the effect of a particular sampling outcome (defined in the case of DFTD by the number of trapped animals that are infected) affects our belief distribution. Based on our understanding of prevalence (Eq. (1)) and detectability (Eq. (2)), the posterior probability of the disease front location, Pt(x), is defined by the equation:

Lðn; ijx; xT Þ  P r ðxÞ ; Lðn; ijx; xT Þ  Pr ðxÞ  dx x¼0

PtðxÞ ¼ R 1

ð3Þ

Fig. 2. Prior belief distribution about the disease front. This prior belief distribution uses our current, limited understanding of the disease spread dynamics to predict the current location of the front. The origin (x = 0) is situated at the last verified infected location on the transect.

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where Pr(x) is the prior belief, and L(n, i | x, xT) is the likelihood of finding i infected individuals out of the n that were trapped at location xT (relative to the disease front at location x). The definition of the likelihood, and the derivation of Eq. (3) are described in Appendix A. As a conceptual illustration, Fig. 3 shows the ways in which a trapping expedition could affect our belief in the disease front location. The thick grey line indicates the prior belief distribution (as in Fig. 2). If the trapping expedition were placed 8.5 km to the west of the site of known infection (the vertical line), there are three possible, qualitative outcomes: first, a small proportion of trapped individuals are infected; second, a large proportion of the trapped individuals are infected; and third, no trapped individuals are infected. Any of these outcomes would alter our belief about the location of the front. If the sample turns up any infected devils, we know for certain that the chosen trapping location is behind the true front. However, we are not sure how far behind the front we are – although if we find a large proportion of infected devils, we would assume that we are further behind the front than if we had only found a few infected devils. Using Eq. (3), we can quantify this change in our belief based on the likelihood of trapping a given number of infected devils (Fig. 3). If we found that a small proportion of the trapped devils were infected, our posterior belief in the location of the devil front would change to the dotted line. If the trapping sample contained higher proportion of infected devils, the posterior would move further westward – to the dashed line. On the other hand, if the trapping expedition does not find any infected devils, there are three possible explanations. (i) The chosen trapping location might be in front of the disease front; the expedition did not find any infected devils because the local population has not yet been exposed to DFTD. (ii) The result is a false negative. Trapping expeditions do not completely sample the population, and the low prevalence of the disease makes it unlikely to be detected in only 5 days. (iii) The result is a false negative, because the disease has a high prevalence in the local population, but the trapped devils were co-incidentally not infected. The majority of devils at any one time are not infected with DFTD, even in populations where the disease is well-established. Taking these possibilities into account, the posterior belief in the disease front location after a single negative trapping result is closer to the known disease front, but not markedly (the solid black line in Fig. 3). A negative result (the probability of observing zero infected devils in the sample is determined by Eq. (2) does not change the managers’ belief as much as a positive result, while the inference

arising from a positive result will depend on how many trapped individuals are infected. 2.4. Choosing the optimal trapping locations As discussed, our objective is to choose trapping locations to maximise the amount of information obtained by each survey. This amounts to choosing locations that minimise the standard deviation of the posterior belief distribution, once the two trapping expeditions are completed. However, given the stochastic nature of imperfect sampling, we can only minimise the expected standard deviation of the posterior distribution. We can determine the optimal locations for these expeditions using either an exhaustive search, or preferably Stochastic Dynamic Programming (SDP; see Appendix). These methods are both dynamic, as they must address the fundamentally sequential nature of our decision: the optimal location of the second trapping expedition will depend on the outcome of the first expedition. For example, if our first expedition were 10 km from the known extent of infection, and our trapping uncovered multiple diseased individuals, we would be more likely to send the second expedition further from the original site of known infection. If, on the other hand, we did not find any diseased devils, we would be inclined to move the second trapping expedition closer to the site of known infection. Optimal search decisions for the two-expedition DFTD front search are shown in Fig. 4, for both high (triangular markers) and low (circular markers) density areas. The location of the first trapping expedition is based solely on the prior belief distribution (shown by a dashed line), and occurs slightly behind the most likely front location. The placement of the second expedition depends on how many trapped devils at the first location were infected. The x-axis position of the markers indicates the optimal location of the second trapping expedition, given the number of individuals trapped at the first location that were infected, indicated by the y-axis position. If we do not find any infected devils, our second expedition is placed slightly closer to the site of known infection. The more informative our prior distribution, the smaller the distance moved towards the location of known infection after a negative observation. The more infected devils we trap, the further from the site of known infection that the second expedition should be located. Note that the density of devils in the landscape has little effect on the optimal placement of the next trapping expedition, given the same number of infected animals trapped. However, given that more infected animals can potentially be trapped in the

Fig. 3. Belief distributions before and after a single trapping expedition. The informative prior distribution is indicated by the thick grey line, (a). The single trapping expedition took place 10 km from the known extent of the disease, indicated by the vertical line, (c). The dotted line, (d), indicates the posterior belief distribution if 5 of the 25 trapped devils were infected. The dashed line (e) indicates the posterior belief distribution if all the trapped devils were infected. The solid line (b) indicates the posterior distribution if no diseased individuals were trapped.

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Fig. 4. Optimal location for the two trapping expeditions. The dashed line shows the prior belief distribution (scale at right) of the front location; the most likely front location is 8.5 km from the site of known infection. The first trapping expedition is sited at the solid markers below the x-axis. The site of the second expedition depends on how many trapped individuals were diseased, indicated by the markers (scale at left). Triangles correspond to high devil density (we expect to catch 20 devils), circles correspond to low density (we expect to catch five devils). For example, if we caught 20 devils, of which 15 were diseased (indicated by the arrow), we should site the second trapping expedition at 15 km from the site of known infection. If only two were diseased, we would site the second expedition at approximately 10.5 km.

high density landscape, the site of the second trapping expedition can be much further from the first location. 2.5. Relative performance of the decision theory approach To assess the performance of a decision theory approach, we compare it with currently used methods for determining the extent of a disease. The Australian Veterinary Emergency Plan advises that, to determine the extent of disease spread, ‘‘animals could be sampled in a radial pattern, at fixed distances from the known infected location” (Australian Veterinary Emergency Plan 2005, p. 60). Since the restricted budget of the devil managers only allows for two trapping expeditions along each transect, we follow these guidelines by placing the expeditions at the 99% confidence interval of our prior distribution (13 km west of the site of known infection), and halfway to this point (6.5 km west). We call this the ‘‘fixed-distance” approach to front location. The posterior belief distribution that will result from the fixeddistance approach has an expected standard deviation of r = 1.81, reduced from r = 2, the standard deviation of the prior distribution. In comparison, for the same sampling budget the decision theory approach yields an expected belief distribution with an expected standard deviation of r = 1.65, a substantial improvement compared with a fixed-distance approach. (These results are based on expeditions that trap 20 devils over 5 trapping days.) Note that framing the front detection problem in a decision theory context has provided the additional benefit of allowing alternative sampling strategies to be compared. In doing so, it also allows managers to predict the likely outcomes of a particular sampling budget. For example, the current budget allowed for sampling the DFTD front will reduce our uncertainty about the location of the devil front by approximately 17%. By predicting the likely outcome of optimal trapping, managers can decide whether the budget allocated to boundary detection is sufficient to achieve their desired goals. 3. Discussion We have proposed a general method for optimally determining the location of a conservation boundary with a limited sampling budget. The approach applies a sequential state-dependent deci-

sion algorithm, SDP, within a Bayesian belief framework. To illustrate the method, we have used it to calculate an optimal sampling strategy for locating the DFTD front in the endangered Tasmanian devil population. The methodology is conceptually straightforward, yet can be applied to quite complex conservation situations. For example, in the case of DFTD, the conservation boundary in question is moving continually across the landscape, the presence of disease is relatively hard to identify, and the transition between infected and uninfected populations is not discrete (i.e., prevalence declines with proximity to the front). Our results show that applying a decision theory approach yields a more precise understanding of the front location, given a fixed budget. In addition to a more precise understanding of the boundary location, there are three other benefits of decision theory approaches which are often overlooked. First the explicit nature of the problem definition helps define the problem’s various facets, and their expected impact on management interventions. Using such explicit methods, we are more likely to acknowledge and consider factors that are important to decision making, compared with intuitive choices. The sequential nature of the boundary location problem is an example of this. Given that the results from the first trapping expedition will improve our understanding of DFTD’s extent, the location of both trapping expeditions should obviously not be decided simultaneously. Second, a decision theory approach yields decisions that can be rationally defended, based as they are on our best understanding of the ecological situation, and an explicit acknowledgement of the problem’s underlying uncertainty. Finally, the transparent nature of a decision theory approach means that interested parties can assess and critique the various pieces of information which have informed the analysis (e.g., the function which describes the increase in DFTD prevalence through time), and how these factors have been integrated to provide a management intervention. If intuitive decisions made by different experts conflict, it is difficult to determine what factors have driven the divergence of opinion. Transparency and defensibility are important considerations in the management of Tasmanian devils, considering the cultural and economic importance of the world’s largest extant marsupial carnivore (McCallum and Jones, 2006), and the substantial domestic and international media interest in the species’ predicament.

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The use of SDP is clearly more mathematically involved than intuitive approaches, but it is well-understood, and by no means computationally intractable. It is also quite easy to consider additional complexity using this method. For example, individualbased simulation modelling (or population level mathematical models of the epidemic) will yield a more accurate prior belief distribution, but including such results would not complicate the methodology outlined here (the only change required would be a new prior belief distribution; Fig. 2). Considering the benefits of additional trapping expeditions will only increase the run-time of the algorithm linearly. Managers might also wish to consider alternative distributions of trapping resources (e.g., two trips of 5 days, or five trips of 2 days). The SDP algorithm could consider such questions with little difficulty, and can provide quantitative comparison of resulting gains and losses in order to identify the optimal strategy. Formulating the problem within a Bayesian framework allows the uncertainty surrounding the location of the boundary to be considered explicitly and quantitatively. In addition, the uncertainty that surrounds other aspects of the problem is acknowledged and incorporated (e.g., the potential for false-negative samples). Despite the fact that the Tasmanian Devil is Australia’s best known marsupial carnivore, many of the parameters underpinning this case study are uncertain, a problem is magnified by the population’s response to a new disease. By acknowledging and quantifying this uncertainty, a Bayesian framework allows managers to respond to, rather than ignore uncertainty (Clark and Gelfand, 2006). Of course, ongoing research that will resolve this uncertainty is vital. For the DFTD front problem in particular, the rate at which the disease front moves could be better understood, as could our understanding of how prevalence increases through time. A Bayesian decision theory framework permits managers to deal with these ever-present uncertainties in a coherent manner, while allowing them to include relevant new information. There is room in our DFTD results for a more thorough treatment of some aspects of uncertainty (e.g., the rate at which prevalence increases through time is assumed to be certain and deterministic). Future refinements should treat such assumptions as uncertain. Although the search for the location of the DFTD front will proceed along a set of one-dimensional transects (Fig. 1), some conservation boundaries will need to be searched for in two dimensions. An important example of this is the initial spread of an invasive species from a single point of escape. Although such a search could be framed one dimensionally – with a number transects aligned radially from the likely location of escape – it would be more appropriate to freely search for the boundary throughout the domain. Defining an optimal boundary location strategy in two dimensions is the obvious next step for this methodology. This additional dimensionality would not greatly complicate the numerical bayesian updating of the belief distribution (Eq. (3)). It would, however, make the computation of the optimal search strategy more demanding, with many more discrete system states required to define the potential belief distributions. The conservation boundary location problem that we addressed involves a simple and distinct ecological transition. In reality, ecologists and conservation biologists routinely deal with more complicated problems. For example, a transition of interest might be between habitat types defined by multiple factors (Fortin et al., 2000). The cost of sampling at different locations may also vary, depending on road proximity, or the negotiability of the terrain. Despite the resultant increase in complexity, these factors should still be dealt with in a decision theory framework. Indeed, the location of more complicated conservation boundaries will be even more difficult to consider without a rigorous mathematical framework. Faced with increasing complexity, intuitive approaches be-

come less justifiable. The ability of a manager’s ecological (and economic) intuition to optimise complex, sequential and uncertain boundary location problems is severely limited. Decision theory approaches offer a rational and transparent solution. Acknowledgements We thank H. McCallum and N. Beeton for advice on the application to Tasmanian devils. L. Waller and two anonymous referees provided insightful advice on the manuscript. Numerous staff members and volunteers of the Save the Tasmanian Devil Program who helped gather the field data that guided this work. The research question was defined during a workshop funded by the Applied Environmental Decision Analysis Centre, a Commonwealth Environmental Research Facility Hub. Appendix A In this appendix we outline the basic equations governing the updating of a prior belief distribution about the front location with sequential, uncertain sampling data. We also briefly run through the process of optimally locating a given number of sequential sampling locations. Efficiently searching for ecological boundaries is conceptually quite simple. We have some a priori information about the location of the boundary, contained in the prior distribution, Pr(x). Each of our sequential data gathering expeditions will refine this belief distribution, with certain trapping locations being more likely to result in particular outcomes (e.g., areas with low prior probabilities of infection are likely to result in negative trapping outcomes). We can search through the potential outcomes of proposed search locations, and choose to trap at sites which are likely to result in the most precise posterior understanding of the disease front location. Putting this idea into mathematical practice is slightly more complicated, and given that the belief distributions will generally be non-canonical, it must be implemented numerically. The prevalence, q (defined as the proportion of individuals infected), of devil facial tumour disease (DFTD) at a particular location depends on the distance to the disease front. If the trapping location is any distance ahead of the front, the prevalence is zero; if trapping is behind the front, the prevalence of the disease is positive. We assume that the prevalence of the disease increases linearly with this distance, to a maximum of qmax. The prevalence at location xT, given that the front is at xF, is therefore equal to:

qðxT jxF Þ ¼

8 > < mq ðxT  xF Þ > :

if xF  qmax =mq  xT < xF

qmax

if xT < xF  qmax =mq

0

if xT  xF

ðA1Þ

where qmax is the maximum prevalence the disease reaches in a population, xF is the location of the disease front, and mq is the linear rate of prevalence increase with distance. We can use this prevalence function to calculate a likelihood distribution for the ‘‘true” location of the disease front, given the outcome of a trapping expedition. The likelihood of finding i infecteds out of the n individuals trapped at xT, given the prevalence q is drawn from a binomial distribution:

Lðn; ijxF ; qÞ ¼ n C i  qðxT jxF Þi  ð1  qðxT jxF ÞÞni :

ðA2Þ

We can use this equation to calculate the likelihood of the front being at xF, L(n, i|xF, xT), by substituting Eq. (A1) into Eq. (A2). By updating our prior information with this likelihood function, we can calculate our posterior belief in the location of the disease front, given a particular trapping outcome:

M. Bode et al. / Biological Conservation 142 (2009) 1333–1339

PtðxÞ ¼ R 1 x¼0

Lðn; ijx; xT Þ  Pr ðxÞ ; Lðn; ijx; xT Þ  Pr ðxÞ  dx

ðA3Þ

where Pt(x) is our posterior belief that the disease front is located at x. The integral in the denominator can be constrained to the positive domain, given that we know that the DFTD front is beyond the location of the last verified infection (the origin). Operationally, we easily choose the two trapping locations by finely discretising the trapping transect, and exhaustively determining the outcome of every combination of sampling locations, and every sampling outcome. Although robust, this process can be computationally expensive as the number of trapping expeditions increases, and so a simple Stochastic Dynamic Programming algorithm (Bellman and Dreyfuss, 1962) can be applied to determine the optimal trapping locations in a computationally efficient manner. Belief distributions are approximated by skew-normal probability distribution functions. System states are each defined by the distributions’ first three moments, at a high resolution discretisation. State transition probabilities are defined by the application of Eqs. (A2) and (A3) to each discrete distribution. The potential management decisions correspond to trapping locations xT, with the aim of minimising the terminal value function, which is simply the variance of each probability distribution. A more full description of the SDP methodology in a conservation context can be found in Bode and Possingham (2007) or McDonald-Madden et al. (2008). References Australian Veterinary Emergency Plan, 2005. Wild Animal Response Strategy (Version 32). Primary Industries Ministerial Council, Canberra. Bellman, R.E., Dreyfuss, E., 1962. Applied Dynamic Programming. Princeton University Press, Princeton. Berg, A., Ehnstrom, B., Gustafsson, L., Hallingback, T., Jonsell, M., Weslien, J., 1994. Threatened plant, animal and fungus species in Swedish forests, distribution and habitat associations. Conservation Biology 8, 718–731. Berger, J.O., 1985. Statistical Decision Theory and Bayesian Analysis. Springer, New York. Bode, M., Possingham, H.P., 2007. Can culling a threatened species increase its chance of persisting? Ecological Modelling 201, 11–18. Clark, J.S., Gelfand, A.E., 2006. A future for models and data in environmental science. Trends in Ecology and Evolution 21, 375–380. Connolly, S.R., Roughgarden, J., 2001. A latitudinal gradient in Northeast Pacific intertidal community structure: evidence for an oceanographically based synthesis of marine community theory. The American Naturalist 151, 311–326. Dobson, A.P., Rodriguez, J.P., Roberts, W.M., Wilcove, D.S., 1997. Geographic distribution of endangered species in the United States. Science 275, 550– 553. DPIW, 2006. Devil Facial Tumour Disease Newsletter, March 2006. [WWW document] . Fagan, W.F., Fortin, M., Soykan, C., 2003. Integrating edge detection and dynamic modeling in quantitative analyses of ecological boundaries. Bioscience 53, 730– 738. Fortin, M., 1994. Edge detection algorithms for two-dimensional ecological data. Ecology 75, 956–965.

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