Prevent, search or destroy? A partially observable model for invasive species management

Authors

  • Tracy M. Rout,

    Corresponding author
    1. School of Botany, University of Melbourne, Parkville, Vic., Australia
    2. School of Biological Sciences, University of Queensland, St Lucia, Qld, Australia
    Current affiliation:
    1. School of Biological Sciences, University of Queensland, St Lucia, Qld 4072, Australia
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  • Joslin L. Moore,

    1. Australian Research Centre for Urban Ecology, Royal Botanic Gardens, c/o School of Botany, University of Melbourne, Parkville, Vic., Australia
    2. School of Biological Sciences, Monash University, Vic., Australia
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  • Michael A. McCarthy

    1. School of Botany, University of Melbourne, Parkville, Vic., Australia
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Summary

  1. The extensive impact of invasive species has motivated a growing field of research combining ecological and economic models to find cost-effective management strategies. Ecological systems are rarely perfectly observable, meaning decision-makers are usually uncertain about the current extent of an infestation and even whether an invasive species is present or absent. We show how to account for this uncertainty when providing decision support for invasive species management.
  2. We constructed the first partially observable model to analyse the trade-off between all three facets of invasive species management: quarantine, surveillance and control. We use a partially observable Markov decision process (POMDP) to determine how to allocate resources between these actions when the extent of an invasion is uncertain. We use a case study of potential black rat Rattus rattus invasion on Barrow Island, Western Australia.
  3. Our model shows it is often better to manage based on an uncertain belief in species presence than to spend money trying to confirm the presence or absence through surveillance. While it was never optimal to invest solely in surveillance to reduce uncertainty, it was often optimal to combine surveillance with quarantine or control. These mixed strategies, where multiple actions are implemented simultaneously, were more often optimal than for similar decision models where the extent of the infestation is known, suggesting an element of risk spreading.
  4. Optimal investments in each action were driven by their estimated efficacy, and the difference in the estimated impact of a localized and widespread invasion. For example, in our case study, it was often optimal to invest solely in control due to the low estimated efficacy of quarantine and the relatively small impact of a localized incursion.
  5. Synthesis and applications. Our analysis shows that the cost of reducing uncertainty through surveillance is not always accompanied by an improvement in management outcomes. By carefully analysing the benefits of surveillance prior to implementation of invasive species management strategies, managers can avoid wasting resources and improve management outcomes.

Introduction

Invasive species cause major economic and environmental damage (Pimentel, Zuniga & Morrison 2005), the scale of which is expected to increase into the future (Lodge et al. 2006). This has motivated a growing body of research combining ecological and economic models to find cost-effective strategies for managing invasives and their impacts (Epanchin-Niell & Hastings 2010). While many studies focus on allocating control effort across time and space (Hauser & McCarthy 2009; Epanchin-Niell & Hastings 2010), there is a huge interest in how management effort should be allocated between actions that target different levels of infestation, namely between quarantine, surveillance and different forms of control (Leung et al. 2002; Finnoff et al. 2007; Mehta, Haight & Homans 2007; Bogich, Liebhold & Shea 2008; Burnett et al. 2008; Moore et al. 2010; Polasky 2010; Rout et al. 2011). While investing in quarantine can prevent invasions at the outset, investing in surveillance increases the chance of early detection and a less costly eradication (Rejmanek & Pitcairn 2002). If all else fails, control actions can at least reduce the environmental and economic impacts of an infestation.

In previous analyses, the optimal allocation of resources between quarantine, surveillance and control has been state dependent. That is, the recommended management strategy depends on the current state of the infestation, often defined by the species' population size (Mehta, Haight & Homans 2007), population density (Leung et al. 2002) or spatial coverage (Bogich, Liebhold & Shea 2008). These studies commonly use state-dependent optimization methods, such as stochastic dynamic programming (e.g. Leung et al. 2002; Finnoff et al. 2007; Moore et al. 2010; Rout et al. 2011), to find the optimal management strategy. However, rarely are ecological systems perfectly observable, so decision-makers are usually uncertain about the current state of the system they are managing (Parma et al. 1998). Uncertainty about the state of the system obviously presents a challenge when applying state-dependent decision models to ecological management problems.

Observational uncertainty can be accounted for using a modelling technique born out of operations research and artificial intelligence: the partially observable Markov decision process (POMDP). POMDPs can be applied to problems where the decision-maker does not know the state of the system, but makes observations that are probabilistically linked to this underlying state (Monahan 1982; Littman 2009). Interest is now growing in this method as a decision support tool for ecological management, with recent applications to vegetation restoration (White 2005), watershed management (Tomberlin 2010), threatened species conservation (Chadés et al. 2008; Mc-Donald-Madden et al. 2011; Nicol & Chadés 2012), and disease and invasive species management (Haight & Polasky 2010; Chadés et al. 2011; Regan, Chadés & Possingham 2011).

We outline the first partially observable model to analyse the trade-off between quarantine, surveillance and control. We use a model of island invasion as the basis for our analysis and find cost-effective levels of investment in quarantine, surveillance and control for a species where the extent of invasion is uncertain. This allows us to analyse how optimal decisions are affected by the size of invader impacts and the effectiveness of management actions, and to examine the trade-off between reducing uncertainty and taking action. To illustrate our approach, we use the example of managing potential black rat Rattus rattus incursions on Barrow Island, Western Australia.

Materials and methods

The model

The manager of an island (or other isolated habitat) is concerned about the possibility of a particular species invading. If the species invades the island, it may be present as a localized population, or it may grow to be widespread across the island. We assume that widespread invasions are always detected, while localized populations may not be. How should a manager uncertain about the presence and extent of this island invader invest their resources in the different actions of quarantine, surveillance and control? We find optimal strategies for allocating across these activities, depending on the manager's belief about the extent of the invader.

These optimal strategies minimize the net expected cost, which includes the cost of management as well as the potential cost of impact of the invasive species. Let the cost of impact of a localized population be CL and the cost of impact of a widespread population be CW, where CkCW and 0 ≤  1. We assume these cost parameters represent the combined cost of all impacts associated with the invasive species including economic, environmental, social and opportunity costs. There is no cost of impact when the species is absent from the island.

Partially observable Markov decision process requires a description of the underlying dynamics of the system (Fig. 1), as well as a description of the manager's observation of the system. Both of these will be driven by the manager's decision d, which is the set of allocations to quarantine, surveillance and control (xQ, xS, xC). The amount of effort invested in quarantine will affect the annual probability of incursion, as given by:

display math(eqn 1)

where xQ is the amount spent on quarantine, p0 is the probability of incursion in the absence of quarantine, and α is the effectiveness of quarantine. This exponential equation describes a simple ‘diminishing-returns’ relationship, where incidences of quarantine failure over time occur as a Poisson process, that is, incursions occur randomly in time, at a constant rate α.

Figure 1.

System states and transition probabilities, where PI is the annual probability of incursion, PL is the annual probability of eradicating a localized population, g is the probability a localized population will become widespread, and PW is the annual probability of eradicating a widespread invasion. Probabilities of transition are functions of the amount spent on quarantine (xQ), surveillance (xS) and control (xC).

The amount of effort invested in control will affect the probabilities of successfully eradicating localized and widespread invasions. The annual probability of eradicating a localized population is:

display math(eqn 2)

where xC is the amount spent on control, and λL is its effectiveness in eradicating a localized population. If not eradicated, localized populations have probability g of growing to become widespread. The annual probability of successfully eradicating a widespread invasion is:

display math(eqn 3)

where λW is the effectiveness of control in eradicating a widespread population. We assume that it takes less effort to successfully eradicate a localized population than a widespread population, that is, λλW.

We assign numbers to the system states: (1) absent, (2) localized and (3) widespread. The transition matrix, which describes the underlying system dynamics (Fig. 1), is then:

display math

where each element mi, j is the probability of transitioning from state i at time t to state j at time t + 1. For example, m1,2 = PI(xQ) is the probability of transitioning from the species being absent (state 1) to being localized (state 2).

The amount of effort invested in surveillance will affect the probability a localized population is detected:

display math(eqn 4)

where xS is the amount spent on surveillance and β is its effectiveness. This probability of detection, based on a simple random encounter model (Garrard et al. 2008; McCarthy et al. 2013), will inform the manager's perception of the system as described by the observation matrix:

display math

Each element of the observation matrix math formula is the probability of observing state z when the system is in state j at time + 1 and decision xQ, xS, xC was made at time t. For example, math formula is the probability of observing that the species is absent (state 1) when it is in fact localized (state 2) and xs was invested in surveillance.

In each year, a decision is made, state transitions occur, observations occur, costs are incurred and the belief state is updated (Fig. 2). This belief state is a probability distribution over the system states, which summarizes the decision-maker's previous experience and uncertainty (Kaelbling, Littman & Cassandra 1998). Each element of the belief state corresponds to a possible system state and contains the probability the system is in that state. For example, the belief state b= (0·2, 0·3, 0·5) means that at time t, the probability the species is absent is 0·2, the probability it is localized is 0·3, and the probability it is widespread is 0·5. We can also refer to each of these elements as bt(i), the probability that the system is in state i. The order of events (Fig. 2) is the standard POMDP formulation (Monahan 1982), and in this context, it means there is a chance to detect (and subsequently eradicate) a new incursion before it grows into a widespread invasion.

Figure 2.

Order of events in each time step t of the POMDP (adapted from Monahan 1982).

Finding optimal management decisions

In each year, a fixed amount of management effort can either be invested in a single action (prevention, surveillance or removal) or split between two different actions. We explored four possible ratios for the mixed strategies, with 80%, 60%, 40% or 20% of management effort invested in one action, and the remainder invested in the other. There is also the option to do nothing, which incurs no management cost. To limit the number of possible allocations to 16 alternatives, we did not consider splitting management effort between all three actions.

In each scenario, we found the optimal management decision for each possible belief state bt, with the objective of minimizing the total expected cost over a finite time horizon T. The expected cost of making decision d (the set of allocations xQ, xS, xC) at time t, when the system is in state i is:

display math

where C(i, j, d) represents the immediate cost of decision d when the underlying system is in state i and moves to state j (Monahan 1982). For this model, this is the cost of the management decision d (the sum of the allocations xQ, xS, and xC), plus the cost of impact of the end state j (i.e. 0 for absent, CL for localized or CW for widespread).

The optimal strategies for each belief state through time are found using the Bellman principle of optimality (Bellman 1957), which also underlies the stochastic dynamic programming algorithm frequently used to optimize ecological management problems (e.g. McCarthy & Possingham 2007; Rout et al. 2011). First, the expected cost of being in each belief state in the final time step T is calculated as:

display math

where bT is the manager's belief that the system is in each possible state and bT(i) is the belief that the system is in state i. The optimal decision in the final time step T is the decision with the lowest expected cost summed across all possible states i. This equation minimizes the instantaneous cost – future expected costs are not included because it is the final time step of the management programme.

The algorithm then steps back to the previous time step T – 1 to calculate the expected cost of each decision, assuming the optimal decision is made in time step T. It continues to step backwards, repeating this process and finding the optimal decision for each belief state in each time step t. The expected cost of being in belief state b at time step t is calculated as:

display math

Instantaneous costs are included as before, but future expected costs are also added. These are summed over all possible current states i, states in the next time step j, and observations z. EC (bt+1) is the expected cost of being in the resulting belief state bt+1, after decision d and observation z. This resulting belief state is calculated from the previous belief state using Bayes' rule:

display math

where the denominator math formula. This probability is a normalizing factor, independent of the new state j, that causes the new set of beliefs across all possible states to sum to one (Kaelbling, Littman & Cassandra 1998).

Calculating the expected cost of a belief state may seem strange, as the decision-maker can incur a lower cost by believing they are in a good state. However, these belief states are not just the subjective belief of the decision-maker – they represent the actual probability that the system is in each state. POMDPs assume the observation, and transition matrices are accurate and precise and that beliefs are updated logically in response to observations. Therefore, the belief state is the actual probability of being in each state given the observations that have been made. This means the cost function is the actual expected cost for the decision-maker (Kaelbling, Littman & Cassandra 1998). Calculating the expected cost function for each decision across belief states is a complex task; hence, there are a range of algorithms for finding optimal solutions to POMDPs (Monahan 1982; Lovejoy 1991; Kaelbling, Littman & Cassandra 1998). We used the incremental pruning algorithm (Zhang & Liu 1996) in the ‘pomdp-solve’ program Version 5.3 (Cassandra 2005). We optimized decisions over a management time horizon of 10 years (= 10), with no discounting of future expected costs. For the starting belief state of the POMDP, we specified a uniform distribution over the three possible states.

Example scenarios

We created a range of example scenarios to allow us to examine the trade-offs between investment in each action and the general situations in which different actions are optimal. We assumed $250 000 is available each year for management. Although managers can also choose to do nothing, the management budget does not accumulate if unspent. We explored three levels of impact: C= $500 000 (two times the management budget), C= $2 500 000 (10 times the management budget) and C= $25 000 000 (100 times the management budget). We also considered three possible levels of effectiveness for management actions: low (α, β, λW, λ= 1·0 × 10−6), medium (α, β, λW, λ= 3·0 × 10−6) and high (α, β, λW, λ= 5·0 × 10−5) (Fig. 3).

Figure 3.

The probability of management success under different levels of investment, for surveillance and control (solid black line) and quarantine (grey dashed line). For quarantine, management success refers to prevention of a localized incursion; for surveillance, success refers to the detection of a localized incursion; for control, success is eradication. Parameter values are given in Table 1.

Case study

To illustrate the application of this model, we used a case study of potential black rat invasion on Barrow Island, which lies off the coast of Western Australia. Barrow Island has both natural and commercial assets: it is a Class A Nature Reserve with 24 endemic species (Environmental Protection Authority 2003), while being home to an oilfield and marine terminal for crude oil trading vessels (Morris 2002). Construction of a large liquefied natural gas (LNG) plant has begun on the island, with plant start-up planned for late 2014 (Chevron Corporation 2013). The resulting increase in the movement of goods and workers between the mainland and the island increases the risk of invasion by non-native species (Environmental Protection Authority 2003). Chevron has expanded their quarantine and surveillance program, investing AU$271 million in quarantine and AU$4 million in surveillance over the 4-year construction period (Moore et al. 2010). The quarantine program covers all activities involving movement of material, personnel, aircraft and vessels to Barrow Island and its surrounds (Moore et al. 2010). Approximately 25% of the surveillance budget is directed towards detecting vertebrates, with a targeted surveillance program for rats.

To model this case study, we used the parameter estimates from Moore et al. (2010) and Rout et al. (2011), reproduced in Table 1. The estimated cost of impact of a widespread rat invasion is extremely wide-ranging (AU$2·9 million – AU$4 billion per year) due to the difficulty of quantifying these impacts monetarily (Rout et al. 2011). For practical reasons, we have used the lower limit of this estimate. We fit the effort curve for localized eradication by assuming the estimated cost of successfully eradicating a localized population of black rats (AU$96 667, Moore et al. 2010) was the minimum to guarantee successful eradication (i.e. P= 1–4 decimal places).

Table 1. Model parameters
ParameterSymbolDefault valuesCase study estimates
Base probability of incursion p 0 0·990·99
Effectiveness of quarantineα

Three levels of effectiveness:

high = 5·0 × 10−5

medium = 3·0 × 10−6

low = 1·0 × 10−6

2·07 × 10−6
Effectiveness of surveillanceβAs for α1·57 × 10−5
Effectiveness of control when widespreadλWAs for α4·944 × 10−6
Effectiveness of control when localizedλLAs for α1·03 × 10−4
Probability of spread from a localized to widespread population g 0·50·5
Cost of impact of widespread population C W

Three levels:

low = $500 000

medium = $2 500 000

high = $25 000 000

$2 900 000
Ratio of impact cost of localized and widespread populations k 0·01 (C= $29 000)

We again assumed AU$250 000 is available each year for management. This amount is much lower than the current expenditure on Barrow Island but was chosen because altering the percentage of this amount allocated to different actions significantly changes their probabilities of success, given the estimated effectiveness of actions (Fig. 3). Setting this limited budget, which is insufficient to obtain high probabilities of management success, allows us to examine the trade-offs between different management actions.

Results

Example scenarios

Widespread invasions are easily detected, while localized incursions cannot be detected without surveillance. The main problem for the island manager is therefore uncertainty about whether the species is absent or localized. We present general results for the range of belief states between these two possibilities, where the belief there is a widespread invasion is zero. We also focus only on optimal strategies in the first time step, as these incorporate all future possibilities and costs over the 10-year time horizon.

When all three management actions are equally effective (α, β, λ= medium, λ= low, see Table 1), it is best to spend a limited budget on quarantine, control or a mixture of both; depending on the probability, there is an undetected localized population on the island (Fig. 4). If the impact of an invasion is expected to be small (CW is twice the management budget), and the probability it is already localized is less than around 0·5, it is better to do nothing than invest in management (Fig. 4a). It can be optimal to spend money on quarantine if a localized incursion will have as great an impact as a widespread invasion, but only if it is certain, the species is not already present (Fig. 4a).

Figure 4.

Decisions that minimize total expected costs over 10 years when all management actions are equally effective (α, β, λ= medium, λL = low, see Table 1). Shown as a function of the probability the species is localized and the ratio of impact costs k, which is the cost of impact of a localized incursion as a proportion of the cost of impact of a widespread invasion (CL/CW). The cost of impact of a widespread invasion is: (a) CW = $500 000, two times the management budget, and (b) CW = $2·5 million, ten times the management budget. In (a), ‘All in quarantine’ is optimal where = 1 and the probability the species is localized is ≤ 0·01.

If the impact of an invasion is expected to be large (CW is ten times the management budget), it is always optimal to spend the management budget (Fig. 4b). It is best to invest in quarantine, if the species is likely to be absent, in control if it is likely to be present as a localized population, and in a mixture of both for the belief states in between. These strategies are reasonably stable to the magnitude of impact caused by a localized incursion compared with a widespread invasion.

When quarantine is more effective than other management actions (α = high, β, λ= medium, λ= low, see Table 1), optimal strategies are insensitive to the impact of the invasive species (Fig. 5a,b). This is because spending as little as 20% of the management budget on quarantine gives a high probability of successfully preventing an incursion and thus preventing all impacts (Fig. 3). It is only optimal to spend more than this amount on quarantine, if the manager is certain, the species is absent (Fig. 5a,b). Again, spending solely on control is optimal when the species is likely to already be present as a localized population.

Figure 5.

Decisions that minimize the total expected cost over 10 years when: (a, b) quarantine is the most effective action (α = high, β, λ= medium, λ= low, see Table 1), (c, d) surveillance is the most effective action (β = high, α, λ= medium, λ= low, see Table 1), (e, f) control is the most effective action (λ= high, α, β λ= medium, see Table 1). Shown as a function of the probability the species is localized and the ratio of impact costs k, which is the cost of impact of a localized incursion as a proportion of the cost of impact of a widespread invasion (CL/CW). The cost of impact of a widespread invasion is: (a, c, e) C= $500 000, two times the management budget, and (b, d, f) C= $2·5 million, ten times the management budget. In (a) and (b), ‘All in quarantine’ is optimal where the probability the species is localized = 0), and in (c) ‘All in quarantine’ is optimal where = 1 and the probability the species is localized is ≤ 0·01.

These strategies change markedly when surveillance is the most effective management action (β = high, α, λ= medium, λ= low, see Table 1) (Fig. 5c,d). Despite it being more effective than other actions, it is never optimal to spend the management budget solely on surveillance. However, it can be optimal to combine surveillance with other actions. If the impact of an invasion is small, it is optimal to invest in a combination of surveillance and control when unsure if the species is absent or localized (Fig. 5c). If the impact of an invasion is large, it is optimal to invest in a combination of surveillance and quarantine when there is a benefit to catching an incursion early (the impact of a localized incursion is small compared with a widespread invasion), and the species is likely to be absent (Fig. 5d). As in the previous results, we see it can be optimal to do nothing when the impact of an invasion is small (Fig. 5c), that it is optimal to invest solely in quarantine when the impact of an invasion is large and the species is likely to be absent (Fig. 5d), and it is optimal to invest solely in control when the species is likely to be present as a localized population (Fig. 5c,d).

When control is the most effective management action (λ= high, α, β, λ= medium, see Table 1), the range of belief states for which it is optimal to invest solely in control actually decreases (Fig. 5e,f). This is because a high probability of success can be achieved with only a small investment, making it more beneficial to combine control with other actions. If the impact of an invasion is large (Fig. 5f), it can be optimal to invest in a combination of control and surveillance, but only when the manager is uncertain whether the species is absent or localized, and there is a benefit to early detection (the impact of a localized incursion is small). It seems surprising here that the proportion allocated to control decreases as the probability of a localized incursion increases. This is driven by the utility of surveillance – it is more useful to survey when the state of the system is more uncertain (i.e. for intermediate probabilities), so more is invested in surveillance.

Case study

For the Barrow Island example, we examined optimal strategies for the full range of belief states (Fig. 6), and how strategies change through time (Fig. 7). Spending solely on control is optimal for a wide range of states (Fig. 6). Mixtures of control and surveillance are also optimal, although it is never optimal to spend all available management effort on surveillance. Most effort should be allocated to surveillance when the manager is unsure whether the species is absent or localized (i.e. midway along the hypotenuse in Fig. 6). When the species is most likely to be localized, spending on surveillance decreases as the belief that the species is widespread increases (i.e. moving right to left across the top of Fig. 6). It becomes better to spend less on surveillance and more on control, to increase the chance of eradication. Also, the species can be detected without surveillance when widespread, so less effort in surveillance is needed as the belief there is a widespread invasion increases. Because of the small impact of a localized incursion compared with a widespread invasion, it is never optimal to spend money on quarantine – when the manager is sure the species is absent, it is more cost-effective to do nothing.

Figure 6.

Decisions that minimize the total expected cost over 10 years for different belief states, for the Barrow Island parameter estimates (Table 1). On the x-axis is the probability the species is absent, and on the y-axis is the probability the species is localized. Probabilities must add to one, so for any point on this graph, the complement is the belief that the species is widespread. Optimal decisions are indicated by colour, with the legend showing the percentage of the total management budget ($250 000) invested in each action. Actions not listed in the legend were not optimal for any belief state.

Figure 7.

Decisions that minimize the total expected cost over 10 years for different belief states through time, when the belief the species is widespread = 0. Shown for (a) the Barrow Island parameter estimates (Table 1), and (b) the Barrow Island parameter estimates but with = 0·5 (CL = $1 450 000). Optimal decisions are white: no investment, light grey: surveillance and control, mid grey: quarantine and control, dark grey: quarantine only, and black: control only. Lines and markers show a simulation of how the probability the species is localized would change through time when the optimal action is taken and there are no observations of the species. In both cases, the simulation starts with 10 years remaining with a 0·5 probability that the species is localized and 0·5 probability it is absent.

Again constraining our results to the case where the probability of a widespread invasion is zero, we can examine how optimal strategies change through time and simulate how the state of the island would change due to the management strategies implemented (Fig. 7a). In this figure, the optimal strategies when 10 years remain correspond to the edge of the longest side in Fig. 6. These optimal strategies are reasonably stable through time, favouring a mixture of surveillance and control. However, as the end of the management horizon approaches (1 or 2 years remaining), it becomes optimal to invest instead in a mixture of quarantine and control. Surveillance has no immediate benefit – it is used to better inform the management decision in the following year. Therefore, as the future management horizon decreases, the benefit of surveillance decreases and quarantine is favoured.

We simulated an optimal trajectory through time, beginning with a probability of 0·5 that the species is localized, and a probability of 0·5 that it is absent. The first optimal strategy – a combination of surveillance and control – decreased the probability of a localized incursion from 0·5 to 0·04. With a low probability of presence and a low cost of incursion (Table 1), it was then optimal to do nothing. This lack of quarantine action (and high probability of incursion, Table 1) leads to a 0·96 probability that a localized population was present in the following year. The high probability of incursion and small impact of a localized population combine to give an optimal strategy that alternates between minimizing impact through surveillance and control, and minimizing management cost by doing nothing. In contrast, if the impact cost of a localized population was larger (e.g. half the cost of a widespread invasion, Fig. 7b), it is almost always optimal to invest some effort in quarantine. However, the high annual probability of incursion means that despite this investment, the probability there is a localized incursion of rats on the island remains around 0·5 for the duration of management.

Discussion

When is it better to survey to reduce uncertainty, and when is it better to just take action? Our model suggests it is often better to manage based on an uncertain belief in species presence rather than spending money trying to confirm presence or absence. Across the scenarios we explored, it was never optimal to invest solely in surveillance to reduce uncertainty. However, it was often useful to combine surveillance with quarantine or control, particularly when the manager's belief is split between whether the species is absent or present as a localized population. A combination of surveillance and control was a particularly useful strategy. In our model, control is implemented first and then surveillance used to assess its success, with both actions working together to increase the probability that the pest is absent.

Incorporating uncertainty in the system state seems to promote these mixed strategies (i.e. investing in more than one action simultaneously). The prevalence of mixed strategies in our optimal results was much higher than for similar decision models with a known system state (Moore et al. 2010; Rout et al. 2011). This reflects the results of decision models that aim to maximize the chance of meeting a set performance threshold – investing in two actions simultaneously deals with uncertainty by spreading the risk of choosing a poor action (McCarthy et al. 2010). Despite our objective of minimizing expected cost, incorporating uncertainty in the system state seems to have a similar effect. Mixing control and quarantine can be a particularly effective way to achieve minimum expected cost despite uncertainty in the state of the system. Quarantine is useful when the species is absent, and control is useful when the species is localized or widespread. Investing in both therefore ensures that some useful action is taken regardless of the actual abundance of the species. Of course, our model assumes there is the option to control when presence is uncertain, which makes sense for our case study of black rats where control is implemented by dropping bait or traps. Controlling when pest presence is uncertain would not be possible for species where individuals need to be located to be removed (e.g. weeds that are treated through mechanical removal or treatment with herbicide).

In creating a simple model of invasion management, we have made several assumptions that affect the optimal results. First, we have restricted the set of possible management options to those where a set budget is allocated to one or two management methods. Without a fixed budget, we would expect to see a decrease in the total amount invested in management as the impact of a localized incursion (i.e. k) decreases, with a reduced investment in quarantine (Rout et al. 2011). If options where the budget is allocated between all three management methods were included, we would expect these options to be optimal where the manager's belief is split between the three states of the system. POMDPs are somewhat restricted in the number of options that can be assessed, with each option adding to the complexity and difficulty in analysing the problem.

Secondly, we made choices when constructing our transition and observation matrices that have implications for the biology of the system. We have assumed that a widespread invasion targeted with control efforts transitions to absent if the efforts are successful and remains widespread if the efforts are unsuccessful. An alternative option would be for a failed eradication to reduce the invasion from widespread to localized. We chose the first option because we found that under the second, any investment in control could achieve a reduction in the invasion and its impact. It was therefore most often optimal to invest a tiny amount in control to achieve this reduction rather than aiming for eradication. A third option in constructing the transition matrix would be to include all three transitions as outcomes of control: from widespread to widespread (total eradication failure), widespread to localized (partial success) and widespread to absent (complete success). The effect of this change would depend on how each transition probability is related to the amount invested in control.

Our transition matrix also assumes that localized incursions and widespread invasions will not go extinct independently. We considered this to be a reasonable assumption for our case study and for most highly invasive species, but it may not apply depending on the size and extent at which localized and widespread invasions are defined. Our observation matrix assumes that a localized population cannot be detected without active surveillance, an assumption that was motivated by our case study. Barrow Island is not a highly populated area, being home to only a few hundred workers at one time. We therefore assumed limited opportunities for passive community-driven surveillance of rats. However, a probability of detection through passive surveillance could easily be added to the model if applicable. Inclusion of passive surveillance would likely decrease the prevalence of targeted surveillance in the optimal solutions. We assumed in our example scenarios that the probability of incursion without investment in quarantine was very high, with p0 = 0·99, the value estimated for our case study. Using this value for the example scenarios allowed the relationship between investment and probability of success to be similar for all three actions (Fig. 3). If this probability was reduced, we would expect investment in quarantine to decrease as the marginal benefit of investment in quarantine is decreased.

Thirdly, the majority of the results we analysed were optimal decisions for the first year of a 10-year management horizon. Infinite time solutions could not be calculated, so we chose this short horizon to reflect the time considerations of a real management program. Our case study showed that the effect of time on optimal decisions can vary greatly depending on the parameter estimates used. Being unable to represent all possibilities, we chose the first time-step solutions as most representative as they show the optimal decisions given all future possibilities in the next 10 years. Finally, the extent of an invasion is a continuous property that we have represented with discrete states: absent, localized and widespread. While standard POMDP methods cannot accommodate continuous states, heuristics have been developed to solve partially observable continuous state problems (e.g. Nicol & Chadés 2012) and these would be well suited for applications in invasion management.

Natural systems cannot be observed perfectly, and this makes it difficult for invasive species managers to make cost-effective decisions. Our model provides a framework for managers to prioritize investment between different types of management actions when they are uncertain about the presence and extent of an invasion at a managed site. It shows that reducing uncertainty through surveillance does not always improve management performance. Assessing the management benefits of surveys before they are undertaken can avoid unnecessary expense and improve the efficiency of management.

Acknowledgements

Many thanks to Iadine Chadés, David Roberts and Andrew Solow for helpful comments, and to Michael Bode and Iadine Chadés for technical advice and assistance. This research was supported by an Australian Research Council (ARC) Discovery Grant to T.M.R. and J.L.M. (DP110101499), and an ARC Future Fellowship to M.A.M. (FT100100923).

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