1. Failing to account for uncertainty in the detection of invasive plants may lead to inefficient management strategies and wasted resources. Smart strategies to manage plant invasions requires consideration of the economic costs and benefits, and plant life-history characteristics as well as imperfect detection.
2. We develop a partially observable Markov decision process (POMDP) to provide optimal management actions when we are uncertain about the presence of invasive plants. The optimal strategy depends on the probability of being in a particular state. We ask the question, ‘When is it preferable to use a less efficient, less costly action to a more efficient, more costly action?’ We apply the POMDP to branched broomrape Orobanche ramosa, a parasitic plant species at the centre of a national eradication campaign in South Australia.
3. The optimal strategy depends on the ability to detect the invasive species and the location of the infested site. For high detection rates, if the site is a satellite infestation, management should employ the more efficient, more costly action (i.e. soil fumigation) the year the weed is detected followed by monitoring. When the detection probability is low, then it is optimal to employ the less efficient, low cost action (i.e. host denial) in the years the species is not detected. For sites in the centre of the infestation, management should employ the less costly, less efficient action. While the optimal strategy is insensitive to colonization, the likelihood of local eradication diminishes as colonization probability increases, highlighting the importance of limiting colonization if eradication is to be achieved.
4.Synthesis and applications. Providing decision support for managing ecological systems is a key role of applied research. Formulating this support within a decision theory context provides a framework for good decision-making. The POMDP model is a novel decision support tool for optimal sequential decision making when invasive plants are difficult to detect. The model can determine the best management action to employ based on the location of the infestation and can inform when to switch to alternative management actions that buffer against imperfect detection.
Invasive plants are a major threat to natural and managed systems. They are notoriously difficult to control or eradicate and require large amounts of effort and resources to manage effectively (Pimental 2002; Panetta & Timmins 2004). With the world wide economic impact of invasive plants estimated as US$560 billion annually (Pimental et al. 2001; Pimental, Zuniga & Morrison 2005), it is crucial that resources are directed towards the most effective management activities and not wasted on management actions that are inefficient. Yet deciding the best course of action for managing invasive plants can be excruciatingly difficult due to the complex interaction of factors such as the extent of the invasion, the ecology of the species, the dynamics of the system, and how the species responds to different management actions (Taylor & Hastings 2004). The decision process is exacerbated further by our inability to observe the invasion perfectly. Imprecise survey techniques, the cryptic nature of some species, and persistent seed banks make it difficult to verify whether the imposed management actions are successful or not.
Formal decision theory tools provide a useful avenue to investigate these complex interactions by systematically incorporating them in a transparent and consistent manner. These tools can calculate the optimal management actions given a specific objective and any constraints imposed on the system (i.e. cost) (Possingham 2001). Methods exist for determining optimal strategies and several have been applied to ecological systems (Richards, Possingham & Tizard 1999; McCarthy, Possingham & Gill 2001; Westphal et al. 2003; Field et al. 2004; Tenhumberg et al. 2004), and specifically in invasive species management (Shea & Possingham 2000; Taylor & Hastings 2004; Regan et al. 2006; Hauser & McCarthy 2009; Rout, Salomon & McCarthy 2009). These optimizing methods generally describe the system of interest as a Markov chain; a set of mutually exclusive states and transitions between states determined by the different processes governing the system. In the case of an invasive plant the states are often either some measure of abundance, number of sites, or occupancy, and the processes are generally colonization, germination, and the persistence of the adult population and the seed bank. In addition, any management imposed on the system can alter the rate of one or more of these processes and hence their influence on the state. The method uses a reward system that incorporates the cost of the action and the benefit of being in a particular state. Optimization algorithms, such as stochastic dynamic programming (SDP), a backward iterative procedure that finds the optimal solution for a stochastic system, can be applied to the Markov chain to determine the optimal state-dependent strategy based on the objective of maximizing the net expected benefit (Mangel & Clark 1988). The combination of the Markov chain (the model representing the system) and the decision process (objective, management options, costs and rewards) are commonly known as a Markov decision process (MDP) (Bellman 1957).
While Markov decision processes are very useful for representing dynamic systems, choosing the best action depends on the assumption that the state of the system is known precisely. This may not always be the case when managing ecological systems. While there is some underlying truth about the state of the system at any particular point in time, it is often not known precisely because of imperfect detection. Individuals may go undetected during a survey, and for plants, seeds may be viable in the seed bank and go unnoticed for a number of years. Despite the fact we rarely know the state of any ecological system perfectly, examples of using optimization methods that deal with imperfect detectability are limited in the ecological literature (but see Regan et al. 2006; Chades et al. 2008; Rout, Salomon & McCarthy 2009). However in Operations Research and Artificial Intelligence there is a thriving literature on developing optimization methods that specifically address the issue of incomplete knowledge of the state of the system (Monahan 1982; Cassandra 1998; Kaelbling, Littman & Cassandra 1998). This general methodology is a special case of Markov decision processes, aptly named partially observable Markov decision process (POMDP). In addition to the MDP formulation, POMDP defines observed states of the system and uses information about the probability of the true state given what is known. For example, if an adult invasive plant has not been observed in a site for 3 years, what is the chance a seed bank remains? Rather than the optimal solution being state dependent, as in the MDP case, for the POMDP case, the optimal solution is dependent on the probability of being in a particular state at a given time (Cassandra 1998).
In this research we develop a POMDP model to determine optimal management strategies for an invasive plant species where the states of the system are not known perfectly. We investigate how the optimal solution changes depending on the value of eradicating the species, the likelihood the site is re-colonized, and our ability to detect adults or seeds. We illustrate this approach using a case study of branched broomrape, Orobanche ramosa L., a parasitic species that is currently targeted by a national eradication program in South Australia (Jupp, Warren & Secomb 2002). While we tailor the problem to the case study, the approach is general enough to be applicable to a wide range of agricultural and environmental invasive plants as well as threatened plant species that are cryptic in nature and/or have persistent soil seed banks.
Materials and methods
Branched broomrape is an annual obligate parasitic plant that attaches itself to the root system of its host plant. It originates from Mediterranean Europe, the Middle East, and northern Africa. It was first discovered in Australia in 1911 but disappeared and was rediscovered in 1992 in southern Australia (Jupp, Warren & Secomb 2002). Branched broomrape is a threat to production crops with reported losses in biomass of up to 75%–90% (Linke, Sauerborn & Saxena 1989). Its biggest threat, however, is to export markets that prohibit the import of goods that are likely to carry broomrape seed, potentially costing the Australian industry $1·7 billion (Wilson & Bowran 2002).
Much of the life cycle of branched broomrape occurs underground. Seeds germinate and attach to hosts in response to chemical triggers from susceptible host roots. Dust-like seeds set within 3–4 weeks after plant emergence and have the potential to survive up to 12 years in the soil (Panetta & Lawes 2007). Plants are short lived and only visible for a very short time, making detection difficult (Kebreab & Murdoch 2001). A single plant can produce over 50,000 seeds that can disperse by sticking to animals and equipment that come in contact with the plant. Seeds can also be dispersed by wind and water, soil, fodder, on footwear and clothing, and can remain viable after passage through digestive tracts.
A national campaign to eradicate branched broomrape in Australia commenced in 2000. Currently the species is contained within a quarantine zone approximately 70 × 70 km, which restricts the movement of fodder, crops, livestock, machinery and soil out of the area. The quarantine area is a large continuous area that encompasses the earliest known infestations. In addition there are several satellite infestations surrounding the main quarantine area where the species has been detected subsequently.
Determining the most suitable course of action requires consideration of the trade-offs between the cost and efficiency of alternative actions. Here we investigate when employing a less expensive, less efficient action is preferable to a more expensive, more efficient action. In the case of branched broomrape, it is thought that the key to eradication is to attack the seed bank (Cooke 2002). Soil fumigation attacks the seed bank directly, killing all seeds on contact. However it is extremely expensive and can be highly toxic, thus it is only applied to areas where branched broomrape is known to be present. Consequently there is the potential to miss areas that have broomrape seed. The soil fumigation option involves the use of methyl bromide. Surveys suggest that the application of methyl bromide is 79·6% effective. That is, adult plants were detected the following year in 20·4% of the fumigated sites (N. Secomb, pers. comm.).
The second management action is to promote depletion of the seed bank through host denial. For this method, a non-host crop is planted and any plants that might serve as hosts are controlled so that the seed bank decays naturally through time. The host denial action is much less expensive than soil fumigation, is much less ecologically damaging and has no human health risks, but it is also much less efficient than soil fumigation. Lastly we include a ‘do nothing’ action that has minimal cost and efficiency. This action is included to highlight the situations where the other direct management actions are suboptimal. Since ‘do nothing’ still requires ‘doing something’, we represent the ‘do nothing’ action as pasture but with no other management imposed to eradicate the species, so there may be other plants within the site that can serve as hosts. All three management actions assume some monitoring is implemented during the flowering season to determine presence or absence of the species.
The POMDP model
We model branched broomrape within a single management unit. In this case the management unit is a paddock, a fenced area used to grow crops, pasture or graze livestock. For other situations it may be a patch or any other unit where small scale direct management is focused. The system is represented by a Markov chain with three mutually exclusive states; , where se represents an empty paddock, free of broomrape, ss is a paddock with broomrape seeds, and sw is when the paddock has both seeds and adult plants. In any particular year, the paddock can only be in one of these states. The ability of the system to move from one state to another is governed by processes such as colonization, c, seed bank decay, ρ, and germination, g. These rates may alter depending on the action imposed on the system. The set of management actions is .
Transitions between states are calculated as the conditional probability of being in state s’ in the next time step given state s and action a in the previous time step. This is represented mathematically as:
To determine the optimal solution requires defining a reward function for the states and actions of the system. The reward function is a combination of the relative benefit for being in a particular state of the system and the relative cost of the management action:
To take into account the imperfect detection of the species we also define a finite set of observations: where za is an observation where broomrape was absent and zp represents an observation where broomrape was present. An observation function, also called the detection rate (d), relates the observations to the real states of the system. The probability of observation z, given the system is in state s can be represented mathematically as
A schematic representation of the POMDP with three real states and two observed states is shown in Fig. 1. The processes governing the transitions between states are shown in Table 1.
Table 1. State transition probabilities for the three alternative management strategies
State to State
Action1 (Do nothing)
Action2 (Soil fumigation)
Action3 (Host denial)
(1 - c)
c(1 - g)
(1 - ρ1)*(1 - g)
(1 - ρ1)*g
(1 - ρ2)*(1 - g)
(1 - ρ2)*g
We use stochastic dynamic programming to find optimal solutions. Since the real state of the system is not known precisely, belief states are used to summarize and overcome the difficulties of incomplete detection. Aström (1965) has shown that belief states are sufficient statistical tools to summarize all the observable history of a POMDP without loss of optimality. A POMDP can be cast into a framework of a fully observable Markov decision process where belief states represent the continuous but fully observable state space. Here, a belief state b is defined as a probability distribution over the three real states of the system (empty, seeds, and adults and seeds). In our case, solving a POMDP is finding a strategy π:B × τ → A such that the strategy π maps a current belief state (b B) and a time-step (t τ) on to a management action, aA. An optimal strategy maximizes the expected sum of rewards over a finite time horizon, T. This expected summation is also referred to as the ‘value function’ and essentially ranks strategies by assigning a real value to each belief state, b. Using the Bellman principle of optimality (i.e. subdividing the problem into simpler manageable parts) (Bellman 1957) and the POMDP parameters, we can calculate the optimal t-step value function from the (t - 1)-step value function:
where Pr(s|b) represents the probability of being in state s given a belief state b, and baz is the belief state assuming action a and observation z. Equation 1 maximizes the expected sum of instantaneous rewards when there is no time left to manage the species. Similarly when there are t-steps to go, Equation 2 maximizes the instantaneous rewards and the future expected rewards for the remaining t - 1 steps.
Three main processes govern the system dynamics; germination (g), seed bank decay [ρ1 and ρ2 (see below for description)] and colonization (c). Since most of the life-cycle for branched broomrape occurs underground, germination in this situation is calculated as an emergence event and incorporates several processes such as conditioning, seed stimulation, germination of seeds, host penetration and emergence.
We estimate germination rates from land use data from years 2001 to 2004 and knowledge of the known infestations (N. Secomb, unpublished data). Since broomrape seed can remain viable for many years, we assume that if a paddock was infested any time in the previous 4 years (i.e. the length of the available survey data for this study), it is still infested in the current time step, unless some additional management has been imposed on the area. The probability of germination is calculated as the proportion of paddocks in a particular year where the land use was pasture and where broomrape was found, given that it had been found there any time in the past. Pasture in this case was identified based on the known management in each paddock. This resulted in an average germination rate of 0·27 per paddock. Since detection of an infestation is not perfect, the germination rates have been scaled by the detection rate. Thus the emergence probability used in the model is 0·38 per year. Little information is available to estimate the effectiveness of host denial other than it is thought to be much less effective than soil fumigation. To investigate the trade-offs between the different management strategies we set the effectiveness of host denial as a 50% reduction in the background germination rate (i.e. 0·19). Given the uncertainty associated with this action we perform sensitivity analysis to investigate the importance of this estimate.
There are two types of seed bank decay for branched broomrape. The background decay probability (ρ1) is the probability that the seed bank decays given that no new seeds have entered the seed bank in the current time step. It is thought that the seed bank will persist for approximately 12 years if no new seeds are added (Panetta & Lawes 2007). This equates to a decay probability of ρ1 = 0·08. The second decay probability (ρ2) is post seed set decay and implies that new seeds have entered the seed bank in the current year. We take a precautionary approach and assume that this decay rate is negligible and set it to zero. There are no estimates available for the colonization rate even though there is management imposed to prevent the dispersal of seed between sites. The specific rates will depend on the vector of seed dispersal and the distance from the infested site. Since the POMDP model is not spatially explicit, estimating this parameter becomes difficult. Rather than ignoring this process, we look at the likely impact of colonization through investigating a range of values that could be representative of broomrape dispersal vectors that are not addressed through management provisions (i.e. wind). We initially assume colonization rate is zero and then increase it incrementally to 0·15.
Data indicate that if soil fumigation is implemented then the probability that a paddock in the adult state will remain in the adult state is 0·204 (N. Secomb, unpublished data). We use this value as the effectiveness of soil fumigation but scale it by the detection factor. Once the fumigant is applied it kills all seeds on site. In general, however, only sites known to be infested with broomrape are fumigated. It is not economically feasible to fumigate areas where the plant has not been detected in the past, thus we constrain the model so that the soil fumigant can only be applied in the year broomrape is observed. Transition probabilities for each action are given in Table 1.
We assume that monitoring occurs every year around flowering time. The probability of detecting adult branched broomrape is calculated based on double blind surveys within the quarantine area (Kuhnert & Possingham 2003). We use an average detection probability of 0·7 (the probability an adult plant is detected given it is present). As this parameter is highly uncertain we investigate scenarios where the detection rate ranges from 0·1 to 1·0.
The reward function is expressed as a ratio between the benefit (or reward) for being in a particular state and the cost of the management option imposed. In this case, soil fumigation amounted to an economic cost on average of $18 000 per paddock (N. Secomb, unpublished data). This cost includes basic application but does not include environmental and human health costs thus perhaps underestimating the true cost of soil fumigation. For the management action ‘host denial’ we assume that there is some value in the crop itself in that it can be sold (usually within the quarantine area) or used for feed for livestock. For the purposes of illustration we assume the benefit from selling the non-host crop is equal to the cost of planting, maintaining and harvesting the crop. For the ‘do nothing’ action, there is a cost of managing the paddock as pasture but we assume there is no value in the pasture itself. We set the cost for managing the paddock as pasture at $2000. The monitoring costs are assumed to be equal across the three alternative management strategies.
In this case, the benefits of being in a particular state are independent of the action employed. Given the objective is to eradicate branched broomrape from the site, there is no benefit for the system to be in a seed or adult and seed state. However for circumstances where control is the objective, there may be some benefit for being in one of the other two stages. This can easily be incorporated into the POMDP formulation by adjusting the rewards for the alternative states based on the objective.
The reward for being in an empty state may depend on the number of remaining infested sites. If the site of interest is the last known infested area then the benefit for being in an empty state will be very large as it will constitute complete eradication of the species. Alternatively, the reward for a site being in an empty state will be less if there are other infested sites in the area. It is plausible that the reward could also depend on the proximity of the site to other infestations within in the area. Satellite infestations or sites on the edge of the main infested area may have a high reward if they become empty as they are likely to be deemed locally eradicated. On the contrary, empty sites in the interior of the infestation may not be deemed eradicated until the neighboring sites are also clear. Thus there may not be a very large associated benefit with this outcome. We apply a range of rewards to the empty state to assess how the location and circumstances of the paddock changes the best management strategy. The rewards function includes a discount rate of 4% (Sumaila & Walters 2005).
The optimal strategy graphs in Fig. 2 map the optimal action for each time step and observation, present or absent, over the entire management time horizon (20 years). For example, in Fig. 2b where the cost ratio (i.e. reward of being in an empty state/cost of the action) is 1·0, colonization rate = 0 and detection probability = 0·7, if there are 20 years remaining and branched broomrape is detected (i.e. time since detection, t = 0), the optimal action to employ is soil fumigation. In subsequent years if we continue to not detect any plants (i.e. time since detection, t ≥ 1), it is optimal to switch to host denial until year 4 after which the ‘do nothing’ action becomes optimal. Once broomrape is detected the optimal action depends on how many years remain in the management time horizon. For instance if broomrape is again detected in the 12th year the optimal action with eight years remaining is to employ the host denial action.
The optimal course of action for managing branched broomrape depends on the reward for being in an empty state (Fig. 2). For areas where the cost ratio is ≥10, such as a satellite infestation or a site on the edge of the main infested area, when the species is detected, the optimal action is to perform the most efficient and costly action (i.e. soil fumigation). In subsequent years, the optimal strategy when adult plants are not detected is to ‘do nothing’ unless there are only 4 years left to manage, then host denial is optimal (Fig. 2c,d). For areas of low relative reward (i.e. R/cost = 0·5), such as sites in the centre of the infestation, the less efficient and less costly action (i.e. host denial) is the best strategy (Fig. 2a). For areas where the cost ratio is equal to 1, the best strategy depends on the management time horizon: when the species is detected at the early stage of the management period, the more efficient action is optimal. The less efficient action becomes optimal only when the management time horizon is less than nine years. Under other conditions ‘do nothing’ is optimal (Fig. 2b).
When colonization is a factor, the optimal strategies do not change. For a reward/cost ratio of 100 the optimal management involves soil fumigation in the year plants are detected followed by the ‘do nothing’ action in subsequent years when plants are not detected (Fig. 3). While the optimal set of management actions are stable as the colonization rate increases, the probability of eradicating the weed from a site diminishes. When colonization is zero (c = 0), the probability of eradication approaches 1·0 in approximately 15 years. However if the colonization rate is c = 0·05 the best that can be achieved in the management time horizon is a probability of eradication of 0·86. When the colonization rate increases then the likelihood of eradication becomes less. At a colonization rate of 0·15 the best possible outcome is a probability of eradication of 0·62. Under these colonization scenarios even if the management time horizon were extended (>20 years), it would not increase the probability of eradication by a substantial amount as all the curves reach an asymptote by year 20.
The difference between the MDP and the POMDP formulation is the incorporation of imperfect detection. This has an effect on the optimal course of action as well as the probability of eradication within the management time horizon (Figs 4 and 5). For the MDP formulation where detection is perfect (d = 1·0), for a reward/cost ratio = 100 and colonization = 0, the optimal course of action is to fumigate the year the weed is detected and then ‘do nothing’ when it is not detected. Under this scenario the probability of eradication approaches 1·0 in approximately 12 years (Fig. 5). This is consistent with the current philosophy of branched broomrape management. However if the ability to detect the weed is imperfect, not only does the probability of eradication change but so does the optimal course of action. For high detection probabilities (0·5–0·7) the optimal course of action is the same as the perfect detection scenario (Fig. 4c,d) but the probability of eradication diminishes. The probability of eradication approaches 1·0 within the 20-year management time horizon but at a much slower rate (Fig. 5). For a detection probability of 0·7, the probability of eradication approaches 1·0 around 15 years after detection, while for a detection probability of 0·5, the probability of eradication approaches 1·0 around 20 years after initial detection. When dealing with low probability of detection (0·1–0·3), the optimal course of action remains, i.e. to fumigate in the year the weed is detected but it is now followed by host denial in the years when the weed is absent (Fig. 4a,b). This results in a probability of eradication of 0·82–0·90 at the end of 20 years for the two detection rates respectively. The ‘do nothing’ action at the end of the management time horizon in these scenarios is an artefact of the time horizon constraint. Since there are only a couple of years left to manage, the gain from doing a more costly action (host denial in this case) will not improve the overall situation, thus a less costly option becomes optimal in the final stages of management.
Data for the effectiveness of host denial were unavailable for this study. Sensitivity analysis was performed for two situations, high cost ratio and low detection (Fig. 6a) versus low cost ratio, high detection (Fig. 6b). The effectiveness of host denial was set at 50% the background germination rate. In the sensitivity analysis the germination rate was decreased from 50% to 100% of the background rate. Results revealed the model to be relatively insensitive to changes in the effectiveness of host denial (EHD) with only small changes in the switching time from host denial to the ‘do nothing’ strategy at the end of the management time horizon in Fig. 6a. The largest difference in the probability of eradication occurred at around nine years where it ranged between 0·45 and 0·7 for decreases in the germination rate of 50% and 100% respectively. There were also small changes in the optimal time to switch from host denial to the ‘do nothing’ action.
One of the main roles of applied research is to provide decision support for managing ecological systems (Buckley 2008). Formulating this support within a decision theory context provides a framework for good decision making (Possingham 2001; Ayaz, Leung & Miao 2008; Franklin et al. 2008). The adoption of a POMDP model as a decision support tool provides many benefits that a more informal approach to decision making may lack. One of the major challenges in managing ecological systems effectively is appropriately accounting for multiple processes and decision factors. The POMDP model explicitly incorporates imperfect detection of an invasive plant, as well as the interacting processes at play (i.e. dispersal, germination, seed bank decay, colonization). It also includes important management factors such as the probability of successful management, benefits, costs and management time horizons. The POMDP model informs on the best management action to employ under imperfect detection. It also informs on where, when, and for how long different actions should be implemented. The POMDP framework can quantify the likelihood of eradication which is useful when deciding when to stop managing the invasive species. It is unlikely and difficult to verify whether a more informal decision process takes into account of all these factors appropriately and consistently. For our case study, branched broomrape, current management prescriptions include some direct management via host denial or soil fumigation followed by monitoring in subsequent years. If the weed is not detected at the site for 12 consecutive years (i.e. the longevity of the seed bank) then the weed is deemed eradicated from the site (Panetta & Lawes 2005). Our model concurs with this line of management only in the situation where the colonization rate is zero and the detection probability is very high. While managers acknowledge that colonization and imperfect detection are important factors in managing invasive species, they are not necessarily accounted for adequately in the informal decision process.
The POMDP model demonstrates that ignoring imperfect detection of invasive plants may result in inappropriate management. In circumstances when detection is imperfect, if an MDP formulation of the model is adopted instead of the POMDP model, it may lead to poor management decisions in the years where the species is not detected. In the MDP a non-detection assumes an empty state, so the optimal action will always be ‘do nothing’. In the POMDP model, a non-detection is used to update the probability of being in an empty state. If this translates to a low probability of being in an empty state the ‘do nothing’ action may not be optimal and adopting alternative actions to buffer the uncertainty in non-detection years may be needed. In our case, we constrained soil fumigation, the most costly action, to instances when the species was detected, thus the benefit from using a POMDP is when the host denial action is optimal in the years the plant is not observed. Constraining management actions to times of detection is common in invasive species management (i.e. spraying or removal of adults). If, however, there are multiple alternative actions that could be performed when the species is not detected, such as placing barriers to hinder dispersal, or other methods for treating the seed bank, the usefulness of the POMDP will be even greater.
The POMDP model resulted in several useful general management recommendations that are dependent on the location of the infestation relative to other infected sites. For areas with low relative benefit when empty, such as sites within the interior of the infestation that are unlikely to be declared eradicated until sites around it are also clear, the use of a less efficient and inexpensive action (i.e. host denial in our case study) is preferred. However once the benefits of being in an empty state outweigh the cost of the most efficient action (e.g. soil fumigation in this case), the most efficient, and more costly action should be employed the year the species is detected. This would be the case for satellite infestations or newly infested sites on the edge of the main infestation. This result is robust to increases in the value of the site so exact estimates for these values are not important. In our case study, the cost of applying the soil fumigant did not incorporate human health costs explicitly. An optimal solution incorporating human health costs can be inferred from the results by using relative measures of human health costs versus the value of an empty site.
When management areas are at risk of being re-colonized by surrounding infected areas, the likelihood of eradication diminishes as the colonization rate increases, even though the optimal set of actions does not change significantly. This highlights the importance of limiting colonization from surrounding areas if eradication is ever to be achieved, otherwise, management can only endeavour to achieve a level of control rather than eradication. These results emphasize the challenge of managing invasive plants with high colonization rates under imperfect detection. Understanding the magnitude of additional potential seed dispersal vectors and how they can be effectively limited should be considered a research priority.
The POMDP model is based on the data available, in this case presence, absence data on the species. For situations where alternative actions based on different abundances are warranted, then increasing the number of states may be advantageous. However the complexity of solving the POMDP prevents us from using a detailed state space when looking for exact solutions. Thus POMDP is appropriate when the system can be summarized into a small number of states.
We present several graphs of the probability of eradication under optimal management illustrating that while the optimal management actions may not change with changes in specific parameter values, the effectiveness of eradicating the weed can change. This pattern was particularly evident when the colonization rate increased. The probability of eradication at the end of the time horizon ranged from 1·0 to 0·62 across the colonization rates explored (0-0·15). While it is tempting to use these graphs for deciding when to declare eradication of a site, we do not recommend it. Instead the decision to declare eradication involves consideration not only of the probability of eradication but also the trade-off between the cost of continued monitoring and the consequences should the invasive species escape, colonize new areas and causes more damage if eradication is declared too soon. For sites where the cost of escape is relatively low, such as species where the potential damage is likely to be small, it may not be worthwhile to continue monitoring even if the probability of eradication is relatively low. Alternatively, for sites that have a high consequence of escape, managers may be advised to continue monitoring to a point where the probability of eradication approaches 1·0 to avoid the high cost of escape and damage. We recommend that when managers are deciding when to declare eradication of a particular site they should apply the method outlined in Regan et al. (2006) that makes the trade-offs between potential errors explicit.
In summary the POMDP formulation is a convenient and novel model for optimal sequential decision making when invasive plants are partially detectable. While the management strategies are fairly robust to uncertainty, if the invasive plant has a low probability of detection, then an alternative course of action should be adopted. The model also allows an explicit investigation into how management may change when there are alternative rewards for being in an empty state. This led to several useful recommendations on how best to manage an invasive species depending on the location of the infestation. These types of recommendations are difficult to deduce without a formal decision support tool such as the POMDP model. The model presented is not spatially explicit even though we address the potential threat of colonization from neighbouring areas. It is plausible that the optimal course of action will also depend on the state of neighbouring sites. This is an active area of research in the utility of the POMDP formulation.
We thank Dane Panetta, Yvonne Buckley and two anonymous reviewers for providing useful comments on earlier versions of this paper. We are grateful to Anthony Cassandra for insightful discussions on POMDP and the branched broomrape eradication team for support and provision of data for the model. TJR and HPP are financially supported by AEDA and the ARC. IC is financially supported by INRA, CSIRO, MASCOS, AEDA and ACERA. A grant from the GRDC and support from the Weeds CRC enabled this work to be completed.