Ecology Letters (2010) 13: 528–541
We review studies that address economically optimal control of established invasive species. We describe three important components for determining optimal invasion management: invasion dynamics, costs of control efforts and a monetary measure of invasion damages. We find that a management objective that explicitly considers both costs and damages is most appropriate for determining economically optimal strategies, but also leads to large challenges due to uncertainty in components of the management problem. To address uncertainty, some studies have included stochasticity in their models; others have quantified the value of information or focused on decision-making with limited information. Our synthesis shows how invasion characteristics, such as costs, damages, pattern of spread and invasion and landscape size, affect optimal control strategies and goals in systematic ways. We find that even for simple questions, such as whether control should be applied at the centre of an invasion or to satellite patches, the answer depends on the details of a particular invasion. Future work should seek to better quantify key components of this problem, determine best management in the face of limited information, improve understanding of spatial aspects of invasion control and design approaches to improve the feasibility of achieving regional control goals.
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Invasive species are one of the world’s largest immediate environmental threats (Olson 2006). Whether measured in terms of direct or indirect ecological impacts, loss of ecosystem services, economic damages or costs of control, their impact has been very large (Olson 2006). Examples of invasive species that have had widespread ecological and economic impact include kudzu (Pueraria montana), gypsy moths (Lymantria dispar) and zebra mussels (Dreissena polymorpha).
Sequential stages of an invasion include arrival, establishment, spread and saturation (Shigesada & Kawasaki 1997). Not all introduced species establish, but after establishment they may spread and cause ecological or economic damages. Although control can be targeted at any stage of an invasion, this review focuses on economically optimal control of established invaders. Focusing on earlier control also is important, but prevention efforts never are 100% successful and high levels of effort for prevention can be very costly. Furthermore, appropriate prevention levels depend on the expected costs and damages of established invaders, so identifying optimal management of established invasions is important for choosing optimal levels of prevention and detection effort (Leung 2002).
Established invaders lead to both high potential damages and high control costs. Determining how much effort should be put into controlling an invasion is difficult and decisions are often made arbitrarily or based on available budgets allotted for control. It has been recognized that explicitly incorporating economics into conservation planning is important and not doing so makes implicit assumptions about costs that may not be justified (Naidoo et al. 2006). Here we consider studies on controlling established invasions that incorporate varying extents of economics.
We review the types of management objectives pursued by existing studies and the extent to which they consider the costs of control and costs of damages from an invasion. We discuss the economic implications of the different management objectives. We describe the biological and economic components for determining economically optimal management of established invaders and then review how studies have assessed the value of information and accounted for stochasticity and information constraints when determining optimal management. We synthesize the existing literature to show how various features of invasions, including economic factors, spread patterns, invasion and landscape size and likelihood for reinvasion, affect when, where and how much control optimally should be applied. We find that future research should better quantify key components of this management problem, determine best management in the face of limited information, improve understanding of spatial aspects of invasion control and design approaches to facilitate achieving regional control goals.
Approaches for determining best management strategies
Research on control of established invasions falls into two broad categories: methods and strategies. Methodological research has focused on identifying and improving methods that kill or prevent reproduction or spread of invasive species, such as physical or chemical control, biological control agents, mating disruption techniques etc. (e.g. Clout & Williams 2009). In contrast, research on strategies for control, our focus, addresses the question of how to best allocate resources to invasion control: when, where and how much control should be applied.
Determining strategies for invasion management requires prediction of the effect of control efforts on the extent of the invasion over time, which is derived from dynamic models of species spread. Bioeconomics, in turn, combines economics and models of species dynamics to identify economically optimal control strategies that account for the impact of current control choices on future conditions (Clark 1990). Bioeconomic approaches (Box 1) have been developed primarily in the context of fisheries management (e.g. Clark 1990), but also are important for invasive species management.
Like fisheries management, invasive species control is a dynamic problem that involves intertemporal trade-offs: current control efforts affect the future extent of the invasion. An objective of minimizing the sum of control costs and invasion damages over time, by choosing when, where and how much to control, relates most closely to the objective of profit maximization in fisheries management (Box 1). This approach seeks to identify the strategy that provides the greatest overall payoffs – the optimal control strategy. Evaluating this management objective requires that damages and control costs be measured in the same currency so that costs and damages can be summed over time; however, placing a monetary value on ecological damages is very difficult (Olson 2006). Nonetheless, the objective of greatest overall payoffs has been addressed by some invasion management studies (e.g. Sharov & Liebhold 1998; Eiswerth & Johnson 2002; Olson & Roy 2002; Potapov et al. 2007; Wilen 2007; Olson & Roy 2008; Potapov & Lewis 2008).
Alternatively, some studies have sought to minimize the costs of eradication or containment of an invasion (e.g. Menz et al. 1980; Higgins et al. 2000; Taylor & Hastings 2004), which avoids the need to estimate damages associated with the invasion. The objective of minimizing the cost of eradication implicitly assumes that the damages from invasion spread are so high that eradication is the best management approach and that damages accrued during the eradication phase are negligible and can be ignored. Similarly, studies that minimize the cost of containment assume that containment, rather than continued spread or eradication, is the best strategy.
Some other studies have sought to maximize the amount of control achieved subject to a budget constraint or to minimize the time to achieve a given level of control given a budget constraint (e.g. Hof 1998; Wadsworth et al. 2000; Grevstad 2005). These objectives implicitly assume that damages are so high that all available resources should be applied to control. Again, this approach avoids the need for damage value estimates.
Each of the objectives employed in invasion management studies makes different implicit and explicit assumptions about the costs and benefits of control by choosing which values to consider explicitly. Recognizing these different implicit and explicit assumptions provides the way to understand and compare existing studies and their findings. Three key components for evaluating management strategies for established invasions are: (1) invasion dynamics and the effect of control on those dynamics, (2) damages caused by the species and (3) the costs associated with control. All three components are needed to evaluate the objective of minimizing the total costs and damages of an invasion, whereas alternative objectives may require only a subset of these components. Additionally, these components will be affected greatly by stochasticity and uncertainty.
Invasion dynamics and the effects of control
Understanding population and spread dynamics of an invasion is essential for determining optimal management, because invasion dynamics and the effect of control efforts on those dynamics, determine present and future invasion extent and damages. Here we briefly describe biological models that can be used to describe spread (Hastings et al. 2005) and we discuss their features and applicability for evaluating invasion management strategies. Control efforts can be incorporated into many of these models as changes in a model parameter, such as the growth or spread rate or as a direct change in the number of individuals or extent of the invasion. Here we describe models in generally increasing complexity.
The most basic models of invasive species growth are population growth models that focus only on the numbers of individuals or some other measure of population size, ignoring any spatial description. These models are limited to addressing non-spatial management strategies, such as controlling the density or numbers of an invasive species (e.g. Saphores & Shogren 2005; Yokomizo et al. 2009) and are important in designing management strategies for species that disperse widely relative to the potential area of interest, such as a fish species confined to a lake.
Exponential growth models are among the simplest population growth models and can be extended to include age or stage structure if such structure is important. For example, controlling different life stages of a species can have different costs or different effects on species growth (Buhle et al. 2005). Alternatively, density-dependent growth models may be appropriate. Negative influences of density can result in compensatory or overcompensatory density-dependent growth. Allee effects (Courchamp et al. 2008) or lower or negative population growth rates at low population densities, also can be important for understanding invasion dynamics because invasive species are often at low density (Taylor & Hastings 2005) and Allee effects can influence the effectiveness of control (Liebhold & Bascompte 2003).
Predicting spread, in addition to growth, is important for addressing management of most invasions. Spatially implicit models consider the proportion or area of habitat that is occupied by a species, while ignoring the spatial arrangement of occupied habitats, as in the classic metapopulation model of Levins (1969) or other metapopulation formulations (e.g. Bogich & Shea 2008). These types of models may be appropriate for species that have high densities in the parts of the habitat that they occupy and are absent elsewhere or for species that disperse relatively widely. The simplest spatially implicit models of species spread have the same form as the non-spatial models previously described, such as exponential or density-dependent growth, except that they are used to predict the area or fraction of area occupied by the species rather than the numbers or density of individuals (e.g. Eiswerth & Johnson 2002).
None of the models described above explicitly consider space and thus cannot be used to develop control strategies that depend on spatial aspects of the invasion or landscape. For example, management with barrier zones aims to reduce rates of spatial spread by focusing control efforts at the growing edge of an invasion (Sharov & Liebhold 1998), so explicit inclusion of space is needed. Spatially explicit population spread models vary in their assumptions about space, movement and population growth, so they are useful in different contexts. In these models, the pattern of spread affects both the rate of spread and the location of invaded areas within a landscape (Tobin et al. 2007).
Spatially explicit spread models can employ a continuous description of space, as in reaction-diffusion or integro-difference equation models (Kot et al. 1996; Hastings et al. 2005). Reaction-diffusion models assume that movement occurs continuously through habitat by random short-distance dispersal and predict a constant rate of invasion spread. Integro-difference equations instead use kernels to describe movement, which can be more general, allowing for jumps in spread and predicting constant or accelerating spread rates. These two classes of models provide good descriptions of observed spread patterns for a variety of species (Hastings et al. 2005).
Space also can be defined as discrete patches to describe a variety of landscape types and spread patterns, including dispersal among heterogeneous patches in a lattice or dispersal across a network. The latter would be appropriate for aquatic invasions spreading in a complex of lakes (e.g. Facon & David 2006; Potapov & Lewis 2008). These discrete space models could be formulated as cellular automata (With 2004) and control effort could be modelled in a variety of ways, including clearing of patches, prevention of spread or alteration of habitat quality. If species can only spread to adjacent suitable habitats, percolation models could be used to predict movement and to develop strategies that control spread by altering the arrangement of suitable habitat or reducing the amount of suitable habitat below a percolation threshold (With 2004).
Many species exhibit occasional long-distance dispersal (Mundt et al. 2009), either through natural processes, as in wind dispersed seeds and insects or aided by human actions. These dispersal events affect the pattern and rate of invasion spread and can be very important for determining optimal management for some species. Because of the unpredictability and rarity of these dispersal events, these models are inherently stochastic (Lewis & Pacala 2000), which makes solving for optimal control strategies more difficult.
For species whose spread is primarily human-mediated, dispersal is generally non-random because it is driven by human behaviour; this non-random dispersal can be explicitly modelled. Gravity models assume that spread is influenced by the ‘attractiveness’ of different destinations and have been used to model dispersal of the emerald ash-borer by firewood transport (Muirhead et al. 2006) and zebra mussels between lakes by boaters (Bossenbroek et al. 2001). Economic models of human-behaviour, such as random utility models, also can be used to predict invasion spread and to model the effects of policies that alter spread rate by changing human behaviour (e.g. Macpherson et al. 2006).
Although multi-species spread models are less common than single species models, they are important for understanding the impact of invasive species on other organisms in the invaded landscape (Eppstein & Molofsky 2007). For example, models of competitor displacement in space highlight impacts of competitive interactions (Okubo et al. 1989). The spread of exploiters, including diseases or predators, can reduce or eliminate other species (Gaucel et al. 2005). Similarly, models of spread or growth of ecosystem engineers can help to elucidate impacts of these species on ecosystems (Cuddington & Hastings 2007).
Choosing among models requires determining which model features are most appropriate for the specific invasion that is considered. Criteria include the model’s ability to represent relevant features of the invasion, including patterns of growth and spread and potential control approaches, while maintaining simplicity (Levin 1966). There are trade-offs between the complexity and realism of a model and the ability to solve for optimal control strategies that fully account for intertemporal trade-offs. Consequently, most studies that have employed complex, detailed models to predict spread, such as individual-based models, have used simulation approaches to choose among a predetermined set of control options, rather than solving for dynamic optimal control policies (e.g. Higgins et al. 2000; Wadsworth et al. 2000; Grevstad 2005).
Invasion damages and control costs
Damages and control costs must be quantified to evaluate management objectives that depend on these values, such as cost minimization, minimizing total costs and damages or maximizing control subject to a budget constraint. Estimating these costs and damages can be a very difficult or nearly impossible statistical problem (Olson 2006; McIntosh et al. 2009). Damages are often measured as the impact per area of land invaded, per individual or as a function of density, whereas costs can be measured as the cost of applying control to a given area or number of individuals or as the cost to achieve a given amount of control.
The cost of applying control can be relatively straightforward to define, since these values are often observable. However control effectiveness can be uncertain, making estimation difficult and invasion control can cause unintended damages (i.e. non-target effects) that need to be accounted for in the costs. In addition, the per unit cost of control may vary depending on how much control is applied (e.g. Sharov & Liebhold 1998). For example, controlling a small area of an invasion may be cheaper per amount controlled than controlling a large area because the manager can selectively treat the patches that are easiest to access or least costly to control (Clout & Williams 2009). Similarly, if an invasion must be controlled within a short time window (e.g. as dictated by the invasive species’ phenology), then controlling a small amount of the invasion may be feasible, whereas it may be much more costly to control all of it within the necessary time frame. In contrast, large upfront or fixed costs to control may lead to economies of scale that result in lower costs per unit area when larger total areas are controlled. Control costs also may vary dependent on total invasion size because it may be cheaper to remove or cull a small number of plants or animals when total populations are larger because search costs are lower (e.g. Burnett et al. 2007b).
Damages are potentially much more difficult to estimate than costs. Damages include direct and indirect economic impacts such as impacts on ecosystem services (e.g. the productivity of the land, water availability, native species diversity, recreation, etc.). Some impacts may be relatively straightforward to quantify monetarily, such as the damages from zebra mussel to power plants from clogged water pipes (Leung 2002). For most other damages, such as those caused by altered fire regimes or changes to native communities, determining monetary values is much more challenging (McIntosh et al. 2009). Even determining the ecological impact of invasions on natural communities is very difficult (Tilman 1999; Stohlgren & Schnase 2006).
Existing studies have taken a variety of approaches to estimate damages caused by invasions and no single methodology has been accepted as correct (Olson 2006). For example, studies have calculated damage values based on statistical estimates of people’s willingness to pay to avoid the damage (e.g. Burnett et al. 2007b; McIntosh et al. 2009) or from direct, measurable costs incurred from the species, such as the cost of a hospital visit to treat a bite from an invasive snake (Burnett et al. 2008). Damages have also been estimated as lost revenues caused by the invasion, such as from crop or forage loss (e.g. Eiswerth & van Kooten 2007). Valuing damages that have no observable monetary value, such as biodiversity loss, is particularly difficult (McIntosh et al. 2009). Furthermore, damages also can be stochastic, as the effects of invasive species interact with other environmental and economic conditions to cause variable damages across time (Deen et al. 1993). As with control costs, the damages per unit of invasion may depend, in part, on the total area invaded (e.g. Cacho et al. 2008).
Consideration of uncertainty
Various forms of uncertainty, including both lack of information and stochasticity, can affect the consequences and optimality of control. The growth and dispersal of the invading species, the effectiveness of control efforts and the damages imposed by a species can all be stochastic processes (Deen et al. 1993; Olson & Roy 2002; Cacho et al. 2008; Melbourne & Hastings 2009). Information constraints take a variety of forms. For example, parameter estimates are needed to apply models to specific management issues, but their estimation remains difficult even though progress has been made using Bayesian methods (e.g. Cook et al. 2007). Prior to or early in an invasion, little may be known about the future pattern of spread, the potential damages that a novel invader might cause, or how to control it (Horan et al. 2002; Carrasco et al. 2009). Accurate detection and measurement of invasions also is difficult. These issues greatly hinder the ability to determine optimal control strategies.
Studies have taken different approaches for accounting for various types of uncertainty. Some have incorporated randomness into their model of invasion growth or spread (e.g. Higgins et al. 2000; Eiswerth & van Kooten 2002; Olson & Roy 2002; Grevstad 2005; Bogich & Shea 2008; Hyder et al. 2008). With the simplest of these models, the optimal control strategy could be solved for using stochastic dynamic programming methods or analytical techniques (e.g. Eiswerth & van Kooten 2002; Olson & Roy 2002; Bogich & Shea 2008; Hyder et al. 2008), whereas others employed simulation methods to solve for preferred management strategies. Olson & Roy (2002) found that the maximum magnitude of disturbances in growth and spread is an important value for determining the optimality of eradication.
A more complicated issue is decision-making in the face of limited knowledge, as is typical early in an invasion. Some studies have tried to quantify the value of information or to identify approaches for determining best management in the absence of information. Yokomizo et al. (2009) showed that optimal control depends on knowing the shape of the invasion’s damage function and Saphores & Shogren (2005) examined the value of learning bioeconomic parameters and showed that accounting for uncertainty is important. Eiswerth & Van Kooten (2007) compared two approaches, learning models and stochastic dynamic programming, for accommodating model uncertainty and found that the learning models may be more flexible for accounting for real world uncertainties without deviating too far from optimal policy prescriptions. In addition, they showed that expert opinion data can be used to analyse control options when hard data is unavailable (Eiswerth & van Kooten 2002). In contrast, Simberloff (2003b) argues that very little information may be needed to initially control an invasion and waiting to acquire information delays control, reducing the likelihood for successful eradication.
The studies just described address limitations and issues associated with uncertainty when controlling invader density, but none focus explicitly on spreading invasions. Active adaptive management has been suggested as a useful approach for managing spreading invasions in the face of uncertainty (Shea et al. 2002). A study by Edwards & Leung (2009) suggested approaches for determining the feasibility of eradication in the face of limited information, accounting for species dispersal. Information-gap theory also has been used to test the robustness of optimal control policies in the presence of uncertainty about spread rates, control effectiveness and the probability of achieving complete eradication (Carrasco et al. 2009). This study focused on control of Colorado potato beetle (Leptinotarsa decemlineata) in the UK and found that optimal policies that mandated eradication tended to be robust under uncertainty because eradication efforts slow invasion spread even if eradication is unsuccessful.
Monitoring or measuring current invasion levels, which are imperfectly known, is important for deciding and applying optimal control. Thus, some studies have considered the tradeoffs between controlling an invasion and gaining information about its state. D’Evelyn et al. (2008) point out that if the population size of the invader is not known, then management can provide the dual benefits of information gains and control. Similarly, an important component of management is determining how much resources should be applied to detection, monitoring or gaining information about the system (e.g. Cacho et al. 2007; Mehta et al. 2007; Bogich et al. 2008) and deciding when to declare an invasion eradicated (e.g. Regan et al. 2006; Rout et al. 2009).
Economic and ecological characteristics of invasions affect optimal management goals
Goals for the control of an invasive species can range from abandonment to eradication. Given this range, what control policy is optimal? Eradication is often the presumed goal for invasions, whether or not a formal analysis of costs and damages has been conducted (e.g. Menz et al. 1980; Higgins et al. 2000; Wadsworth et al. 2000; Taylor & Hastings 2004; Grevstad 2005). Eradication is often desirable, despite potentially large removal expenses, because it eliminates the invasion and the need for long-term control costs (Sharov & Liebhold 1998). Nonetheless, despite the benefits of ending damages and costs in finite time, eradication may not be feasible or optimal for some invasions and less intensive management may be preferable (Sharov & Liebhold 1998; Olson & Roy 2002; Edwards & Leung 2009). Containment prevents further spread of damages and slowing the spread of an invasion provides benefits by delaying damages. Also, controlling some invasions may not be worthwhile (e.g. widespread invasions that have low damages or high control costs), because the costs of control outweigh the benefits of control (Sharov & Liebhold 1998; Wilen 2007).
Some studies have sought to determine how much resources optimally should be applied for controlling specific invasions by employing estimates of real costs, damages and spread patterns into an optimization framework (e.g. Sharov & Liebhold 1998; Eiswerth & Johnson 2002; Burnett et al. 2007a; Hyder et al. 2008). Others have employed theoretical models to derive a general understanding of optimal control principles and the factors that influence them (e.g. Sharov & Liebhold 1998; Eiswerth & Johnson 2002; Olson & Roy 2002; Potapov et al. 2007; Wilen 2007; Olson & Roy 2008; Potapov & Lewis 2008).
Some of the general principles that emerge from existing studies are not surprising. For example, species that produce high damages per area invaded generally justify higher control efforts to offset damages (Sharov & Liebhold 1998; Wilen 2007; Carrasco et al. 2009; Lewis et al. 2009). Nonetheless, the intensity of optimal control also depends on how damages vary with the area invaded and on relationships between damages and other characteristics of an invasion (Olson & Roy 2008; Lewis et al. 2009).
Higher control costs lead to lower optimal levels of control. For example, long-lived seed banks are associated with higher costs of species removal, thereby reducing the value of eradication (Odom et al. 2005; Cacho et al. 2008). As with damages, the relationship between control costs and invasion size and area controlled affects the optimal control policy (Burnett et al. 2007b; Olson & Roy 2008). If eradication is optimal and the total control costs are proportional to the amount of invasion removed (i.e. constant marginal costs, see Box 1), then eradication should be conducted in one time step because there is no penalty for high rates of control. Instantaneous rather than slow eradication removes damages more quickly and prevents further spread. On the other hand, if the per unit control costs increase with the total area controlled, slower eradication may be justified (Sharov & Liebhold 1998).
For species whose per unit control costs increase as the total area invaded decreases, eradication is less likely to be optimal because the costs of eliminating the last few individuals may be greater than the costs and damages from continual, long-term control and damages (Burnett et al. 2007b). If eradication is not optimal and marginal control costs are high relative to marginal damages when populations are low, delaying control until the invasion has grown can be optimal, as suggested for Miconia calvescens on some of the Hawiian islands (Burnett et al. 2007a; Olson & Roy 2008).
The effectiveness of control efforts influences optimal control policy by affecting the cost to achieve different levels of control and the ability to reduce populations to low levels. Wadsworth et al. (2000) found that high control efficiencies were necessary to eradicate two invasive riparian plants and that near-perfect effectiveness was required for one of the species because of compensatory seed production following control. In contrast, for species that exhibit an Allee effect, even inefficient control actions may achieve eradication (Liebhold & Bascompte 2003; Liebhold & Tobin 2008).
A high discount rate shifts optimal control policies towards lower control effort (e.g. Sharov & Liebhold 1998; Olson & Roy 2002; Wilen 2007; Lewis et al. 2009), because high discount rates deemphasize future damages, reducing the perceived benefits of control. Adjusting the time horizon over which control efforts are optimized has similar effects as adjusting the discount rate; shorter time horizons reduce the perceived benefits of control (Blackwood et al. 2009; Lewis et al. 2009). Some studies have employed short time horizons to simplify the optimization problem or to reflect time scales of management (e.g. Whittle et al. 2007), but the policies derived from this approach may be misguided if the optimization is not set up to otherwise capture the future values of the control.
In addition to economic factors, biological characteristics of the invasion influence optimal policy. High rates of spread cause more rapid accumulation of damages, increasing the benefits of control. Consequently, higher rates of spread can increase the likelihood for eradication to be an optimal policy (Olson & Roy 2008). For example, a species that is spreading radially, rather than linearly, accumulates damages at a greater rate and therefore is more likely to be optimally eradicated (Sharov & Liebhold 1998). Olson & Roy (2002) found that in some situations eradication of small invasions is optimal if the growth rate is larger than the discount rate. Nonetheless, species with higher spread rates can be more costly to control, so the effect of spread rate on optimal policy can be ambiguous. For example, invasive species that exhibit Allee effects may be more efficiently contained (Lewis et al. 2009).
The pattern of growth also can affect optimal control policies by affecting how current levels of control affect future damages. If the spread rate of an invasive species depends on the area occupied then the benefit of a marginal increase in control (i.e. the effect of an additional unit of current control on current and future damages) depends on the species growth function. For example, if the growth rate increases with invasion size, control has the additional benefit of reducing future growth rate (e.g. Eiswerth & Johnson 2002; Burnett et al. 2007b). Also, the existence of a density threshold below which population growth is negative can increase the prospects of eradication as an optimal control goal (Liebhold & Bascompte 2003).
The size of the invasion and the size of the potentially invadable area influence optimal control policy. Sharov & Liebhold (1998) showed that the optimal control goal can shift from eradication to slowing the spread to abandonment with increasing size of an invasion. Larger invasions cost more to eradicate, decreasing the returns from eradication and reducing the likelihood for eradication to be an optimal policy (Olson & Roy 2002; Carrasco et al. 2009). Also, as the invaded area increases in size, there remains less uninvaded landscape to protect from damages, reducing the benefits from control. As a result, under the assumption of constant marginal damages, optimal levels of control effort tend to decrease as the amount of uninvaded landscape declines (Sharov & Liebhold 1998; Wilen 2007). Similarly, species that can spread and cause damage across a large range warrant higher levels of control than similar species whose potential range is more limited (Sharov & Liebhold 1998; Eiswerth & Johnson 2002).
When the chances of reinvasion are high, the benefits of eradication are reduced (Simberloff 2003a). Nonetheless, eradication may still be warranted. In fact, some very damaging invasive species that frequently re-enter the USA are managed by applying regular area-wide control measures at likely entry locations in order to eradicate potential invasions before they can spread (Liebhold & Tobin 2008).
The combination of the influences of landscape size and reinvasion probabilities on optimal control choices have consequences for control choices by managers of land parcels that occupy just a fraction of an invasion’s potential range (Wilen 2007; Epanchin-Niell et al. 2009). Most landscapes are divided among multiple managers, who each make control choices on their own land based on their private incentives. With increasing landscape subdivision individual managers are likely to conduct less control, because the likelihood of reinvasion from uncontrolled neighbouring infestations can be high and because their perceived landscape size is smaller (Epanchin-Niell et al. 2009). In these cases, an individual’s optimal level of control may be much lower than the optimal level of control from society’s perspective (Wilen 2007). Politics also can affect the control of invasions through regulations that mandate specific control efforts or that influence private incentives to control.
Budget constraints can hinder the control of invasive species and increase the total costs and damages from an invasion by preventing the optimal levels of control from being applied, as predicted by optimization theory and evidenced by a number of studies (Taylor & Hastings 2004; Odom et al. 2005; Cacho et al. 2008). Furthermore, when budget constraints are imposed, the resulting optimal strategy for controlling an invasion can change drastically from the unconstrained optimum (Taylor & Hastings 2004; Cacho et al. 2008). Thus, guaranteed and known budgets are important in planning control strategies (Taylor & Hastings 2004; Odom et al. 2005).
Delaying control efforts also increases the total costs and damages from an invasion by reducing control options and increasing eradication costs (Smith et al. 1999; Higgins et al. 2000; Taylor & Hastings 2004). In addition, increasing control delays leads to larger invasion sizes, which shifts optimal control goals towards less aggressive control policies (e.g. from eradication to slowing; Sharov & Liebhold 1998; Wilen 2007).
Numerous bioeconomic factors affect the optimal control policy for invasions, including discount rate, marginal damages and costs, budgetary constraints, spread pattern and rate, the size of the invasion and the potential range of spread. In addition, factors such as seed bank dynamics and the efficiency of available controls affect optimal control choices. Thus, although it is relatively clear how various factors can shift optimal control policies, the optimal control strategy depends on both the species and the situation.
Invasion characteristics and control goals affect optimal spatial control allocation
Understanding where to allocate control effort is important for on-the-ground invasion management. Strategies for dealing with spatially spreading species often apply controls heterogeneously in space, either implicitly or explicitly. The choice of where to allocate control effort can be based on factors such as stage or density class, characteristics of the underlying environment or characteristics of the invasion (e.g. Taylor & Hastings 2004; Grevstad 2005; Hastings et al. 2006). For example, when a species spreads by stratified diffusion such that dispersal from a core population results in many new, small populations, how should control be allocated between core and satellite invasions? More generally, how should control efforts be allocated spatially?
The question of how control resources should be allocated between the main and outlier populations was first addressed by Moody & Mack (1988), but no definitive answer has been obtained. Studies on the subject have arrived at conflicting conclusions. Taylor & Hastings (2004) pointed out that even theoretical frameworks without economic considerations support different strategies: ‘the population biology approach suggests that, in general, outliers contribute the most to range expansion and should be removed first, whereas the metapopulation approach suggests prioritizing core populations that supply most of the new propagules’.
Conclusions by previous studies about the choice of patch to prioritize for removal clearly depend on the assumptions of the study’s model. Key issues are assumptions about costs of removal and the description of population dynamics. The approach by Moody & Mack (1988) was to model the spread of an invasion consisting of a main core population and multiple smaller satellite populations each exhibiting constant radial growth. They asked whether it is most effective to remove some fraction of the satellite populations or a fraction of the main focal population and found that control of the satellite populations should be prioritized because this increased the time for the total area of the satellite populations to exceed the total area of the main population. However, this measure of effectiveness does not consider the costs of control or the damages associated with the invasion and there exist many scenarios in which shorter crossover times could produce faster or cheaper containment than scenarios with longer crossover times (Taylor & Hastings 2004). If costs of removal increase with invasion density, removal of low density, juvenile populations might be the most cost effective (e.g. Higgins et al. 2000). If costs are equal, but young or low density patches have higher rates of spread because they are not density constrained, again prioritizing low density satellite populations might be favoured, as shown for Spartina alterniflora in a study by Grevstad (2005) and by Taylor & Hastings (2004) under most budget situations.
In contrast to these results, there are scenarios where allocating control efforts to high density main populations was preferred. Taylor & Hastings (2004) found that with a weak Allee effect exhibited by S. alterniflora and a high budget constraint, prioritizing the high density main population reduced the cost and risk associated with eradication, reflecting the benefits provided by the reduced seed production from removing high density patches. Wadsworth et al. (2000) found that focusing control within more contiguous, larger patches was more effective than prioritizing small satellite populations for minimizing the time to eradication under a budget constraint. Their study compared six different control prioritizations based on invasion patch characteristics, assumed that the control cost per area of treated invasion was constant across patch characteristics and did not consider damages. Strategies that prioritized upstream or high density patches led to treatment of more contiguous invaded areas and performed better than strategies based on age class, lending credence to the strategy of prioritizing high density, contiguous areas. Their finding may reflect the greater ‘bang for the buck’ of treating dense populations. Menz et al. (1980) compared the cost of several spatial control strategies for controlling an invasion, including control from the inside-out (i.e. from the high density interior toward the low density exterior). They found that in some circumstances with large budgets and relatively low spread rates, control from the inside-out was cheaper than immediate eradication. This result seems to have been driven by the combination of lower marginal costs of controlling high density infestations and lower search costs associated with this strategy.
Whittle et al. (2007) repeated Moody & Mack’s (1988) study in a framework that considered total costs of control over time and damages from any invasion remaining at the end of the finite time horizon. They found that most control effort should focus on the largest population, for which control per unit area was cheaper, but that some efforts also should be allocated to controlling small patches. They also found that control should be delayed for as long as possible, a finding that may depend on their assumptions that control of large invasions was cheaper and damages only accrued at the end of the management horizon. Another recent study also found that control should be applied to both the core population and outlying satellite populations (Blackwood et al. 2009).
Based on the assumptions and conclusions of these studies, the relative merits of the two prioritization approaches (targeting core vs. satellite populations) appears to depend on dispersal and growth rates in core vs. satellite populations and relative control and search costs. None of these studies, except Whittle et al. (2007) and Blackwood et al. (2009), considered damages, which could markedly change conclusions. Key relationships among bioeconomic parameters have not yet been identified to guide the prioritization of control efforts between core and satellite populations. In fact, this debate may be too strongly focused on a single best strategy because appropriate prioritization may vary across situations and in many situations it may be optimal to simultaneously control both satellite and core areas. This type of strategizing needs to be considered within the broader context of the optimal control goal for an invasion. For example, if eradication is optimal, focusing control only on core or satellite populations will fail to achieve the desired outcome. Likewise, applying no control effort could be optimal for some invasions. This question of prioritization is just one component of the control choice of when, where and how much control to apply.
Barrier zones are an alternative form of spatial control strategy in which control efforts are focused along the growing edge of an invasion (Sharov & Liebhold 1998). Barrier zones have been employed for controlling a number of pests including gypsy moths and boll weevils (Sharov & Liebhold 1998). Morgan et al. (2006) found that barrier zone control also was relevant for maintaining a low density of possums within an area surrounded by an uncontrolled possum population that could disperse into the controlled centre. They found that using traps set along the border between the controlled and uncontrolled populations to reduce immigration could be less costly than regular applications of control to the central region, for maintaining possum populations below a threshold density. Barrier zones also may be created by altering the arrangement of suitable habitat for an invasion (With 2004).
Potapov & Lewis (2008) examined optimal control of an invasive species spreading among lakes. They modelled spread between lakes using a gravity model and found that when lakes are clustered, if some lakes in a cluster become invaded, it may be optimal to abandon protection of that cluster and focus efforts on preventing spread to new clusters. This mimics a type of barrier zone between clusters of lakes. Similarly, Blackwood et al. (2009) found that reducing patch connectivity could greatly reduce management costs.
Grevstad (2005) and Hof (1998) also have contributed to understanding spatial allocation of control strategies. Hof used linear programming to look at spatial control of a pest on a grid, with the goal of minimizing pest numbers summed across time, subject to the constraint of controlling a fixed number of pests in each time period. Similar to barrier zones, Hof found that controlling in a border around the invasion was most effective in some cases. Hof also found that using control to move an invasion to the edge of the landscape can facilitate eradication since individuals disperse out of the system at the landscape boundaries, reducing the amount of control needed. Similarly, Grevstad (2005) found that the most effective strategy for controlling Spartina alterniflora in a simulation model focused control to reduce the growing edge of the invasion and away from edges currently bounded by habitat barriers.
Studies incorporating bioeconomic principles have considered a variety of management objectives, including minimizing the cost to achieve a particular control goal, minimizing the extent of an invasion subject to a budget constraint and minimizing total control costs and damages from an invasion. By employing bioeconomic principles to determine optimal control levels, these studies account for the interdependencies of control effort and invasion spread across time, but each management objective makes different implicit and explicit assumptions about control costs and damages. We find that an objective that explicitly considers both costs and damages is most appropriate for determining economically optimal control strategies (Naidoo et al. 2006; Keller et al. 2009), but also leads to large challenges due to the uncertainty in all components of the management problem (Naidoo et al. 2006).
Strategies based on minimizing the cost of eradicating a species do not require damage estimates, but are likely to allocate too few or too many resources to control. If damages are high, control efforts beyond the minimum cost strategy could reduce the overall costs and damages that are incurred during the eradication period. On the other hand, eradication may not be an appropriate objective for an invasion if the cheapest eradication strategy has higher costs and damages than a strategy that slows or contains the invasion. In this case, regulations that mandate eradication are inefficient.
Similar drawbacks hold for studies that minimize the invasion subject to a budget constraint. These studies provide guidance for how to manage an invasion when a budget falls short of the optimal allocation and can provide an important reality check in situations where eradication is mandated but the budget falls short of what is needed to achieve this outcome. This objective, however, does not dictate economically optimal control, which may require greater control expenditures. Also, in the unlikely situation that the allotted control budget is higher than the amount needed to implement the optimal control policy, spending the entire budget on control will result in excessive control effort and higher total costs and damages than applying economically optimal control levels. Identification of the unconstrained optimal control policy, with consideration of both costs and damages, is necessary to provide information about the level of expenditures that are warranted and the consequences of a limited budget, which can help guide more efficient budget allocations (Odom et al. 2005).
Clearly, quantifying the key components of the invasion management problem is a major challenge to determining optimal control levels. In particular, estimating damages and predicting invasion spread are especially challenging because these components are not readily observed until a species has begun to spread and spread rates can be highly uncertain even in the simplest ecological settings (Melbourne & Hastings 2009). In addition, damages and control costs need to be quantified in the same currency, which generally requires putting a monetary value on damages.
Several studies have examined management of spreading invasions in the face of uncertainty about key factors, such as control effectiveness and spread rates, for particular invasions (e.g. Eiswerth & van Kooten 2007; Carrasco et al. 2009; Edwards & Leung 2009). However, identifying methods for determining best management strategies in the face of limited information remains a critical gap in current understanding. For example, developing frameworks for determining optimal management of newly established, novel invaders whose potential damages are unknown, is a necessary area for future research.
There also has been little work explicitly examining how spatial characteristics of the invasion or landscape affect optimal control policies or where to apply control efforts in heterogeneous landscapes. In fact, heterogeneity in the underlying landscape or invasion has received little attention in invasion management, but is clearly important for on the ground management of invasions. Better understanding of how to implement optimal invasion control in a landscape managed by many decision-makers also is important for invasion management and requires determining control goals and designing regulations and incentives that adequately account for human behaviour and invasion characteristics (Epanchin-Niell et al. 2009).
In summary, our synthesis of existing research demonstrates that optimal invasion control policies depend on the particular species and situation because many characteristics of the invasion affect the long-term costs and damages of management, including economic factors, spread patterns, invasion and landscape size and likelihood for reinvasion. To improve the decision making process regarding optimal control strategies for invasive species, future work should focus on improving models of invasion spread and gaining better knowledge of impacts, identifying strategies for invasion control decision-making in the face of limited information and high stochasticity, better understanding spatial aspects of invasion management and designing regulatory frameworks to improve regional invasion control. Developing appropriate models and improving our knowledge of how to best manage biological invasions will benefit from greater communication and collaboration between ecologists and economists. The challenges are large but the rewards in terms of a better ecological future are larger.
Box 1: bioeconomic principles
Bioeconomics is the study of the dynamics of living resources using economic models, with particular emphasis on applying optimal control methods to mathematical models of species dynamics (Clark 1990). Here, we summarize ideas from bioeconomics that are useful for understanding studies of optimal management of spreading invasive species, beginning with a description of bioeconomic principles in the context of fisheries management. We then describe assumptions that underlie the economic analysis, explaining ideas of costs and of discounting. Finally, we discuss the general methods used for solution, which in turn influence model setup.
Fisheries management is a classic application of bioeconomics and provides an ideal framework for understanding basic bioeconomic principles. The simplest objective in fisheries management is in the case of a single entity (single owner) controlling the system. In this case the fisheries manager wants to choose the level of fishing effort across time and space in order to maximize long-term profits, which are calculated by summing the profits (costs minus revenue) expected across time for a given harvesting strategy. Profits in each time period depend on harvesting effort and the fish population size at that time. Consideration of fish population dynamics thus is a necessary component of this management problem, because current harvesting effort affects future profits by affecting the future abundance of fish. In this way, optimal fisheries management involves making trade-offs across time and space that account for species dynamics (Sanchirico & Wilen 1999). Bioeconomic approaches account for these intertemporal trade-offs to determine best management.
When costs or benefits are summed or compared across time, monetary values must be converted into equivalent units (their present value) through discounting (Clark 1990). Economists generally assume that current benefits are valued more than future benefits, because current benefits could be invested to increase their value through time and can provide more opportunities than if those benefits were delayed. For example, with invasive species, money for immediate control may need to be diverted from other investments and damages occurring now may be of more concern than similar levels of damage occurring in the future. Thus discounting is used to weight current costs and damages more heavily than future costs and damages.
The cost of applying an additional unit of control is an important measure in economics and is referred to as the marginal cost and equals the derivative of the cost function with respect to the amount of control (Clark 1990). In some situations, the cost per unit of control may not depend on the total amount of control, as with a linear cost function; in this case marginal costs are constant. On the other hand, if controlling a few hectares of an invasion is cheaper per acre than controlling a larger area, marginal costs increase with the amount of control applied. Economists often assume increasing marginal costs, based on the idea that resources are limited and cannot be obtained infinitely at the same price (e.g. Eiswerth & Johnson 2002). However, there also can be economies of scale, such as when there are large upfront costs to control, which lead to decreasing marginal control costs (Clout & Williams 2009). Analogous to marginal costs, the marginal benefit of control is the monetary damage avoided by applying an additional unit of control. Marginal benefits can be increasing, decreasing or constant, depending on how the amount of damages varies with the extent of invasion.
Understanding marginal costs and benefits is important because in typical economic problems that seek to minimize total costs and damages, the optimal control policy is the choice of control effort that sets the marginal cost of control equal to the marginal benefit of control (Fig. 1a; Clark 1990; Pindyck & Rubinfeld 2001). In other words, it is optimal to apply increasing amounts of control, up until the point where the cost of applying the next unit of control would be greater than the damages it would prevent. When marginal cost is less than marginal benefit, it is worthwhile to invest in more control because it reduces damages by more than the cost of control and vice versa (Fig. 1a).
In dynamic management problems, such as for fisheries or invasion management, this relationship between marginal costs and marginal benefits must be computed in a way to account for the effect of current control choices on future costs and damages. Optimal management still invests in control up until the point where paying for another unit of control is not cost effective, but in dynamic problems this occurs when the marginal cost of control equals the sum of current and future benefits (avoided damages) from the additional unit of control (Fig. 1b; Olson 2006).
A variety of optimization methods exist to solve a bioeconomic problem after the key components and the management objectives have been clearly defined (Olson 2006). Among these methods are dynamic programming (e.g. Clark 1990), optimal control theory (e.g. Clark 1990; Eiswerth & Johnson 2002) and genetic algorithms (e.g. Taylor & Hastings 2004), which can be used to identify control strategies that fully account for intertemporal trade-offs. An alternative solution approach, which requires less computation, but does not fully account for intertemporal trade-offs imposed by control choices, simulates invasion spread under various predetermined strategies and chooses the preferred strategy by comparing predicted outcomes (e.g. Higgins et al. 2000; Wadsworth et al. 2000; Grevstad 2005). It is also much more difficult to deduce general principles from simulation studies, and general principles may be especially useful in cases with limited data. However, simulation approaches can accommodate much more complicated models and assumptions.
RSE and AH are grateful for support from USDA PREISM 58-7000-7-0088 (PI Wilen) and NSF EF-08-27460, respectively. We thank three anonymous reviewers and P. Epanchin for helpful manuscript advice.