1In conservation decision-making, we operate within the confines of limited funding. Furthermore, we often assume particular relationships between management impact and our investment in management. The structure of these relationships, however, is rarely known with certainty – there is model uncertainty. We investigate how these two fundamentally limiting factors in conservation management, money and knowledge, impact optimal decision-making.
2We use information-gap decision theory to find strategies for maximizing the number of extant subpopulations of a threatened species that are most immune to failure due to model uncertainty. We thus find a robust framework for exploring optimal decision-making.
3The performance of every strategy decreases as model uncertainty increases.
4The strategy most robust to model uncertainty depends not only on what performance is perceived to be acceptable but also on available funding and the time horizon over which extinction is considered.
5Synthesis and applications. We investigate the impact of model uncertainty on robust decision-making in conservation and how this is affected by available conservation funding. We show that subpopulation triage can be a natural consequence of robust decision-making. We highlight the need for managers to consider triage not as merely giving up, but as a tool for ensuring species persistence in light of the urgency of most conservation requirements, uncertainty and the poor state of conservation funding. We illustrate this theory by a specific application to allocation of funding to reduce poaching impact on the Sumatran tiger Panthera tigris sumatrae in Kerinci Seblat National Park.
Decisions about the management of threatened species are made in the face of considerable uncertainty. This uncertainty arises from a lack of knowledge about many aspects, for example; population dynamics, impacts of threatening processes, and the usefulness of conservation actions. Ignoring this uncertainty may lead to inferior decision-making (Regan et al. 2005). Further to this, making optimal decisions in conservation is a science that must acknowledge limited budgets (Possingham 2001; Possingham et al. 2001; Wilson et al. 2006). Given these limited resources, and a finite time available to implement management due to the urgency of conservation concerns, our capacity to substantially increase knowledge and reduce uncertainty is constrained. As conservation scientists we are not only charged with pressing for increases to environmental funding by heightening awareness of the importance of conservation endeavours but also with aiding responsible conservation decision-making within this current climate of limited funding and limited knowledge.
There are a number of forms of uncertainty that have been highlighted in ecological and conservation studies (Regan, Colyvan & Burgman 2002). There may be uncertainty around our estimates of parameters, which lead us to question the output of our models (Caswell 2001). Sensitivities that are commonly considered are how do our estimates of survival and fecundity affect our estimate of population growth? (Caswell 2001), or how do estimates of extinction risk vary with changes in estimated parameters? (McCarthy, Burgman & Ferson 1995). We may also have ‘model uncertainty’, i.e. uncertainty about the actual model we have chosen to represent our system (Chatfield 1995; Regan, Colyvan & Burgman 2002; Runge & Johnson 2002). Model representations of systems are used extensively in threatened species conservation, pest control and in conservation planning for predicting future events, for answering questions about the response of a system to explicit scenarios, and for selecting among scenarios (e.g. Possingham, Lindenmayer & Norton 1993; Punt & Smith 1999). Although averaging methods exist in both poles of the statistical spectrum, Bayesian and frequentist, to incorporate model uncertainty into the model selection process (Burnham & Anderson 2002; Wintle et al. 2003; Link & Barker 2006), the impact of such uncertainty on the performance of subsequent management decisions remains largely unresolved (see Runge & Johnson 2002).
Knowing how sensitive a decision is to uncertainty or using model averaging to incorporate model uncertainty does not enable decision-makers to assess how uncertainty would change a decision, information that may be beneficial, given there is limited time and money available in which to make decisions. This is the basis of information-gap decision theory (info-gap), finding the strategies that are most immune to failure due to uncertainty (Ben-Haim 2001, 2006). Unlike the conventional sensitivity analysis approach, there are no underlying assumptions about the distribution of uncertainty using this robust optimization technique: the amount of uncertainty is unknown and unbounded (Ben-Haim 2006). In data-poor situations, as is often the case in conservation, the exact form of uncertainty is often very uncertain in itself. The info-gap modelling approach presents itself as a suitable means of assessing the impact of uncertainty, in any form, on the decision-making process (Ben-Haim 2006).
The Sumatran tiger Panthera tigris sumatrae (Pocock 1929) is under threat from numerous anthropogenic processes (Linkie et al. 2006) in the Kerinci Seblat region of Sumatra. Considered a level 1 ‘tiger conservation unit’ (Wikramanayake et al. 1998), Kerinci Seblat is patrolled with the intent of decreasing one of the threatening processes, illegal poaching. Linkie et al. (2006) identified four core areas of suitable habitat and the level of poaching protection (and therefore cost) necessary to maintain tiger populations in these areas. Using information from this work, McDonald-Madden, Baxter & Possingham (2008) were able to assess the effect of resources on yearly local extinction probability of a subpopulation (core habitat area). Employing this relationship, they derive a framework for determining how many subpopulations of a threatened species should be managed given budgetary constraints, costs of managing, and the impact of management. Sensitivity analysis showed that the number of subpopulations to manage varies when confronted with changes to key parameters in this relationship; however, structural uncertainty in this relationship is not explored (McDonald-Madden, Baxter & Possingham 2008). We extend this work in order to elucidate how many subpopulations should be managed under differing levels of uncertainty in the relationship between conservation effort and the probability of extinction of an individual subpopulation (extinction–investment curve). Further, we explore these conceptually simple yet vital questions for conservation management: how should we manage when we have both a degree of uncertainty surrounding our system and a set amount of money to spend on managing this system? How will our actions change as our degree of uncertainty changes or the funding available to management changes? Should we manage all the remaining subpopulations of a threatened species (a common approach to threatened species management)? Or should we funnel resources to a few subpopulations, invoking the concept of triage? Furthermore, how do these two fundamentally limiting factors in conservation management, money and knowledge, interact to determine the number of subpopulations we can feasibly manage?
1a mathematical process model that delivers a measure of performance as a result of management
2a performance requirement, below which we consider our performance unacceptable, and
3a model describing uncertainty.
performance measure and requirement
To formulate our problem, we first need to assume a mathematical representation of our ecological system. In this case, we have a relationship between resource allocation and probability of extinction of a threatened species subpopulation (extinction–investment curve). McDonald-Madden et al. (2008) assumed the probability of extinction of a subpopulation to be a monotonically decreasing function of the budget allocated to it for the planning period. In this study, we consider uncertainty in the functional form of this relationship, suggesting that it could be a number of other forms, for example, a decreasing sigmoidal function. We approximate the model given by equation 1 in McDonald-Madden et al. (2008) with the more general function:
where P0 is the annual probability of extinction of the species in a subpopulation that is not managed and x is the budget allocation to the subpopulation in a year. The value of k specifies what budget is required to halve the initial annual probability of extinction (when unmanaged), that is, the half-saturation point of the relationship, while θ alters the shape of the function. For a given value of k, large values of θ indicate that the initial benefit of increasing the budget allocated to the subpopulation is small, while small values of θ mean that there is a large initial benefit to budget increase. Varying k and θ changes the relationship between probability of extinction and resources, thus altering the specified model of the system (see Fig. 1). Using this sigmoidal function enables the curve to vary widely, for example, from an exponential-like decline in extinction probability with increasing resources, to an almost negligible change in extinction probability until significant investment, after which extinction probably declines severely (i.e. approaching a step-function; see Fig. 1). This relationship also contains the probability of a subpopulation going extinct when no management action is taken, P0. Parameter estimates for this value can significantly affect the possible recovery of a species and, depending on the species, may be highly uncertain.
We assume that each time step, the entire budget will be divided equally among a number of managed subpopulations, n, and therefore, that the intensity of management per subpopulation decreases as the number of subpopulations managed increases. Hence, the probability of extinction of a single subpopulation in one year is a function of n:
Following McDonald-Madden et al. (2008), we use the number of extant subpopulations remaining at the end of each time step as a measure of performance. The expected number of extant subpopulations (E) in the next time step is the sum of the number of subpopulations that are managed that persist, and the number of subpopulations that are not managed that persist:
E = n(1 – P(x/n)) + (N – n)(1 – P0)(eqn 3)
As it is assumes that all subpopulations are the same, P0 and P(x/n) are equal for all subpopulations. We modify McDonald-Madden et al. (2008) to incorporate time over which extinction is considered, t:
E = n(1 – P(x/n))t + (N – n)(1 – P0)t(eqn 4)
The number of management options is the number of subpopulations existing plus an additional strategy to manage nothing, that is, one could choose to manage anything from no or one subpopulation, up to all existing subpopulations, N. This management decision is implemented over the entire period over which extinction is considered.
We assume that the ultimate objective of threatened-species managers is to maximize the number of extant subpopulations remaining at the end of the management period. Using info-gap, however, managers can specify a ‘performance requirement’ they wish to achieve and thus choose the most robust strategy that ensures this goal is attained at the highest level of (unknown) uncertainty. In info-gap theory, this approach is known as robust satisficing (Ben-Haim 2006). If the magnitude of uncertainty is known, a state not often the case in ecological systems, the manager can choose the action that maximizes the number extant subpopulations (performance) for a given uncertainty level. In info-gap theory, this is considered the worst-case analysis (Ben-Haim 2006). In conservation, an idealistic performance requirement would be to save all subpopulations of our threatened species; however, this may often be unrealistic due to limitations in funding and the need to distribute resources between subpopulations. Here we consider the minimum acceptable requirement or critical performance requirement (Ec) of keeping at least one population, Ec ≥ 1, as well as a more risk-averse requirement of ensuring at least two remaining subpopulations, Ec ≥ 2. The first paradigm ensures that we just maintain the species while the other considers the chance of catastrophic risks or unaccounted for stochastic variation in the environment and incorporates a back-up subpopulation to reduce the impact of such events. Both are commonly considered notions in conservation theory (Bascompte, Possingham & Roughgarden 2002; McCarthy, Thompson & Possingham 2005).
We know that the extinction–investment curve assumed is likely to be incorrect, and that there are in fact a range of possible values of K, θ and P0 that will express a multitude of different functional forms (see Fig. 1). The range of possible values around the best estimate value of k, θ and P0, k̃, and0 respectively, are a function of the uncertainty, α, known as the horizon of uncertainty.
We use an ellipsoid bound info-gap model of uncertainty for vectors of our variables k, θ and P0 (Ben-Haim 2006):
The ellipsoid-bound model enables us to consider overall uncertainty in the model specification by varying k, θ and P0 at different rates around the ellipsoid. The distribution of uncertainty between k, θ and P0 can be described using azimuthal (0 ≤ ω ≤ 2π) and polar (0 ≤ τ ≤ π) angles:
The true value of α is unknown (Ben-Haim 2004), as we do not know how uncertain we are of the best estimates of k, θ and P0. The greater our uncertainty, the higher the value of α and the larger our ellipsoid of uncertainty. Thus, the set U(α) of possible values for k, θ and P0 becomes more inclusive as α increases. Consequently, info-gap theory refers to nested subsets of model uncertainty for the assumed extinction–investment curve.
Using info-gap, we are not trying to find the strategy that maximizes the expected number of extant subpopulations but determine a robust management strategy that guarantees a minimally satisfactory level of performance, Ec. To ensure we find this minimal point of satisfaction, we calculate the minimum number of extant subpopulations within the ellipsoid for each value of α. We do this numerically by sampling angles ω and τ and calculating the points k̃, and0 around (and within) the ellipsoid based on equations 6, 7 and 8. We use these values to calculate our performances, and thus, identify the minimum performance for that horizon of uncertainty. Hence, for each management strategy, we assess how wrong we can be about our assumed model of the system (how large α can be) while still satisfying our performance requirement. In info-gap theory, such a precautionary approach is known as the robustness function (Ben-Haim 2004). This maximum value of α () that satisfies the performance requirement(s) is referred to as the ‘robustness’ of the management option. Thus, we can determine the strategy that has the greatest robustness for a given performance requirement, Ec, and therefore, the most robust strategy.
We investigate the effect of budget, x, and time over which extinction is considered, t, on the choice of the most robust management strategy. Further, we investigate the most robust strategies given budget and uncertainty for a simple adaptive scenario where resources can be reallocated biennially following reassessment of the system state.
How does uncertainty affect the minimum-performing extinction–investment model for each strategy? As model uncertainty increases, the minimum-performing curves for each management strategy move further from the assumed curve (Fig. 2a–d). As uncertainty increases, then the values of , k̃ and P0 that give the worst performance increase. That is, the initial benefit of increased resources is less, the budget required to reduce the initial probability of extinction by half increases, and the probability of extinction when no management occurs increases with increasing uncertainty, α. Also for larger budgets and increased uncertainty, the difference among management strategies increases (Fig. 2).
The minimum performance attainable by each strategy decreases as uncertainty increases (Figs 3 and 4). The most robust strategy to model uncertainty depends not only on performance level required but also on the resources available to the conservation program, x (Fig. 5), and the time horizon over which extinction is considered, t (Fig. 6).
If we could increase the available budget, we see that the most robust strategy changes and that the performance (expected number of extant subpopulations, E) increases (Fig. 3). We find that for low budgets, the species goes extinct within 10 years (t = 10) no matter how it is managed, with E < 1 even when we are certain of our model structure (Fig. 3a). That is, we can never reach a minimum performance criteria, Ec, even when Ec = 1. If we increase the budget, we can ensure that one subpopulation is saved from extinction by managing two subpopulations but only with minimal robustness (a ≈ 0·25) to model uncertainty (Fig. 3b). In this case, if our performance criterion was to retain at least two subpopulations, then for this budget, we can never attain our goal and no robust strategy can be found. As we increase the budget further, we increase our ability to attain the higher performance criteria (Ec = 2) and the management of more subpopulations becomes a more robust strategy (Fig. 3c,d). However, this higher performance criterion can only be reached with minimal robustness to uncertainty, no matter what strategy we decide to implement. If the manager is happy to consider a less risk-averse strategy, then we can reach our goal with more uncertainty about our extinction–investment curve (Fig. 3c,d).
If we fix the budget and investigate the impact of the time horizon, t, over which we consider extinction, we also see changes in the most robust strategy and the decision based on the performance criterion (Fig. 4). If we consider 1 year as the extinction horizon, then any strategy will enable us to attain either performance criteria with a reasonable level of model uncertainty (Fig. 4a). If we increase the time horizon, however, the expected number of extant populations decreases. When we consider extinction over a 5- or 10-year horizon, a less risk-averse manager may retain at least one subpopulation for all strategies but only under relatively low levels of uncertainty; however, if the manager is risk-averse, wishing to conserve two subpopulations, then their goal can only be achieved under very minimal uncertainty (Fig. 4b,c). Over these time horizons as model uncertainty increases, no strategy reaches either of the required performance level (Fig. 4b,c). As we increase the time horizon to 20 years (t = 20) for extinction, we will never attain Ec = 2, and only for low level model uncertainty can we achieve a performance requirement of Ec = 1 (Fig. 4d). Thus, for longer-term management objectives, one should focus resources on fewer subpopulations to reach performance requirements while staying immune to uncertainty.
Over a large budget-uncertainty parameter space, we can observe the impact of these factors on robust decision-making for the Sumatran tiger (Fig. 5a). We see that the option to manage all tiger subpopulations is only robust for large budgets and under minimal model uncertainty. By far, the most robust option under a range of budgets and reasonable levels of model uncertainty is to manage two or three subpopulations. In most conservation programmes, precise understanding of the system is limited, and thus, model uncertainty around the extinction–investment curve is high and the budget available for management is small; then, the optimal approach is not to manage all subpopulations but to concentrate efforts in one subpopulation. Similarly, when we investigate robust decision-making for different time horizons in which we consider extinction, that is we consider a short-term or long-term approach to management, we see that again the most robust management involves triage (Fig. 5b). Our optimal approach is not to manage all subpopulations but to manage two or three subpopulations. In both these scenarios, the minimum performance obtained, in terms of the number extant subpopulations, varies depending on uncertainty, budget and time. The greatest performance is achieved for low uncertainty and high budgets and for low uncertainty and short management horizons (Fig. 5).
If resources could be reallocated every 2 years based on re-assessing the number of extant subpopulations, the most robust strategy again involves some level of triage (Fig. 6). As the number of subpopulations remaining declines, the strategy to manage all subpopulations remaining is still only robust for high budgets and low levels of uncertainty. Again by far, the most robust option under a range of budgets and reasonable levels of model uncertainty is to manage fewer subpopulations than remain extant.
Decisions are made about how to conserve our environment in the face of considerable uncertainty. Further to this, limitations of funding, time and the scale of environmental issues we are confronted with, have led to a lack of knowledge on population dynamics, impacts of threatening processes, and the benefits and efficiencies of conservation actions (Balmford & Cowling 2006). Given this lack of valuable decision-informing data, the need to evaluate the impact of uncertainty on the performance of management options is considerable (Regan et al. 2005). Testing hypotheses and modelling the impact of management on threatened species recovery has enabled managers to make more measured decisions about management practices (Milner-Gulland 1997; McCarthy, Possingham & Gill 2001; Regan et al. 2001; Tenhumberg et al. 2004). In doing this, however, assumptions are made about relationships and interactions in the real world; relationships that, although logical, are unknown and uncertain. In this study, we address the impact of uncertainty in the functional representations of biological systems on conservation decision-making. We have been able to elucidate the impact of this type of uncertainty, ‘model uncertainty’, on decision-making. Furthermore, we explore two key elements limiting conservation decision-making, funding and knowledge, and how they interact to affect the best management strategy for a threatened species.
The money available to conservation programmes is a major limiting factor to the effectiveness of such programmes; here we discover that it also reduces a management strategy's robustness to our incomplete understanding of the systems we are trying to save. As we increase our budget, we enable more inclusive policies to be robust, for example, with the Sumatran tiger, we can manage more subpopulations, and we can thus increase the performance of our conservation programme. However, this benefit is only clear for marginal levels of uncertainty. When uncertainty is great, our ability to achieve high performance is minimal for even a large budget. In these situations, the most robust strategies are those that take a triage approach and distribute financial resources towards fewer subpopulations rather than attempt to manage all or even most of our remaining subpopulations. In taking this approach, the managers would be acting in a risk-averse manner; they may not achieve the conservation of all subpopulations but can at least save one or two subpopulations and ensure species persistence, which is a reasonable minimum acceptable goal for most conservation programmes.
We also show that the time horizon over which a conservation programme is designed impacts on the management strategy that gives the most robust performance under model uncertainty. That is, short-sighted policies may achieve higher performance levels over those brief timeframes with more inclusive management actions but, as our goals become longer term in nature, the most robust strategies are those that again take a triage approach – allocating money towards fewer subpopulations. This assumes that the strategy implemented over this time horizon is not changed, that is, the allocation rule does not give the best strategy to implement in each year of the time horizon but the best overall action for continual implementation over the entire time horizon. In most conservation programmes, the state of the system is unknown: we do not know how many subpopulations are persisting at any one time. However, unless money is invested in a monitoring programme to understand the state of each subpopulation, the reality is that management actions will be implemented assuming a known set of subpopulations are persisting at the beginning of the management horizon. If the number of extant subpopulations in the system could be assessed every few years, then the strategy can be adapted accordingly. We found that even if a simple adaptive procedure was in place involving reassessment of the state of the system every 2 years, a triage approach to allocation, where resources are distributed to fewer subpopulations than are known to be extant, remains the most robust conservation strategy. Using this simple approach, we assume that funding is available for monitoring and does not detract from management funding, and that monitoring can perfectly detect the state of each subpopulation, an assumption unlikely to hold in most conservation programmes. Indeed, if the funding for precise and reliable monitoring were to come out of the management budget, the need for triage would be increased. A comprehensive analysis of the problem that accounted for state-dependent decision-making under uncertain states, detection issues and budget trade-offs with monitoring would require reformulation of the problem as a Markov decision process accounting for partial observability (Cassandra, Littman & Zhang 1997; White 2005).
In this study, we assume a relationship between the resources available to management and the probability of extinction of a threatened species (extinction–investment curve). Recognizing and incorporating our uncertainty in the functional form of this relationship using info-gap dramatically changes the way in which we manage a species. Indeed, there is a major change in the shape of this extinction–investment curve and the extinction probability of an unmanaged subpopulation from the initial assumed model. As our model uncertainty increases, the initial impact and benefit that we have assumed from our conservation funding decreases. Our ability to decrease the probability of extinction declines, and in fact overall, our conservation resources have become less effective. Further, when our uncertainty in the relationship is high, the probability of extinction of an individual subpopulation under a scenario of no action increases.
Even if we considerably increase the resources available to the conservation of the Sumatran tiger in Kerinci Seblat National Park, we may fail to perform well under even marginal model uncertainty if we choose to manage all subpopulations in the area. Given that most conservation problems are limited by budget and biological understanding, the common strategy of managing all known subpopulations of a threatened species is unlikely to be the most robust. Indeed, this approach may only be tenable for species where significant research has been undertaken, probabilities of extinction are very low and response to management considerable, and where there is a huge amount of funding available to implement management. Such species are probably not threatened. We believe that under many scenarios of current funding and uncertainty, subpopulation triage may indeed be required to ensure we conserve threatened species from global extinction. By investigating the impact of model uncertainty on robust decision-making in conservation and how this is affected by available conservation funding, we show that triage may in fact be not only an important consideration in conservation but also the most robust approach for managers to take to reduce the risk of losing the species.
The aim of this study was to confront optimal decision-making for the management of threatened species with multiple subpopulations with uncertainty. McDonald-Madden et al. (2008) provide such a decision-making framework, showing how to allocate funding between the isolated subpopulations of a threatened species to get the greatest number of these subpopulations persisting through time. That paper assumes an analogous model of monetary investment and extinction risk but does not confront the known uncertainties in model structure and parameter estimates. In McDonald-Madden et al. (2008) and in this study, two key assumptions about the system are made: (i) all subpopulations are equivalent and thus their initial probability of extinction without management, P0, and their response to management, are the same; and (ii) money is allocated evenly between subpopulations when a decision is implemented. These two assumptions are coupled as proportional allocation of funding would only be justifiable if there where distinguishing factors of subpopulations to drive this allocation. The assumption of equality is made for simplicity, to enable exploration of an already complex problem. Our method could be expanded to incorporate the inequality of subpopulations in real-world scenarios, as long as these inequalities can be recognized and quantified by system managers.
This study focuses on decision-making that accepts uncertainty and limitations in funding, and builds a framework that enables robust decisions to be made in the light of these limitations. Uncertainty is not, however, always beyond our control. We can reduce uncertainty by diverting funding from management to collect data on our systems – an adaptive management approach. However, monitoring entails further costs that must be considered given the limited funding in conservation. Thus, the amount of resources to expend on monitoring becomes an additional decision to be made by managers – a decision we take as given in this study. We believe our work focuses attention on the need for optimal decision-making that incorporates learning to reduce our uncertainty. Our work also highlights the importance of considering funding limitations and cost in any decision-making framework for reducing uncertainty. We believe such an approach is a necessary advancement in conservation resource allocation.
This is the first work in conservation to assess the impact of model uncertainty based on the shape and structure of the models we assume within an info-gap framework. It allows us to assess our lack of understanding about biological systems on how we make decisions with respect to the conservation of our environment. It clearly shows the need for managers to consider a triage approach to threatened species management not as a process of giving up but as a tool for ensuring species persistence in the light of the urgency of most conservation requirements, the realities of limited biological understanding and the poor state of conservation funding.
We thank H. Yokomizo and O. Venter for helpful comments and input to the formulation of this work. Information on the Sumatran tiger was provided by M. Linkie, University of Kent, Canterbury. E.M.M.'s PhD Studentship was supported by the Invasive Animals Cooperative Research Centre within the Detection, Prevention and Mitigation Program and by a MYQRS scholarship from the University of Queensland. H.P.P and P.W.J.B. were supported by grants from the Australian Research Council (ARC) and a Commonwealth Environmental Research Facility (CERF) grant. We thank two anonymous reviewers for their comments on the paper.