1. Invasive species threaten biodiversity, and their eradication is desirable whenever possible. Deciding whether an invasive species has been successfully eradicated is difficult because of imperfect detection. Two previous studies [Regan et al., Ecology Letters, 9 (2006), 759; Rout et al., Journal of Applied Ecology, 46 (2009), 110] have used a decision theory framework to minimize the total expected cost by finding the number of consecutive surveys without detection (absent surveys) after which a species should be declared eradicated. These two studies used different methods to calculate the probability that the invasive species is present when it has not been detected for a number of surveys. However, neither acknowledged uncertainty in this probability, which can lead to suboptimal solutions.
2. We use info-gap theory to examine the effect of uncertainty in the probability of presence on decision-making. Instead of optimizing performance for an assumed system model, info-gap theory finds the decision among the alternatives considered that is most robust to model uncertainty while meeting a set performance requirement. This is the first application of info-gap theory to invasive species management.
3. We find the number of absent surveys after which eradication should be declared to be relatively robust to uncertainty in the probability of presence. This solution depends on the nominal estimate of the probability of presence, the performance requirement and the cost of surveying, but not the cost of falsely declaring eradication.
4. More generally, to be robust to uncertainty in the probability of presence, managers should conduct at least as many surveys as the number that minimizes the total expected cost. This holds for any nominal model of the probability of presence.
5.Synthesis and applications. Uncertainty is pervasive in ecology and conservation biology. It is therefore crucial to consider its impact on decision-making; info-gap theory provides a way to do this. We find a simple expression for the info-gap solution, which could be applied by eradication managers to make decisions that are robust to uncertainty in the probability of presence.
Invasive species are one of the main threats to the world’s biodiversity (Primack 2006). In numerous parts of the world, exotic predators have contributed to declines of native species, and weed invasion threatens native plant species and communities (Hunter 2003; Primack 2006). Control and eradication of exotic species is often difficult once they have established, so eradication of newly invaded species is desirable whenever possible.
It is difficult to be sure that a species has been eradicated because detection is imperfect, especially for small populations (Usher 1989; Reed 1996; Regan et al. 2006; Morrison et al. 2007). How much should we spend surveying for an invasive species before declaring it eradicated? If we mistakenly declare a species eradicated and stop surveying, its population may grow undetected, incurring large economic and environmental costs. On the other hand, continuing surveys after a species has been eradicated involves unnecessary survey costs. Regan et al. (2006) balanced these opposing risks by stopping surveys when the total expected cost is minimized.
Decisions that optimize an expected outcome may not be robust to uncertainty (Ben-Haim 2006). These decisions depend critically on an accurate system model, but ecological models are often very uncertain. Regan et al. (2006) decision-making framework involves calculating the probability that the invasive species is present after several surveys without detection, using a two-parameter system model. The parameters are (i) the probability that the species persists at a site from year to year and (ii) the probability of detecting the species when it is present. Both of these are likely to be uncertain. Sometimes these parameters can be estimated probabilistically (e.g. Wintle et al. 2004), but often bounds on their values are the best that can be achieved (Regan et al. 2006).
As well as being uncertain about model parameters, we may be uncertain about model structure. For example, how does the probability of presence decline with increasing numbers of surveys without detection? Rout, Salomon, & McCarthy (2009) explored an alternative method for calculating the probability of presence, which used only the sighting record of the species. They found that using a different method changed the optimal number of surveys without detection before declaring eradication (Rout et al. 2009). The probability of presence could potentially be calculated in many ways, with different assumed system models.
Regan et al. (2006) found the optimal time to declare eradication of an invasive species, based on the number of consecutive surveys in which the species is not found (hereafter referred to as ‘absent surveys’). The net expected cost (NEC) of declaring eradication after d absent surveys is:
where Cs is the cost of a survey, Ce is the expected cost of escape and damage (the expected cost of falsely declaring eradication) and p(d) is the probability that the species is still present after d absent surveys. To calculate this probability, Regan et al. (2006) use the model:
( eqn 2)
where q is the annual probability of detection of the invasive species and h is the probability the species persists from year to year. They then minimize eqn 1 to find the optimal number of consecutive absent surveys after which monitoring should stop and eradication should be declared.
Within this framework, the probability of presence p(d) is subject to considerable uncertainty and ignoring this uncertainty may lead to suboptimal solutions. Instead of minimizing NEC(d), the info-gap approach imposes the performance requirement:
NEC(d) ≤ Nc.
This means the NEC must not exceed some maximum acceptable limit Nc. Info-gap aims to find the value of d that permits p(d) to depart as much as possible from its nominal value while satisfying this constraint. The nominal value is our ‘best guess’ for p(d), and we calculate this using the model from Regan et al. (2006) (eqn 2).
We express uncertainty surrounding p(d) with the info-gap model:
( eqn 3)
where is the nominal model for p(d) and α is the horizon of uncertainty. This means that, for a particular value of α, we explore a possible range for p(d) between and , with the additional constraint 0 ≤ p(d) ≤ 1 required for probabilities. When α is zero, this info-gap model for p(d) reduces to the nominal model .
The robustness of the model is the largest α for which the performance constraint is still met. For this info-gap model, the robustness is:
The maximum NEC given uncertainty in p(d) is (from eqns 1 and 3):
It can be seen in this equation that the NEC will increase as α increases, which means the maximum α occurs at the highest possible NEC within the performance constraint, i.e. where NEC = Nc. Therefore,
This equation gives us the relationship between the robustness and the performance constraint. We can then find the robust–optimal solution , which is the value of d where the robustness is maximized.
Our example case study is the invasive weed Helenium amarum (Raf.) H. Rock, which was declared eradicated from a small site in Queensland Australia in 1992, after 5 years without detection (Tomley & Panetta 2002). This eradication programme was unsuccessful – the species was found at the site in March 2007, and survey and control activities are currently underway (Csurhes & Zhou 2008). The data we use are the same as Regan et al. (2006), gathered before this recent re-emergence. We also generalize our results by finding solutions for arbitrary functions for the nominal probability .
With eqn 2 as the nominal model for p(d), the robust–optimal solution is (see Appendix S1 for details of the solution):
where r = (1–q)h. Superficially, this bears no resemblance to the optimal solution d* [treating the estimated value of p(d) as true] obtained by Regan et al. (2006):
Most notably, eqn 5 does not depend on the expected cost of escape Ce, which was a critical parameter in the optimal solution (eqn 6). However, a close correspondence can be seen between the optimal and robust–optimal solutions by noting that the latter is predicated on the assumption that the horizon of uncertainty α is non-negative, which leads to the constraint:
The right-hand side of this inequality is the optimal solution d* (eqn 6). The optimal solution therefore provides a lower bound on the robust–optimal solution.
For H. amarum, the parameter r is 0·136 and the cost ratio Cs/Ce is 1/354·55. The optimal solution for H. amarum (eqn 6) is to stop surveying after 3·29 absent surveys. This occurs at a total cost (relative to the cost of surveying, NEC/Cs) of 2·79. This optimal solution therefore places a lower bound on the optimal–robust solution and a lower limit on the ratio Nc/Cs.
As the relative cost constraint Nc/Cs is increased, the robust–optimal solution is to survey for longer (Fig. 1) with a corresponding increase in the robustness of the decision (Fig. 2). The robustness is the proportional error in p(d) permissible while satisfying the cost constraint. This necessitates a trade-off between performance (in the form of cost minimization) and robustness of decisions, which is a general characteristic of info-gap analysis (Regan et al. 2005). The optimal solution has the lowest possible expected cost (Fig. 1), but also has zero robustness to uncertainty (Fig. 2).
In this example, r is sufficiently small that eqn 5 gives (as 1/ln r ≈ −0·5 is small enough to be ignored), making the robust–optimal expected total cost of monitoring after the last observation of the invasive species () approximately equal to the total cost that is deemed to be satisfactory (Nc). This is an intuitive result: when an invasive species is not observed we should keep surveying until the total cost of the surveys becomes unsatisfactorily large.
A general result
Equation 5 is the robust–optimal solution if we use the model from Regan et al. (2006) as our nominal model . However, there are other models that could be used for . It is therefore useful to make some general observations regarding the relationship between the robust–optimal solution for a particular nominal model, and the solution optimizing without uncertainty (optimal solution d*).
If we differentiate eqn 4 implicitly with respect to d, we obtain:
The robust–optimal solution occurs when is maximized, where by definition . Therefore, at eqn 7 becomes:
We know that is a decreasing function: the more surveys without detection, the lower the probability of presence. Also, for a minimum (rather than maximum) expected cost to exist, must be a convex function, that is, the rate at which it decreases also decreases (). Therefore, eqn 11 implies
that is, the robust–optimal solution is greater than or equal to the optimal solution, for any plausible (convex) nominal model , whether it exists now or is developed in the future. If there is any uncertainty, the robust–optimal solution will be greater than the optimal solution (as in our case study, Fig. 1). They are only equal when there is no uncertainty (α = 0).
If managers are confident that the model analysed by Regan et al. (2006) accurately predicts the probability of presence, then they should use the solution of Regan et al. to minimize the total expected cost of declaring eradication. However, if they are uncertain about this model and wish to minimize the chance of unacceptably large costs, they can calculate the robust–optimal number of surveys with eqn 5. If another model represents the system more accurately than the model used by Regan et al., it could be substituted as the nominal model in a similar info-gap analysis. For any nominal model, to be robust to uncertainty managers should conduct at least as many surveys as the optimal calculated with the nominal model.
Info-gap theory has been advocated for use in ecology and conservation biology because it provides a framework to examine the sensitivity of a model’s results to assumptions, explicitly acknowledging uncertainty (Regan et al. 2005; Halpern et al. 2006). For ecological management in the face of uncertainty, managers may use info-gap to gain some protection against catastrophic outcomes by answering the question: how wrong could this model be before outcomes are unacceptably bad?
Although info-gap theory is relevant for many management problems, two components must be carefully selected: the nominal estimate of the uncertain parameter, and the model of uncertainty in that parameter. If the nominal estimate is radically different from the unknown true parameter value, then the horizon of uncertainty around the nominal estimate may not encompass the true value, even at low performance requirements. Thus, the method challenges us to question our belief in the nominal estimate, so that we evaluate whether differences within the horizon of uncertainty are ‘plausible’. Our uncertainty should not be so severe that a reasonable nominal estimate cannot be selected.
Similarly, the uncertainty model must be chosen carefully. It must be asked: precisely which components of the model are uncertain, and how is this uncertainty structured?
For example, in our problem we chose to model uncertainty in the probability of presence. We could have modelled uncertainty in both the parameters used to calculate the probability of presence (the detectability and annual probability of persistence of the species). We found that if uncertainty was included in these more fundamental parameters, info-gap proposed the same robust–optimal solutions for low cost thresholds (Nc/Cs < 10), but that the results diverged when the cost threshold was larger (see Appendix S2). The results of an info-gap analysis can therefore differ depending on the level at which the uncertainty is modelled.
The structure of uncertainty assumed by managers is also crucial. In our uncertainty model (eqn 3), we assumed that uncertainty is proportional to the probability of presence, implying that we are more uncertain about high probabilities than low probabilities. This uncertainty could plausibly be modelled in other ways. We tested a model where uncertainty is proportional to one minus the probability of presence, implying that we are more uncertain about low probabilities than high probabilities. We found once again that the robust–optimal solutions differed from those of our original model (see Appendix S2). The particular choice of uncertainty model greatly affects the robust–optimal solution, so efforts must be made to choose a form that is sensible, and appropriate to the system being examined. The structure of uncertainty can be quite an abstract concept, so this could prove difficult in many cases. However, this does not diminish the relevance or applicability of info-gap – it simply means that it is not a standard formula to be mindlessly applied, but must be carefully tailored to each problem.
The probability that the invasive species is present may not be the only parameter in this decision-making framework subject to significant uncertainty. The cost of escape describes the possible future outcome of declaring eradication when the invasive species is still present. This cost could also be uncertain, as the example of H. amarum illustrates. Regan et al. (2006) estimated the cost of escape as the impact to the dairy industry on H. amarum becoming widespread. As it turns out, H. amarum was declared eradicated when still present, but was rediscovered before escaping the site of infestation (Csurhes & Zhou 2008). The consequences of declaring eradication prematurely were therefore not as dire as predicted. In info-gap theory the same α (horizon of uncertainty) can be used to describe uncertainty in multiple parameters. Info-gap theory could therefore be used to address uncertainty in both the probability of presence and the cost of escape simultaneously. However, the uncertainty model must be chosen carefully so that the uncertainty (a single variable) is scaled appropriately for each parameter. This would be particularly important in this case, as the two parameters are measured on very different scales.
Regan et al. (2006) and subsequently Rout et al. (2009) explored a systematic approach to declaring eradication of invasive species. Both studies addressed uncertainty in the parameters of their models through sensitivity analysis. While sensitivity analyses explore how optimal decisions are altered by changes in parameter values, they often do not quantify the corresponding change in performance. For example, the optimal decision under one possible parameter value might be twice as costly as the optimal decision under another possible value. An info-gap analysis is fundamentally different from a sensitivity analysis as it finds the decision that enables maximum uncertainty while still meeting a minimum performance requirement. The focus of the whole optimization is shifted from maximizing performance to maximizing permissible uncertainty in the parameters. Info-gap also allows decision makers to view the trade-off between minimum acceptable performance and the robustness of decisions. Uncertainty is pervasive in decision-making in ecology and conservation biology, so addressing it explicitly helps find robust decisions that avoid catastrophic outcomes.
Many thanks to Yakov Ben-Haim and Dane Panetta for helpful advice and discussions. Also thanks to Michael Bode, Mark Burgman, Peter Baxter and three anonymous reviewers for comments on this manuscript. This research was supported by an Australian Postgraduate Award, the Commonwealth Environment Research Facility (AEDA), the Australian Centre of Excellence for Risk Analysis and an Australian Research Council Linkage Grant to MMcC (LP0884052).