Cost-effective surveillance of an invasive species: application of info-gap theory to the Asian House Gecko Hemidactylus frenatus

Invasive species can lead to community-level damage to the invaded ecosystem and extinction of 19 native species. Most surveillance systems for the detection of invasive species are developed 20 based on expert assessment, inherently coming with a level of uncertainty. In this research, info- 21 gap decision theory (IGDT) is applied to model and manage such uncertainty; surveillance of the 22 Asian House Gecko, Hemidactylus frenatus Duméril and Bibron, 1836 on Barrow Island, is used 23 as a case study. Our research provides a novel method for applying IGDT to determine the robust 24 population threshold ( K ) to trigger detection, robust-optimizing surveillance costs rather than 25 minimizing surveillance costs. We demonstrate that increasing the population threshold for 26 detection increases both robustness to the errors in the model parameter estimates, and 27 opportuneness to lower surveillance costs than the accepted maximum budget. This paper 28 provides guidance for decision makers to balance robustness and required surveillance 29 expenditure. IGDT offers a novel method to model and manage the uncertainty prevalent in 30 biodiversity conservation practices and modelling. The method outlined here can be used to 31 design robust surveillance systems for invasive species in a wider context, and to better tackle 32 uncertainty in protection of biodiversity and native species in a cost-effective manner. 33

Asian House Gecko, Hemidactylus frenatus Duméril and Bibron, 1836 on Barrow Island, is used 23 as a case study. Our research provides a novel method for applying IGDT to determine the robust 24 population threshold ( K ) to trigger detection, robust-optimizing surveillance costs rather than 25 minimizing surveillance costs. We demonstrate that increasing the population threshold for 26 detection increases both robustness to the errors in the model parameter estimates, and 27 opportuneness to lower surveillance costs than the accepted maximum budget. This paper 28 provides guidance for decision makers to balance robustness and required surveillance 29 expenditure. IGDT offers a novel method to model and manage the uncertainty prevalent in 30 biodiversity conservation practices and modelling. The method outlined here can be used to 31 design robust surveillance systems for invasive species in a wider context, and to better tackle 32 uncertainty in protection of biodiversity and native species in a cost-effective manner. 33

Introduction 34
Invasive species can contribute to the extinction of native species, decrease species diversity, and 35 cause considerable ecological and economic loss 1 . Following initial introduction, invasive 36 species have a higher likelihood of being eradicated when detected early with a subsequent rapid 37 response 2 . While early detection requires effective surveillance to detect small numbers of 38 individuals 3 , often with limited resources. 39 Info-gap decision theory (IGDT) is a non-probabilistic theory that has been used in a range of 40 areas 4 . Such robust decision making methods are often desirable in ecological systems 41 characterized by Knightian uncertainty 5 , without considering the probability or frequency of 42 outcomes. IGDT offers an alternative of Wald's maximin to quantify the confidence in realising 43 specified aspirations but enable a balance between them as a robust-satisfying method 6 . 44 Knightian uncertainty may exist in parameter estimates, regarded as parameter uncertainty. 45 IGDT has been used previously to manage invasive species (e.g. 7,8,9 ), however, this project is 46 the first application of IGDT in determining the population threshold ( K ) for detection of an 47 invasive species in order for it to be successfully eradicated. The population threshold ( K ) 48 represents a population number low enough to enable successful eradication without major 49 impact, but high enough that it can be detected without significant surveillance costs. Given that 50 the population threshold of a species is based on the risk tolerance of managers and the biology 51 of the species in question, it is difficult to provide actual values against this parameter. Despite 52 this, such management parameters are widely used (e.g. 10,11 ). Because of the direct relation 53 between surveillance cost and population size 12 , these population threshold estimates could have 54 significant cost consequences if underestimated. 55 in the model parameters with lack of data and other potential and subtle independent 78 influences 20 , and result in exceeding budget limits. 79 This research was conducted to determine the robust population threshold required to trigger 80 detection of invasive species at a point that will reduce the likelihood of environmental damage. 81 Our study provides a new method for designing robust surveillance, contributing to efficient 82 application of limited resources and reducing financial and environmental costs. 83

Data 85
The Surveillance System for terrestrial vertebrates on BWI has undergone a large number of 86 reviews (e.g. 8 ) since the commencement of the Gorgon Project. The reviews take into 87 consideration the operations being undertaken at the time. These include changes to the volume 88 of cargo, number of personnel travelling to BWI and the ports of origin. Our study is based on 89 revised input data from the most recent studies (e.g. 21 ). 90 Our model considers six primary or secondary points of entry (Supplementary Table S1, Fig. 1). 91 In addition, three preferred habitat types were the focus of island surveillance for AHG 92 (Supplementary Table S2), as opposed to surveying the entire island. AHG are rarely found in 93 natural areas in their introduced range and prefer anthropologically modified areas, particularly 94 where there is artificial lighting that attracts insect prey 22 . 95 For the purpose of this model, BWI has been spatially separated into four zones (See Fig. 1 Table S4). 101 The characters of these SSCs at different locations and zones are summarized in Supplementary 102 Table S5. The surveillance period designed for this model is one year since the probability of 103 AHG being detected changes due to seasonal elements within a year 24 . 104

Model methods 105
Info-gap analysis requires three primary components, each of which builds on the last; a system 106 model, the performance requirement and the uncertainty model 4 . Following Jarrad et al. 19 and 107 Whittle et al. 24 , the quantity of SSC i to be used in each zone j , j i N ( 0 j i N  ), could be 108 calculated as equation (1): Where i is the type of SSC (Supplementary Table S4); j is the risk zone; j R indicates the 111 probability of occupancy in risk zone j ;  is the probability of a type 2 error (the probability of 112 falsely declaring the invasive species to be absent); j i  is the detection probability of SSC i in 113 zone j given invasive species present in the footprint; j i F is the sampling fraction covered by 114 one SSC unit of i in zone j (i.e. footprint/total target area); K is the population threshold for 115 detection; i C is the cost per unit SSC i . 116 7 Therefore, the system model in this research, i.e. the total surveillance cost used at all locations, 117 the total area of zone j (see Supplementary Section S1 for details).  Table S5 R for clarity or simplicity. We have initial parameter estimates 135 denoted by the known 40-vector x . The only additional information available is that the 8 parameters are all non-negative, and all parameters other than K are bounded above by one. We 137 have no further information. For instance, probability distributions for uncertainty in these 138 parameters are lacking. In face of this severe uncertainty we will use fractional-error info-gap 139 model of uncertainty (equation (2)). Analogous to deviations around the mean in Bayesian 140 approach, info-gap uncertainty model (equation (2)) measures the fractional deviation between 141 the parameters and the estimates, but without probabilities assigned 6 . 142 Robustness can be mathematically explained as the greatest horizon of uncertainty up to which 144 the system model obeys the performance requirement,

9
The predicted surveillance cost based on the initial estimates of uncertainty parameters ,, FR  156 decreases hyperbolically with an increasing AHG population threshold (Fig. 2A). The estimated 157 surveillance cost with the initial estimate for ( 8) KK= and ,, j j j ii FR  is AU$10,060, which 158 seems to be a relatively small cost. However, these initial estimates are subjectively based on the 159 information experts had available at the time, and may not be necessarily accurate. To solve this 160 problem, the robustness is taken into consideration. 161 Fig. 2A illustrates that the surveillance cost is very large when K is four or less, but the curve 162 flattens out when it is between six and twenty. This indicates that a K value of between six and 163 twenty would be more economically sustainable. We therefore evaluated the robustness and 164 opportuneness using K estimate values of 6, 8, 10 and 20. 165 Robustness curve when , , , are uncertain. 166 The step-wise continuity of these robustness curves is a result of the discrete values attainable by 167 N and K , as expressed by the "ceil" function in Supplementary equation (S12). There are two 168 properties that should be noted, including zeroing and the trade-off property 4 . 169 The zeroing property affirms that predicted outcomes have zero robustness to uncertainty. Each 170 curve reaches the horizontal axis (for which robustness is zero) precisely when the required 171 performance equals the predicted value (Fig. 2B). For instance, consider the black curve, 172 calculated with an estimated surveillance cost of AU$10,060, for which the robustness equals 173 zero (Fig. 2B). The trade-off property asserts that the robustness improves (increases) as the 174 performance requirement worsens (increases). This is reflected in the robustness curves of Comparing four robustness curves in Fig. 2B, even though the minimized cost on curve ' 20 K = 178 ' is the lowest one, there is no guarantee that ' 20 K = ' is the best choice for the underlying 179 inaccuracy in the parameter estimates. Curve ' 20 K = ' is robust-preferred to the other three 180 curves at the same value of critical budget. For example, the largest error in the estimation of 181 parameters (i.e. robustness), ± 51% (0.51) would guarantee the total surveillance cost to be no 182 more than required AU$40,000 when using ' 20 K = '. 183 Opportuneness curve when , , , are uncertain. 184 Robustness and opportuneness curves converge on the horizontal axis at the point reaching the 185 predicted surveillance cost, i.e. zeroing property (Fig. 2B). Zeroing for opportuneness means that 186 no uncertainty is needed in order to enable the expected outcome. Opportuneness decreases 187 along with the required surveillance cost. Opportuneness shows the immunity to windfall 188 success, thus the smaller it is the better. Opportuneness curve ' 20 K = ' is lower than the other 189 three curves (Fig. 2B) and is thus preferred. We see that robustness and opportuneness are 190 synchronous in this example: Any change in K that improves robustness, also improves 191 opportuneness. 192

Discussion 193
Ecological systems, and the species within, are highly complex and variable. This complexity 194 makes it difficult for model input parameter values to be derived from empirical data or for 195 uncertainty in these parameters to be parameterized probabilistically. This results in Knightian 196 uncertainty underlying ecological analysis and consequently may skew policy outcomes. For 197 example, in our study, in a scenario in which the performance requirement is no more than a 198 critical budget value of AU$40,000 (Fig. 2B), if any of the parameter estimates (when K is 199 eight) err by more than ± 37%, the total surveillance cost will be considerably over AU$40,000. 200 To avoid the burden of excessive surveillance costs, it is critical to have a robust approach to 201 prevalent uncertainty, such as that outlined in this research. 202 Our research shows that the estimated cost for surveillance for a very small population threshold 203 (less than four) of AHGs is large ( Fig. 2A). This is due to the fact that low population densities 204 reduce the probability of detection success, consequently for detection to occur, more resources 205 will be required 12 . An example of this is the annual trapping cost for gypsy moth in North 206 America, which Bogich et al. 26 showed there was a linear function of trap density and total area, 207 with more traps required when the population size is small 19 . Counter to this, robustness 208 increases as size of the budget becomes larger (Fig. 2B). 209 Importantly, an increase in robustness allows for more mistakes in the parameter estimates to be 210 tolerated when making decisions, which is beneficial in pest management scenarios. One way to 211 improve robustness is by increasing the size of the budget (Fig. 2B). The other is by increasing 212 population threshold within the possible population threshold range, which simultaneously 213 improves opportuneness. The robustness gained with increasing the population threshold from 214 six to ten is similar to increasing the population threshold from ten to twenty at the same level of 215 required performance (Fig. 2B). Unrestrained increase in population threshold is not 216 recommended, as it could result in delay in detection, and subsequent failure to eradicate. How 217 big the population threshold is, is a matter of risk appetite. For robustness, 'bigger is better', 218 however, 'big is bad' for opportuneness 4 . A lower value associated with the opportuneness 219 function means less uncertainty has to be tolerated, i.e. higher opportunity to seek lower 220 surveillance cost. In the case of AHG, the scenario in which ' 20 K = ' (Fig. 2B) is not only 221 robust-dominant but is also opportuneness-dominant at the same level of required surveillance 222 cost, thus would be preferred by many decision makers. Davidovitch et al. 8 pointed out that 223 decision makers are prone to being more risk averse rather than seeking windfalls. 224 Our methodology demonstrates how robustness/opportuneness curves may be applied to 225 determine if the size of the budget is 'sufficiently robust/opportune'. Determining a suitable 226 value of robustness/opportuneness is dependent upon the trade-offs deemed appropriate by 227 decision makers. If the decision-makers aim to reach ambitious goals (i.e. low critical 228 surveillance cost), they may need to compromise on the level of confidence in realizing them 6 . 229 Conversely, if decision-makers ideally have high confidence in realizing specified targets, then 230 the targets could be moderated by ambitiousness (i.e. they may need to improve the critical 231 surveillance cost). Such compromises in pest management requirements could be compensated 232 by achieving greater robustness against error in the parameter estimates. 233 The IGDT-based method explored in this research does not optimize (minimize) the cost spent 234 on surveillance but concentrates on identifying the robust population threshold against the 235 uncertainty existing in , , , K F R  for implementing eradication of a known pest, while 236 simultaneously satisfying a cost requirement. The method can easily be tailored to specific 237 requirements associated with varying degrees of uncertainty. As uncertainty parameters may not 238 be restricted to the parameters analysed in this research, more uncertainty parameters could be 239 included, e.g. the surveillance cost-effectiveness, population growth rate, discount rate that may Quarantine invasion risk map of the Asian House Gecko on Barrow Island. Zone 1 is the area with the highest occupancy probability, where the majority of the surveillance budget should be spent. This zone consists of areas where high risk activities for incursion are undertaken. These activities include the unloading of cargo (laydown areas), the receipt of planes (airports) and marine vessels (marine ports).
Zone 2 is the secondary introduction area (100 m buffer area around Zone 1) and is considered the lower risk boundary for a species dispersing out of Zone 1. Zone 0 is the buffer area at Material O oading Facility (MOF) (i.e. X-Blocs area). This is because the habitat at Zone 2 at the MOF is highly different to that at all other locations, making the characters of Surveillance System Components (SSCs) in Zone 2 at the MOF (i.e. X Blocs) different from that in Zone 2 at other locations. Zone 3 is the remaining island area where the AHG is less likely to establish prior to detection thus with no SSCs allocated