Allocating surveillance effort in the management of invasive species: A spatially-explicit model

Allocating surveillance effort in the management of invasive species: A spatially-explicit model

Environmental Modelling & Software 25 (2010) 444–454 Contents lists available at ScienceDirect Environmental Modelling & Software journal homepage: ...

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Environmental Modelling & Software 25 (2010) 444–454

Contents lists available at ScienceDirect

Environmental Modelling & Software journal homepage: www.elsevier.com/locate/envsoft

Allocating surveillance effort in the management of invasive species: A spatially-explicit model Oscar J. Cacho a, *, Daniel Spring b, Susan Hester a, Ralph Mac Nally b a b

School of Business, Economics and Public Policy, University of New England, Armidale NSW 2351, Australia Australian Centre for Biodiversity, School of Biological Sciences, PO Box 18, Monash University, Clayton, Victoria 3800, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 January 2009 Received in revised form 25 August 2009 Accepted 31 October 2009 Available online 1 December 2009

Invasive organisms often exist at low densities at the beginning and end of eradication programs. As a consequence, such organisms are often difficult to find, particularly if they are dispersed long distances to unknown locations. In such circumstances, large amounts of money can be spent searching for invasive organisms without finding any. However, chance encounters between invasive organisms and private citizens can occur even when invasive organisms exist at low densities. Reports of these ‘passive detections’ may be a critically important source of information for public pest management agencies. Rates of reporting may be improved using bounty payments and increasing public awareness about the presence of the invader. To explore the importance of passive surveillance in general, and its interaction with active surveillance by pest management agencies, we developed a simulation model of the spread of an invasive species. Simulations conducted under alternative scenarios for detection rates and search effort applied demonstrate that even small increases in detection or reporting rates substantially reduced eradication costs and increased the probability of eradication. In circumstances where resources are insufficient to achieve eradication, the simulation model provides useful information on the minimum expenditure required to contain the invasion. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: Eradication Search theory Passive surveillance Economics Detection

1. Introduction Invasive species can cause significant damage to natural environments, agricultural systems, human populations and the economy as a whole (Liebhold et al., 1995; Liebman et al., 2001; Pimentel et al., 2005; Sinden et al., 2005), and they are recognised as an important threat to global biodiversity (Vitousek et al., 1996). Biological invasions are complex stochastic, dynamic systems with many sources of uncertainty and generally exhibit strong geographical variation. Modelling the spread of invaders can assist in mitigating the impacts of biological invasions by allowing us to identify strategies that are likely to be most effective in slowing or reversing their spread. Peck (2004) argues that simulation modelling is the most effective way to study complex systems such as this, by allowing us to perform a large number of computer experiments to test hypotheses and improve our understanding of the system. Early detection of a new invasive population is vitally important if eradication is to be a realistic goal (Rejmanek and Pitcairn, 2002).

* Corresponding author. Tel.: þ61 (0)2 6773 3215; fax: þ61 (0)2 6773 3596. E-mail address: [email protected] (O.J. Cacho). 1364-8152/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.envsoft.2009.10.014

Once detection occurs, knowledge about the full extent of the invasion is a fundamental prerequisite for successful eradication (Panetta and Lawes, 2005). In most non-agricultural situations, the main constraint to eliminating invaders is not killing them but finding them (Cacho et al., 2006). Once an invasion is found it can usually be successfully treated and destroyed. Nevertheless, early detection and accurate delimitation of the invasion can be difficult to achieve because invasive organisms often exist at low densities at the beginning of eradication programs. Individual organisms can be difficult to find, particularly if some organisms are dispersed long distances to unknown locations, and as a result large amounts of public money can be spent searching for invasive organisms without finding any. As an option to avoid spending large amounts of money searching for invaded sites over a large area, pest management agencies use ‘passive surveillance’; reports from members of the public of encounters with pests (MAFBNZ, 2008), to assist in surveillance and control. Passive detections have often been the method by which an invader is first recognised in a country or region. For example, the initial detection of the European wasp (Vespula germanica) in Western Australia occurred following a private submission of a wasp for identification (Davis and Wilson,

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1991). This initial report subsequently led to the discovery of five nests and an eradication campaign. Another example of the public’s role in early detection is the initial discovery of the red imported fire ant (Solenopsis invicta) in Australia. The presence of this invasion in Brisbane was confirmed following submission of samples of the ant by members of the public, leading to the establishment of the Red Imported Fire Ant Eradication Program (Jennings, 2004). Passive surveillance during eradication programs has also proven to be very important in the success of such programs. Records from the campaign to eradicate the European wasp from Western Australia show that the public are responsible for finding 90% of the infestations in new areas (Davis and Wilson, 1991). Reporting of fire ants by the public has resulted in detections of half the outlying populations of the ant (Jennings, 2004). Australia’s Four Tropical Weeds Eradication Program obtained information for more than a quarter of the locations of weeds in the program due to detection by members of the public (Brooks and Galway, 2008). For organisms that remain at one location for an extended period, reports of their location can assist pest management agencies in finding and removing them. Such reports can result from chance encounters with invasive organisms or active search by pest management agencies. Both sources of detections – that is, from passive detection and active search – are likely to result in a higher probability of eradication and a shorter duration of eradication programs in cases where pest populations are sufficiently small to be eradicable. Here we use a spatially-explicit model to simulate the spread of an invasive population in a heterogeneous landscape and study the interactions between active and passive surveillance. Models of spatial spread of invaders have a long history (With, 2002). The most common approaches in continuous time and continuous space consist of diffusion models based on partial differential equations (Gilbert et al., 2004). The most common approaches in discrete time and continuous space are based on integrodifference models that treat dispersal and population growth as separate stages (Kot et al., 1996). Extensions that consider population structure (life stages) in addition to growth have also been reported in the literature (Neubert and Caswell, 2000). At the centre of most spread models is a dispersal kernel, k(i,j), showing the probability that a propagule originating in point i will land in point j based on the distance between i and j. The main use of these models has been to predict the speed of the invasion front (Hastings et al., 2005). More detailed spread models that account for spatial heterogeneity in the landscape are usually discrete in both time and space; with space being represented as a grid (lattice) of square cells. Recent studies that use grid-based models to improve the management of invasions include Hyder et al. (2008) and Pitt et al. (2009). The model we develop in this paper is grid-based and accounts for habitat heterogeneity. To our knowledge this is the first spatially-explicit model that incorporates search theory and considers passive surveillance. In addition to providing policymakers with a tool to assist in managing new invasions, our analysis provides insights into the effective combination of passive and active surveillance as a means of improving the control of invasive species in general. 2. Method 2.1. Model description The landscape is represented as a grid of nr rows and nc columns containing n ¼ nr  nc square cells of area a. Variables associated with this grid are represented as column vectors of dimension n with corresponding grid cells identified by index i ¼ 1,.,n; numbered down the rows and then across columns. This arrangement simplifies numerical calculations and allows cells that do not belong to the map to be

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excluded from calculations (i.e. it accommodates non-rectangular maps). This arrangement also reduces computer memory requirements to run the model. These vectors are converted into matrices of dimension nr  nc for display as maps. The state of cell i is given by its invasion status, represented by the binary variable xi, (1 ¼ presence, 0 ¼ absence). The state of the invasion is contained in column vector x, with elements xi. Cells have the following attributes: Habitat suitability (ai), contained in column vector a, represents the probability that a propagule landing on a cell will become established (0  ai  1), thereby changing the state (xi) of uninvaded cells from 0 to 1. Detectability (li), contained in column vector l, is the effective sweep width (see below) measured in m from the search path. It depends on target characteristics, environmental conditions and searcher ability. Search speed (si), contained in column vector s, is the speed (in m h1) at which the cell can be traversed following standard search procedures. It depends on characteristics of the environment, such as slope and roughness. Ownership type (oi), contained in column vector o is a binary variable (1 ¼ private, 0 ¼ public).

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Landscape attributes can be read from GIS raster map layers if available, but in the example below we use a hypothetical landscape generated by a fractal algorithm rather than actual maps. 2.1.1. Dispersal An invasion can be introduced in random locations or seeded on selected cells. An invaded cell produces w propagules per time period, and these propagules spread to neighbouring cells. The distance between cells (dij) determines the proportion of propagules from cell i that reach cell j according to a dispersal kernel. A Cauchy kernel was assumed, where the probability that a propagule originating in cell i and moving in direction q will land on cell j is given by:

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kij ðqÞ ¼ PD

l ¼ 0 dil

(1)

where D is the maximum dispersal distance and

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1 dij pffiffia2

(2)

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where g is a dispersal parameter. For convenience in the mapping process dij is measured in terms of number of cells and converted to metres in (2) based on the area (a) of a cell. A Cauchy kernel was selected because it is easy to compute and has heavy tails that capture long-distance spread (Kot et al., 1996). Any kernel can be applied in the model simply by changing a function call in the code. The dispersal kernel assumed here is presented in Fig. 1A. Upon model initialisation, an adjacency matrix (A) of dimensions n  n is created, whose element Aij represents the probability that a propagule originating in cell i will land on cell j. Each row Ai is created by identifying the cells that are located within a circle of radius D from i and applying the kernel (1) to each cell in this set. The resulting P values are then normalised so that Aij ¼ 1. Fig. 1B shows a graphical representation of a row of A rearranged for mappingj in two dimensions. The original cell (i ¼ 12,340) is mapped to pixel (52,96) in the figure. The probability values within the coloured circle add up to 1.0. The probability that a given site will be invaded depends on both the number of propagules landing on it and its habitat suitability. In matrix notation: p ¼ 1  expða0 +nÞ 0

n ¼ ðx wÞA

(3) (4)

The invasion probability vector p contains a probability map that incorporates the combined effect of invaded sites (x) and habitat suitability (a). To represent dispersal, p is compared to a vector of uniform random numbers r (0  ri < 1) and the new state of each cell is set according to the rule: xi ¼ 1;

if ri  pi

(5)

otherwise xi remains in its current state. Long-distance dispersal can occur with probability pL independently of the dispersal kernel, as may occur when propagules are transported by road, water, or other means. This is represented by drawing a vector of uniform random numbers v (0  vi < 1), with the same number of elements as the number of invaded sites. A long-distance jump occurs for each case where: vi  p L

(6)

The destination of these jumps is selected randomly within x. Stochastic dispersal simulation through time is executed by applying (3) to (6) iteratively, resulting in an invasion state trajectory xt.

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Distance from epicenter (cells) Fig. 1. (A) The dispersal kernel showing the probability that a propagule originating in cell 1 and travelling in a given direction will land on other cells. (B) Dispersal probabilities contained in the adjacency matrix for an arbitrary point on the map; darker pixels represent higher probabilities, the probabilities in the shaded area sum to 1.0. 2.1.2. Surveillance We consider two broad forms of surveillance: active and passive (Table 1). The latter refers to surveillance by members of the public and the former refers to surveillance conducted by the agency in charge of controlling the invasion. Active surveillance includes three types of search, all of which involve applying a prescribed amount of search effort (m) per cell. We distinguish between search types for operational reasons. The total amount of effort allocated to supplementary search per time period (M) is fixed in advance, whereas the amount of effort allocated to repeat search and follow-up search is not. While the agency controls the radius of the buffer area searched around detections and the number of repeat visits to sites that have been previously treated, it has no control over the number of detections that occur through time -this is a stochastic variable, and therefore effort required to respond to new detections is not known in advance. Note that supplementary search represents ‘new’ sites surveyed, not associated with reports by the public or previous finds. We use the term active search throughout the paper to refer to the three types of search as a group. We use passive surveillance as a general term that includes any detections reported by the public. We recognise that passive surveillance may have an active component when monetary incentives are introduced and members of the public go out of their way to find a pest, so we use the term passive in a loose sense. A passive detection may not be reported but for simplicity here we assume all detections are reported. When a detection is made by a private citizen, an intensive search is conducted in all parcels within a specific radius (rm) of the detection site. This is a form of adaptive cluster sampling which is applied in searches for rare organisms (Philippi, 2005; Smith et al., 2003; Thompson, 1990). An invaded cell can be detected through passive surveillance with probability (qi) or through active surveillance with probability (zi). In passive surveillance the public detects an invader and reports it to the relevant agency. The probability of passive detection (q) depends on the ownership attributes of cells; for any cell i: qi ¼ pp ðoi Þ þ pu ð1  oi Þ

(7)

where pp and pu are passive detection probabilities in private, and public, land, respectively. Each year, search effort is invested in the following activities: (i) searching sites where treatment has occurred in the recent past (repeat search); (ii) in response to reports from the public (follow-up search); and (iii) through independent surveillance in public land not previously searched during (i) or (ii) (supplementary search). The order of priority of these activities can be controlled by the user; in this application we apply them in the order given above. The probability of detection through any type of search (follow-up, repeat or supplementary) is calculated based on search theory (Cacho et al., 2006, 2007). The probability that an invasion in cell i will be detected depends on the search effort applied mi (h per cell), the speed of search si (m h1), effective sweep width li (m) and the area of the cell a (m2):    sml zi ¼ 1  exp  i i i a

(8)

The expression within the inner brackets in (8) measures coverage: the numerator is the area effectively searched (m2) as the product of distance traversed (si  mi) times effective sweep width (li); the denominator is the area of the cell (m2). A plot of Eq. (8) is presented in Fig. 2A (see Cacho et al., 2006 for details). Effective sweep width (li) measures the detectability of the target. It is derived from a lateral range curve which shows the probability that a target will be detected as a function of its lateral distance from the searcher (Fig. 2B). The efficiency of search per unit of distance covered is given by the area under the curve. The value of l is computed by constructing a rectangular box of height 1 and lateral range equal

to that at which the number of missed detections (b) within this range equals the number of detections (a) outside the range (Fig. 2B). We could replace area b with area a and have the same number of total detections, thus a standard rectangle can characterise detectability for a given search method applied in a given environment. Effective sweep width is the width of the box in Fig. 2B. We assume that, to encourage passive detections, the public is offered a bounty payment (CB) for each detection reported to the relevant agency. The total cost of the operation is given by the number of reports by the public, the amount of surveillance undertaken by the agency and the cost of treatment. Total cost in terms of present value is:

C ¼

22 3 3 t 1 X X   44 Npt CB þ Nat mCm þ ATt CT þ Nas Cm 5ð1 þ rÞt 5 t

(9)

s ¼ tSR

where Npt is the number of cells where a passive find is reported in year t; Nat is the number of cells where search takes place; AT is the area treated as a result of the three types of search (repeat, follow-up and supplementary); r is the discount rate; and CB, Cm and CT are the bounty payment ($ per report), the cost of searching ($ ha1) and the cost of treatment ($ ha1), respectively. The second summation term in (9) represents the cost of repeat searches required as a result of detections in the previous SR years. 2.2. Numerical model In order to test the model and design decision tools it is useful to have a ‘world’ with realistic attributes but which can be manipulated to represent any landscape pattern of interest. In our model the world is defined in terms of the four maps described earlier: habitat suitability, detectability, search speed and ownership type. To create a world, these attributes can be allocated using a fractal algorithm that generates random worlds which possess realistic landscape structure. The midpoint-displacement method (Saupe, 1988), a simple and popular algorithm used for this purpose (i.e. Tyre et al., 1999), is a two-dimensional stochastic process defined by two parameters: variance, which determines the spread of values generated, and H, which determines the fractal dimension of the landscape. H can take values in the range of 0.0 (weak spatial autocorrelation) to 1.0 (strong spatial autocorrelation). Table 1 Definitions of surveillance and search actions used in this paper. Term

Definition

Active surveillance

The general action of searching for invaders by a biosecurity agency or pest-control agency. Includes three types of search: follow-up, repeat and supplementary. The action of searching for pests by an agency in response to reports by the public. The action of searching for pests by an agency in areas where pests were previously located. Independent search by an agency targeting areas not covered in response to public reports or previous finds. Any encounter with a pest by members of the public that is reported to the relevant authority. The action of detecting a pest by a member of the public.

Follow-up search Repeat search Supplementary search

Passive surveillance Passive detection

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Fig. 2. (A) The detection curve represented in Eq. (8). (B) The lateral range curve used to calculate effective sweep width (l) for a given target. We used a midpoint-displacement algorithm (implemented based on the pseudocode presented by Saupe, 1988) to create hypothetical worlds. Fig. 3 shows two ownership maps and two habitat-suitability maps generated by controlling the fractal dimension of the map. The range of floating-point values generated by the algorithm was linearly scaled to the range of values appropriate for each map. This approach allows us to generate worlds that contain the same frequency distribution of attribute values but different levels of clustering, and hence enables us to study whether certain landscape features affect optimal decision rules. Our world had dimensions 128  128 (nr ¼ nc ¼ 128), with each cell representing one hectare (a ¼ 10,000 m2); therefore the total number of cells is the same as the number of hectares (n ¼ nr  nc ¼ 16,384). The dimension of the world was arbitrarily decided as a compromise between creating a world large enough to contain interesting spatial features but small enough to allow fast solution of the dispersal model. The map in Fig. 3A was used in the base simulations reported below, the other maps in Fig. 3 were later used in sensitivity analysis. Other world attributes were assumed to be homogeneously distributed across the landscape to test the model in the absence of confounding factors. The model is implemented in Matlab (The Mathworks, 2005). A simulation run proceeds as explained in the Appendix. Results from simulations were converted to cumulative distribution functions (CDFs) and relevant summary statistics were calculated. The parameter values used in base-case simulations are presented in Table 2. The simulations do not represent any specific species, but they could apply to an insect that produces colonies (ants or wasps) or a plant. The parameter values are plausible and biologically meaningful, and they are later subjected to sensitivity analysis. The effective sweep width parameter (l) was arbitrarily set a 5 m as a conservative assumption regarding the detectability of pests. The speed of search (S) is influenced by the vegetation and terrain at the site and was set at 1000 m h1, the same value used by Cacho et al. (2006, 2007). The habitat-suitability parameter (a) implies that a propagule landing on a cell has 0.02 probability of surviving. This value, combined with the assumption that each invaded site produces 100 propagules (w) means that, on average, each infested cell produces two new infestations. The maximum dispersal distance associated with the kernel (D) is 10 cells, or 1 km. Spread rates reported in the literature for Hymenoptera (ants and wasps) range between 1 and 500 km per year (Liebhold and Tobin, 2008); whereas for perennial plants values between 76 m y1 and 750 m y1 have been reported (Higgins and Richardson, 1999; Lonsdale, 1993). Based on these values, our kernel could be argued to represent a relatively slow invasion, but our model also includes long-distance dispersal (pL), which ultimately drives the speed of an invasion (Higgins and Richardson, 1999). Neubert and Caswell (2000) showed that probabilities of longdistance dispersal as low as 1012 can influence invasion wave speed; therefore our assumption of pL ¼ 0.02 is not conservative. The cost of the bounty (CB) was based on the amount paid as part of the RIFA eradication program in 2008 (QDPIF, 2008). The cost of search (Cm) is within the typical range of wages per hour for field work in Australia. Treatment costs on a per-hectare basis (CT) can vary widely depending on the labour-intensity of the technique used. For example, manual removal techniques are usually more expensive than those where machinery is used to remove plants or apply chemicals. We used an arbitrary value of $100 ha1 for this parameter. The Passive detection probabilities (0.1 for public areas and either 0.3 or 0.7 for private areas) were set at plausible values appropriate as a starting point for the analysis and allowing us to cover from moderately detectable to highly detectable pests. These values are later varied as needed to explore circumstances of interest. The model was run for 500 Monte Carlo iterations and a planning horizon (T) of 15 years for each scenario tested. Let Xt ¼

X

xit

(10)

i

represent the total area invaded at time t for a single iteration of the model. Xt is a measure of performance used to calculate the probabilities of control and

eradication. Eradication was defined as absence of invaded sites after T years of simulation (i.e., where XT ¼ 0). Control was defined as a situation where XT  X0. Fig. 4 shows a sample of six random invasions at time of discovery using the parameters from Table 2. Discovery of the invader can occur in any randomly selected pixel in those figures. The total area invaded (Xt) is equivalent to the number of black pixels.

3. Results The model was initially run for the base parameter values (Table 2). Four scenarios were considered, consisting of combinations of low (L) or high (H) passive detection probability (pp ¼ 0.3 or pp ¼ 0.7) and low or high supplementary search effort (M ¼ 0 or M ¼ 6554 hours). In the cases where supplementary search was zero (M ¼ 0), search was applied only in response to detections associated with passive detections and repeat searches. Additional search was applied if the follow-up to those detections (the search of radius rm) resulted in additional detections, but no supplementary search was applied. Results show that the invasion would never be controlled in the LL case (Fig. 5A), as the area invaded exhibits an increasing trend from year 2. This is because the infestations that are missed in year 1 spread out of control to sites with low probability of passive detection. When pp increases (the HL case) or supplementary search is introduced (the LH case), control is achieved in the sense that XT < X0 (Figs. 5B and C respectively), but eradication will not be achieved. Combining both high passive detection probability and high supplementary search results not only in control of the invasion, but also in a positive probability of eradication as shown by the intersection of the HH curve with the horizontal axis (Fig. 5D). Summary statistics for the cases illustrated in Fig. 5 are presented in Table 3. In the absence of supplementary search (M ¼ 0), increasing pp caused the expected final area invaded to decrease substantially, from 470 ha to 87 ha, and the expected cost of the control program to decrease from $4.8 million to $2.2 million, (compare LL and HL in Table 3). The overall cost of eradication was lower at higher pp because the pest was found (and controlled) earlier, so invasions were smaller and program duration was shorter. Introducing supplementary search (M ¼ 6554) caused the expected final area invaded to also decrease substantially, from 470 ha to 84 ha (LL vs. LH) and from 87 to 4 ha (HL vs. HH) depending on whether passive detection probability was low or high. 3.1. Analysis of probability functions Analysis of cumulative distribution functions (CDF) of final area invaded provides further insights (Fig. 6). The mean initial area invaded is shown as a vertical dotted line for reference. The positions of the CDFs relative to this line clearly indicate that control will never be achieved under LL (which is to the right of the

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Fig. 3. Hypothetical worlds created to test the model: (A) and (B) represent ownership structure, where black cells are private and white cells are public; (C) and (D) are habitatsuitability maps, where white cells represent a ¼ 0 and darker shades represent higher values of a to a maximum value of 0.04. The spatial autocorrelation of the maps was controlled by the fractal parameter H with values of 0.01 (A), 0.5 (B), 0.2 (C) and 0.8 (D).

reference line), whereas the other three strategies (LH, HL and HH) are highly likely to achieve control (they are to the left of the reference line). Fig. 6 also shows that eradication is unlikely in all cases except HH, where the curve intercepts the vertical axis at a positive value. Although the probability of eradication with HH is relatively low (0.29 according to Table 3), the probability of the final area invaded being below 5 ha is 0.612. This reflects that the last remaining invasive organisms are the most difficult to find and are unlikely to be found by actively searching for them. 3.2. The role of passive surveillance To determine how passive detection probability may affect expected success rates, we used the base parameter values from Table 2 but varied pp from 0.1 to 0.9, with supplementary search effort (M) held at either 0 or 6554 as before. As already established above, the probability of passive detection has a marked negative effect on the expected final area invaded, but now we see that the effect decreases at a decreasing rate (Fig. 7). The introduction of supplementary search causes the curve to shift down and the variance to decrease. This means that although the LH and HL strategies results in similar median area invaded at the end of the simulation (84 ha vs. 91 ha) the variance is considerable larger for HL than for LH (651 vs. 1032). Given that the strategies involving no

supplementary search (the upper curve in Fig. 7) produce poor results, now we focus on the lower curve and evaluate other performance measures. Three useful measures of performance are plotted against pp in Fig. 8: the probability of containment (PC), the probability of eradication (PE) and the total cost (C). For the base parameter values containment is virtually guaranteed with pp values of 0.4 or higher, complemented by M ¼ 6554, but eradication is unlikely to be achieved even at high levels of pp (Fig. 8A). The total program cost decreases at a decreasing rate as pp increases (Fig. 8B). The overall cost decreases because the pest is found and controlled earlier than in the absence of passive surveillance. The vertical difference between the points labelled LH and HH in Fig. 8B is the cost saving that would occur as a result of achieving the given increase in pp, and these funds could be used to increase public awareness as discussed below. 4. Sensitivity analysis The analysis so far is based on plausible but arbitrary parameter values. Now we undertake sensitivity analysis on the key parameters that define the spread of the invasion and the effectiveness of control options. Four different hypothetical worlds were tested, representing all the combinations of two levels of clustering in the

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datasets (4 worlds  4 scenarios) with 5000 observations each. The data were analysed through linear regression of the model:

Table 2 Parameter values used in the base simulations. Parameter

Value

Description

y ¼ b0 þ b1 lnðwÞ þ b2 lnðlÞ þ b3 lnðpk Þ þ b4 lnðpL Þ þ b5 lnðtD Þ

Environmental and biological assumptions: w 100 propagule pressure a 0.02 habitat suitability l 5 effective sweep width (m) 0.98 treatment effectiveness pk pL 0.02 probability of long-distance jump 5 time period when invasion is discovered tD s 1000 search speed (m h1) g 3.95 dispersal kernel parameter D 10 maximum dispersal distance (number of cells)

þ b6 I1 þ b7 I2 (11) where y represents the performance measure (either total cost or final area invaded); I1 and I2 are dummy variables representing clustering of private parcels, and habitat suitability, respectively and b’s are parameters estimated statistically. When the corresponding map exhibits low clustering (Figs. 3A and C) Ii takes a value of 0, and when the corresponding map exhibits high clustering (Figs. 3B and D) Ii takes a value of 1. The analysis was first undertaken for each world separately, with I1 and I2 forced to zero, and then for the data pooled across worlds (for 20,000 observations per scenario). Only the latter results are reported. Regression results are presented in the Appendix. All the signs of the coefficients are as expected (see Appendix and discussion below) and most of them are statistically significant (p  0.05). The regression results were converted to elasticities for interpretation. The elasticity Eyx is defined as the proportional change in y caused by a change in x. In our case, given Eq. (11):

Invasion management assumptions: 0.3–0.7 probability of passive detection, (private) pp 0.1 probability of passive detection, (public) pu M 0–6554 total effort available (h) m 2 minimum search effort per cell (h) 5 search radius for reported sites (no. of cells) rm 3 number of repeat searches SR Economic assumptions: 500 CB 30 Cm 100 CT r 0.06 a 10,000 T 15

cost of bounty ($ per find) cost of search ($ h1) cost of treatment ($ ha1) discount rate cell area (m2) planning horizon (y)

Eyx ¼ ownership map (Figs. 3A and B) and two levels of clustering in the habitat map (Figs. 3C and D). For each world, the five parameters tested were: number of propagules produced by invaded cells (w), effective sweep width (l), treatment effectiveness (pk); probability of long-distance jumps (pL); and time to discovery (tD). For each parameter 5000 random values were drawn from a lognormal distribution with mean and standard deviation as shown in Table 4. These values were used as inputs to execute Monte Carlo runs for each of the four base scenarios. These simulations resulted in 16

b x b vy x ¼ x ¼ x x y y vx y

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(12)

Since the elasticity varies with the value of y it is customary to report values at the mean (Table 5). In comparing the results from the base simulations (Table 3) with those from sensitivity analysis (Table 5) two factors must be kept in mind: (1) the mean parameter values were the same in both cases, but in the latter case the actual values were selected randomly for each iteration of the model; and (2) the mean habitat suitability was also the same in both cases (0.02), but it was

20

20

449

20

40

60

80

100

120

20

40

40

Fig. 4. A sample of simulated invasions at the time of discovery, when the control program commences.

60

60

80

80

100

100

120

120

450

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A

Area invaded (ha)

B

Scenario LL

600

Scenario LH

600

500

500

400

400

300

300

200

200

100

100

0

0

0

2

4

C

6

8

10

12

2

4

D

Scenario HL

600

0

14

500

400

400

300

300

200

200

100

100

8

10

12

14

12

14

Scenario HH

600

500

6

0

0 0

2

4

6

8

10

12

0

14

2

4

6

8

10

Year Fig. 5. Time trajectories of simulated invasions for four scenarios, solid lines are medians and dotted lines are the 5th and 95th percentiles from 500 simulation runs. Scenarios denote different combinations of passive detection probability and supplementary search effort; refer to Table 3 for details.

homogenous in the base case whereas it varied spatially in the sensitivity analysis (see Figs. 3C and D). These differences resulted in probabilities of containment being generally lower (from a maximum of 0.99 in Table 3, to a maximum of 0.77 in Table 5 for the HH case) and probabilities of eradication being generally higher (from 0.29 in Table 3 to 0.34 in Table 5 for the HH case). This reflects the higher variability in the simulations indicated by the standard deviations of final area invaded and total cost (Table 5 vs Table 3). Clustering of private parcels had no statistically significant effect on the final size of the invasion when pp was low (LL and LH cases in Table 5). But it had a significant positive effect when pp was high. Table 3 Mean results for base scenarios.

On average, the size of the final invasion increased by 112 ha for the HL scenario, and by 57 ha for the HH scenario when private parcels exhibited high clustering. In these scenarios passive detection probability in private parcels was considerably higher than in public parcels (pp ¼ 0.7 vs. pu ¼ 0.1). Higher clustering implies that, when an invasion is detected in a private parcel the neighbouring sites are also likely to be private. This reduces the likelihood that follow-up search (which occurs within a fixed radius of a detection) will cover public areas that have low passive detection probability. In the HH case, this problem is partially overcome by supplementary search which occurs in public areas. Clustering of habitat suitability had a significant positive effect on final area invaded in all scenarios (Table 5). Given that most dispersal occurs to neighbouring areas, this result reflects that once

Scenario LL

LH

HL

HH

Final area invaded (ha) Standard deviation Total cost ($M) Standard deviation Reward cost (%) Search cost (%) Treatment cost (%)

LH HH

0.3 0.1

0.7 0.1

0.7 0.1

0 0

6554 0.2

0 0

6554 0.2

0.002 0.000

0.918 0.000

0.882 0.012

0.999 0.290

470 49.3

84 25.5

87 32.1

4 4.6

4.798 1.039

4.716 0.936

2.192 0.802

2.981 0.577

12.51 86.62 0.87

4.80 94.85 0.35

14.86 84.38 0.76

5.95 93.76 0.30

LL HL

0.8 mean of initial invasion

0.3 0.1

Results (means of 500 iterations) Probability of containment Probability of eradication

1.0

Probability

Assumptions Passive detection probability private land, pp public land, pu Supplementary search effort hours (M) proportion area covered

0.6 0.4

0.2

0.0 0

100

200

300

400

500

600

700

Final area invaded (ha) Fig. 6. Cumulative distribution functions of final area invaded for the base scenarios in Table 3. Results from 500 Monte Carlo iterations.

O.J. Cacho et al. / Environmental Modelling & Software 25 (2010) 444–454

Final area invaded (ha)

600 LL

500

400

300

200 HL

100 LH

HH

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Probability of passive detection Fig. 7. Area invaded at T ¼ 15 years as affected by the probability of passive detection, with either no supplementary search (M ¼ 0 in upper curves) or with supplementary search (M ¼ 6554 in lower curves). Solid lines are medians and dotted lines are the 95th and 5th percentiles of 500 simulation runs. Labels identify the four scenarios from Table 3; other parameters were set at values from Table 2.

a suitable site is infested, the neighbouring sites are more likely to be infested because they are more likely to also be suitable. Elasticity results had all the expected signs: propagule pressure (w), probability of long-distance jumps (pL) and time to discovery (tD) were positively related to total cost; whereas detectability of the invader (l) and treatment effectiveness (pk) were negatively related. Propagule pressure exhibited the strongest positive effect, with the mean elasticity indicating that a 1 percent increase in the number of propagules produced per invaded cells will result in a 1.28 percent increase in the total cost of the program (Table 5). Whereas treatment effectiveness exhibited the strongest negative effect, the mean elasticity indicates that a 1.27 percent reduction in the total cost of the program could be achieved for a 1 percent increase in the probability that a treatment application will kill the pest (Table 5). 5. Discussion We derived three useful measures of performance: probability of containment (PC), probability of eradication (PE) and total cost (C). PC is based on a simple definition of control which measures the

A 0.9

6

LH

5

0.8

Total Cost ($M)

Probability of eradication

net area invaded but gives no indication of the gross area invaded (Rejmanek and Pitcairn, 2002). It is possible that a smaller net area invaded at the end of the simulation could be more scattered across the landscape than the initial invasion. If so, the invasion may be more expensive to contain or eradicate. Notwithstanding this limitation, XT  X0 offers a convenient criterion that can help us eliminate inefficient surveillance strategies. Our results show that increasing passive detection probability can produce substantial cost savings. But enhancing passive detections would probably require public information campaigns and the costs associated with this were not considered in our analysis. We do not have information to determine the relationship between expenditure and enhanced passive surveillance. This relationship would be driven by the behaviour of individuals and community groups, and their response to information and monetary incentives. In the absence of such information, however, we can still estimate the minimum amount that an agency should be willing to spend on a public awareness campaign. In Table 3, which includes only the costs of search, control and bounty payments, a lower bound can be placed on this expenditure. The difference in cost between LL and HL is $2.6 million, the difference between LH and HH is $1.7 million; these would be the minimum amounts the agency should be prepared to spend to achieve the prescribed increase in passive detection probability. Similar comparisons can be undertaken for results reported in Fig. 8 or for sensitivity analysis results in Table 5. Arguably, the cost differences calculated above represent lower bounds on the amount that should be invested in awareness campaigns to achieve a given level of passive detection probability; because the total budget would remain the same as in the initial situation, but the probabilities of containment and eradication would increase (see Fig. 8) and this has value in terms of reduced damage. The cost saving (in search, control and bounty payments) and investment of these funds in awareness campaigns is also likely to underestimate the value of the investment because improved public awareness could increase passive detections in public as well as private land (for example, detections made in public parks), and this possible benefit has not been evaluated. When passive detection probability was high, introducing supplementary search caused expected cost to increase, from $3.73 million to $4.38 million (HL vs. HH in Table 5) while mean final area invaded decreased from 407 ha to 223 ha. These results indicate that the marginal cost of reducing the final invasion size was approximately $3,533 ha1 in present-value terms. This was accompanied by increases in probability of containment (from 0.50 to 0.77) and probability of eradication (from 0.10 to 0.34). Whether

B

containment

1.0

0.7 0.6

eradication

0.5 0.4 0.3

HH

0.2

451

4

HH

3 2 1

0.1

LH

0.0 0.0

0.1

0.2

0.3

0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0. 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Probability of passive detection Fig. 8. Effect of passive detection probability on probability of eradication and probability of containment (A); and total cost of controlling the invasion over 15 years (B) for scenarios involving supplementary search (M ¼ 6554). The labels (LH and HH) refer to the base scenarios in Table 3.

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O.J. Cacho et al. / Environmental Modelling & Software 25 (2010) 444–454

Table 4 Parameters used in sensitivity analysis. Parameter

Mean

Standard deviation

Minimum

Maximum

w

100 4.987 0.980 0.020 4.953

30.1 1.971 0.008 0.008 2.032

34 1.115 0.948 0.004 1

336 19.065 1.000 0.081 19

l pk pL tD

this is considered a good investment depends on the benefits of control, or the damage caused by the pest in the absence of control. Two additional pieces of information that would be required if benefits were to be considered are a function linking pest presence to economic impacts and flows of environmental services, and a function describing the demand for these environmental services. These functions can be expensive and time-consuming to estimate when non-market valuation methods, such as choice-modelling or contingent-valuation surveys, are required (Amigues et al., 2002; Loomis and White, 1996; Van Bueren and Bennett, 2004). An agency considering whether to attempt eradication may decide to devote part of their budget to non-market valuation in order to take a more informed decision. The absence of benefit measures, however, does not invalidate the utility of models designed for conducting cost-effectiveness analysis, such as ours. Given the urgency of responding to newly discovered biological invasions, pest management agencies often must undertake control projects before a full economic evaluation can be undertaken. In any event, the inclusion of benefits into the model would simply involve modification of Eq. (9) if the benefit parameters were available. Our elasticity results can be useful in policy analysis and program planning. To help interpret these results we present the cost implications of changes in each parameter (Table 6). It is interesting to note that a one-year delay in time to discovery can increase cost by almost one million dollars and an improvement in detectability by one metre can result in cost savings of over 700 thousand dollars in present-value terms. If the probability of longdistance jumps can be reduced through quarantine measures, the result would be savings of approximately 540 thousand dollars for a decrease of 0.01. We found that habitat clustering made the invasion more difficult to eradicate, because it was easier for infested sites to spread to neighbouring cells. However, clustering of habitat suitability would not be a problem if the pest agency had knowledge of

Table 5 Results of sensitivity analysis, each column represents results of 20 000 Monte Carlo iterations; elasticities were estimated based on regression results from the Appendix.

Probability of containment Probability of eradication Final area invaded (ha) S. D. Total cost ($M) S.D.

LL

LH

HL

HH

0.25 0.04

0.56 0.15

0.50 0.10

0.77 0.34

1,111 1,193

467 704

407 506

223 399

4.89 3.70

5.74 3.92

3.73 3.64

4.38 3.50

0 16

112 43

57 51

1.08 0.75 1.24 0.19 0.81

1.54 0.81 1.36 0.29 1.24

1.22 0.79 1.20 0.17 0.98

Clustering effects on final area invaded: Private parcels 0 Habitat suitability 68 Elasticity of cost with respect to: Propagule pressure (w) Detectability (l) Treatment effectiveness (pk) Prob. of long-distance jump (pL) Time to discovery (tD)

1.29 0.68 1.27 0.26 0.93

Mean

4.69

1.28 0.76 1.27 0.23 0.99

Table 6 Monetary effects of changes in parameter values, calculated from elasticity estimates in Table 5. Parameter

Parameter change

Cost of change ($)

Propagule pressure (w) Detectability (l) Treatment effectiveness (pk) Prob. of long-distance jump (pL) Time to discovery (tD)

from 100 to 101 from 5 m to 6 m 98% to 99% from 2% to 3% from 5 y to 6 y

60,157 710,180 60,530 538,688 927,955

the habitat map and could use it to target search effort allocation. Although supplementary search was allocated to public sites, that had a low passive detection probability, we implicitly assumed that the habitat suitability map and the dispersal kernel were not known, and therefore did not influence search allocation decisions. As a program progresses and information is gathered, Bayesian techniques could be used to allocate search effort based on probability maps that consider known invaded sites, the dispersal kernel and habitat suitability (Hooten and Wikle 2007). We assumed that passive detection probability was determined by land ownership; with private parcels having higher values (pp) than public parcels (pu). There are other factors that may affect passive detection probability, such as population density, frequency of visits by the public to recreational areas and land cover types. These features of the landscape could be introduced into the model or used in advance to derive a passive detection probability map for a given application. Our purpose was to allow passive detection probability to vary spatially and we abstracted away from additional complications. This simplification does not invalidate any of our findings. The model is based on a binary state variable (presence/absence) rather than population density per site. This has advantages in terms of solution speed, but it imposes some limitations. For example, the number of propagules produced per infested site are constant in the model, which means that propagule pressure increases when more sites are infested, but not as the density of the invasion grows within a site, because this increase is not modelled. Extending the model to include population density per site, rather than considering only presence/absence, will result in a better representation of propagule pressure and will reduce the sensitivity of the model to spatial scale (i.e. reduce the error introduced as the area per cell in the map increases). Population density will also affect the probability of detecting an infested site and this can have important implications for effort allocation. The simple binary state we use is appropriate for the type of general analysis we have undertaken and allows us to avoid introducing additional assumptions regarding growth rates and density dependence. However, the benefit of including local growth and pest density in the model is obvious for application to actual invasions. We consider this extension of the model an important task for future work. To our knowledge this is the first application of a spatiallyexplicit model to study passive surveillance and there are some issues that need to be considered in future work. Two broad approaches to stimulating public interest are educational (e.g., public information campaigns) and the provision of financial incentives (e.g., bounty schemes). The application of bounty payments poses some well known difficulties. Bounty schemes have been used for centuries to control unwanted species (PohjaMykra¨ et al., 2005; Knowlton et al., 1999; Louisiana Department of Wildlife and Fisheries, 2003), although these schemes have not been very effective in helping to eradicate such species (Taylor and Edwards, 2005). In some cases, bounty schemes have even encouraged opportunistic behaviour such as deliberate transport of invasive organisms to within control areas (see Pracy, 1962 for an example). The typical focus of bounty schemes, though, has been on

O.J. Cacho et al. / Environmental Modelling & Software 25 (2010) 444–454

the removal of organisms rather than reports of their presence, particularly if the target organisms are highly mobile. A remaining question that has not been explored here is the possibility that stimulating passive surveillance may lead to increasing false positives. Checking on these reports by the public could be costly, but to include this cost in the model would not require any major modifications. The cost of false positives can be calculated if we know the proportion of incorrect reports by the public as a function of passive detection probability. Then we simply multiply the number of passive detections by the inverse of this proportion. It is logical to expect that pest-control agencies would maintain a record of responses to reports by citizens, therefore it should be relatively straightforward to account for this cost as the control program progresses. 6. Concluding comments In this paper we developed a spatially-explicit simulation model of the spread of an invasive species and applied it to a hypothetical pest invasion. The model incorporates both active and passive surveillance and provides an effective framework for testing the cost effectiveness of different surveillance strategies. Importantly, we have shown that when improvements in passive surveillance occur, the area invaded by a pest may be reduced to eradicable levels. Provided resources are available to invest enough effort in active surveillance, the probability of eradication increases and total costs of managing the invasion are reduced considerably as passive detection probability increases. The probability of passive detection may be enhanced through awareness campaigns and bounty schemes that provide an incentive to members of the public to search for and report detections. While we do not know the cost of the public awareness campaign, we showed how to calculate the minimum expenditure that would be economically efficient to undertake. This paper has focused on describing the model and presenting a relatively simple set of scenarios that demonstrates innovative ways of planning and evaluating the allocation of passive and active surveillance. Notwithstanding the simplicity of the application presented our analysis demonstrates that the model has considerable potential as a management tool. Complex situations can be summarised in a few variables, and landscape attributes can be read from GIS maps.

4. An invasion is started at a random location and allowed to spread undetected (applying Eqs. 3 to 6) until the time of discovery tD. The initial invasion state (xt) for t ¼ 1 is generated (i.e. the time counter starts upon discovery of the invasion). 5. Passive detections are generated by comparing a vector of random numbers (r) to the probability of detection of invaded sites (pp+xt). An invasion is detected for cell i if ri  ppi xit. A proportion pB of passive detections are reported. The parcels included in the reported set are selected randomly from the passive-detection set. 6. A search area of radius rm is drawn around each reported cell. 7. Search commences in the areas identified in step 6 by applying the search Eq. (8) to each cell within these areas; if additional invasions are found these are included in the reported list and step 6 is repeated. 8. Repeat searching of sites previously treated up to SR years ago is undertaken. 9. When supplementary search occurs, a cell is randomly selected from the valid active-search set (initially the entire set of publicly owned cells minus cells already searched in previous steps) and searched by applying effort m. 10. If an invasion is detected in steps 8 or 9, a search area of radius rm is drawn around each reported cell and searching within this area commences according to Eq. (8). 11. The amount of supplementary search effort available and the valid active-search set are updated based on the searches executed in steps 9 and 10. If supplementary search effort is still available, steps 9 to 11 are repeated. Otherwise step 12 is executed. 12. Control is applied to all cells where invasions were detected; these invasions are eliminated with probability pk. The state of the invasion xt is updated. 13. The time counter is increased. If t < T, the invasion spreads (applying Eqs. 3–6) and the state of the invasion xt is updated. If t ¼ T, results are saved, the time counter is restarted and the simulation returns to step 4 for the desired number of Monte Carlo iterations. Table A1. Regression results of Eq. (11) where y is total cost, t-ratios are shown in parentheses.

w

Acknowlegements This research was undertaken with funding from the Australian Centre for Excellence in Risk Analysis (ACERA). The authors gratefully acknowledge that this work has benefited from discussions with Paul Pheloung, Dane Panetta and participants in a workshop for ACERA project 0806: Application of search theory to invasivespecies control programs. D. Spring and R. Mac Nally acknowledge financial assistance from the Australian Research Council under Project DP0771672. We also acknowledge useful comments by 4 anonymous referees. Appendix

li pk pL tD Ownership effect Habitat effect R2 N

LL

LH

HL

HH

6.31 (115.72) 3.33 (80.39) 6.19 (3.14) 1.28 (30.61) 4.55 (111.51) 0.12 (3.89)

6.22 (107.87) 4.30 (98.15) 7.10 (3.41) 1.09 (24.53) 4.62 (107.23) 0.11 (3.34)

5.77 (101.99) 3.03 (70.36) 5.09 (2.49) 1.09 (25.11) 4.64 (109.55) 0.04 (1.09)

5.33 (98.89) 3.45 (84.12) 5.24 (2.69) 0.76 (18.37) 4.31 (106.82) 0.14 (4.41)

0.08 (2.63)

0.04 (1.31)

0.15 (4.50)

0.19 (5.94)

0.63 20 000

0.63 20 000

0.59 20 000

0.59 20 000

Table A2. Regression results of Eq. (11) where y is final area invaded, t-ratios are shown in parentheses.

w

li

A simulation run proceeds as follows: 1. A ‘world’ is created by generating the attribute vectors a, l, s, u, o, the state vector x and the adjacency matrix A. 2. The passive detection probability vector (pp) is created based on ownership attributes o. 3. Other demographic (w, pL, SR), economic (d, CB, Cm, CT) and logistic (M, m, pB, rm, tD) parameters are initialised according to the scenario to be tested.

453

pk pL tD Ownership effect Habitat effect R2 N

LL

LH

HL

HH

1795.2 (114.4) 2147.5 (179.8) 1754.2 (3.1) 161.4 (13.4) 336.0 (28.6) 13.9 (1.5)

1062.4 (94.1) 1060.2 (123.4) 871.4 (2.1) 79.7 (9.2) 215.2 (25.5) 4.3 (0.6)

707.6 (93.4) 833.4 (144.6) 647.5 (2.4) 61.5 (10.5) 172.3 (30.4) 111.9 (25.2)

564.9 (83.5) 563.9 (109.6) 169.5 (0.7) 30.5 (5.9) 136.2 (26.9) 56.9 (14.4)

67.5 (7.3)

15.8 (2.4)

43.4 (9.8)

50.5 (12.7)

0.70 20 000

0.56 20 000

0.61 20 000

0.51 20 000

454

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References Amigues, J., Boulatoff (Broadhead), C., Desaigues, B., Gauthier, C., Keith, J.E., 2002. The benefits and costs of riparian analysis habitat preservation: a willingness to accept/willingness to pay contingent valuation approach. Ecological Economics 43, 17–31. Brooks, S.J., Galway, K.E., 2008. Processes leading to the detection of tropical weed infestations during an eradication program. In: Van Klinken, R.D., Osten, V.A., Panetta, F.D., Scanlan, J.C. (Eds.), 16th Australian Weeds Conference, ISBN 9780646488196, pp. 424–426. Cacho, O.J., Hester, S.M., Spring, D., 2007. Applying search theory to determine the feasibility of eradicating an invasive population in natural environments. Australian Journal of Agricultural and Resource Economics 51, 425–433. Cacho, O.J., Spring, D., Pheloung, P., Hester, S., 2006. Evaluating the feasibility of eradicating an invasion. Biological Invasions 8, 903–917. Davis, P.R., Wilson, P.L., 1991. Report on European Wasps in Western Australia with Special Reference to the 1990–91 Season. Department of Agriculture, Western Australia. Gilbert, M., Gregoire, J.-C., Freise, J.F., Heitland, W., 2004. Long-distance dispersal and human population density allow the prediction of invasive patterns in the horse chestnut leafminer Cameraria ohridella. Journal of Animal Ecology 73, 459–468. Hastings, A., Cuddington, K., Davies, K.F., Dugaw, C.J., Elmendorf, S., Freestone, A., Harrison, S., Holland, M., Lambrinos, J., Malvadkar, U., Melbourne, B.A., Moore, K., Taylor, C.M., Thomson, D., 2005. The spatial spread of invasions: new developments in theory and evidence. Ecology Letters 8, 91–101. Higgins, S.I., Richardson, D.M., 1999. Predicting plant migration rates in a changing world: the role of long-distance dispersal. The American Naturalist 153 (5), 464–475. Hooten, M.B., Wikle, C.K., 2007. A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove. Environmental and Ecological Statistics 15, 59–70. Hyder, A., Leung, B., Miao, Z., 2008. Integrating data, biology, and decision models for invasive species management: application to leafy spurge (Euphorbia esula). Ecology and Society 13 (2), 12. Jennings, C., 2004. A brief history of the red imported fire ant eradication program. The Australian Journal of Emergency Management 19 (3), 97–100. Knowlton, F.F., Gese, E.M., Jaeger, M.J., 1999. Coyote depredation control: an interface between biology and management. Journal of Range Management 52, 398–412. Kot, M., Lewis, M.A., van den Driessche, P., 1996. Dispersal data and the spread of invading organisms. Ecology 77 (7), 2027–2042. Liebhold, A.M., Macdonald, W.L., Bergdahl, D., Mastro, V.C., 1995. Invasion by exotic forest pests: a threat to forest ecosystems. Forest Science Monographs 30, 49. Liebhold, A.M., Tobin, P.C., 2008. Population ecology of insect invasions and their management. Annual Review of Entomology 53, 387–408. Liebman, M., Mohler, C.L., Staver, C.P., 2001. Ecological Management of Agricultural Weeds. Cambridge University Press, Cambridge, UK. Lonsdale, W.M., 1993. Rates of spread of an innvading species – Mimosa pigra in northern Australia. Journal of Ecology 81 (3), 513–521. Loomis, J.B., White, D.S., 1996. Economic benefits of rare and endangered species: summary and meta-analysis. Ecological Economics 18, 197–206. Louisiana Department of Wildlife and Fisheries, 2003. Louisiana coastwide nutria control program (2003–04). Available from URL: http://www.nutria.com/site10. php (accessed 20.10.08.). MAFBNZ (Ministry of Agriculture and Forestry Biosecurity New Zealand), 2008. Review of the current state of the biosecurity surveillance system. Available from:

http://www.biosecurity.govt.nz/files/pests/surv-mgmt/surv/mafbnz-surv-strategy-current-state.pdf. Neubert, M.G., Caswell, H., 2000. Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology 81 (6), 1613–1628. Panetta, F.D., Lawes, R., 2005. Evaluation of weed eradication programs: the delimitation of extent. Diversity and Distributions 11, 435–442. Peck, S.L., 2004. Simulation as experiment: a philosophical reassessment for biological modelling. Trends in Ecology and Evolution 19, 530–534. Philippi, T., 2005. Adaptive cluster sampling for estimation of abundances within local populations of low-abundance plants. Ecology 86 (5), 1091–1100. Pimentel, D., Zuniga, R., Morrison, D., 2005. Update on the environmental and economic costs associated with alien-invasive species in the United States. Ecological Economics 52 (3), 273–288. Pitt, J.P.W., Worner, S.P., Suarez, A.V., 2009. Predicting Argentine ant spread over the heterogenous landscape using a spatially explicit model. Ecological Applications 19, 1176–1186. Pohja-Mykra¨, M., Vuorisalo, T., Mykra¨, S., 2005. Hunting bounties as a key measure of historical wildlife management and game conservation: finnish bounty schemes 1647–1975. Oryx 39, 284–291. Pracy, L.T., 1962. Introduction and Liberation of the Opossum (Trichosurus vulpecula) into New Zealand. New Zealand Forest Service, Wellington, Information Series No. 45 28 pp. QDPIF (Queensland Department of Primary Industries and Fisheries), 2008. National fire ant eradication program progress report 2007–2008. Oxley, Qld. Rejmanek, M., Pitcairn, M.J., 2002. When is eradication of exotic pest plants a realistic goal? In: Veitch, C.R., Clout, M.N. (Eds.), Turning the Tide: the Eradication of Invasive Species. IUCN SSC Invasive Species Specialist Group, Gland, Switzerland and Cambridge, UK. Saupe, D., 1988. Algorithms for random fractals. In: Peitgen, H.-O., Saupe, D. (Eds.), The Science of Fractal Images. Springer-Verlag New York Inc, New York. Sinden, J., Jones, R., Hester, S., Odom, D., Kalisch, C., James, R., Cacho, O., Griffith, G.R., 2005. The economic impact of weeds in Australia. Plant Protection Quarterly 20, 25–32. Smith, D.R., Villella, R.F., LeMarie´, D.P., 2003. Application of adaptive cluster sampling to low-density populations of freshwater mussels. Environmental and Ecological Statistics 10, 7–15. Taylor, R., Edwards, G. (Eds.), 2005. A Review of the Impact and Control of Cane Toads in Australia with Recommendations for Future Research and Management Approaches. A Report to the Vertebrate Pests Committee from the National Cane Toad Task Force The Mathworks 2002. Using Matlab. The Mathworks Inc, Matick, MA. The Mathworks, 2005. Using MATLAB. The Mathworks, Matick, Massachusetts. Thompson, S.K., 1990. Adaptive cluster sampling. Journal of the American Statistical Association 85, 1050–1059. Tyre, A.J., Possingham, H.P., Lindenmayer, D.B., 1999. Modelling dispersal behaviour on a fractal landscape. Environmental Modelling and Software 14 (1), 103–111. Van Bueren, M., Bennett, J., 2004. Towards the development of a transferable set of value estimates for environmental attributes. Australian Journal of Agricultural and Resource Economics 48, 1–32. Vitousek, P.M., D’Antonio, C.M., Loope, L.L., Westbrooks, R., 1996. Biological invasions as global environmental change. American Scientist 84, 468–478. With, K.A., 2002. The landscape ecology of invasive spread. Conservation Biology 16 (5), 1192–1203.