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Porsche redesigned the parts yet again for Pictute is NOT of the bike stolen, ours didn't have mud guards or a drinks holder Carrera axle limited edition mens bike bought only a couple of weeks ago. More generally, challenges in estimating range limits can arise when there is a complex interplay between species reproduction, dispersal rates and habitat suitability.
An agent-based approach In previous work 7 , we developed an agent-based model to reconstruct a history of the Brisbane fire ant invasion, or more precisely to sample multiple plausible histories from a posterior distribution using a Markov chain Monte Carlo MCMC technique. This approach combined features of all of the above methods. The available data included: extensive records of individual nest detection points, as in utilization methods; records of search actions and estimates of detection probabilities by targeted search and by public reporting in urban and rural environments, as in monitory methods; environmental variables in the form of a habitat suitability map, as in correlative methods; and a detailed model of population dynamics, including a distribution of founding distances, reproductive rate and a complete phylogenetic tree for all detected and putative undetected nests, as in mechanistic methods.
While it is not possible to infer the exact number, locations or lifespans of undetected individuals, our method does simulate multiple plausible invasion histories at that level of detail. We typically sample such histories to explore the space of plausible histories consistent with the data. Full details of the model and the Markov chain Monte Carlo technique we used to sample from it are provided in Keith and Spring 7 , primarily in the Supplementary Information. Our approach addressed many of the limitations identified above.
In particular, it can be applied in circumstances where complex spatio-temporal dynamic processes create substantial gaps in occupied regions and irregular boundary shifts over time, using data obtained with imperfect and incomplete survey methods. However, one of our outputs involved processing the sampled histories to produce a time series of heat maps showing the expected areal abundance of fire ant nests.
As we point out in our discussion of monitory methods above and in our earlier paper , a time sequence of such heat maps can create an illusion of expansion due to increasing uncertainty regarding the location of undetected nests. Each sampled point set includes known locations of detected individuals and putative locations of undetected individuals.
In practice, we generate such point sets using our published agent-based method 7. The polygons are selected from a polygon family, thus constraining the polygons to have properties deemed desirable for a specific application, such as convexity or connectedness.
In our examples we use chi-shapes - simple polygons constructed using an algorithm of Duckham et al.
Alternative polygon families could be used, for example to allow polygons with holes. To illustrate the new method we estimated the boundary of an invasive species that is subject to an eradication program.
The method can also be readily applied to estimate boundaries of native species that are contracting or shifting due to environmental change, harvesting pressure or demographic variability. The program we consider is aimed at eradicating a fire ant invasion in South East Queensland, Australia.
We estimated the boundary of the invasion at the end of April , to inform a decision on whether to continue program funding, based on historical data regarding where fire ants were detected and where efforts were made to remove them.
We compared our most conservative estimate to the operational boundaries in use by the eradication program at that time. We found that the outer operational boundary at the end of April that is, the outer limits of the region monitored by remote sensing corresponded over most of its length to our most conservative inferred boundary. On this basis, we concluded that the invasion had been successfully delimited, subject to modest extensions being made to the operational boundary in a few identified locations.
Methods The method takes as input multiple sets of points that is, map locations in a two-dimensional landscape, representing the locations of both detected and undetected individuals. These points may represent habitations, or alternatively the notional centre of range for each individual.
Note that undetected individuals do not have known locations, and even the number of undetected individuals is unknown. Plausible locations for undetected individuals must therefore be imputed via some algorithm. We assume that multiple alternative sets of points are available, each containing locations of all detected individuals, but differing in the number and locations of imputed undetected individuals.
In principle, such sets of points do not have to be generated within a Bayesian framework: any algorithm capable of imputing missing data will suffice. However, the probabilistic interpretation that we give to the polygons constructed here assumes that the multiple sets of points have been sampled from a posterior distribution. In the examples presented below we use an MCMC algorithm that we developed 7 to sample from a posterior distribution over plausible histories of a biological invasion.
Input to the method The input consists of the following items: 1. A set Q of reference points distributed throughout the region of interest. In this paper, all polygons are chi-shapes defined below or modified chi-shapes. Each of the point sets P1, P2, …, PN includes a subset of observations common to them all, representing known locations of individuals. The point sets differ in the number and locations of undetected individuals, imputed by some appropriate method.
Here we use the posterior sampling method of Keith and Spring 7. The set Q of reference points provides a convenient discretization of the geographic region of interest. In principle it can be any collection of points scattered throughout the region, but in this paper we use the centres of cells in a square tiling.
In that case, the locations of all reference points can be determined by supplying map coordinates of one reference point in some specified coordinate system aligned to the tiling and the side length of the tiling. However, convex polygons have the disadvantage of resulting in potentially substantial overestimation of the species boundary when actual boundaries are nonconvex.
This family includes all convex polygons, but chi-shapes may also be non-convex. Only the longest such edge is removed, necessarily creating two new external edges and one new external vertex. This process is iterated until no external edges satisfying these criteria remain see Fig.
A A point set. B The corresponding Delaunay triangulation. The external edges form the polygonal boundary of the convex hull of the point set. C The chi-shape obtained by removing the longest external edge of B, thus creating two new external edges and one new external vertex. The external edges of C now form a polygon that is no longer convex, but remains simple.
D The chi-shape obtained by removing the longest external edge of C. The longest external edge of D cannot now be removed because the external edges of the remaining triangles would not form a simple polygon.
Full size image In this paper, all polygons are either chi-shapes or modified chi-shapes in which we relax criterion 2. The properties of this algorithm should be analysed in future work; here we merely note that by removing the other external edges in the same triangle, it becomes possible to form disjoint polygons.
Figure 2 Constructing modified chi-shapes. C The chi-shape obtained by removing the longest external edge of B. The remaining triangles include all vertices. D The chi-shape obtained by iteratively removing the longest external edge, along with any other external edges in the same triangle, until two disjoint components are formed.
Full size image Numerous other polygon families are available, for example, the families of polygons produced by LoCoH 18 or by parametric kernel density estimation We do not claim that chi-shapes or modified chi-shapes are preferable to these alternatives; a comparison is a potential direction for future research. Inferring boundaries Our proposed method consists of the following steps: 1.
For each reference point, count the number of point sets for which the polygons constructed in Step 1 contain that reference point in their interior or on their edge.
If a high resolution is desired, the number of reference points may be large. In that case, Step 4 can be computationally intensive. The computational efficiency of Step 4 can be improved if the reference points are centres of cells in a square tiling, as in all our examples below.
In that case, we can first identify boundary reference points. Step 4 then consists of constructing a polygon only for these boundary reference points. If the polygons are chi-shapes or modified chi-shapes, and the length L used in their construction is sufficiently large relative to the spacing between reference points, the polygon will be the same as if all of the reference points identified at Step 3 had been used.
Using the centres of a square tiling as reference points also facilitates an alternative visualization. The counts obtained at Step 2 or alternatively the proportions obtained by dividing these counts by N form a data matrix that can be visualized using a heat map. This heat map is of interest in its own right, and we present an example below Fig.
Figure 3 An example of a heatmap indicating the number of point sets out of for which the corresponding chi-shape contains each reference point. Reference points are at the centres of cells of side length m in a square tiling. There are reference points covering a km by km region. The values shown in this heat map were used to construct the polygons in the December subplot of Fig.
Full size image The proposed method can be interpreted as averaging classifiers built from multiple point sets. That is, one can interpret the polygons built at Step 1 as classifying space into infested regions interior and non-infested regions exterior , with the above-mentioned heat map being essentially an average of these classifiers. In this respect, our method resembles range bagging The resemblance is somewhat superficial however, as range bagging is primarily a computational technique for building classifiers in high-dimensional environmental space by averaging over classifiers in one or two dimensions.
Moreover, range-bagging generates multiple point sets via sub-sampling observations rather than by imputing locations to undetected individuals. For the analysis presented below, we experiment with square tilings having spacings of 50 m and m. Results Simulation study To test the capacity of the method to infer the geographic range of a species, and in particular to quantify the likely locations of undetected individuals, we used a simulated data set that we had previously generated to mimic a biological invasion and eradication program 7.
The simulation involved constructing an entire detailed history of a hypothetical invasion, starting with an initial introduction, recording individual founding events, including time of founding and location of all individuals, and also simulating management efforts to identify which individuals were detected and thus available for inference, and which nests were killed by treatment. Here the relevant points are the following: 1 We sampled plausible histories of the invasion from a posterior distribution.
From each of these we extracted the known for detected nests and imputed for undetected nests locations of all individuals alive during the second last month of the modeled period.
We chose the second last month so that the imputed locations of undetected individuals would be informed by detections made in the final month. This produced point sets. From this we extracted the true locations of all individuals alive during the second last month of the modeled period. Figure 4 Inferred boundaries of the simulated invasion in the second last month of the modelled period.
True locations of all individuals alive in the second last month are shown as grey circles, and the locations of detections that occurred during that month are shown as black crosses. Note that all of the detections are inside the 0. As the history of this eradication program underscores the importance of accurately delimiting an invasion, we provide the following summary.
During the early years of the NFAEP, control efforts were focused primarily on known infestations and nearby areas, with relatively little surveillance around those areas. This strategy can be slow in achieving delimitation when infestations exist well beyond the boundary of the managed area. It was subsequently determined that an invasion had been spreading undetected from a point in or near Amberley for an extended period. This realization was a major setback for the eradication program, which had been operating with apparent success since In previous modeling 7 , we estimated that eradication was close to being achieved by , but that the population subsequently recovered, in large part due to delimitation failure.
Our results indicated that Amberley was not the only delimitation failure — there were undetected areas of spread in the eastern part of the invasion at around the same time, and these contributed to the recovery after It was decided that continued funding of the program beyond June would depend partly on the invasion being successfully delimited by 30 June To undertake this task, low cost monitoring methods involving remote sensing and citizen monitoring were applied.
These methods have substantially lower detection probabilities than conventional surveillance methods, including ground surveillance with trained personnel, but enable large areas to be rapidly surveyed at affordable cost. At the time this analysis was performed, we had data on detections and interventions to the end of May We decided to assess whether the invasion had successfully been delimited by the end of April , so that the inference would be informed by one month of subsequent detections.
We first inferred a complete history of the invasion using a Bayesian agent-based model previously developed for reconstructing the Brisbane RIFA invasion 7 and summarized in Appendix 1. The remote sensing efficacy ie. We therefore performed five separate MCMC runs with: 1.
Remote sensing efficacy 0.