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Revista chilena de historia natural

Print version ISSN 0716-078X

Rev. chil. hist. nat. vol.82 no.3 Santiago  2009

http://dx.doi.org/10.4067/S0716-078X2009000300003 

Revista Chilena de Historia Natural 82: 347-360, 2009

RESEARCH ARTICLE

 

An evaluation of methods for modelling distribution of Patagonian insects

Una evaluación de los métodos para modelizar la distribución de insectos patagónicos

 

 

MARCELO F. TOGNELLI1, SERGIO A. ROIG-JUÑENT1, ADRIANA E. MARVALDI1, GUSTAVO E. FLORES1 & JORGE M. LOBO2,*

1 Laboratorio de Entomología, IADIZA-CONICET, CCT-Mendoza, CC 507, 5500 Mendoza, Argentina

2 Departamento de Biodiversidad y Biología Evolutiva, Museo Nacional de Ciencias Naturales (CSIC) c/ José Gutiérrez Abascal 2, 28006 Madrid, Spain

* Corresponding author: mcnjll7@mncn.csic.es


ABSTRACT

Various studies have shown that model performance may vary depending on the species being modelled, the study área, or the number of sampled localities, and suggest that it is necessary to assess which model is better for a particular situation. Thus, in this study we evalúate the performance of different techniques for modelling the distribution of Patagonian insects. We applied eight of the most widely used modelling methods (artificial neural networks, BIOCLIM, classification and regression trees, DOMAIN, generalized additive models, GARP, generalized linear models, and Maxent) to the distribution of ten Patagonian insect species. We compared model performance with five accuracy measures. To overeóme the problem of not having reliable absence data with which to evalúate model performance, we used randomly selected pseudo-absences located outside of the polygon área defined by taxonomic experts. Our analyses show significant differences among modelling methods depending on the chosen accuracy measure. Maxent performed the best according to four out of the five accuracy measures, although its accuracy did not differ significantly from that obtained with artificial neural networks. When assessed on per species basis, Maxent was also one of the strongest performing methods, particularly for species sampled from a relatively low number of localities. Overall, our study identified four groups of modelling techniques based on model performance. The top-performing group is composed of Maxent and artificial neural networks, followed closely by the DOMAIN technique. The third group includes GARP, GAM, GLM, and CART, and the fourth best performer is the BIOCLIM technique. Although these results may allow obtaining better distributional predictions for reserve selection, it is necessary to be cautious in their use due to the provisional nature of these simulations.

Key words: Expert opinión, model evaluation, Patagonia, pseudo-absence data, species distribution modelling.


RESUMEN

Varios estudios han mostrado que el desempeño de los modelos de distribución puede variar dependiendo de la especie modelizada, el área de estudio o el número de localidades de presencia utilizadas, sugiriendo que es necesario evaluar cuál es la mejor técnica de modelización en cada situación concreta. En este estudio evaluamos distintas técnicas de modelización para la distribución de los insectos patagónicos. Hemos aplicado ocho de los métodos más ampliamente usados (redes neuronales, BIOCLIM, árboles de clasificación y regresión, DOMAIN, Modelos Aditivos Generalizados, GARP, Modelos Lineares Generalizados y Maxent) a los datos de distribución de diez especies de insectos patagónicos, comparando su efectividad mediante cinco medidas diferentes. Para evitar el problema de la carencia de datos de ausencia fiables con los que evaluar los modelos, hemos utilizado pseudoausencias seleccionadas al azar fuera de un área poligonal definida por taxónomos expertos. Nuestros análisis muestran diferencias significativas entre los distintos métodos de modelización dependiendo de la medida de validación utilizada. Maxent es el método que ofrece mejores resultados para cuatro de las cinco medidas de validación utilizadas, aunque su precisión no difiere de la obtenida con redes neuronales. Cuando se examina la efectividad para cada una de las especies, Maxent resultó también uno de los métodos más fiables, especialmente en el caso de aquellas especies con un pequeño número de localidades. En conjunto, este estudio identifica cuatro grupos de técnicas de modelización. El de mayor fiabilidad sería el compuesto por Maxent y las redes neuronales, seguido de cerca por DOMAIN. El tercer grupo incluiría GARP, GAM, GLM y CART, mientras que el cuarto estaría formado por BIOCLIM. Aunque estos resultados pueden permitir obtener mejores predicciones de distribución con capacidad para ser utilizadas en la planificación de reservas, es necesario ser cauto en su utilización debido a la naturaleza provisional de estas simulaciones.

Palabras clave: Datos de pseudoausencia, evaluación de modelos, modelos de distribución de especies, opinión de expertos, Patagonia.


 

INTRODUCTION

One of the central problems in ecology is understanding how organisms are distributed on earth. In the absence of a complete inventory of where species occur, predictive models of species distribution are an alternative that is increasingly being explored to produce detailed distribution and habitat suitability maps. Species distribution models examine associations between general environmental characteristics and the known occurrences of a particular species (Guisan & Zimmermann 2000, Scott et al. 2002, Guisan & Thuiller 2005). These models allow us to project the geographic distribution of a species into unexplored regions, or into scenarios of future or past climatic conditions. In recent years, this approach has been widely applied to address issues in ecology (Anderson et al. 2003, Vetaas 2002), biogeography (Coudun et al. 2006, Luoto et al. 2006), evolution (Peterson et al. 1999, Graham et al. 2004, Martínez-Meyer & Peterson 2006), conservation biology (Ferrier 2002, Araújo et al. 2004, Cabeza et al. 2004; Sánchez-Cordero et al. 2005), species invasión (Peterson 2005, Fitzpatrick et al. 2006), and the effects of climate change on species (Skov & Svenning 2004, Thomas et al. 2004, Thuiller 2004, Araújo et al. 2006, Thuiller et al. 2006). However, many studies have not expressly addressed a very important part of the modelling process: selecting the most appropriate modelling method to address a particular question (Pearson et al. 2006).

Numerous modelling methods and tools have been developed in the last decade (Guisan & Thuiller 2005, Elith et al. 2006). These can be roughly divided into those that only use records of species presence (e.g., bioclimatic envelopes and distance-based measures) and those that use presence-absence data (e.g., general linear and additive models and decisión trees; Guisan & Zimmermann 2000). As absence data are frequently lacking, an alternative to presence-absence techniques involves generating "pseudo-absences" from background áreas (Zaniewski et al. 2002, Engler et al. 2004). Generally, pseudo-absences are randomly selected from the background environment (Stockwell & Peterson 2002) or by using environmentally weighted random sampling (Zaniewski et al. 2002, Engler et al. 2004). The comparative performance of different modelling methods has been assessed with varying results (Muñoz & Felicísimo 2004, Segurado & Araújo 2004, Elith et al. 2006, Hernández et al. 2006, Pearson et al. 2006). Overall, results show that model performance varies according to the taxa being modelled, the studied región, and the sampie size (Le., number of locality records). Thus, before generating predictive distribution models for a specific taxonomic group and región, it is necessary to first examine the comparative performance of the different possible techniques.

Predictive models of species distribution have been generally developed for plants and vertébrate species (see Guisan & Thuiller 2005 and references therein). Only a few studies have modelled the potential distribution of insect species, and these have dealt mostly with vectors of human diseases (Kornar et al. 2005, López-Cárdenas et al. 2005, Peterson et al. 2005) and introduced species (Roura-Pascual et al. 2005, Fitzpatrick et al. 2006). Although arthropod species comprise approximately 80 % of total species (Rupert et al. 2004), the lack of reliable taxonomic and distributional information for insects has prevented the use of species distribution models for conservation purposes (but see Meggs et al. 2004, Chefaoui et al. 2005). As conservation prioritization is generally based on higher-level taxa (Le., vertebrates), although they may not be effective surrogates for invertebrates, our general aim is obtaining guidelines for the further use of distribution models in the design of conservation plans for Patagonian insects. Using the data of ten species of insects belonging to different families of southern South América for which we only have presence data, we assessed here the comparative performance of eight of the most widely used modelling techniques. Predictions of profile presence-only methods and discrimination modelling techniques which use environmentally weighted pseudo-absences were compared. As absence information is unreliable, model predictions were evaluated using absence áreas based on expert opinions.

METHODS

Species and climate data

We studied a región of southern South América extending from 46°15' to 75°66' W and 23°98' to 55°98' S. Species locality data included presence data for 10 insect species (Table 1). These species were selected because they are good representatives of their taxonomic groups in the studied región. The minimum number of presence localities for a species was 19, the máximum number was 94, and the median was 39.

A set of 25 climatic variables was initially considered (Table 2). Twenty-one climatic variables were extracted from the WorldClim datábase (Hijmans et al. 2005), and four variables were extracted from the Climate Research Unit (New et al. 2002). The spatial resolution of environmental variables was 0.04° (approximately 4.6 x 4.6 km). These variables were standardized to eliminate measurement-scale effects (with a mean of 0 and a standard deviation of 1). To select variables that better represent the environment of the región, we used the so-called Jolliffe's principal component method (Rencher 2002). First, a Principal Component Analysis (PCA) was carried out including all variables, and five non-correlated factors with eigenvalues > 1 were obtained that explained 87.13 % of the climatic variation across the región. For each one of the five PCA factors, the variable with the highest factor loadings (which measure the correlations between the original variables and the factor axes) was selected (> 0.8). The five selected variables were annual mean temperature, isothermality, mean diurnal range, precipitation during the driest month of the year, and precipitation during the wettest quarter of the year. Relative humidity was also included as a predictor variable because it was the only one that was not significantly correlated with any of the formerly mentioned PCA factors. In total, these six variables were considered to be the most representative of the climate in southern South América.

 

 

Modelling methods and evaluation

We tested eight of the most commonly used modelling methods. The type of data these techniques use distinguishes them: those that use presence-only data (BIOCLIM, DOMAIN, GARP, and Maxent), and those that require presence-absence data (Artificial Neural Networks, Classification and Regression Trees, Generalized Additive Models, and Generalized Linear Models). Within the first group, two of the methods (GARP and Maxent) use a sample of the background environment as pseudo-absences. BIOCLIM and DOMAIN were run with DIVA-GIS (www.diva-gis.org) using default specifications. GARP models were run using the best subset (Anderson et al. 2003) module of OpenModeller (www.openmodeller.sf.net), and máximum entropy models were run with Maxent (Phillips et al. 2006). Artificial Neural Networks (ANN), Classification and Regression Trees (CART), Generalized Additive Models (GAM), and Generalized Linear Models (GLM) were fitted in R (www.r-project.org) using the mgcv, nnet and tree packages.

For each species, we randomly separated the occurrence localities into 10 partitions. Each partition was created by selecting 50 % of the data to train the models, and the remaining 50 % to test the models. For those modelling methods that required presence-absence data to run, we generated environmentally weighted pseudo-absences (Lobo et al. 2006). These pseudo-absences were selected calculating the máximum and minimum scores of the six selected environmental variables for the presence localities of each species. Then, we calculated the multidimensional envelope defined by the scores of the localities in which the species was recorded. Pseudo-absences were then selected outside the environmentally suitable región (10 times the number of training presences).

Several discrimination Índices are frequently used to test model performance (Fielding & Bell 1997). These are usually derived from the two by two confusión matrix, which describes the frequency of correctly and incorrectly predicted data from the known presences and absences. However, when reliable absence data are lacking and pseudo-absences selected across environmentally unsuitable regions are used, model absence predictions should not be validated using these pseudo-absences; high absence success rates would only indicate successful forecasting of the locations under unfavourable environmental regions. Thus, when only presence data are available, commonly applied Índices, such as Kappa and Área Under the Curve (AUC) of the ROC plot, cannot be calculated. In this instance, to overeóme the problem of not having reliable absence data with which to evalúate model performance, we used the taxonomic expertise of some of the authors (SARJ, AEM, and GEF), who are specialists in the taxa being modelled. These experts defined a polygon área based on their collecting experience and knowledge of where the species might occur, including those probable áreas inhabited by the species for which no presence information exist. We then generated 10,000 random pseudo-absences (validation pseudo-absences or VSA) outside the polygon área defined by the taxonomic experts to calcúlate performance evaluation Índices.

To assess the agreement between the presence-VSA data and the predictions obtained by the different modelling methods, we calculated the AUC, máximum Kappa valué, and the true skill statistic (TSS; Allouche et al. 2006). Área Under the ROC Curve (AUC) is one of the most widely used approaches to evalúate model performance (Fielding & Bell 1997) whose results should not be used in species' model comparisons when the ratios between the extent of occurrence and the whole extent of the territory under study differ (Lobo et al. 2008). In our case, we have decided to maintain AUC results because all the species considered differ slightly in their occurrence área, and the analyzed territory is the same. AUC measures the ability of a model to discriminate between sites where a species is present and sites where a species is absent. Valúes of AUC range from 0.5 for models with predictive discrimination abilities no better than random to 1 for models with perfect predictive ability (Fielding & Bell 1997). The Kappa statistic measures the proportion of correctly predicted sites after accounting for the probability of chance agreement (Moisen & Frescino 2002). It requires a suitability cut-off threshold, which is generally arbitrarily selected. Alternatively, one can choose the máximum valué for the Kappa score obtained from varying the threshold from 0 to 1 (Guisan et al. 1998). We calculated and used this máximum score (max Kappa) for each modelling method. The True Skill Statistic (TSS) has been recently introduced to ecology as an alternative measure of model accuracy (Allouche et al. 2006). In addition to having the advantages of Kappa (i.e., takes into account omission and commission errors and corrects for chance agreement), TSS also does not depend on prevalence (Allouche et al. 2006). It is generally used in weather forecasting and compares the number of correctly classified forecasts, excluding those attributable to random guessing, to that of a hypothetical set of perfect forecasts (Allouche et al. 2006). TSS ranges from -1 to +1, where valúes of 0 or less indicate a model performance no better than random, and a valué of +1 indicates perfect performance (Allouche et al. 2006). In addition to these three accuracy measures, we calculated sensitivity (the proportion of true positives correctly predicted) and specificity (the proportion of true negatives correctly predicted). To calcúlate these two last accuracy measures we used the point of the ROC curve where the sum of sensitivity and specificity is maximized as a cut-off criterion to convert continuous model predictions to binary classifications (presence/absence). This threshold has the advantage of giving equal weights to the probability of success of both presences and absences (Manel et al. 2001). It is one of the most appropriate methods to correctly derive a binary variable from continuous probabilities when species presence-absence distribution data are unbalanced (Liu et al. 2005, Jiménez-Valverde & Lobo 2006).

We used the Friedman test, which is a non-parametric versión of the one-way repeated measures ANOVA (Sprent & Smeeton 2001), to test for differences in modelling performance among the different modelling methods. We then performed post hoc tests of múltiple comparisons (Dunn test) to determine which methods differed from each other. In addition, we assessed model performance on a per species basis and evaluated effects due to sample sizes Models for species 1 to 5 were trained with < 18 sites, whereas models for species 6 to 10 were trained with 22 to 47 sites (Table 1).

RESULTS

Comparisons across modelling methods

Performance differed significantly among modelling methods for all accuracy measures considered (AUC, max Kappa, TSS, sensitivity, and specificity; Table 3). In general, the rank order of model performance varied considerably for each measure assessed (Fig. 1a-e). However, múltiple comparisons tests revealed that not all rank-order differences were significant (Table 4). Overall, we found that three methods, ANN, DOMAIN and Maxent, consistently performed better than the remaining techniques in four of the five accuracy measures. However, when it carne to predict percentage of absence evaluation points, they lagged behind BIOCLIM in accuracy (Fig. 1E; Table 4). GARP was the next best ranked method, performing very well in three out five accuracy measures. The remaining modelling methods were less consistent in their performance across accuracy measures, and also showed higher variability within accuracy measures.

 

In general, model performance at the species level showed similar trends to the pooled species models. ANN, DOMAIN, and Maxent performed very well on a per-species basis for all accuracy measures, except for Specificity, followed by GARP, which performed well in three of the five accuracy measures (Fig. 2A-E).

To assess how model performance varied with sample size (i.e., the number of presence records), we plotted the median and interquantile range valúes for each species, and for each modelling method (Fig. 3). In general, there was not a clear trend between sample size and model performance. Only a few modelling methods showed a trend for some, but not all, of the accuracy measures. For instance, BIOCLIM and GAM improved their performance with larger sample sizes when assessed by AUC, TSS, and Sensitivity, whereas GARP showed this trend for AUC, TSS, and Specificity (Fig. 3A-E). ANN, DOMAIN, Maxent, and GARP showed relative low within and across species variation when assessed by AUC, TSS, and Sensitivity.

The lack of a clear trend between sample size and model performance is also reinforced by correlation analyses between sample size and each accuracy measure. Although significant, Spearman's rank correlations were very low (Table 5), and showed only a slight increase in the valúes of performance measures with sample size, with the exception of specificity, which decreased with sample size.

 

 

 

DISCUSSION

Previous studies have shown that model performance varies among different modelling methods, depending on the species modelled. the study área, and the sampling size (Manel et al. 1999, Dettmers et al. 2002, Moisen & Frescino 2002, Thuiller et al. 2003, Segurado & Araújo 2004, Elith et al. 2006, Hernández et al. 2006, Pearson et al. 2006, Tsoar et al. 2007). Not surprisingly, our analyses also showed significant differences among predictions generated by various modelling techniques. However, since the goal of this study is to use predictive models of species distributions for prioritizing áreas for conservation, it is crucial that we evalúate which modelling method is better for the taxa being studied and for our área of study. Although museum records combined with predictive models of species distribution have great potential valué for the conservation of biodiversity (Graham et al. 2004), uncritical use of these models may misdirect conservation actions. Poor model choice may lead to misrouted conservation efforts (Loiselle et al. 2003, Johnson & Gillingham 2004).

Model performance not only differed among different modelling methods but also within methods used to evalúate different species. This variability in predictions makes it difficult to identify the 'best' modelling technique (Segurado & Araújo 2004, Pearson et al. 2006). It has been suggested that the best modelling methods are those that reduce the omission error rate (Anderson et al. 2003). The argument is that predicting unsuitable habitat where a species is known to be present is a clear error, whereas predicting suitable habitat where there is no record of a species' presence may be due to insufficient sampling or other non-climatic factors that limit its distribution (Anderson et al. 2003, Pearson et al. 2006). This approach, however, may be adequate when predicting the ranges of invasive species that have not yet colonized all suitable environments (Peterson 2003). In contrast, Loiselle et al. (2003) suggest that if models are used to predict the distribution of a particular species for conservation purposes, they should not over-predict áreas of distribution so that unsuitable sites for the protection of the species are included. Thus, model assessments should weigh the costs of making a false positive prediction versus the costs of making a false negative prediction depending on the intended use of the model (Fielding & Bell 1997). Múltiple measures of model accuracy are needed to evalúate relative model performance, besides separately assessing their ability to predict presences and absences (Fielding 2002, Segurado & Araújo 2004, Bulluck et al. 2006, Hernández et al. 2006).

In all, our results showed that Maxent was one of the best performing methods, as was ANN. Maxent ranked first in four out the five measures used to assess model accuracy; although, its performance was not significantly different from that of ANN. When assessed on a per species basis, Maxent outperformed the other methods in three out of the five accuracy measures. In addition to its strong performance, with the exception of max Kappa valúes, it was very stable across species with different sample sizes. Most importantly, Maxent performed well with small sample sizes, which constituted the majority of the species records in our datábase. These results support the findings of previous studies (Elith et al. 2006, Hernández et al. 2006, Pearson et al. 2007) that compared several modelling techniques and found that Maxent was one of the best performing methods, even with sample sizes as low as five positive observations (Hernández et al. 2006, Pearson et al. 2007).

ANN also performed well in this study. However, other researchers comparing the performance of different modelling methods have found contradictory results regarding the accuracy of ANN. Whereas Segurado & Araújo (2004) found ANN to be the highest-performing method, Manel et al. (1999) found ANN model performance to be comparable to that of GLM. Unfortunately, in the most comprehensive model comparison study to date in which the predictive performance of 16 modelling techniques was assessed (Elith et al. 2006), ANN was not included. Besides performing slightly better, the advantage of Maxent over ANN is that the former method does not need absence data to genérate a predictive distribution model. It has been suggested that the success of Maxent may be due to its regularization procedure that prevents or allows overfitting depending on whether a few or many training data points are used, respectively, to build the model (Dudik et al. 2004, Phillips et al. 2004, Phillips et al. 2006, Hernández et al. 2006). Thus, given its strong performance and simplicity of use, we believe that Maxent is the best option for our larger project.

Another method that performed relatively well in this study was DOMAIN. Its performance was among the best, as judged by three of the five measures of accuracy. However, although it correctly predicted a high proportion of presences, it did not predict absences as well. A similar situation occurred with GARP, which had the lowest valué for specificity. This method performed relatively well in three of the five accuracy measures, but it performed the worst according to the remaining two. In general, GARP performed somewhat better than GAM, and the latter outperformed GLM and CART. Previous model comparison studies have found GAM to perform better than GLM (Franklin 1998, Pearce & Ferrier 2000, Thuiller et al. 2003). Segurado & Araújo (2004) found CART to perform better than GLM, but not significan ti y better than GAM. Finally, although BIOCLIM correctly predicted a high proportion of the absences, in general, it was the poorest performing method, particularly for species with a low number of locality records. This finding is consistent with other studies in which BIOCLIM had a poor performance compared to other methods (Elith et al. 2006, Hernández et al. 2006).

Overall, we can broadly identify four groups of modelling techniques in our study of model performance. First, the top-performing group is composed of Maxent and ANN, followed closely by the second group, DOMAIN. The third group includes GARP, GAM, GLM, and CART which is followed by BIOCLIM. These results should be viewed as recommendations only applicable to the taxa modelled (insects), and for the study área under consideration. Importantly, model techniques cannot replace good data when the purpose is to delimit the realized distributions of species (Lobo 2008). Distribution models are rarely used in the case of invertebrates, but this group of species harbour most of the recognized species and their data are rarely used for conservation purposes. Lack of data, survey bias, and restricted distributions are common characteristics in the case of insects and distribution simulations, although of provisional nature, are the only available procedure able to quickly provide relatively reliable distributional estimations that can be validated in the future.

However, the recommendations provided by our study should be considered with caution because some caveats can influence our results. First and importan tly, modelling methods differ in their capacity to represent the potential-realized distribution gradient (Jiménez-Valverde et al. 2008) and comparisons among techniques must consider both this question and the quality of the data used as dependent variable. When reliable absences are lacking modelling exercises genérate geographical representations located between the potential and realized distribution of species. The models obtained that require presence-absence data (i.e. ANN, CART, GAM, and GLM) were calculated using pseudo-absences outside the environmental domain of presences. This procedure artificially increase the statistical power of group discrimination techniques (Guisan & Zimmermann 2000, Graham et al. 2004), allowing obtaining geographic representations closer to the potential distribution of species (Jiménez-Valverde et al. 2008) and so, able to include the species in some not colonized localities. This can lead to high valúes of AUC and specificity but also high commission errors (Lobo 2008). Second, our models have been validated using a different kind of pseudo-absences previously defined by an expert taxonomist. Ideally, validation data should be independent of the data used to genérate the model (Fielding & Bell 1997), although this is practically impossible for Natural History CoUection data (Graham et al. 2004). Our procedure is not exempt of error since the suggestions based on the knowledge and collecting experience of taxonomists can be biased (see Seoane et al. 2005). These drawbacks outline the fact that distributions models are simply hypotheses of the real distributions of the species and cannot replace a well designed field coUection of species data.

ACKNOWLEDGEMENTS

This study was supported by a Fundación BBVA grant (Project: Diseño de una red de reservas para la protección de la biodiversidad en América del Sur austral utilizando modelos predictivos de distribución con taxones hiperdiversos), and Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET-CCT-Mendoza).

 

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JOHNSON CJ & MP GILLINGHAM (2005) An evaluation of mapped species distribution models used for conservation planning. Environmental Conservation 32: 117-128.        [ Links ]

KOMAR O, BJ O'SHEA, AT PETERSON & AG NAVARRO-SIGÜENZA (2005) Evidence of latitudinal sexual segregation among migratory birds wintering in México. Auk 122: 938-948.        [ Links ]

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THUILLER W, MB ARAÚJO & S LAVOREL (2003) Generalized models vs. classification tree analysis: Predicting spatial distributions of plant species at different scales. Journal of Vegetation Science 14: 669-680.        [ Links ]

THUILLER W, S LAVOREL, MT SYKES & MB ARAÚJO (2006) Using niche-based modelling to assess the impact of climate change on tree functional diversity in Europe. Diversity and Distributions 12: 49-60.        [ Links ]

TSOAR A, O ALLOUCHE, O STEINITZ, D ROTEM & R KADMON (2007) A comparative evaluation of presence-only methods for modelling species distribution. Diversity and Distributions 13: 397-405.        [ Links ]

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Associate Editor: Rodrigo Ramos-Jiliberto

Received August 6, 2008; accepted June 5, 2009

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