Different approaches can be used to model the spread of invasive species. Here we demonstrate the use of survival regression, an approach that can be used to study a variety of events, not just death, to model the time to colonization. The advantage of survival regression to study colonisation of new areas is that information on those areas that have not been invaded by the end of a study can be included in the analysis, thus potentially increasing the accuracy of parameter estimation. We use proportional hazards regression (PHR; a type of survival regression) to model the spread of the common waxbill Estrilda astrild in Portugal. The species invaded Portugal in two peaks of invasion between 1964 and 1999. We built a PHR model with the information available up to the first invasion peak, then used this model to predict the pattern of invasion in the second peak. PHR had useful forecasting capabilities: areas that were actually colonised by 1999 had significantly higher hazards of colonization based on information from the first wave of invasion than areas that were not colonised. We then built a final model of expansion of the common waxbill that combined all available data up to 1999. Among climate variables, the most important predictor of colonization was temperature, followed by relative humidity. We used this model to estimate the invasion potential of the species under climate change scenarios, observing that an increase of 18C in mean annual temperature increased the risk of a new invasion by 47%. Our analyses suggest that survival regression may be a useful tool for studying the geographical spread of invasive species. However, PHR was conceived as a descriptive technique rather than as a predictive tool, and thus further research is needed to empirically test the predictive capabilities of PHR.
This study examined the interplay of spatial and environmental effects shaping the range margin of the red-backed shrike ( Lanius collurio ) in northern Portugal. The occurrence of shrikes in 10 × 10 km UTM squares was related to three sets of explanatory variables, reflecting environmental effects (climate and habitat), largescale spatial trends, and neighbourhood influences (considering an autologistic term); spatial variables were used as surrogates for historical and demographic factors. Multiple logistic regression models were built for each set, and then variation partitioning based on partial regressions isolated the unique and shared components of explained variation. The environmental model revealed a dominant influence of climate effects, with the occurrence of shrikes increasing with frost and thermal amplitude, declining with insolation, and responding unimodally to rainfall. There was a weaker influence of habitat conditions, though shrikes were more likely with increasing cover by annual crops and pastures, and decreasing forest cover. Only a relatively small proportion of explained variation was due to a 'pure' environmental component (10.4%), as most variation explained by environmental factors appeared spatially structured (51.9%). The unique contributions of spatial variables to the overall model were also small, though the neighbourhood effects appeared relatively stronger than large-scale trends. Taken together, results suggested that the southwestern range margin of the red-backed shrike was largely determined by spatially structured environmental factors. Nevertheless, there were also 'pure' environmental factors determining some isolate occurrences irrespective of any spatial structure, and 'pure' spatial factors that appeared to favour the occupation of squares surrounding the core distribution areas irrespective of environmental conditions. These results add to the growing evidence that both environmental and spatial factors need to be considered in predictive modelling of species range margins.
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