Modelling the spread of European buckthorn in the Region of Waterloo

Abstract

Established invasive species, such as European buckthorn (Rhamnus cathartica), pose a challenging problem for land managers who must decide when and how to control them. In order to make an informed decision land managers need to be able to predict the spread of these invasive species at local scales and without the need for excessive sampling. Our approach uses a hybrid model, combining habitat suitability and the presence of the invasive in neighbouring cells to predict the probability of a cell being invaded over time. A generalized linear mixed-effects model was used to create a habitat suitability model and a spread model. The habitat suitability model predicts the presence of buckthorn based on environmental characteristics and the results are used in the spread model. The spread model indicates that the invasion of buckthorn is influenced by the suitability of habitat and the presence of buckthorn in neighbouring cells. The success of the spread model suggests that this approach can be used to create a spatiotemporally explicit model with limited sampling effort.

Appendix 1: Extended model results

See Fig. 6.

Fig. 6

Overview of modeling decisions. For descriptions of variable names see Tables 1 and 3.

Habitat suitability

The univariable logistic regressions indicated that all explanatory variables had a significant relationship (p < 0.25) with the presence of buckthorn, except for NearPropLine (distance to the nearest property line) and CTI (compound topographic index). Therefore NearPropLine and CTI were excluded from subsequent analyses.

The formula for the maximal model was:

$$Buckthorn\;Presence \sim Material + Permea + log10\left( {PatchDist + 1} \right) + Land\;Cover + Land\;Cover:Permea$$

where Material indicates surficial geology, Permea is permeability of the surficial material, PatchDist is the distance from the centre of a cell to the edge of the nearest wooded patch, and Land Cover is land cover class.

The maximal model failed because of collinearity between Material and Permea. Since the univariable logistic regressions indicated that the Silt level of Material did not fit well, Material was removed from the model. Additionally, the interaction term was removed at this point. Although a likelihood ratio test showed that the model with the interaction provided a better fit than the model without the interaction, the coefficient for the interaction term had an excessive standard error (−12.20 ± 240).

The model simplification indicated that the minimal acceptable model was a model that included Permea, PatchDist, and Land Cover as explanatory variables. The Wald test showed that all coefficients in the minimal acceptable model were significant, except the Swamp level of Land Cover and the Low level of Permea (Table 2). Based on the results of the leave-one-out cross-validation, the AUROC for this model was calculated to be 0.63, which is considered poor discrimination (Hosmer and Lemeshow 2000).

The correlograms for the minimal acceptable model showed that some spatial autocorrelation was present within sampling areas. To account for spatial autocorrelation, we tried including a spherical correlation structure term that produced a model with a range of 25 m and a nugget of 0.95. This model effectively produced the same coefficients as the model without the spatial autocorrelation term (coefficient estimates within one standard error from each other; results not shown). The correlograms of the Pearson residuals for the model with spatial autocorrelation term are very similar to the model without the spatial autocorrelation term. An exponential correlation structure was tested with similar results. Since the results did not indicate a substantial reduction in the Pearson residuals through inclusion of within sampling areas spatial correlation structure, the model without a spatial autocorrelation term was used for the remaining analyses.

Spread model

Visual investigation of the GAMs showing the relationships between the number of first and second order neighbours with buckthorn and the probability of buckthorn invasion, suggested a non-linear positive effect of the number of first order neighbour cells on the probability of buckthorn invasion. The largest change in invasion probability occurred when the number of first order neighbours with buckthorn increased from zero to one. Further increases of neighbour cells with buckthorn resulted in less pronounced changes to the invasion probability (Table 3). Neither the presence of second order neighbour cells with buckthorn nor their exact number had a significant effect (Table 3). Accordingly, to account for the positive effect of buckthorn presence in first order neighbours on buckthorn invasion probability, we introduced a binary variable of buckthorn presence in first order neighbour cells.

Habitat suitability and year also had significant effects on the probability of buckthorn invasion (Fig. 3). Using the explanatory variables mentioned above, the semivariogram for the spatial autocorrelation in the spread model showed the presence of weak (nugget-to-sill ratio of approximately 0.045:0.060) residual autocorrelation. Additionally, the semivariogram for the temporal autocorrelation indicated the presence of nonstationarity in the model residuals with respect to time. The GLM model coefficients were within one standard error of the GLMM coefficients so we were able to use the GLM to produce cumulative residuals to evaluate the GLMM model (results not shown). The cumulative residual plots showed that the observed cumulative residuals are in the expected range for all the variables but not for the model results (Fig. 4). The model presented good discrimination with an AUROC of 0.88.