Supporting Information
Appendix S1 Methods
Calculation of the observed rates of range expansion
To calculate the range of mink at sequential time steps we used the nonparametric kernel method of
adaptive local convex hulls (a-LoCoH) (Getz et al., 2007) to construct home ranges around records of
mink presence. Local convex hull methods are particularly useful when constructing home ranges
from data that are influenced by the geometry of the landscape within the range of the study animal
(Getz et al., 2007). The a-LoCoH method constructs local hulls (polygons) from the maximum
nearest neighbouring points, to a root point, with the sum of the distances from the nearest
neighbours being less than or equal to a. ‘a’ was the maximum distance between any two points in
the dataset. In our analyses, the value of a was 300 km and was kept constant across all calculations
at all time intervals. The 95% home range (HR) for each time step was calculated so as to exclude
outlying observations that were potential false positives, reflecting, for example recorder confusion
with similar species such as pine martens (Martes martes) or ferrets (Mustela putorius). For each
time interval, t to t+1, the distribution of mink was taken as an accumulation of the distributions in
previous time steps e.g. t + t-1. The newly occupied area at each time step was calculated (in km2) as
the difference between the current distribution (t+1) and the previous distribution (t + t-1).
The method of calculating radial invasion was adapted (as in White et al., 2012) to account for the
observation that range expansion did not occur in all directions (because of geographical
boundaries) or equally in those directions that it did occur. A quadrant was placed over North East
Scotland, which was then split in to three 30⁰ segments (Figure 1). The radial range for each sector
was calculated as √(A*12)/π, where A was the newly occupied area for each time interval, and was
plotted against time. The radial expansion rate was calculated by dividing the radial range by the
number of years in each time interval. Expansion on the west coast and the far northeast coast was
measured latitudinally, as a Euclidean distance from the southernmost to northernmost point of the
range for each time period. The pattern of expansion in west Scotland was assumed to be
approximately linear, with populations in the northwest originating from a southwest source (Fraser
et al., 2013). The rate of area expansion based on the total area colonised at each time interval was
calculated by dividing the newly occupied area at each time interval by the number of years in each
time interval.
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Modelling habitat suitability within the Scottish landscape
Using a subset of the same mink presence data as used previously, we modelled the probability of
occurrence of mink across Scotland using a method developed by Royle et al. (2012). The resultant
estimated occurrence probability was used as a summary of habitat suitability, an association
endorsed by Royle and Dorazio (2008). This model predicted the probability of occurrence of a
species from presence-only data, using conventional likelihood methods, based on the assumptions
that the space is randomly sampled and species detection probability is constant (Royle et al., 2012).
The logistic regression model was run using the ‘maxlike’ package (Royle et al., 2012) in R (R Core
Team, 2013). Mink presence records had a location precision of 1 km or less (n = 4141) and
environmental covariates were created to parameterise the model at a 2 km resolution. A 2 x 2 km
cell size was deemed appropriate because it is within the range of reported home range lengths for
female mink (e.g. Gerell, 1970, Harrington et al., 2009, Helyar, 2005, Melero et al., 2008). Habitat
covariates were derived from Land Cover Map (LCM) 2007 data for the UK (Morton et al., 2011) and
presented as the area, in hectares, of each habitat type within each 4 km2 cell. These habitat types
were: improved grass, other grass (rough grass + acid grass), heath, montane, arable, bog, littoral
rock, littoral sediment, supralittoral rock and supralittoral sediment. Mean elevation (metres above
sea level) and length of coastline (m) per 4 km2 were also included as covariates. All environmental
covariates were standardised to make them comparable. Several covariates were found to be
correlated (Spearman’s rank correlation, rs ≥ 0.6), so only the covariates with the broadest and most
general applications were included (e.g. elevation and montane habitat, elevation was preferable).
We compared models using AIC following backward step-wise selection of individual covariates. If
the AIC of the model decreased by ≥ 2 units, the simpler model was taken as the best descriptor of
the data. Starting models included parameters for elevation, improved grass, other grass, heath,
bog, supralittoral sediment, and either coast length or littoral rock. These two options for the
coastal descriptor were correlated and therefore not included in the same model, but we tested
both separately to see which was better at explaining mink presence. The model was run at two
different scales, firstly at the whole Scotland scale, and secondly only on a portion of west Scotland
to see if factors that predicted the occurrence of mink in western Scotland were different to those
for the whole of Scotland. To define suitable versus unsuitable habitat within the model predictions
we calculated the optimal threshold for determining the probability of occurrence using the
‘required sensitivity’ method (PresenceAbsence package in R, Freeman and Moisen, 2008). This
method allowed us to set a sensitivity, x, whereby the threshold of the model had to account for x%
of the presence data in the model. Cells with a value greater than or equal to this threshold were
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considered suitable habitat, whereas cells with a value less than this threshold were deemed
unsuitable.
Testing the habitat suitability model
To test the accuracy of the ‘maxlike’ model in predicting the probability of occurrence of mink and
hence habitat suitability, we applied the model to another area where we had access to the relevant
habitat and mink presence data via an island mink eradication project (Scottish Natural Heritage,
2012). Model estimates from the ‘maxlike’ model were used to calculate a probability of occurrence
for each 2 km x 2 km cell in the island extent and this was compared to the actual presence of mink
across the island. The presence data (n = 445) were a subset of results from six years of systematic
trapping including only females (because females tend to be more settled than males in territories
throughout the year (Dunstone, 1993)) caught after the first year of the eradication between
December and August (excluding juvenile dispersal (Dunstone, 1993)). The proportion of female
mink assigned to cells representing unsuitable habitat in the island area was compared to the
proportion of mink records assigned to unsuitable habitat in mainland Scotland to verify whether
habitat predictions made for the Scottish mainland were appropriate.
Relating observed rates of range expansion to habitat suitability
The area (km2) of suitable habitat within each newly occupied area in the observed range was
calculated. This was compared to the total area available and also used to calculate the rate of
expansion according to the two methods of radial range and total newly occupied area as previously
described.
Testing scenarios for spatial variation by simulating range expansion
To investigate scenarios that might explain mink invasion patterns in Scotland we used a recently
developed individual-based model (Bocedi et al., 2014) to predict the range expansion of mink at
time intervals matching those used to estimate actual mink range. We tested the hypotheses that
habitat heterogeneity caused variation in expansion rate and direction; that all fur farms contributed
to the establishment of feral mink populations in Scotland and alternatively that introduction points
in west Scotland did not contribute to colonisation of west Scotland. The model scenarios varied in
habitat suitability, the distribution of introduction points and the mean dispersal distance of
individuals. The series of simulated distributions were visually compared with observed
distributions.
3
This was a female-only model with one individual per 4 km2 cell. Only juveniles dispersed and
dispersal distance was extracted from a single dispersal kernel (a probability density function from
which the distance that individuals travel is randomly drawn (e.g. Kot et al., 1996)). The expected
fecundity (the mean of a Poisson distribution from which the actual number of young was drawn)
was scaled linearly with habitat suitability which was derived from the predictions of the ‘maxlike’
model. We modelled dispersal assuming individuals settle stochastically at a distance from their
natal site drawn at random from a negative exponential kernel. In cases where the dispersal
distance drawn failed to displace the individual from its natal cell, the distance was redrawn. The
dispersal distance was resampled if the initial distance drawn did not result in the individual being
displaced from its natal cell. Thus, slightly more than half of all dispersal movements were greater
than loge(2) of the mean dispersal distance parameter value. Model parameters for fecundity, adult
survival and mean dispersal distance were based on estimated values in the literature but were
optimised by testing a range of values to find the closest match between the simulated output
distribution and the observed modelled distribution in the first seven years of range expansion. The
range of values trialled for fecundity was 0.4 – 0.8 female offspring per adult female, with a juvenile
survival probability of 0.9 in their first year. Adult survival probability was tested from 0.6 – 0.9 and
mean dispersal distance from 5 – 20km.
Parameters for survival, fecundity, and dispersal (the kernel mean) were gained using available
empirical estimates and optimised by testing a range of values to best fit the observed modelled
distribution in the first seven years of range expansion
For all model runs, cells outside of the coastal boundary i.e. the sea, were inaccessible such that any
dispersal movements ending in the sea were repeated until the arrival cell was within the coastal
boundary. Using optimal parameters, the model was first run using a homogeneous landscape. All
cells within the landscape raster were set to the maximum value of habitat suitability (99%) and the
model was run with two variants of fecundity probability – the maximum and the mean (calculated
from mean habitat suitability values). The purpose of this was to look at the potential expansion of
mink where the range was only restricted by geographical boundaries. The landscape structure was
then modified to match the habitat suitability as predicted by ‘maxlike’ modelling, where all habitat
cells were deemed suitable (range: 6 - 99%). Fecundity was scaled linearly with habitat suitability.
The landscape was further modified to make the habitat availability more realistic. Habitat cells with
values below the suitability threshold were deemed unsuitable which prevented mink from settling
in very low quality areas. If dispersing individuals landed in unsuitable habitat, they would move to a
randomly selected neighbouring cell of suitable habitat, or die if none were available. The purpose
4
of this was to simulate the idea that mink are unlikely to choose to settle, and hence reproduce, in
low quality habitat, but if they did they would likely starve to death. Fecundity was still scaled
linearly with habitat suitability above the threshold. In the final version of the model, unsuitable
habitat was given the same value as the sea, and hence was inaccessible. This forced dispersing
individuals in to suitable habitat, thus preventing loss of individuals in unsuitable habitat areas.
Each model was replicated 100 times with optimal parameters and was initialised using records of
mink presence in 1964. We also ran the model with the locations of all known mink farms in 1962 as
starting points to test the hypothesis that all farms acted as sources of feral mink populations, and
without introduction points in west Scotland to test the hypothesis that these did not contribute to
colonisation of west Scotland.
For each model scenario (with the exception of scenario viii, for which no observed range is
available), the predicted range from 100 RangeShifter (RS) simulations was compared with the
observed range (in 2012 except for scenarios iv and v) using the true skill statistic (TSS) (Allouche,
Tsoar & Kadmon 2006). This takes into account both the sensitivity (probability of correctly
classifying a true presence) and specificity (probability of correctly classifying a true absence) of the
model, whilst being independent of prevalence.
The TSS requires binomial observed and predicted data in the form of a 2 x 2 confusion matrix, and
therefore it was necessary to apply a threshold value to the RS output data, which, for each 2 km x 2
km square, returned the percentage of replicate simulations in which the square was occupied by at
least one mink at the end of the simulation period (usually 48 years). Following Aben et al. (2014),
all possible thresholds from 1 to 99% were examined. For each threshold, each square in the RS
predictions was classified as predicted absence (< threshold) or presence (≥ threshold), the TSS
calculated on that basis, and the maximum TSS for the scenario and the corresponding threshold
recorded.
5
Appendix S2 Results
Results of ‘maxlike’ model testing
Figure S1. Model predictions for probability of occurrence of mink on the island, with records of
females caught between December and August, 2008-2012 shown as black dots.
True Skill Statistic calculations
For all but scenarios i, vii and ix the optimum threshold leading to the highest TSS was very low, at
5% or less, i.e. a square should be regarded as predicted presence provided that only a small fraction
of RS simulations predicted mink presence within it. Applying the TSS evaluation criteria reported by
Eskildsen et al. (2013), all scenarios would be regarded as a ‘good’ fit to the observed range, except
for scenarios vi, vii, ix and x which would be regarded as ‘poor’.
6
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