Specifying a model for lag time distributions
I modelled lag times as drawn from both a Weibull and normal
distribution. The normal distribution closely approximates the Weibull
distribution for certain parameter values (e.g., Fig 1D) but differs in
being a symmetric distribution with an upward accelerating hazard
function that steepens more rapidly than the Weibull hazard function,
thus allowing for a wider range of shapes for the lag time distribution
than modelled by the Weibull alone.
I specified truncated distributions for both the normal and Weibull
because the data were censored: for species introduced in yearYi , we can only observe the lag times of species
that have naturalised up to the present year Yp .
Other species introduced in year Yi could
naturalise in the future, meaning the observed distribution of lag times
is truncated at an upper limit Yp –Yi . I aimed to estimate the full distribution of
lag times by modelling the truncated portion of the distribution using
the observed lag times. The number of species likely to naturalise in
the future (the invasion debt) can then be estimated from that portion
of the full distribution that extends beyond the present (Fig. 2 and
Appendix S3). I set the present year Yp to the
year 2000, which was the most recent year of recorded naturalisation for
species in the dataset. The most recent year of first introduction was
1960, meaning I estimated the invasion debt associated with plant
species introduced to Britain between 1500 and 1960.
I compared the fit of ten models to the data (Table 1) to assess how
well the data matched theoretical predictions (Fig. 1 and Appendix S1)
and to identify a best-fitting model to infer the full distribution of
lag times. Models 1-5 modelled lag times as drawn from a truncated
normal distribution, with the five models differing in how the mean and
standard deviation were specified: as either constant or changing over
time (as predicted by theory), and whether lag times differed among
life-forms (tree/shrub, perennial herb and annual/biennial herb; Table
1). Models 6-10 had the same specifications for the mean and standard
deviation, but modelled lag times as drawn from a truncated Weibull
distribution.