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 YpYi . 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.