FIGURE CAPTIONS
Figure 1 . Hazard functions (inset boxes) and the associated probability density distributions (main panels) where lag times follow a Weibull distribution (see Appendix S1). Parameter k determines the shape of the hazard function: A if k = 1, the hazard is constant over time and the lag time distribution is negative exponential; B if 1 < k < 2 the hazard function is downward accelerating and the lag time distribution shifts to unimodal for all k > 1; C ifk = 2 the hazard function is linear; D if k> 2 the hazard function is upward accelerating. Each panel shows three hazard functions and associated lag time distributions given the shape parameter k , with distributions having a mean lag time of approximately 100 (red lines), 200 (blue lines) or 300 (black lines) years. In each of B-D , a steeper hazard function results in the lag time distribution having a smaller mean and variance (see Appendix S1 for details).
Figure 2 . Lag time (the lag between introduction and naturalisation in years) plotted against year of first introduction for 708 plant species naturalised in Britain, with marginal histograms for lag time and year of first introduction (shaded grey). The lag times for each introduction year are truncated at the sloping line because we can only observe species that have naturalised up to the present (set to the year 2000). The two inset histograms (shaded blue) show the distribution of lag times for species introduced in the periods 1620-1640 and 1890-1910 (shown by the blue vertical lines). These lag time distributions are truncated at the dotted lines on the histograms. The black curves are normal distributions fitted to the truncated lag time data with the area under the curve beyond the year of truncation (shaded red) estimating the invasion debt (the number of species introduced in each period that are yet to naturalise based on the truncated lag time distribution for species that have naturalised).
Figure 3 . A-C Parameter estimates for model 4 (the best-performing model, see Table 1). Filled circles are the means of the posterior distributions and vertical lines show the 95% credible intervals. A shows the estimated mean and standard deviation of the expected lag times for trees and shrubs introduced in 1500 AD;B shows the rate at which the mean and standard deviation of the lag time distribution changed per year after 1500 AD; Cshows the mean lag times of perennial herbs and annual/biennial species relative to trees/shrubs, which were set as the reference class. All parameter estimates are on the loge scale. D-FHazard functions by life-form for model 4 for the years 1650, 1750, 1850 and 1950, showing the increase in the hazard (a number proportional to the probability a species will naturalise in the next small time step given it has not yet naturalised) over time following introduction.
Figure 4 . The observed distribution of lag times (grey histograms) for: A all species and B-F species introduced in different centuries (starting in 1500 through to 1900). For each century, the observed lag times are truncated at a maximum value shown as dashed lines on each histogram. The black lines show the mean predicted distribution of lag times based on the actual dates of introduction and 10000 simulations of each species lag time derived from model 4 (see text), with the red shading showing the 95% quantiles from the simulations.
Figure 5 . Histograms with 20 year bins for year of introduction and year of naturalisation for 708 naturalised plant species in Britain grouped by life-form (A, D = trees/shrubs; B, E = perennial herbs; C, F = annual/biennial species). The most recent year of introduction for naturalised species in the data (1960) is shown as a dashed line in A-C , and the most recent year of recorded naturalisation (2000) is shown as a dashed line inD-F . The black curves in D-F show the expected number of species naturalising in each 20 year interval based on the actual dates of introduction (A-C ) and the expected lag times estimated using model 4. For each life-form, the area under the curve beyond the year 2000 (shaded red in D-F ) estimates the invasion debt for that life-form (the number of species introduced before 1960 that are yet to naturalise based on the truncated lag time distribution of the species that have naturalised). The black curves in A ,C and E show the trend in the number of species introduced in each 20 year period that will naturalise in the foreseeable future (see Appendix S3 for details).
Figure 1 .