Results
Figure 2 plots lag time as a function of introduction year for the 708 species naturalised in Britain for which data were available, along with marginal histograms for lag time and year of introduction. An apparent spike in the number of early introductions is evident in the decades 1580-1600 and 1620-1640. These spikes occur because many early introductions were first recorded in two compilations of garden plants published during those periods. While these compilations provide the first record of cultivation for many introduced species, the true dates of introduction for cultivation could have been earlier but were undocumented. Simulations using artificial data revealed that errors in the true dates of introduction and naturalisation should not unduly affect the lag time estimates reported in this study, but that dating errors could result in underestimates of the invasion debt (Appendix S5).
Apart from the two early spikes, the number of introductions tended to increase over time, peaking in the early 1800s before declining. This decline is potentially due to the lag between introduction and naturalisation, which could result in a greater proportion of more recent introductions having not yet naturalised. The lag time distribution peaks at just under 100 years with a long tail. The mean and median lag times between introduction and naturalisation were 145 and 122 years, respectively.
Table 1 compares the performance of the ten candidate models. For all models, the estimated effective number of parameters was close to the actual number of parameters, indicating none of the models were badly mis-specified, and plots of the Pareto k diagnostics indicated most models were well behaved with few outlying observations (Appendix S2). Model 4, which specified a truncated normal distribution of lag times with mean and standard deviation changing over time, and mean lag time differing by plant life-form, was the best-performing of the ten candidate models (Table 1). Model 5, which specified that the standard deviation also differed by life-form, was the second-best performing and close in predictive accuracy to model 4. Nevertheless, there seemed no reason to favour the more complex model 5 given that model 4 performed slightly better. Apart from model 5, model 4 performed substantially better than other models: the difference in PSIS-LOO between model 4 and model 10 (the next best-performing after model 5) was nearly twice the standard error of their difference (roughly equivalent to a 95% confidence interval) suggesting that, allowing for model uncertainty, we could be reasonably confident that model 4 performed better. I therefore used model 4 for subsequent inferences about the distribution of lag times. Simulations using artificial data revealed that fitting a truncated distribution to censored lag time data accurately recovered the true parameter values (Appendix S4), and that parameter estimates were not unduly affected by dating errors (Appendix S5).
The parameter estimates for model 4 revealed two features of the lag time distribution. First, as predicted, both the mean and standard deviation of lag times declined over time (Fig 3B) causing the lag time distribution to shift towards zero and become narrower and more peaked for more recent introductions (Fig. 4). The lag time distribution for trees/shrubs introduced in 1500, for example, had a mean of 571 years and a standard deviation of 169 years, while the distribution for trees/shrubs introduced in 1960 had a mean of 70 years and a standard deviation of 26 years, equating to an overall decline in mean lag time of around 100 years per century. The hazard functions associated with model 4 were all upward accelerating and steepened appreciably for more recent introductions (Fig 3D-E).
Second, trees/shrubs had a longer mean lag time than perennial herbs, which in turn had a longer mean lag time than biennial/annual species (Fig. 3C). For species first introduced in 1500, the predicted mean lag times for trees/shrubs, perennial herbs and biennial/annual species were 571, 455 and 377 years, respectively, although these differences narrowed over time: for species introduced in 1960, the predicted mean lag times were 70, 56 and 46 years, respectively.
The distribution of lag times predicted by model 4 fitted the observed distribution of lag times well, both overall (Fig 4A) and when naturalised species were split by their century of introduction (Fig 4B-F). In almost all cases, the actual number of species in the 20 year lag time bins were within the 95% quantiles obtained from the model simulations.
The estimated number of species introduced to Britain per year that have or will naturalise has increased over time for trees/shrubs and perennial herbs (curved lines in Fig 5A, C) but remained relatively constant for annual/biennial species (Fig 5E). The number of species predicted to naturalise every 20 years, based on introductions prior to 1960 and the fitted lag time model, matched the observed data well (Fig 5B, D, F). For each life-form, the area under the predicted distribution that extended beyond the year 2000 (shaded red in Fig 5) estimated the size of the invasion debt. These areas indicated that, in addition to the 708 species already naturalised, a further 84 trees/shrubs (95% credible interval 63-110), 65 perennial herbs (48-86) and 9 biennial/annual species (5-14) are expected to naturalise within the next 150 years; a total of 158 species (116-210). These figures could be underestimates because artificial data simulations revealed that dating errors can downwardly bias estimates of invasion debt (Appendix S5).