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).