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\begin{document}

\title{Time lags and the invasion debt in plant naturalisations}



\author[1]{Richard Duncan}%
\affil[1]{University of Canberra}%


\vspace{-1em}



  
  \date{May 23, 2024}


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\selectlanguage{english}
\begin{abstract}
Ecological processes often exhibit time lags. For plant invasions, lags
of decades to centuries between species' introduction and establishment
in the wild (naturalisation) are common, leading to the idea of an
invasion debt: accelerating rates of introduction result in an expanding
pool of introduced species that will naturalise in the future. Here, I
show how a concept from survival analysis, the hazard function, provides
an intuitive way to understand and forecast time lags. For plant
naturalisation, theoretical arguments predict that lags between
introduction and naturalisation will have a unimodal distribution, and
that increasing horticultural activity will cause the mean and variance
of lag times to decline over time. These predictions were supported by
data on introduction and naturalisation dates for plant species
introduced to Britain. While increasing trade and horticultural activity
can generate an invasion debt by accelerating introductions, the same
processes could lower that debt by reducing lag times.%
\end{abstract}%



\sloppy


\textbf{Introduction}

Many ecological processes are characterised by time lags: a delay
between initiation of a process and its outcome. The concept of an
extinction debt, for example, arises from time delays between the start
of processes that erode biodiversity and when species go extinct
(Tilman\emph{et al.} 1994; Kuussaari \emph{et al.} 2009). Biodiversity
recovery following conservation interventions can likewise be delayed
due to temporal lags in species' responses to those actions (Watts
\emph{et al.} 2020). Time lags are a particular feature of biological
invasions (Williamson 1996; Crooks 2005; Aikio \emph{et al.} 2010). For
example, delays of decades to centuries between introduction and
establishment in the wild (naturalisation) have been documented for many
invasive plant species (Kowarik 1995; Caley \emph{et al.} 2008; Daehler
2009; van Klinken \emph{et al.} 2015), leading to the concept of an
invasion debt. Accelerating global trade and transport, leading to
dramatic increases in the numbers of species introduced to regions
outside their native range (Meyerson \& Mooney 2007; Hulme 2009; Bradley
\emph{et al.} 2012; Seebens \emph{et al.} 2015; Sikes \emph{et al.}
2018), has resulted in an expanding pool of introduced species that will
naturalise in the future, some proportion of which will become
problematic invaders (Essl\emph{et al.} 2011; Rouget \emph{et al.} 2016;
Dehnen-Schmutz \& Conroy 2018; Haeuser \emph{et al.} 2018).

The examples above imply that the consequences of certain actions are
yet to play out, with the trajectory of future ecological outcomes
depending on the nature of time lags associated with those actions.
While we can often identify factors likely to cause time lags in
ecological processes (e.g., Geerts \emph{et al.} 2013;
Tenhumberg\emph{et al.} 2018; Watts \emph{et al.} 2020), we currently
lack a framework for quantitatively analysing lag times and forecasting
their consequences, such as the size of an invasion or extinction debt.
My aim in this paper is to show how approaches from survival analysis
can be used to understand time lags arising from stochastic processes,
focusing on the lag between introduction and naturalisation, and the
associated invasion debt in plant naturalisations. I show how the hazard
function provides an intuitive way to understand how the risk of an
outcome changes over time, with the time-varying shape of the hazard
function determining the expected distribution of lag times.
Consequently, we can use data on lag times to assess how risk has
changed over time, and we can predict the shape of lag time
distributions given expectations about changing risk and evaluate those
predictions against data.

For plant naturalisations, I first outline in theory how the shape of
the hazard function, and hence the risk of naturalisation, should
increase for many species following introduction, and how the hazard
function should steepen for more recently introduced species. These
shifts in the shape of the hazard function imply predictable changes
over time in the distribution of the lag times between introduction and
naturalisation. I test these predictions using data from Britain, show
that the data match the theory well, describe how the approach can be
used to estimate the invasion debt, and consider the implications of
these findings.

\textbf{Materials and Methods}

\textbf{Theoretical distribution of lag times for plant naturalisation}

Globally an estimated 170000 plant species are cultivated in gardens and
parks and most of the world's approximately 13000 naturalised plant
species are garden escapees (Reichard \& White 2001; Niemiera \& Holle
2009; Bradley \emph{et al.} 2012; van Kleunen \emph{et al.} 2018). In
some regions, historical nursery and planting records provide estimates
of the time lag between introduction and naturalisation, which vary
widely among species ranging from years to centuries (Kowarik 1995;
Caley \emph{et al.} 2008; Daehler 2009; van Klinken \emph{et al.} 2015).
What underlies such extreme variation? And what form should lag time
distributions take?

Lag time can be thought of as a random variable resulting from the
stochastic nature of plant naturalisations (Mack 2000; Rejm\selectlanguage{ngerman}ánek \emph{et
al.} 2005). Consider that each plant grown in cultivation has some
probability its propagules will establish a naturalised population per
unit time. The probability a species will naturalise per time interval
should depend on the number of plants of that species in cultivation,
with more plants releasing more propagules into the environment, any one
of which could establish a naturalised population if they encounter the
right conditions (Duncan \emph{et al.} 2014). In line with this,
planting effort is often the strongest predictor of whether plant
species have naturalised or not (Rejmánek \emph{et al.} 2005;
Dehnen\selectlanguage{english}-Schmutz \emph{et al.} 2007; Py\selectlanguage{polish}š\selectlanguage{english}ek \emph{et al.} 2009;
McGregor\emph{et al.} 2012; Maurel \emph{et al.} 2016).

The time lag between introduction and naturalisation should depend on
the probability of naturalisation per unit time, with a higher
probability implying a shorter time lag. For a given species, however,
naturalisation probability could change over time following introduction
if the number of plants in cultivation changes. An increase in the
number of individuals of a species grown in gardens and parks over time,
for example, should cause the naturalisation risk to increase. Borrowing
from survival analysis, the change in naturalisation risk over time can
be quantified using a hazard function, with hazard defined as the
instantaneous rate of naturalisation, a number proportional to the
probability a species will naturalise in the next time step given it has
not yet naturalised (Muenchow 1986; Tableman \& Kim 2003).

The time-varying shape of the hazard function determines the shape of
the lag time distribution (and vice versa, see Appendix S1). If the
hazard for a group of species remains constant or declines over time
following introduction, their lag time distribution will be
monotonically declining (Fig 1A). For species that become popular garden
plants, however, we expect the hazard to increase over time following
introduction as they become widely planted (Mack 2000). If the hazard
increases over time following introduction, lag times will have a
unimodal distribution (Fig 1B-D).

\textbf{Changing lag time distributions}

For popular species, planting effort should increase over time following
introduction. In addition, we expect total planting effort across all
species to increase over time due to increasing horticultural activity
driven by an ever-growing human population and associated demand for
garden and amenity plantings (Lawson 1996; Dehnen-Schmutz \emph{et
al.}2010; Drew \emph{et al.} 2010; Humair \emph{et al.} 2015). Because
of this, relative to a species first introduced in the year 1800, we
would expect a similar species introduced in 2000 to have an initially
faster rate of planting following introduction. Consequently, an
increase in horticultural activity over time should result in steeper
hazard functions for more recently introduced species relative to
earlier introductions. A steepening of the hazard function over time
could also occur through land-use changes that, over the long-term,
result in more modified habitats providing greater opportunities for
plants to establish in the wild (Hobbs \& Huenneke 1996; Essl \emph{et
al.} 2011; Gonz\selectlanguage{ngerman}ález-Moreno \emph{et al.} 2017). Hence, both increasing
horticultural activity and greater habitat modification should increase
the naturalisation risk for more recent introductions, hastening the
rate at which species escape into the wild and establish.

I used a Weibull distribution and its associated hazard function to
explore how changes in the shape and steepness of the hazard function
alter the shape of the lag time distribution. The Weibull distribution
is commonly used to model time-to-event data because it has a flexible
hazard function that can take a variety of shapes (Carroll 2003;
Tableman \& Kim 2003; Fig. 1 and Appendix S1). If the hazard is constant
over time following introduction, the Weibull simplifies to a negative
exponential distribution (Fig 1A). If the hazard increases over time
following introduction, the Weibull can model that increase as downward
accelerating, linear or upward accelerating (Fig 1B-D). In Appendix S1 I
show that, regardless of its shape, as the slope of the hazard function
steepens (the different coloured lines in Fig. 1 panels), both the mean
and variance of the lag time distribution decline. This outcome is
evident in Figure 1B-D, with lag time distributions shifting toward zero
and becoming more peaked as the hazard functions steepen. Consequently,
if the hazard function for introduced species has steepened over time,
we expect species introduced in the more distant past to have, on
average, a longer lag time with greater among-species variation in their
lag times, while more recent introductions should have, on average,
shorter lag times with less among-species variation. I next test these
predictions using data on the lag times of naturalised plants introduced
to Britain.

\textbf{Data collection}

I obtained a list of all non-native flowering plants recorded as
naturalised in Britain from Stace \& Crawley (2015), which is the most
up-to-date published list currently available. I excluded taxa of
uncertain origin and taxa listed as casual or survivors, retaining only
taxa classed as naturalised, which Stace \& Crawley (2015) defined as
self-reproducing in the wild in Britain after 1987 (i.e., the list
excludes species recorded as naturalised before 1987 that no longer
appear to be self-reproducing in the wild). I obtained a list of the
naturalised non-flowering plants (gymnosperms and ferns) from Stace
(2010) and Hill \emph{et al} . (2005). Both lists included hybrids,
cultivars, and subspecies. For convenience, I term each taxon a species.
For each species in the combined list, I searched for information on the
year the species was first recorded as grown in cultivation in Britain
(which I term year of introduction), and the year it was first recorded
as growing outside cultivation in Britain (which I term year of
naturalisation), using Hill \emph{et al} . (2005), Stace \& Crawley
(2015) and The Online Atlas of the British and Irish Flora
(https://www.brc.ac.uk/plantatlas/ accessed in December 2020). In Hill
\emph{et al} . (2005), the year of introduction was sometimes given as a
range when it could not be determined precisely. In those cases, I used
the earliest year of introduction, and I used the earliest recorded year
of introduction and naturalisation when sources differed.

For the analyses in this paper, I included only naturalised species with
data for both the year of first introduction and naturalisation, and I
excluded species introduced before 1500 AD because their introduction
dates were less certain. The final list comprised 708 naturalised
species in 99 plant families. The year in which species were recorded as
first introduced ranged from 1500 to 1960, while the year in which
species were recorded as first naturalised ranged from 1597 to 2000. I
calculated the lag time (in years) as the year of first recorded
naturalisation minus the year of first recorded introduction.

I classified each species into one of three life-form categories:
tree/shrub (213 species), perennial herb (398 species) or
annual/biennial herb (97 species). I did this because we expect that, on
average, lag times should differ among these life-form categories, with
longer-lived, slower-growing species taking longer to reach reproductive
maturity and hence taking longer to produce propagules that could
establish wild populations relative to faster-growing, shorter-lived
species (Wangen \& Webster 2006; Caley \emph{et al.} 2008). The
life-form classifications were determined primarily from information in
The Online Atlas of the British and Irish Flora
(https://www.brc.ac.uk/plantatlas/).

\textbf{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
year\emph{Y\textsubscript{i}} , we can only observe the lag times of
species that have naturalised up to the present year
\emph{Y\textsubscript{p}} . Other species introduced in year
\emph{Y\textsubscript{i}} could naturalise in the future, meaning the
observed distribution of lag times is truncated at an upper limit
\emph{Y\textsubscript{p}} --\emph{Y\textsubscript{i}} . 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
\emph{Y\textsubscript{p}} 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.

\textbf{Model fitting}

I fitted the models to the lag time data in a Bayesian framework in R
version 4.0.3 (R Core Team 2017) using a Metropolis-Hastings algorithm
(modified from the R package MHadaptive; Chivers 2012) to estimate the
posterior distributions of the parameters. Being a Bayesian model, I
specified a prior distribution for all parameters using relatively
uninformative priors: a normal distribution with mean zero and standard
deviation 10. I ran the models in an adaptive phase for 20000
iterations, using the final 2000 iterations to determine the parameters
for the proposal distribution and initial parameter starting values. I
then ran three chains for 10000 iterations using the proposal
distribution and initial parameter values from the adaptive phase, and
checked the resulting parameters for convergence using the Gelman-Rubin
statistic (Gelman \& Rubin 1992), which was less than 1.1 for all
parameters indicating adequate convergence. I identified the best
performing model using the approximate leave-one-out cross-validation
(LOO) criteria (Vehtari \emph{et al.} 2017), which estimates the
predictive accuracy of each model. LOO is considered an improvement on
other information-criterion based model selection measures such as AIC,
WAIC and DIC that are widely used to compare model performance (see
Vehtari \emph{et al.} 2017 for details). I used Pareto smoothed
importance sampling to estimate LOO (PSIS-LOO) and compared models by
calculating the difference in expected predictive accuracy (\selectlanguage{greek}Δ\selectlanguage{english}PSIS-LOO)
between each model and the best-performing model on the deviance scale
using the \emph{loo} package in R (Vehtari \emph{et al.} 2019), with
smaller PSIS-LOO values implying a model with better predictive
accuracy.

\textbf{Assessing model fit and estimating the invasion debt}

I used the best-performing model based on the LOO criteria to model the
full lag time distribution and used simulation to assess how well the
modelled lag time distribution matched the data. To do this, I simulated
a lag time for each species given its actual year of introduction by
drawing a random value from the lag time distribution specified by the
best-performing model. I carried out 10000 simulated draws for each
species, binned the simulated lag times into 20 year intervals,
calculated the mean and 95\% quantiles for the number of species in each
bin, and compared these simulated outcomes with the actual distribution
of lag times in 20 year bins.

For a species introduced in year \emph{Y\textsubscript{i}} , the
modelled lag time distribution for that year allows us to calculate the
probability that the species will naturalise in
year\emph{Y\textsubscript{t}} , where
\emph{Y\textsubscript{t}}\textgreater{} \emph{Y\textsubscript{i}} . The
total number of species expected to naturalise in year
\emph{Y\textsubscript{t}} can be calculated as the sum of the
probabilities that each previously introduced species will naturalise in
year \emph{Y\textsubscript{t}} . We can therefore use this approach to
estimate the number of species expected to naturalise in each year
beyond the present and sum those estimates to calculate the invasion
debt (Fig. 2). The difficulty is that we do not know the total number of
species introduced each year that are going to naturalise because we
only observe species that have naturalised. In Appendix S3, I show how
the fitted lag time distribution coupled with the introduction dates can
be used to estimate the total number of species introduced each year
that will naturalise, and hence estimate the invasion debt. For each
life-form group, I used the methods in Appendix S3 to model the average
number of species introduced per year between 1500 and 1960 that have or
will naturalise in the foreseeable future, and thus calculate the
invasion debt.

\textbf{Simulation check and uncertainty in introduction and
naturalisation dates}

I used simulation to check the methods described above worked as
expected, using known parameter values to generate artificial data
similar to the real data, fitting the lag time model to the artificial
data and estimating the invasion debt, and assessing how well the
methods recovered the true parameter values using repeated simulations.
Full details are in Appendix S4.

The dates of first introduction and naturalisation in this study are
taken from historical records and are subject to uncertainty, notably
because the true dates might not have been documented or the records
have not been uncovered, resulting in first record dates being more
recent than true dates. While dates can be updated as new records are
discovered, any set of dates will be subject to uncertainty due to the
difficulty of determining precisely when past events occurred. The
question is whether this uncertainty could affect the results reported
in this study. To evaluate this, I used artificial data simulation as
described above but added measurement error to the simulated dates. I
then assessed how well the methods recovered the true parameter values
in the presence of dating errors using repeated simulations. Full
details are in Appendix S5.

\textbf{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 \emph{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).

\textbf{Discussion}

Time lags are a feature of many ecological processes. Here, I have shown
how hazard functions underpin the distribution of lag times arising from
stochastic processes, and consequently how data on lag times can be used
to examine changes in the risk of an outcome over time. I applied this
framework to understanding lag times between plant introduction and
naturalisation and the associated invasion debt, but the approach has
wider applicability. It could, for example, be used to estimate the size
of an extinction debt from the pattern of extinctions over time
(Elphick\emph{et al.} 2010).

For plant naturalisations in Britain, the key finding is that lag times
between introduction and naturalisation have a unimodal distribution
with mean and variance that have declined substantially over the last
500 years, implying that, on average: a) for naturalised species,
hazards increase over time following introduction; and b) hazard
functions are steeper for more recently introduced species.

Changes over time in the shape of the lag time distribution have
consequences for the invasion debt, implying that recently introduced
species with the potential to naturalise will realise that potential
faster than species introduced in the past. The total number of species
introduced to Britain for cultivation has likely increased substantially
over the last century in line with global gardening trends (Lawson 1996;
Drew \emph{et al.} 2010; Humair \emph{et al.} 2015; Dehnen-Schmutz \&
Conroy 2018). The analyses presented here suggest that the invasion debt
associated with introductions prior to 1960 has largely been realised:
just over 80\% (708/866) of species purposefully introduced to Britain
before 1960 that were capable of naturalising are estimated to have
naturalised by the year 2000 (Fig 5).

Of the species yet to naturalise, most are predicted to be longer-lived
trees or shrubs. This likely reflects the fact that trees/shrubs have,
on average, a longer lag-phase than shorter-lived perennial herbaceous
and biennial/annual species (Fig 3C; see also Caley \emph{et al.} 2008).
But it also results from a shifting trend in introductions, with the
numbers of newly introduced trees/shrubs capable of naturalising
increasing at a faster rate over time than perennial herbs and
annual/biennial species (compare the curved lines in Fig 5 A, C, E).

The decline in the mean and variance of the lag time distribution (Fig.
4) and the associated steepening of hazard functions (Fig. 3D-F) imply
that the probability of a species naturalising per time interval has
increased over time in Britain. I have argued that a reduction in lag
times is most likely driven by increased horticultural activity
associated with accelerating human population growth. For species that
become widely planted, and hence more likely to naturalise, an increase
over time in the rate at which species are propagated, distributed and
planted will accelerate the rate at which cultivated plants can supply
propagules, increasing the probability that species capable of
naturalising will do so in a given time period. Changes in other factors
that increase naturalisation probability could also contribute to or
account for changes in the lag time distribution. In particular, an
increase over time in the extent and severity of habitat disturbance
could result in more opportunities for plants to naturalise in
increasingly modified environments (Hobbs \& Huenneke 1996; Essl\emph{et
al.} 2011; Gonz\selectlanguage{ngerman}ález-Moreno \emph{et al.} 2017). Changing climate could
also alter naturalisation probability (Dullinger \emph{et al.} 2017;
Haeuser \emph{et al.} 2018) although anthropogenic climate change, whose
impacts have been most pronounced in the last 100 years, is unlikely to
account for the sustained decline in the mean and variance of the lag
time distribution in Britain over the last 500 years (Fig. 4). Finally,
lag times could shorten through a systematic change in recording effort
if, for example, efforts to document newly naturalised species have
increased over time. The simulations assessing the effects of dating
errors reported in Appendix S5 did not consider a systematic shift in
recording effort over time, but such a shift would have to be large to
generate the observed outcomes.

The overall mean lag-phase in this study (145 years) was similar to the
mean lag times calculated for temperate woody species in two previous
studies (Kowarik 1995; Caley \emph{et al.} 2008), but longer than
recorded for introduced grasses in Australia (van Klinken \emph{et
al.}2015) and trees in Hawaii (Daehler 2009). Here, and in Caley
\emph{et al} . (2008), the mean lag time between introduction and
naturalisation differed by plant life-form, with longer-lived species
having longer lag times. Nevertheless, the difference in mean lag time
between trees/shrubs and biennial/annual species (around 190 years for
species introduced in 1500) was substantially less than the decline over
time for trees/shrubs (mean lag time declined by about 500 years from
1500 to 1960). Hence, lag times are not a fixed property of species or
locations but vary markedly in response to factors affecting
naturalisation probability that can change over time, such as planting
effort and habitat modification. While other plant traits have the
potential to influence lag times, the extent to which lag time
distributions have changed over time in Britain suggests that temporally
varying factors may be as or more important than species traits in
determining the length of the lag phase among species capable of
naturalising (see also Mack 2000).

Britain, like other European countries, may be unique in having a long
history of plant introductions and records that allow lag times to be
reconstructed (see also Kowarik 1995). It is unclear whether the marked
shift in lag time distributions over almost five centuries in Britain
would be replicated in other regions with a shorter history of sustained
plant introductions, such as followed European colonisation of the
Americas, Australia, southern Africa and New Zealand. I predict the same
processes and outcomes will be operating, albeit with a compressed
timeframe in regions with a shorter introduction history. Consequently,
the global invasion debt associated with non-native species already
introduced for horticulture may be substantially less than expected
based on previously reported lag times (Kowarik 1995; Caley \emph{et
al.} 2008; Essl \emph{et al.} 2011; van Klinken \emph{et al.} 2015; but
see Daehler 2009). This has implications for preventing future plant
naturalisations. It suggests that, while an existing pool of cultivated
species in a region will contain species capable of naturalising in the
future, that risk may be lower than expected, and lower than the risk
posed by future introductions of new species, which are capable of
naturalising more rapidly than in the past. Models that aim to predict
future invasion trajectories need to account for the changing nature of
lag time distributions (Leung \emph{et al.} 2012; Essl \emph{et
al.}2019).

\textbf{ACKNOWLEDGEMENT}

I am very grateful to Tim Blackburn, Lizzie Wandrag and three reviewers
for insightful and very helpful comments on previous versions of the
manuscript.

\textbf{References}

Aikio, S., Duncan, R.P. \& Hulme, P.E. (2010). Lag-phases in alien plant
invasions: separating the facts from the artefacts. \emph{Oikos} , 119,
370--378.

Bradley, B.A., Blumenthal, D.M., Early, R., Grosholz, E.D., Lawler,
J.J., Miller, L.P., \emph{et al.} (2012). Global change, global trade,
and the next wave of plant invasions. \emph{Frontiers in Ecology and the
Environment} , 10, 20--28.

Caley, P., Groves, R.H. \& Barker, R. (2008). Estimating the invasion
success of introduced plants. \emph{Diversity and Distributions} , 14,
196--203.

Carroll, K.J. (2003). On the use and utility of the Weibull model in the
analysis of survival data. \emph{Controlled Clinical Trials} , 24,
682--701.

Chivers, C. (2012). \emph{MHadaptive: general Markov Chain Monte Carlo
for Bayesian inference using adaptive Metropolis-Hastings sampling} .
https://cran.r-project.org/web/packages/MHadaptive/index.html.

Crooks, J.A. (2005). Lag times and exotic species: The ecology and
management of biological invasions in slow-motion. \emph{Ecoscience} ,
12, 316--329.

Daehler, C.C. (2009). Short lag times for invasive tropical plants:
evidence from experimental plantings in Hawai'i. \emph{PLOS ONE} , 4,
e4462.

Dehnen-Schmutz, K. \& Conroy, J. (2018). Working with gardeners to
identify potential invasive ornamental garden plants: testing a citizen
science approach. \emph{Biol Invasions} , 20, 3069--3077.

Dehnen-Schmutz, K., Holdenrieder, O., Jeger, M.J. \& Pautasso, M.
(2010). Structural change in the international horticultural industry:
Some implications for plant health. \emph{Scientia Horticulturae} , 125,
1--15.

Dehnen\selectlanguage{english}-Schmutz, K., Touza, J., Perrings, C. \& Williamson, M. (2007).
The horticultural trade and ornamental plant invasions in
Britain.\emph{Conservation Biology} , 21, 224--231.

Drew, J., Anderson, N. \& Andow, D. (2010). Conundrums of a complex
vector for invasive species control: a detailed examination of the
horticultural industry. \emph{Biol Invasions} , 12, 2837--2851.

Dullinger, I., Wessely, J., Bossdorf, O., Dawson, W., Essl, F.,
Gattringer, A., \emph{et al.} (2017). Climate change will increase the
naturalization risk from garden plants in Europe. \emph{Global Ecology
and Biogeography} , 26, 43--53.

Duncan, R.P., Blackburn, T.M., Rossinelli, S. \& Bacher, S. (2014).
Quantifying invasion risk: the relationship between establishment
probability and founding population size. \emph{Methods Ecol Evol} , 5,
1255--1263.

Elphick, C.S., Roberts, D.L. \& Michael Reed, J. (2010). Estimated dates
of recent extinctions for North American and Hawaiian
birds.\emph{Biological Conservation} , 143, 617--624.

Essl, F., Dullinger, S., Rabitsch, W., Hulme, P.E., Hulber, K., Jaro\selectlanguage{polish}š\selectlanguage{ngerman}ík,
V., \emph{et al.} (2011). Socioeconomic legacy yields an invasion
debt.\emph{PNAS} , 108, 203--207.

Essl, F., Lenzner, B., Courchamp, F., Dullinger, S., Jeschke, J.M.,
Kühn, I., \emph{et al.} (2019). Introducing AlienScenarios: a project to
develop scenarios and models of biological invasions for the 21 st
century. \emph{NeoBiota} , 45, 1--17.

Geerts, S., Moodley, D., Gaertner, M., Roux, J.J.L., McGeoch, M.A.,
Muofhe, C., \emph{et al.} (2013). The absence of fire can cause a lag
phase: The invasion dynamics of Banksia ericifolia
(Proteaceae).\emph{Austral Ecology} , 38, 931--941.

Gelman, A. \& Rubin, D.B. (1992). Inference from iterative simulation
using multiple sequences. \emph{Statistical Science} , 7, 457--472.

González-Moreno, P., Pino, J., Cózar, A., García-de-Lomas, J. \& Vilà,
M. (2017). The effects of landscape history and time-lags on plant
invasion in Mediterranean coastal habitats. \emph{Biol Invasions} , 19,
549--561.

Haeuser, E., Dawson, W., Thuiller, W., Dullinger, S., Block, S.,
Bossdorf, O., \emph{et al.} (2018). European ornamental garden flora as
an invasion debt under climate change. \emph{Journal of Applied Ecology}
, 55, 2386--2395.

Hill, M., Baker, R., Broad, G., Chandler, P., Copp, G., Ellis,
J.,\emph{et al.} (2005). \emph{Audit of non-native species in England} .
English Nature Research Reports. Centre for Ecolgy \& Hydrology.

Hobbs, R.J. \& Huenneke, L.F. (1996). Disturbance, diversity, and
invasion: implications for conservation. In: \emph{Ecosystem Management:
Selected Readings} (eds. Samson, F.B. \& Knopf, F.L.). Springer, New
York, NY, pp. 164--180.

Hulme, P.E. (2009). Trade, transport and trouble: managing invasive
species pathways in an era of globalization. \emph{Journal of Applied
Ecology} , 46, 10--18.

Humair, F., Humair, L., Kuhn, F. \& Kueffer, C. (2015). E-commerce trade
in invasive plants. \emph{Conservation Biology} , 29, 1658--1665.

Justus, C.G., Hargraves, W.R., Mikhail, A. \& Graber, D. (1978). Methods
for estimating wind speed frequency distributions. \emph{J. Appl.
Meteor.} , 17, 350--353.

van Kleunen, M., Essl, F., Pergl, J., Brundu, G., Carboni, M.,
Dullinger, S., \emph{et al.} (2018). The changing role of ornamental
horticulture in alien plant invasions. \emph{Biological Reviews} , 93,
1421--1437.

van Klinken, R.D., Panetta, F.D., Coutts, S. \& Simon, B.K. (2015).
Learning from the past to predict the future: an historical analysis of
grass invasions in northern Australia. \emph{Biol Invasions} , 17,
565--579.

Kowarik, I. (1995). Time lags in biological invasions with regard to the
success and failure of alien species. In: \emph{Plant invasions: general
aspects and special problems} (eds. Py\selectlanguage{polish}š\selectlanguage{english}ek, P., Prach, K., Rejm\selectlanguage{ngerman}ánek, M.
\& Wade, M.). SPB Academic Publishing, Amsterdam, pp. 15--38.

Kuussaari, M., Bommarco, R., Heikkinen, R.K., Helm, A., Krauss, J.,
Lindborg, R., \emph{et al.} (2009). Extinction debt: a challenge for
biodiversity conservation. \emph{Trends in Ecology \& Evolution} , 24,
564--571.

Lawson, R.H. (1996). Economic importance and trends in ornamental
horticulture. \emph{Acta horticulturae} , 432, 226--237.

Leung, B., Roura\selectlanguage{english}-Pascual, N., Bacher, S., Heikkila, J., Brotons, L.,
Burgman, M.A., \emph{et al.} (2012). TEASIng apart alien species risk
assessments: a framework for best practices. \emph{Ecology Letters} ,
15, 1475--1493.

Mack, R.N. (2000). Cultivation fosters plant naturalization by reducing
environmental stochasticity. \emph{Biological Invasions} , 2, 111--122.

Maurel, N., Hanspach, J., Kuhn, I., Py\selectlanguage{polish}š\selectlanguage{english}ek, P. \& Kleunen, M. van.
(2016). Introduction bias affects relationships between the
characteristics of ornamental alien plants and their naturalization
success. \emph{Global Ecology and Biogeography} , 25, 1500--1509.

McGregor, K.F., Watt, M.S., Hulme, P.E. \& Duncan, R.P. (2012). What
determines pine naturalization: species traits, climate suitability or
forestry use? \emph{Diversity Distrib.} , 18, 1013--1023.

Meyerson, L.A. \& Mooney, H.A. (2007). Invasive alien species in an era
of globalization. \emph{Frontiers in Ecology and the Environment} , 5,
199--208.

Muenchow, G. (1986). Ecological use of failure time
analysis.\emph{Ecology} , 67, 246--250.

Niemiera, A.X. \& Holle, B.V. (2009). Invasive Plant Species and the
Ornamental Horticulture Industry. In: \emph{Management of Invasive
Weeds} , Invading Nature -- Springer Series In Invasion Ecology (ed.
Inderjit). Springer Netherlands, Dordrecht, pp. 167--187.

Py\selectlanguage{polish}š\selectlanguage{english}ek, P., K\selectlanguage{polish}ř\selectlanguage{english}iv\selectlanguage{ngerman}ánek, M. \& Jaro\selectlanguage{polish}š\selectlanguage{ngerman}ík, V. (2009). Planting intensity,
residence time, and species traits determine invasion success of alien
woody species. \emph{Ecology} , 90, 2734--2744.

R Core Team. (2017). \emph{R: A language and environment for statistical
computing} . https://www.R-project.org/.

Reichard, S.H. \& White, P. (2001). Horticulture as a pathway of
invasive plant introductions in the United States. \emph{BioScience} ,
51, 103--113.

Rejmánek, M., Richardson, D.M., Higgins, S., Pitcairn, M. \& Grotkopp,
E. (2005). Ecology of invasive plants: state of the art.
In:\emph{Invasive alien species: searching for solutions} (eds. Mooney,
H., Mack, R., McNeely, J., Neville, L., Schei, P. \& Waage, J.). Island
Press, Washington, D.C., pp. 104--161.

Rouget, M., Robertson, M.P., Wilson, J.R.U., Hui, C., Essl, F.,
Renteria, J.L., \emph{et al.} (2016). Invasion debt -- quantifying
future biological invasions. \emph{Diversity and Distributions} , 22,
445--456.

Seebens, H., Essl, F., Dawson, W., Fuentes, N., Moser, D., Pergl,
J.,\emph{et al.} (2015). Global trade will accelerate plant invasions in
emerging economies under climate change. \emph{Global Change Biology} ,
21, 4128--4140.

Sikes, B.A., Bufford, J.L., Hulme, P.E., Cooper, J.A., Johnston, P.R. \&
Duncan, R.P. (2018). Import volumes and biosecurity interventions shape
the arrival rate of fungal pathogens. \emph{PLoS Biol} , 16, e2006025.

Stace, C. (2010). \emph{New Flora of the British Isles} . Cambridge
University Press.

Stace, C.A. \& Crawley, M.J. (2015). \emph{Alien Plants} . William
Collins.

Tableman, M. \& Kim, J.S. (2003). \emph{Survival analysis using S:
analysis of time-to-event data} . Chapman and Hall/CRC.

Tenhumberg, B., Crone, E.E., Ramula, S. \& Tyre, A.J. (2018).
Time-lagged effects of weather on plant demography: drought and
Astragalus scaphoides. \emph{Ecology} , 99, 915--925.

Tilman, D., May, R.M., Lehman, C.L. \& Nowak, M.A. (1994). Habitat
destruction and the extinction debt. \emph{Nature} , 371, 65--66.

Vehtari, A., Gabry, J., Magnusson, M., Yao, Y. \& Gelman, A.
(2019).\emph{loo: Efficient leave-one-out cross-validation and WAIC for
Bayesian models} . https://mc-stan.org/loo.

Vehtari, A., Gelman, A. \& Gabry, J. (2017). Practical Bayesian model
evaluation using leave-one-out cross-validation and WAIC. \emph{Stat
Comput} , 27, 1413--1432.

Wangen, S.R. \& Webster, C.R. (2006). Potential for multiple lag phases
during biotic invasions: reconstructing an invasion of the exotic
tree\emph{Acer platanoides} . \emph{Journal of Applied Ecology} , 43,
258--268.

Watts, K., Whytock, R.C., Park, K.J., Fuentes-Montemayor, E., Macgregor,
N.A., Duffield, S., \emph{et al.} (2020). Ecological time lags and the
journey towards conservation success. \emph{Nature Ecology \& Evolution}
, 4, 304--311.

Williamson, M. (1996). \emph{Biological Invasions} . Springer Science \&
Business Media.

\textbf{Table 1.} Ten candidate models to describe lag time
distributions for naturalised plants in Britain. Models 1-5 specified a
truncated normal distribution with T() the upper limit for
truncation.\emph{Y\textsubscript{P}} is the present (set to the year
2000),\emph{Y\textsubscript{i}} is the year of introduction of the
\emph{i} th species and \emph{P\textsubscript{i}} and
\emph{A\textsubscript{i}} are variables specifying whether the \emph{i}
th species was perennial (\emph{P\textsubscript{i}} = 1) or not
(\emph{P\textsubscript{i}} = 0), and annual (\emph{A\textsubscript{i}} =
1) or not (\emph{A\textsubscript{i}} = 0). The \selectlanguage{greek}\emph{β} \selectlanguage{english}are the
parameters estimated for each model. Log-transformations of the mean
{[}e.g.,\(\mathrm{\log}\left(\mu_{Y_{i}}\right)\){]} and standard deviation {[}e.g.,
\(\mathrm{\log}\left(\sigma_{Y_{i}}\right)\){]} ensured these remained positive. Models 6-10
specified a truncated Weibull distribution parameterised in terms of a
shape (\emph{k} ) and scale (\selectlanguage{greek}\emph{λ} \selectlanguage{english}) parameter, with \emph{k} and
\selectlanguage{greek}\emph{λ} \selectlanguage{english}related to the mean and standard deviation as shown (Justus
\emph{et al.} 1978). \selectlanguage{greek}Δ\selectlanguage{english}PSIS-LOO is the difference in PSIS-LOO (Pareto
smoothed importance sampling leave-one-out cross-validation) between
each model and the best fitting model, which had \selectlanguage{greek}Δ\selectlanguage{english}PSIS-LOO = 0 (shown in
bold).\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
\begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
Lag time distribution\strut
\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
Model\strut
\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
Mean \(\mathrm{\log}\left(\mu_{Y_{i}}\right)=\)\strut
\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
Standard deviation \(\mathrm{\log}\left(\sigma_{Y_{i}}\right)=\)\strut
\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
\selectlanguage{greek}Δ\selectlanguage{english}PSIS-LOO\strut
\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
Standard error of \selectlanguage{greek}Δ\selectlanguage{english}PSIS-LOO\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\mathrm{\text{Normal}}\left(\mu_{Y_{i}},\sigma_{Y_{i}}\right)\mathrm{T}\left(Y_{p}-Y_{i}\right)\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
1\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{2}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-215.0\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
26.4\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
2\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\beta_{2}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{3}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-161.0\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
22.8\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
3\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\beta_{2}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{3}+\beta_{4}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-52.3\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
14.0\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\textbf{4}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\mathbf{\beta}_{\mathbf{1}}\mathbf{+\ }\mathbf{\beta}_{\mathbf{2}}\mathbf{P}_{\mathbf{i}}\mathbf{+\ }\mathbf{\beta}_{\mathbf{3}}\mathbf{A}_{\mathbf{i}}\mathbf{+\ }\mathbf{\beta}_{\mathbf{4}}\mathbf{Y}_{\mathbf{i}}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\mathbf{\beta}_{\mathbf{5}}\mathbf{+}\mathbf{\beta}_{\mathbf{6}}\mathbf{Y}_{\mathbf{i}}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\textbf{0}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\textbf{0}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
5\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\ \beta_{2}P_{i}+\ \beta_{3}A_{i}+\ \beta_{4}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{5}+\ \beta_{6}P_{i}+\ \beta_{7}A_{i}+\ \beta_{8}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-1.5\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
1.9\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\(\mathrm{\text{Weibull}}\left(k_{Y_{i}},\ \lambda_{Y_{i}}\right)\mathrm{T}\left(Y_{p}-Y_{i}\right)\) \(k_{Y_{i}}=\left(\frac{\sigma_{Y_{i}}}{\mu_{Y_{i}}}\right)^{-1.086}\) \(\lambda_{Y_{i}}=\frac{\mu_{Y_{i}}}{\Gamma\left(1+\frac{1}{k}\right)}\)\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
6\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\(\beta_{1}\)\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\(\beta_{2}\)\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
-116.2\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
21.4\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
7\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\beta_{2}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{3}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-90.9\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
22.4\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
8\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\beta_{2}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{3}+\beta_{4}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-79.5\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
19.3\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
9\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\ \beta_{2}P_{i}+\ \beta_{3}A_{i}+\ \beta_{4}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{5}+\beta_{6}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-72.8\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
19.6\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
10\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{1}+\ \beta_{2}P_{i}+\ \beta_{3}A_{i}+\ \beta_{4}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
\(\beta_{5}+\ \beta_{6}P_{i}+\ \beta_{7}A_{i}+\ \beta_{8}Y_{i}\)\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-21.6\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
12.7\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}



\textbf{FIGURE CAPTIONS}

\textbf{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 \emph{k} determines
the shape of the hazard function: \textbf{A} if \emph{k} = 1, the hazard
is constant over time and the lag time distribution is negative
exponential; \textbf{B} if 1 \textless{} \emph{k} \textless{} 2 the
hazard function is downward accelerating and the lag time distribution
shifts to unimodal for all \emph{k} \textgreater{} 1; \textbf{C}
if\emph{k} = 2 the hazard function is linear; \textbf{D} if
\emph{k}\textgreater{} 2 the hazard function is upward accelerating.
Each panel shows three hazard functions and associated lag time
distributions given the shape parameter \emph{k} , with distributions
having a mean lag time of approximately 100 (red lines), 200 (blue
lines) or 300 (black lines) years. In each of \textbf{B-D} , a steeper
hazard function results in the lag time distribution having a smaller
mean and variance (see Appendix S1 for details).

\textbf{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).

\textbf{Figure 3} . \textbf{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. \textbf{A} shows the estimated mean and standard deviation of
the expected lag times for trees and shrubs introduced in 1500
AD;\textbf{B} shows the rate at which the mean and standard deviation of
the lag time distribution changed per year after 1500 AD;
\textbf{C}shows 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 log\textsubscript{e}
scale. \textbf{D-F}Hazard 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.

\textbf{Figure 4} . The observed distribution of lag times (grey
histograms) for: \textbf{A} all species and \textbf{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.

\textbf{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 (\textbf{A, D} = trees/shrubs;
\textbf{B, E} = perennial herbs; \textbf{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 \textbf{A-C} , and the
most recent year of recorded naturalisation (2000) is shown as a dashed
line in\textbf{D-F} . The black curves in \textbf{D-F} show the expected
number of species naturalising in each 20 year interval based on the
actual dates of introduction (\textbf{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 \textbf{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
\textbf{A} ,\textbf{C} and \textbf{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).

\textbf{Figure 1} .

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\textbf{Figure 2} .

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\textbf{Figure 3} .

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\textbf{Figure 4} .

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\textbf{Figure 5} .

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