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 (Elphicket 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 et al. 2010; Humair 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 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; Esslet al. 2011; González-Moreno et al. 2017). Changing climate could also alter naturalisation probability (Dullinger et al. 2017; Haeuser 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 et al. 2008), but longer than recorded for introduced grasses in Australia (van Klinken et al.2015) and trees in Hawaii (Daehler 2009). Here, and in Caley 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 et al. 2008; Essl et al. 2011; van Klinken 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 et al. 2012; Essl et al.2019).