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 et al.2010; Drew et al. 2010; Humair 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 et al. 2011; González-Moreno 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.