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.