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 et al. 2012; van Kleunen 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 et al. 2008; Daehler 2009; van Klinken 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; Rejmánek 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 et al. 2014). In line with this, planting effort is often the strongest predictor of whether plant species have naturalised or not (Rejmánek et al. 2005; Dehnen‐Schmutz et al. 2007; Pyšek et al. 2009; McGregoret al. 2012; Maurel 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).