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 (Tilmanet al. 1994; Kuussaari et al. 2009). Biodiversity recovery following conservation interventions can likewise be delayed due to temporal lags in species’ responses to those actions (Watts et al. 2020). Time lags are a particular feature of biological invasions (Williamson 1996; Crooks 2005; Aikio 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 et al. 2008; Daehler 2009; van Klinken 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 et al. 2012; Seebens et al. 2015; Sikes 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 (Esslet al. 2011; Rouget et al. 2016; Dehnen-Schmutz & Conroy 2018; Haeuser 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 et al. 2013; Tenhumberget al. 2018; Watts 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.