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.