Expansion speed—understood as the increase in the radius of a coherent circle having the same area as the AOO (area of occupancy)—is here suggested as a descriptive measure of the ability of an alien species to increase its spatial presence in an assessment area (see also Hill et al. 2001; Catford et al. 2016). Expansion speed represents one dimension of invasiveness, as AOO is one of the factors determining the impact of alien species on native biota (in addition to the local densities obtained and the per-capita ecological effects, Parker et al. 1999). Some impact assessment schemes of alien species explicitly incorporate spatial criteria (e.g. Essl et al. 2011; D’hont et al. 2015). Expansion speed, as defined in this paper, is currently used as one of three criteria describing invasion potential in the Generic Ecological Impact Assessment of Alien Species (GEIAA, Sandvik et al. 2019). This method is used to classify alien species in Norway and Sweden according to their environmental impact on native biota (Artsdatabanken 2018; Strand et al. 2018).
Some aspects of the expansion speed deserve further discussion, viz. its definition, its underlying assumptions, its measurement unit, and its estimation. This is done in the following paragraphs.
Expansion speed, as defined here and by some earlier authors (Hill et al. 2001), represents the increase in AOO, irrespective of the mechanisms or pathways involved. In addition to natural dispersal (active or passive), this includes human transport within the assessment area (intentional or unintentional), and even novel anthropogenic introductions into the assessment area. It might, at first sight, seem counterintuitive to lump together natural and anthropogenic factors into a single measure. However, the invasiveness of many alien species is mainly due to these anthropogenic factors. Arion vulgaris (Spanish slug) is a case in point: its expansion is not so much due to the high dispersal velocity of this gastropod, but rather due to its close association to horticultural products, which are transported over huge distances (Zemanova et al. 2016).
Being defined in terms of AOO, expansion speed should not be envisaged as an equivalent to dispersal velocity, to the velocity of an actual invasion front, or to the rate of range expansion. Only under highly unlikely conditions would these measures be identical, viz. if there is exactly one centre of spread (one source population) and if no grid cells behind the invasion front remain unoccupied. For example, in Fig. 3e, f, only minor parts of the species’ EOOs or ranges (the hatched polygons) are actually occupied (filled with circles). It is thus evident that expansion speed measures a different aspect of an invasion process than do other metrics, such as dispersal velocity.
Expansion speed is intended to quantify the increase in the spatial component of impact, which is closely related to AOO, but not necessarily to EOO, range or dispersal distance. As such, expansion speed is neither intended as a replacement of dispersal velocity, nor can it be inferred directly from estimates of dispersal velocity. On the other hand, the estimation of expansion speed is possible even in cases where the dispersal velocity is unknown, since all that is needed is a dataset with the time and place of observations. Such occupancy data are available from a number of sources (e.g. natural history collections in museums, national record databases, monitoring programs).
The model presented (Eq. 4) is valid for all kinds of expansion patterns. However, for its parameters to be estimable, one is bound to make some simplifying assumptions. The simplest set of assumptions—constant expansion speed and detection rate—requires the radius of the AOO to increase linearly, and thus AOO to increase quadratically over time (see Eq. 8 and Fig. 1). An assumption of quadratic increase in AOO is justified in several situations, e.g. whenever a species expands with an approximately bivariate-Gaussian dispersal kernel (e.g., McGeoch and Latombe 2016), independent of the number of source populations and of the proportion of grid cells occupied behind the invasion front.
However, other situations may warrant other functions, i.e. AOO may increase slower or faster than quadratically. For instance, AOO would increase linearly when a habitat specialist expands due to natural dispersal along linear habitats such as rivers, forest edges or coast lines. As an anthropogenic example of an approximately linear expansion one might think of a species that spreads minimally following introduction, so that all expansion is due to novel introductions (given a roughly constant propagule pressure).
In situations where most spread is due to long-distance dispersal (cf. Nathan et al. 2002; Katul et al. 2005), however, AOO would increase approximately exponentially and thus faster than quadratically. This would be the case for distributions whose mode is sufficiently far from the source location (such as a gamma distribution). Fat-tailed or stratified dispersal functions (cf. McGeoch and Latombe 2016) would be intermediate to the quadratic and the exponential situation.
Given that the increase of AOO is known or assumed to be non-quadratic, Eq. 4 would have to be modified accordingly (see modification 6 above). Several factors make it likely that even seemingly non-quadratic expansions can be approximated by the quadratic model (Eq. 8), however. First, even many linear habitats, such as rivers, can branch. Second, dispersal or anthropogenic transport may happen from one linear habitat plot, e.g. a forest edge, to another. In cases of long-distance dispersal, the increase in AOO is exponential only under the boundary condition that grid cells are very unlikely to be colonised from different source cells. In most realistic situations, this will not be the case, and the increase in AOO would be somewhat intermediate to the quadratic and the exponential model. It will differ from case to case whether the increase in AOO can be approximated to be quadratic. In principle, and given sufficient data, this can be tested by fitting different models to the set of observations and comparing the resulting AICC values.
From Which Part of the Expansion Process Should the Speed be Estimated?
If the increase of AOO cannot be approximated as quadratic, expansion speed is no longer constant during the invasion process. Cases in which AOO increases linearly or exponentially have just been described (cf. modification 6); in addition, there may be lag phases at the beginning of an expansion (Crooks 2005; Aikio et al. 2010; Coutts et al. 2018) and/or a deceleration of expansion speed when a species approaches full colonisation of its potential AOO (cf. modifications 4 and 5).
In all these cases, the expansion speed depends on when during the expansion process it is estimated. As a means of standardisation, it is suggested to report the highest realistic value of expansion speed that is measured, estimated or reported. This choice can be rationalised as following directly from the aim of expressing expansion potential as a propensity, and independently from contingent facts such as when the species had been introduced. If the average expansion speed was to be used, an initial lag phase would reduce the estimate. If the current expansion speed was to be used, a species that has (almost) completed its expansion, has no (or a low) expansion; as soon as the species is removed from parts of its AOO, however, it will again have the ability to expand much faster. This is why expansion speed ought to capture the highest potential expansion ability.
When only few years of data are available, a high estimate of expansion speed may be due to observation error, measurement error or very special (and non-representative) conditions. For this reason, the use of an upper confidence limit (e.g. the 75th or 95th percentile) may be more realistic than strictly reporting the maximum value. Furthermore, if the cause of high expansion speed in the past has ceased to exist and is very unlikely to return, estimates should not be based on the historical increase in AOO. For example, many terrestrial plant seeds were spread with the ballast soil of (sailing) vessels. Ballast soil having been replaced by ballast water, the former pathway is not relevant any more.
According to the definition proposed here, expansion speed is the annual increase in the radius of the AOO, thus measured in metres per year (m/a). In principle, different ways to measure, or units in which to express, expansion speed might have been possible, e.g. (i) as an absolute increase in AOO, measured in km2/a; (ii) as an increase in AOO relative to the AOO at the time of assessment (‘current AOO’), measured in percent per year; or (iii) as an increase in AOO relative to the area that might potentially get colonised (‘final AOO’), measured in percent per year. The disadvantage with such measurement units is that expansion speed would become a function of time: the absolute increase in AOO, as well as the increase relative to the final AOO, is positively related to the duration of the expansion; the increase relative to the current AOO is negatively related to the duration of the expansion.
Having said this, knowledge of current and future areas constitutes relevant information that ought to be made accessible to biodiversity management authorities. However, this is best accomplished in the form of separate estimates, rather than by trying to incorporate several measures into one estimate of expansion speed.
Estimates of AOO are always uncertain, because detection rates of occurrences hardly ever reach 100%. Detection rates are a function of sampling effort as well as biological characteristics of the species (its size, habitat, behaviour etc.). The ratio between the total AOO (observed plus unobserved occurrences) and the known AOO (observed occurrences only) is the dark figure of AOO. The dark figure is by definition unknown (unless it equals 1), but it may be estimated using appropriate sampling designs (see, e.g., Wintle et al. 2004; Stanley and Royle 2005; Christy et al. 2010; McCarthy et al. 2013; Kellner and Swihart 2014).
This uncertainty in AOO affects expansion speed, too (Preuss et al. 2014). Since the uncertainty in AOO is directly proportional to the dark figure, uncertainty in expansion speed is proportional to the square root of the dark figure. The model presented in this paper allows an estimation of detection rates (and dark figures) alongside the expansion speed and the start of the expansion. However, simulations have shown that detection rates tend to be somewhat underestimated (and, consequently, dark figures to be overestimated). Where expert judgements of the dark figure exist, these can be used to improve the estimates of expansion speed.
Application of the model to a few real-world datasets resulted in good model fit (Fig. 3), although a comparison with the true values of the estimates is of course not possible in this case. Furthermore, all species tested were terrestrial plants. It would thus be useful to test the model with datasets on more species, including a more diverse set of taxa and habitats.
For four of the species, models with variable detection rates were better supported than models with constant detection rates. For three species, there was a breakpoint at which detection rates increased (Table 3, Fig. 3c). This may have been due to increased sampling effort, e.g. due to a rise in awareness. For the fourth species (R. rugosa), detection rates varied in parallel with the sampling effort for all flowering plants (Fig. 3d). Both findings indicate that estimates can be improved considerably if knowledge about detection rates is made explicit and incorporated into the models.
Expansion speed has several useful properties. It is a rather intuitive measure of the ability of a species to increase its spatial presence in an assessment area. The latter constitutes an essential part of the impact of a species (Parker et al. 1999). Expansion speed is based on the AOO, which is an established measure in conservation biology (IUCN 2017). It is comparatively easy to estimate, in that it only requires occupancy data, rather than prolonged field studies of dispersal. It is a generic measure that allows a direct comparison of expansion speed across taxa and habitats (Artsdatabanken 2018; Strand et al. 2018). Finally, it is a quantitative measure, a property that increases the testability of impact assessments using expansion speed as a criterion (cf. Mace and Lande 1991).