Invasive species can have enormous economic and ecological impacts (Perrings et al. 2000, Pimentel 2011, Simberloff et al. 2005, Simberloff 2013). If we can assess the invasion threat early in the invasion process, we are likely to have more success in controlling the species, and suffer less impact than if we cannot. In general, the rate of spread of an invasive species will influence practical issues around our ability to control its spread and thus its ultimate impact. Targeted research to address the early, ‘post-entry’ stage of invasion is critical to inform management strategies and ultimately to improve biosecurity.

This article focuses on this ‘post-entry spread’ stage of the invasion process, specifically understanding dispersal processes and modelling the spread and establishment potential of a pest or pathogen once it has arrived into a region. We differentiate this from the population dynamics and dispersal of native species, as post-entry pest spread of non-native species has particular features that add to the modelling challenge. These include the requirement for rapid response, data paucity and high levels of uncertainty.

Whilst the majority of pest and pathogen entry today is largely due to anthropogenic pathways (Wilson et al. 2009), mainly from the transport of goods and commodities (Costello and McAusland 2003), once a pest has gained entry the mechanisms of spread can be multiple and diverse. The rate of spread will depend on a range of factors not just relating to the species’ ecology but also relating to host distributions and to potential dispersal vectors; not only human but animal and environmental (such as wind or ocean currents). Ecological factors and landscape context may influence the pest/pathogen, vector and host, either facilitating or inhibiting the dispersal of the species. Likewise, the success of individual dispersal events may be strongly influenced by low probability extreme meteorological events, or by human-induced or other environmental factors.

When integrated with field-based research and surveillance, dispersal models can help inform pest and pathogen outbreak management about a range of processes, such as the rate of spread of a pest (Gilbert and Liebhold 2010), which can lead to better surveillance strategies (Cacho et al. 2010, Demon et al. 2011, Epanchin-Niell et al. 2012), and more effective response strategies (Coutts et al. 2011). Models can also be used to inform policy-makers about the risks posed to target ecosystems (Rutherford et al. 1999), at both immediate and long-term time scales (Kriticos et al. 2003; 2013a). Similarly, integration of dispersal simulation models and economic models can help to inform the design of optimal management strategies (Bogich et al. 2008, Carrasco et al. 2009, Florec et al. 2013, Kriticos et al. 2013b). For example, models can be used to decide when and at what scale a management strategy should be implemented given the progression of an invasion, and to decide whether the costs will outweigh the benefits.

With such diversity of pathways, scales and complexity of dispersal processes for post-entry spread, and with such a wide range of possible applications, there is a parallel diversity of modelling methods. We aim to give an overview here to help guide modellers to select appropriate methods.

Models of pest and pathogen spread post-entry largely occupy one of two categories of model: analytical methods (Hastings 1996, Kot et al. 1996, Neubert and Caswell 2000, Royama 1992) and mechanistic, process-based methods (Higgins and Richardson 1996, Jongejans et al. 2008). Analytical models have been used for many years to study dispersal in ecology, beginning with simple diffusion equations (Skellam 1951). An analytical model can be broadly defined as a deterministic mathematical expression. Such models seek to distil the complexity of a system or process into a single representation of its behaviour under given circumstances. They have the advantage that they tend to be more easily generalised than mechanistic models (Turchin 1998). They incorporate a range of techniques, in particular theoretical or empirical curve fitting models for dispersal kernels and other generalisations of the movement of organisms as a simplified physical process (e.g. travelling wave (Sharov and Liebhold 1998), matrix models (Parker 2000) and diffusion (Kot et al. 1996)). Analytical models have varying data requirements, depending on whether they are developed as purely abstract theoretical models or if they are phenomenological statistical models that are empirically derived. In the latter case data availability often becomes a big issue for modelling incursions (see following section). Such methods generally involve assumptions that include uniformity of the landscape and population, which mean they are simple to implement but can be highly abstract. Criticisms of these models are a lack of complexity and realism that can be key to studying processes such as long-distance dispersal and the influence of landscape heterogeneity. Moreover, long-distance dispersal events are often caused by different mechanisms to short distance dispersal and are highly significant drivers of accelerated population spread (Liebhold and Tobin 2008).

To explore the long distance connectivity of populations, network models and metapopulation models have also been applied to invasion ecology in recent times (Chadès et al. 2011, Drake and Mandrak 2010, Facon and David 2006, Paini and Yemshanov 2012). Whilst these also have the advantage of simplifying complex processes, equally they make their own assumptions about the uniformity and ‘patchiness’ of the landscape.

Mechanistic, process-based simulationmodels are a more recent development for modelling spread post-entry (Turchin 1998), enabled in part by the growing power of computing to support large, complex models. Such approaches to dispersal modelling align with ‘ballistic’ simulations or in physics termed ‘Lagrangian’ models – where individual pathways are traced as they move according to a set of stochastic or behavioural rules (e.g. individuals influenced by wind trajectories). Such models tend to have greater flexibility across spatial scales, and therefore can more easily encompass both short and long distance dispersal events. Consequently, individual-based models (Grimm and Railsback 2005), cellular automata (Travis and Dytham 2002) and trajectory models (Chapman et al. 2010, Nathan et al. 2005) have become part of the ecological modeller’s toolkit over the last few years, although there are relatively few examples of the application of these methods to dispersal modelling for post-entry spread (e.g. Guichard et al. 2012, Kanarek et al. 2012).

It is also possible and can be advantageous for a dispersal model to contain both analytical and mechanistic components (e.g. Nathan et al. 2011). One example is WALD (Katul et al. 2005), which is used to estimate long distance dispersal kernels of wind-dispersed seeds and their escape probability from the plant canopy. A computationally intensive trajectory model that incorporates the effects of canopy turbulence was used to derive an expression for an analytical model, therefore retaining the mechanisms but giving the advantage of analytical simplicity (essentially an inverse Gaussian distribution). More broadly, bringing together the simplicity of an analytical, phenomenological method with mechanistic understanding of processes can be very powerful (e.g. Pitt et al. 2011).

Multiple dispersal vectors add extra layers of complexity (Buckley et al. 2006, Pitt et al. 2009). Many species have multiple dispersal pathways and these can be considered by the model(s), using an integrated multi-modelling method (Harwood et al. 2009).

In addition, species niche models can inform post-entry spread in multiple ways. Firstly, they can inform the total area that can potentially be invaded. This information can define the spatial bounds of the spread modelling, i.e., the model ‘universe’, for both simulation and analytical spread models. Alternatively, a niche model can be used to differentiate between different components of a heterogeneous landscape over which a species may spread, and this can be used by spatially-explicit dynamic dispersal models (e.g., Pitt et al. 2011).

When considering how best to apply these models, understanding the ecology and landscape factors relevant to the population dynamic and dispersal of a pest or pathogen species is critical. Often, not enough consideration is given to an organism’s ecology and behaviour prior to developing a dispersal model, where population dynamics models are commonly separated from dispersal simulation. However, biological processes operating at different spatial and temporal scales are key drivers in the dispersal process, and ideally should be taken into account explicitly.

In selecting a model, there are also important characteristics to consider, such as the sensitivity of the model (the proportion of known spatio-temporal dispersal events modelled correctly) versus the specificity of the model (the proportion of unoccupied sites that are modelled correctly) (Fielding and Bell 1997, Pitt et al. 2011). Where spread models combine highly specific model realisations to create a probability surface for occupancy, they inevitably become less specific through time, eroding their usefulness for addressing long-term strategic questions (Pitt et al. 2011). A good example of the sensitivity-biased effects of applying a stochastic mechanistic modelling method to long-term dispersal scenarios is Robinet et al. (2009). In this paper, the spread of the pinewood nematode was simulated over 23 years in China. A probability surface of nematode presence was generated from a combination of 300 replicate simulations. The fit of the model was assessed by comparing how many of the known locations fell into cells with a positive modelled probability. This commonly applied method ignores the model specificity (the number of cells that had a positive modelled probability, but did not include any reported infestations). As a guide for surveillance activities, poor model specificity could lead to much wasted effort, and for pest risk, an over-estimate of the potential impacts of the pest due to inappropriately high rates of spread. Therefore there is a need to critically consider this effect when developing a model of post-entry dispersal (Fletcher and Westcott 2013), perhaps limiting mechanistic models to short-term tactical applications such as informing regional pest management plans, including activities such as surveillance, eradication and containment strategies, and using far simpler spread models for strategic applications such as pest risk modelling (e.g. Kriticos et al. 2013b).

Knowledge gaps may relate to either a gap in knowledge of how a process is understood and therefore modelled (i.e. model uncertainty), or the uncertainty with which we can estimate the true value of a model parameter (parametric uncertainty). If the knowledge of a critical process is incomplete, it is prudent to be cautious, and to be wary of management imperatives derived from regression-based patterns. The method of multiple competing hypotheses (Chamberlin 1890, Hilborn and Mangel 1997) is starting to gain popular acceptance as a basis for both studying and communicating deep uncertainty in areas as diverse as ecology and intelligence (Beven et al. 2005). An adaptive management, monitoring and modelling framework (A3MF) may be an appropriate method to adopt. In A3MF a model is iteratively updated as new knowledge or data is acquired that shows the model fails to represent the ecological process well (Holling 1978). Potential also exists within A3MF to employ different management strategies in different regions or periods of time to observe the response of the invasive organism to the different strategies and contexts, thereby accelerating our acquisition of knowledge about the organism and its management. However, A3MF requires long-term investment by a team of experts and managers over a time scale akin to the time scales of the invasion process, and the lagged impacts the invasion may have on the invaded agricultural or ecological system. This weakness of A3MF in its fullest sense is one of the reasons why simple, readily applied models have such broad appeal.

Parameter uncertainty, as a knowledge gap, is a function of data paucity and the availability of statistical methods. Model complexity also contributes to parameter uncertainty. On the one hand, highly complex models may contain so many parameters that not all may be known adequately, but on the other hand models that are very simple often contain parameters that are hard to estimate. Commonly, individual parameters are estimated through monitoring or experimental data targeted towards those parameters. These parameter estimates are then used in the model. If uncertainty in the estimates is quantified then the parameter uncertainty can be fed through the model to provide an estimate of parameter sensitivity. Other sources of uncertainty can also be incorporated into models through developing Bayesian posterior confidence intervals, such as measurement error or errors assigned to ad hoc parameter values (Higgins et al. 2003). In general, Bayesian methods have improved greatly with recent advances in computing, and can support a direct fitting of the model to the data, rather than a parameter plug-in approach. Hierarchical Bayesian methods of inference enable population dispersal models to be fitted to the data (Hooten and Wikle 2008, Royle et al. 2007). In lieu of a direct model-fitting procedure such as Hierarchical Bayes then the ad hoc ‘plug-in’ methods of model calibration are required, which may include:

1. garnering parameter values from analyses of the existing literature; or
2. minimising some measure of discrepancy between model output and the limited set of observations available, and which includes Approximate Bayesian Computation (ABC; Marjoram et al. 2003) and the inverse model problem.

Simulation is perhaps the best way to assign ‘prediction’ error or intervals to deterministic models, given uncertain starting conditions of the pest/pathogen population. Posterior prediction intervals can also be derived for stochastic models through cross-validation, and more generally through the use of independent test and training data sets. Generating prediction intervals to be tested against new data sets also falls under the rubric of model validation (e.g. Higgins et al. 2001), which should also include logical tests for the “reasonableness” of model results. If model uncertainty of various management options on the end point of the invasion can be specified then a measure of policy or management activity risk can be developed, which may help determine an optimal risk mitigation strategy.

Common to all dynamic models is a temporal limit in quantifying model error. In this case an error is associated with a single time step, and in iteratively running a model then the error is compounded. The consequence of this compounding error is that long term utility of any dispersal model is dogged by severe and growing uncertainty. Two options are then available: (i) continual updating of results by resetting the model’s initial conditions to the current conditions (e.g., the Kalman Filter process); or (ii) applying a decision method developed for severe uncertainty. Continual updating is consistent with both A3MF and Bayesian model updating, or ‘learning’: as new data arrive then our understanding of the processes, our ability to predict system processes, or belief in our model should also improve. However, continual updating requires ongoing monitoring to feed the model any change in system state as it occurs. Continual updating is most appropriate for developing tactical management responses to invasions, but does nothing to address the inability of these models to address strategic questions in a timely manner.

In contrast, decision making under severe uncertainty is common for long-term strategy development, or where continual updating is a cost-prohibitive option. Several analytical decision frameworks have been developed for dealing with severe uncertainty, with the two most popular being robust optimisation (RO) (Ben-Tal et al. 2009, Hansen and Sargent 2007) and info-gap theory (IGT) (Yakov 2006). The key difference between the two methods is that IGT provides a robust decision only in the local neighbourhood of the best guess parameter value for a model, whereas RO provides a solution that is robust over the entire range of parameter values to worst case scenarios (Sniedovich 2010). Neither framework handles a multivariate decision and parameter space well. A conservative strategy is to limit decision frameworks under severe uncertainty to those few key parameters that contribute most to the variability in model output, as identified through a sensitivity analysis.

There is no single recipe for constructing a model of post-entry spread, due the diversity of policy applications, ecological and landscape contexts, temporal and spatial scales and possible techniques to employ. We have attempted to present some practical guidelines on how to approach model framing and construction for post-entry spread in invasion ecology by identifying what method may be most suitable to apply to particular policy questions, at what spatial and temporal scale, given the available data and knowledge. In recent years, we have seen the evolution of more process-based, mechanistic models that attempt to capture system dynamics and complexity. This trend has been supported (and perhaps encouraged) by the availability of suitable computer platforms capable of processing the immense amount of information required to simulate these processes, as well as the availability of suitable covariate data.

The need for a more rapid response in outbreak situations has resulted in the recent development of fine-scale dispersal models designed to forecast and backcast spread for surveillance and response activities (Guichard et al. 2012) and generic models, such as a General Model of Biological Invasions (GMBI) (Renton et al. 2011, Savage et al. in press), Modular Dispersal in GIS (MDiG) (Pitt et al. 2009, 2011), demoniche (Nenzén et al. 2012) and a model suite for Pest Risk Analysis (Robinet et al. 2012). In the future, we anticipate modelling methods will continue to improve our ability to incorporate complex spatial and temporal dynamics, such as highly mechanistic models of wind dispersal. For example, recent research has simulated seed dispersal using a ballistic method coupled with large-eddy simulations incorporating turbulent airflow (Nathan et al. 2011). As sophisticated, multi-level wind circulation models are improved and made more accessible for a wider range of applications (e.g. NCEP/NCAR reanalysis data (NOAA 2011)), the opportunity to couple mechanistic dispersal models with process-based population dynamics models becomes apparent (Parry et al. 2011).

However, even when armed with limitless computing power and knowledge of a species’ dispersal ecology we cannot forecast far into the future with high precision. We should be wary therefore of applying increasingly sophisticated mechanistic models and running them for long-term forecasts; the results may appear to have a fine resolution, but this should not be confused with reality – in such instances an analytical approach may be preferable, where fewer variables, constrained behaviour and obvious lack of precision make more explicit the model uncertainties and inaccuracies. Overall, there is great value in combining modelling methods; indeed it is likely to be necessary as the multi-dimensionality of the problem of post-entry pest spread will often require an integrated, multi-model, multi-scale approach, aligned with an empirical surveillance programme.

The most pressing limitations to applying spread modelling to post-entry invasion ecology are clearly not methodological. Modellers are spoilt for choice. The biggest constraints concern our knowledge of the rates of spread of organisms in novel landscapes at fine spatial and temporal scales, as well as across the time course of invasions. A clear challenge here is the cost of monitoring the spread of invasive organisms, which typically sees a rapid decline in interest once an organism stops being an eradication target. Options for overcoming this problem include placing more emphasis on the collection of time-stamped location data for invasive species, “crowd-sourcing” initiatives, and the development of a rich library of spread rate data for different organisms.