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Keywords:

  • analytical model;
  • climate change;
  • demography;
  • invasion;
  • population spread;
  • reintroduction;
  • stochastic model

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

1. Improving the understanding, prediction and management of range expansions is a key challenge for ecology. Over recent years, there has been a rapid increase in modelling effort focussed on range expansions and a shift from predominantly theoretical developments towards application. This is especially the case in the field of invasion biology and also in relation to reintroductions and species’ responses to climate change.

2. While earlier models were exclusively analytical, individual-based models (IBMs) are now increasingly widely used. We argue that instead of being viewed as competing methodologies, analytical and individual-based methods can valuably be used in conjunction.

3. We use a mechanistic wind dispersal model to generate age-specific dispersal kernels for the invasive shrub, Rhododendron ponticum. To demonstrate the utility of employing both modelling approaches, this information along with demographic parameters is incorporated into an IBM and an analytical, integrodifference model. From both models, the equilibrium rate of spread is calculated.

4. Estimates of wavespeeds were similar for the two models, although slower rates of spread were consistently projected by the IBM. Further, our results demonstrate the wavespeed to be sensitive to the characterisation of age structure in the model; when few age classes are used, much higher rates of spread are projected.

5. The analytical model is extremely efficient at providing elasticity analysis of the wavespeed, which can provide helpful information for management. We gain qualitatively similar results using the IBM but obtaining the results is time-consuming and, because the model is stochastic, they are noisy and harder to interpret. We argue that analytically derived transient elasticity analyses are needed for the many cases where success of control is measured on a relatively short time horizon.

6. To demonstrate the flexibility of the IBM approach, we run it on a real landscape comprising different habitat types. The comparison of two different control scenarios is an example of the utility of this approach for more tactical applications.

7. As a general conclusion of the study, we emphasise that analytical and individual-based approaches offer different, but complementary, advantages and suggest how their joint use can facilitate the improvement in biodiversity management at a range of spatial scales.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

Range expansions are involved in two key processes increasingly affecting species distribution: the invasion of alien species, which has major consequences for ecosystem processes, biological diversity, economics and human health (Earth Summit, Rio Convention 1992 http://www.cbd.int/; Vitousek et al. 1996; IUCN Council, 2000; Le Maitre, Versfeld, & Chapman 2000), and the response of species to climate change, which, in most cases, involve range shifts to regions of newly suitable climate space (Thomas et al. 2004; Thuiller et al. 2008). Consequently, there is a pressing need to improve our ability to predict and manage range expansions of undesirable invasive species and of species threatened by climate change. This requires the development of modelling approaches that can be effectively applied at a range of spatial scales.

While modelling dynamics of population spread has a long history (Fisher 1937; Skellam 1951), in recent years there has been a rapid increase in the range of methodologies available (see Hastings et al. 2005 and Jongejans, Skarpaas, & Shea 2008 for reviews) including development of new analytical methods and the use of individual-based models (IBMs). The use of these new methods is already making considerable progress; however, most often, the choice of modelling methodology is likely to be as much down to the individual modeller’s preference or skills as it is to a proper consideration regarding the optimal method for the question at hand (Neubert & Parker 2004). Almost all studies make use of either an analytical (e.g., Caswell, Lensink, & Neubert 2003; Buckley et al. 2005; Soons & Bullock 2008) or an IBM approach (e.g., Nehrbass & Winkler 2007; Nehrbass et al. 2007; Harris et al. 2009; Travis, Smith, & Ranwala 2010), and there is a risk that these methods are viewed as competing alternatives. Instead, recognising that both approaches have their strengths and weaknesses (Grimm & Railsback 2005), we should seek a complementary approach whereby the two methodologies are used together to provide the best possible predictions and management advice.

There are a number of potential benefits to ecological modelling of jointly using more than one modelling method; here, we list three. First, the cross-validation of two or more independent models increases the probability of detecting potential errors in their implementations. Second, different model types are differently well suited in terms of the analysis of output, the adaptation to case-specific circumstances or the presentation of results. Third, possible discrepancies between model results can give important insights into the effects the chosen methods can have on the output.

In this contribution, by using both an analytical approach and an IBM to estimate the spread rate of the invasive shrub Rhododendron ponticum, we demonstrate how new insights can be gained benefiting the development and use of both model types. In particular, we consider the utility of the two approaches for determining equilibrium spread rates and sensitivities (or elasticities) of these rates to key parameters. We compare the estimates of spread rates provided by the two methodologies and assess the relative ease with which these are derived. Subsequently, we consider the different strengths of the two approaches for informing management directed at reducing the spread rates. Elasticity analyses of equilibrium wavespeeds (rates of population spread) have already been applied to plants (Bullock, Pywell, & Coulson-Phillips 2008), but here we emphasise the potential importance of transient dynamics and highlight the flexibility of IBMs in terms of modelling spread dynamics over spatially realistic landscapes.

Study organism

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

Rhododendron ponticum is one of the most problematic invasive weed species in Britain and Ireland (Cross 1982; Colak, Cross, & Rotherham 1998; Rotherham 2001). It was introduced into Britain in 1763 (Elton 1958) and has since become well established throughout the British Isles (Rotherham 2001). It is now considered a major invasive weed of considerable conservation and economic concern (Cross 1982; Dehnen-Schmutz, Perrings, & Williamson 2004). Rhododendron ponticum has many of the characteristics that have been identified as good indicators of high invasive potential in a species (Higgins, Richardson, & Cowling 1996; Rotherham 2001), especially the ability of mature R. ponticum to produce hundreds of thousands of seeds every year (Cross 1975). The morphology and small size of the seeds suggest wind as the primary dispersal agent (Brown 1954; Cross 1975; Shaw 1984), and empirical and modelling studies indicate that although most seeds land close (10–20 m) to the parent plant, there is the potential for seeds to disperse much greater distances (Stephenson et al. 2007).

Modelling

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

We will first present a brief description of the modelling approaches we have used before describing their combined use to explore the dynamics of R. ponticum. Both modelling approaches – the analytical as well as the IBM – depend on the same submodel for seed dispersal: the semi-mechanistic model WALD. We chose to include the WALD model over other available dispersal models because its added complexity means that it better predicts long-distance dispersal events (Katul et al. 2005), and for invasive species, such as R. ponticum, it is important to capture these long-distance dispersal events to ensure our predictions are more precautionary in nature (Stephenson et al. 2007). Here, we describe the WALD model before outlining the IBM and analytical approaches, in both cases explaining how output from the WALD model is incorporated.

The seed dispersal model WALD

In brief, this model is derived from a simplified 3-D stochastic dispersion model but retains the essential physics contained within the more computationally intensive mechanistic models (e.g., Soons & Bullock 2008). Reassuringly, following some daunting maths, the analytical model reduces to the following WALD (or inverse Gaussian) distribution (eqn 5b in Katul et al. 2005) that describes the probability density of dispersal distances x1:

  • image(eqn 1)

where inline image and inline image are dispersal kernel parameters that depend only on the wind velocity statistics (both horizontal and vertical components), seed terminal velocity and seed release height. Full details of the WALD model and its derivation are provided in Katul et al. (2005), and a discussion of its application to the dispersal of R. ponticum seeds can be found in Stephenson et al. (2007).

In the IBM, each seed released was dispersed by drawing a displacement distance at random from the probability distribution function generated by the WALD model with parameters for windspeed, terminal velocity and release height drawn at random from appropriate distributions. For the analytical model, dispersal kernels were required as input. These were generated in a set of simulations, in each of which WALD was used to simulate the dispersal of 100 000 seeds. Exactly as in the IBM, each seed was allocated a windspeed, release height and terminal velocity. Kernels were generated in this way for each age class versions of the model (see key output 2). Release height increases with age as plants grow, so each age class had a different dispersal kernel determined by the mean height of plants in that age class (see Fig. 1 and Table 1). Neubert & Caswell (2000) emphasise that this stage-structured approach to modelling spread has greater precision by allowing stage-specific dispersal kernels to be used. To the best of our knowledge, this is the first study to make use of this flexibility.

image

Figure 1.  Illustrative dispersal kernels generated using the WALD model for plants aged 11, 30 and 50. Importantly, older plants produce more seeds, and, on average, because they have greater release height, these seeds travel further (see also Table 1). The difference in the dispersal distributions is substantial.

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Table 1.   Parameter values associated with each of the age classes (4, 8, 20 and 60 age class models). The increased expected dispersal distances of seeds produced by older plants, together with the increased number of seeds produced by older plants, make the proper representation of age structure within a model extremely important
Age classRange of ages in each age classAnnual mortality %Annual fecundityMean dispersal distance (m)
482060482060482060482060
 10–100–100–31101010100000
 211–2011–154–62000101070020·718·5
 321–4016–207–930001094140032·023·1
 441–6021–2510–12450001337274041·727·117·1
 5 26–3013–155 000 5280 30·719·4
 6 31–4016–186 000 149120 35·322·1
 7 41–5019–217 500 560180 40·124·6
 8 51–6022–248 500 2114260 43·227·2
 9  25–279  00  390  29·2
10  28–3010  00  590  31·4
11  31–3311  00  875  33·216·6
12  34–3612  00  1306  35·117·5
13  37–3913  00  1947  36·718·5
14  40–4214  50  2898  38·319·5
15  43–4515  50  4309  39·520·4
16  46–4816  50  64010  40·821·2
17  49–5117  50  95412  41·822·1
18  52–5418  50  142114  42·823·0
19  55–5719  50  211615  43·523·8
20  58–6020  50  315118  44·224·7
25   25   0   34   28·4
30   30   0   67   32·0
35   35   0   129   35·1
40   40   5   251   37·8
45   45   5   488   40·1
50   50   5   948   41·8
55   55   5   1842   43·1
60   60   5   3577   44·2

The individual-based model

A stochastic, spatially continuous and temporally explicit framework was used to model the spread of R. ponticum through a homogeneous landscape. The model setup consisted of a single plant being introduced to a random position at the left-hand side of a linear corridor of suitable habitat 200 m wide. Each year, first the age of each plant was incremented. Then, if the plant was reproductively mature, it produced seeds that were dispersed according to the WALD model. The order in which individuals were updated was random.

Each seed dispersed from a mature plant was randomly allocated a windspeed from a distribution of mean daily windspeeds (mean = 9 ± 6 ms−1) recorded over 6 weeks (corresponding to the R. ponticum dispersal season) from a permanent weather station in southeast Scotland (see Stephenson et al. 2007 for further details of wind data) and a release height appropriate for the age of a given plant (see Harris et al. 2009 for details of age: height relationship). Similarly, each seed was randomly allocated a terminal velocity from a distribution derived from empirical data and described in Stephenson et al. (2007).

Based on our field data (Harris et al. 2009), the minimum age of maturity for R. ponticum was set to 11 years. Regression equations (see Harris et al. 2009) were used to predict the fecundity and height of plants according to their age. These regression equations were only applied to plants up to 60 years old to prevent extrapolation far beyond the empirical data. Although plants were allowed to continue increasing in age, the height and fecundity were capped at the 60-year-old level. There is a lack of empirical data relating to survival probabilities of R. ponticum seeds from dispersal to germination and through to establishment. Rather than disperse very large numbers of seeds from each plant and apply arbitrary germination and establishment probabilities, which would have been very computationally intensive, we chose to disperse only one seed per flower bud [which contain c. 4500 seeds (unpublished data)]. This implicitly accounts for mortality at the germination and early seedling establishment stages. Additionally, as there is some evidence of allelopathy in R. ponticum (Rotherham & Read 1988), each dispersed seed was subjected to a density-dependent establishment probability whereby successful seedling establishment only occurs if the location at which a seed lands is more than 2 m from an already established plant. A 10% probability of mortality was applied annually to seedlings in their first 3 years and a 5% probability for plants of over 40 years. No mortality was applied to plants between these ages. These represent best guesses for age-dependent survival.

To avoid populations growing to a size at which we would encounter computer memory or simulation speed problems, we only considered plants within 300 m of the expanding front. Essentially, we cut off the back of the range-expanding population. This is a useful and pragmatic approach for IBMs where the focus of the study is on the expanding front (Burton & Travis 2008; Burton, Phillips, & Travis 2010). To ensure that this method did not impact on the rates of spread obtained, we ran several simulations where this cut-off point was varied. Using a 300-m cut-off was a robust choice – only if the cut-off was < 200 m did we detect any change in wavespeed. This indicates that seeds very rarely arrive at the expanding front from plants further than 200 m away.

The analytical model

Neubert & Caswell (2000) introduced an analytical approach to modelling population spread, which combines matrix models describing population growth for a structured population with integrodifference equations describing dispersal. Neubert & Caswell (2000) give the mathematics of this approach, and so we give a precis here.

The model is based on eqn (2) that describes how, in one dimension, population density n at each location x evolves from time t to + 1

  • image(eqn 2)

Bn is a stage-structured population projection matrix that describes density-dependent population growth at location y. K(− y) is a matrix of dispersal kernels that describe the set of probabilities of the relocation from y to x of individuals undergoing each demographic transition. The population at location x is given by integrating this process over all locations y. Under this model, a population forms a wave of constant shape that advances at constant speed, and this asymptotic wavespeed c* can be derived analytically. The approach offers considerable flexibility; differences in dispersal ability among stages and realistically complex dispersal kernels can be incorporated, both of which are important in modelling R. ponticum spread. Simplifying assumptions include no temporal variation, and a spatially homogeneous environment such that demography does not vary in time or space, and dispersal from x to y depends only on their distance apart.

To develop from the generality of eqn (2) to model spread of a population into unoccupied space using data, the density-dependent matrix Bn is substituted in eqn (2) by A (=B0), which summarises demography at low density. The demographic R. ponticum data used for the IBM were used to construct age-classified matrices (Caswell 2000) of the general form for n age classes:

  • image(eqn 3)

where the first age class (1) does not reproduce, but all others do by fi. Plants in age class i die, remain in the same age class (lii) or progress to the next age class (lij). The model has an annual timestep, so if the model is constructed with age classes lasting a single year, then lii = 0. In the final age class (n), all surviving plants remain in that age class; this reflects the assumed no change in height, mortality or fecundity after 60 years of age.

Each transition in A is associated with a dispersal kernel, which is the set of dispersal distances generated using the WALD model. Because the dispersal kernels themselves cannot be solved in closed form, each kernel is represented by its moment-generating function (MGF). The MGF in one dimension (to match the dimension of the analytical model) for a set of dispersal distances in two dimensions can be derived using a nonparametric estimator following Neubert & Parker (2004):

  • image(eqn 4)

where I0 is the modified Bessel function of the first kind. There are N seeds dispersed and the ith has the radial (2-D) distance ri from the source. The parameter s describes the shape of the advancing wave and is found using eqn (5).

M(s) is a matrix of the same dimensions as A, which is populated with the MGF for each transition mij(s) in which seed dispersal takes place; for non-dispersing transitions mij(s) = 1. A new matrix H is created by elementwise multiplication.

  • image(eqn 5)

and the dominant eigenvalue of H is ρ. It can be shown (eqn 6) that the asymptotic rate of spread of a population c* is given by

  • image(eqn 6)

Key model outputs

Calculating equilibrium wavespeeds using the analytical model and the IBM

The data relating to R. ponticum fecundity and mortality, together with the dispersal kernels generated in eqn (1) earlier, provided the input required by the analytical wavespeed approach to generate equilibrium wavespeeds, c*, as described earlier. For comparison with the analytical model, a 1-D rate of spread was determined using the IBM. For simplicity, we call this the wavespeed as it is directly comparable with the analytical wavespeed. To estimate the equilibrium wavespeed of R. ponticum invasion, the IBM was run for 250 years. Each year, the distance between the invasion’s origin and the furthest plant was calculated. Twenty replicates of each scenario were run for the calculations of wavespeed. The extent of expansion between years 150 and 250 provided the equilibrium wavespeed for each IBM simulation, and, in every case, we report the mean wavespeed across the twenty replicates.

An important part of our modelling exercise was to compare results gained using different numbers of age classes. Analytical approaches very often use age (or stage)-structured approaches and the choice of the number of classes can be quite arbitrary. To test the sensitivity of wavespeed estimates to the number of age classes, we compared results obtained from the analytical model using 4, 8, 20 and 60 classes with those obtained using the standard IBM approach where age is a continuous variable. Additionally, to separate effects of model type from any age-structure effect, we ran an IBM-style model implemented with age structure. In this version of the IBM model, each individual progresses from one age class to the next, or suffers mortality, with probabilities exactly equivalent to the rates that are described for the analytical model. Similarly age class–specific fecundities were used. In all other respects, the model was run as a stochastic simulation in the same way as the standard IBM. To provide direct comparisons with the analytical model, we obtained IBM wavespeeds using 4, 8, 20 and 60 age classes (the 60 age class model is the standard IBM where plants hold their exact age).

Elasticity analysis and transient dynamics

Elasticity analysis can be used to examine how demographic changes (for example, because of control measures) affect the projected wavespeed. For the 20 age class version of both the analytical model and the IBM, we conducted elasticity analyses. Considering the analytical model, if transitions in the A matrix are represented as aij, the effect of a change in aij on c* is given by (Neubert & Caswell 2000, eqn 27)

  • image(eqn 7)

where hij represents the corresponding entry in the H matrix.

For the IBM, we simulated range expansion for 250 years to allow equilibrium range expansion dynamics to be reached. We then reduced a matrix element (either transition probability, mortality probability or fecundity value) by 20% and for a further 200 years we recorded the location of the plant furthest from the source of invasion. The final 150 years were used to provide a new equilibrium wavespeed, and this was compared with that obtained before the matrix element was changed to provide the elasticity. Additionally, with these IBM results, we considered the more immediate (i.e., transient) impact of making a change to one of the matrix elements. Specifically, we calculated the change in wavespeed in the first 5 years following control and then in the periods 5–10, 10–20, 20–50 and 50–100 years following control.

Spatial analysis of the IBM

To aid understanding of the range expansion dynamics and to gain an insight into which adult plants are most important in driving range expansion, we collected spatial data from the IBM on the location seeds disperse to and of the parents they disperse from. Specifically, we recorded the distance of each adult plant from the invasion core and the distance, after dispersal, of each of plants’ seeds from the core. From these data, we generated a spatial profile that indicates the relative contribution to the invasion process of seeds dispersed by adults according to their distance from the expanding front.

Spatially realistic model of R. ponticum

To illustrate the potential for using the IBM on spatially realistic landscapes, we simulated the spread of a population at a UK location where we are using the model to inform management (owing to sensitivities related to the ongoing management we are not able to provide further details). To initialise the model, we used data on current occurrence. The model included habitat-specific fecundities and age–height relationships (thus, dispersal kernels are different between habitats), as we have sufficient empirical data to parameterise the model effectively for a range of broad habitat types (Harris et al. 2009 in review). While it is possible to control the plants across much of this area, it is not possible to compel people to remove plants from their private gardens. Here, we show results for two scenarios: in the first we assumed no control and in the second we assumed plants were removed from all locations except the private gardens, and we simulated expansion for 20 years from this starting point.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

Equilibrium wavespeeds and model structure

The equilibrium wavespeeds projected by both the IBM and the analytical model are sensitive to the number of age classes modelled; the greater the number of age classes the lower the projected wavespeed (Fig. 2). These differences can be substantial, for example when we run the full IBM (with 60 age classes) we obtain an equilibrium wavespeed of 2·8 ms−1, whereas when we run the model with four age classes the projected wavespeed is 16·0 ms−1, more than four times higher. Exactly, the same effect is seen with the analytical model for which the equivalent wave speeds are 4·2 (for 60 age classes) and 16·8 ms−1 (for four age classes). The other important result illustrated clearly in Fig. 2 is that the analytical model consistently projects higher equilibrium wavespeeds than the stochastic IBM. This difference is greatest when there are 60 age classes; here, the equilibrium wavespeed projected by the analytical model is 50% higher than that obtained from the IBM.

image

Figure 2.  Wavespeed (metres per year) estimated from both the individual-based models (IBM) and the analytical model parameterised with 4, 8, 20 and 60 age classes. Errors for the IBM model are standard deviations derived from 20 replicate simulations.

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Elasticity analysis and transient dynamics

The analytical model provides very clear results for the elasticity analysis (Fig. 3). The equilibrium wavespeed is most sensitive to changes in the survival and growth transitions in the lower age classes (Fig. 3). The results are similar whichever number of age classes is used, but here we use the 20 age class version for illustration. While the wavespeed is also sensitive to fecundity, the elasticity values associated with fecundity are all substantially lower than those associated with survival and growth. Indeed, increasing mortality of the seven oldest age classes to 90% in the 20 age class model reduced c* by only 0·5%, while increasing mortality of age class 2–50% (assuming it is very difficult to detect the seedlings in age class 1) reduced c* by 27%. In contrast, reducing fecundity of all age classes by 50% reduced c* by only 10%.

image

Figure 3.  The elasticities of the analytical model’s wavespeed, c*, to transitions, aij, of the population projection matrix A (eqn 3) for 20 age classes. Key: elasticities for fecundity values (grey squares), elasticities for the survival of plants staying in the same age class (white circles), elasticities of progressions to the next age class (black triangles).

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The results we obtained using the stochastic IBM (with 20 age classes) to generate elasticities are considerably less clear than those generated by the analytical approach (Fig. 4). This is true even with 100 replicates, and the average response to changing a matrix element across the 100 runs was calculated (an extremely time-consuming process). Despite the noise in the results, it is apparent that they are at least qualitatively consistent with those provided by the analytical method; the greatest reductions in equilibrium wavespeed were obtained when we reduced the transition probability of a lower age class, and changing the fecundity of a lower age class makes a greater difference than changing the fecundity of older age classes. Results from the IBM indicate that the greatest short-term impact on spread rate is not necessarily achieved by modifying the same matrix elements to which equilibrium wavespeed is most sensitive. For example, although the data are extremely noisy (see Fig. 4), results suggest that the greatest reduction in wavespeed over the first 10 years post-intervention may be obtained by reductions in transition probabilities in age classes 9–11, while changes in fecundity in age classes 10–14 may result in the most effective reductions in the extent of range advance over the same period.

image

Figure 4.  A representation of the elasticities of the individual-based models wavespeed to each transition, aij, of the population projection matrix A (eqn 3) for 20 age classes. The resulting mean wavespeed (averaged over 20 replicates) from reducing the transition probability or fecundity of each age class by 20% is represented as a proportion of an unperturbed mean wavespeed (solid black line). Key: mean equilibrium wavespeed following a 20% perturbation in transition probability (black circles), mean wavespeed in the first 10 years following a 20% perturbation in the transition probability (white triangles), mean wavespeed in the first 10 years following a 20% perturbation in fecundity (grey sqaures).

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Spatial analysis

Output from the IBM clearly identified which plants are driving the range expansion, and viewing the data in this way reveals that range expansion is mainly driven by the dispersal of seeds from plants some distance from the front. The recently matured plants close to the expanding front are not driving the next stage of range expansion. Sixty years after the introduction, the oldest (and therefore tallest) plants close to the point of introduction are those that are dispersing seeds furthest ahead of the established range of the plant (see Fig. 5a). By 100 years, the picture has changed (Fig. 5b), with plants some distance away from the core contributing most of the seedlings at the leading edge as a result of some of their seeds leapfrogging over the established range. This general pattern is maintained through time, and when we look at the ages of plants producing the seeds that are advancing the range, we find that they are predominantly older than 20 years (corresponding to classes 7–20 in our 20 age class model). Note that the greatest immediate impact on wavespeed was obtained by reducing the transition probabilities or fecundities of individuals in this range of age classes while equilibrium wavespeeds are most effectively lowered by reducing transitions of age classes that are not yet mature.

image

Figure 5.  The spread of the invasion in one axis from the point of the first established plant (top left corner of each plot). A colour scale below indicates the log number of seeds landing at locations relative to the core (y-axis) that were dispersed from parent plants at different locations relative to the core (x-axis). For example, the blue point marked (a) on the figure at 60 years depicts a small number of seeds that have dispersed c. 250–300 m from the core originating from adult plants located c. 20 m the core. The density of seed rain from any given x-coordinate represents the dispersal kernel for seeds produced by adult plants at that location. For example, the short line highlighted by (b) depicts the dispersal kernel of a recently matured plant very close to the range front. Note that as this plant is short and produces few seeds, its dispersal kernel has much more limited extent than those of adults closer to the core, and its seeds are not responsible for the range expansion. The dashed line on each plot represents the location of the adult plant furthest from the core (i.e., the centre of the dispersal kernel on the far right of each plot) any seeds that land beyond this line represent expansion of the invasion front.

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Spatially realistic modelling with IBM

Running the IBM on a complex real landscape can provide informative projections related to the likely future distribution of R. ponticum under alternative control scenarios. Here, for illustration, we show two possibilities: one where there is no control of plants (Fig. 6a) and a second where there is an immediate one-off removal of all plants except those in private gardens (Fig. 6b). While the projected coverage of R. ponticum 20 years from now is substantially reduced by control, being unable to control in the gardens allows the plant not only to persist, but also to re-invade cleared areas. It is also worth noting that the IBM is readily adapted to incorporate spatial complexity; in this case, we can incorporate empirical estimates for different growth rates and fecundity in the different habitats present (see Fig. 6). The spatial output can also be extremely helpful in terms of identifying particular risks; in this example, we might be particularly concerned about the potential spread of the invasive into deciduous woodland. The stochastic nature of the invasion dynamics is demonstrated by the new pockets of invasion (mostly in deciduous woodland) some distance away from the current front (Fig. 6).

image

Figure 6.  Individual-based models predictions of spread over a real landscape without management (a) and following removal of all plants with the exception of those in private gardens (b) Green = current, red = in 20 years time, black = boundary line for gardens.

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Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

While there has been considerable progress in developing a range of new tools to improve our understanding, prediction and management of range expansions (Hastings et al. 2005; Jongejans, Skarpaas, & Shea 2008), and consideration has been given to the benefits and disadvantages of analytical and IBM approaches (Grimm & Railsback 2005), there has been little work to date exploring how these emerging methodologies may complement each other. We believe this represents a missed opportunity as the combined use of two or more methods may considerably improve the robustness of a modelling exercise. In this paper, we have used both an IBM and an analytical approach to investigate the spread dynamics of the invasive shrub R. ponticum. Our experience highlights that the joint use of these modelling approaches provides much more than simply an opportunity for intermodel comparison. Below we consider in turn our model development and results relating to equilibrium spread rates and elasticity analyses. In both sections, we emphasise how one modelling methodology informed our development of the other and also how the results from each aided interpretation of results from the other. Subsequently, we outline some key areas relating to range expansions that we believe would greatly benefit from the joint application of multiple modelling methodologies.

Equilibrium spread rates

We have shown that the equilibrium wavespeed obtained from our parameter-rich stochastic IBM is similar, but consistently lower, to that obtained using an analytical approach. Several authors have considered the question of whether rates of spread are affected by stochasticity (Lewis 2000; Clark, Lewis, & Horvath 2001; Snyder 2003). While some have suggested that the only effect of stochasticity is a relatively minor slowing of the invasion (e.g., Snyder 2003), others have suggested the effect may be much greater, even turning an accelerating invasion into one of constant speed (Clark, Lewis, & Horvath 2001). Snyder (2003) compared results on spread rate from a very simple individual-based simulation with the deterministic theoretical expectation. Like us, she used the position of the furthest-forward individual to estimate spread rates in simulations. Similarly, she obtained mean spread rates by averaging over numerous runs of the simulation, in her case 2000 replicates were used. Owing to computational limitations, Snyder (2003) was only able to run her model for 18 generations by which point the rate of advance was 3·37 myr−1 compared to the theoretical expectation of 3·8. However, Kot et al. (2004) subsequently showed that the equilibrium wavespeed in Snyder’s simulation eventually reaches very close to the theoretical expectation.

Our model for R. ponticum invasion is far more complex than that used in the theoretical exercises conducted by Snyder (2003) and Kot et al. (2004). Indeed, as far as we are aware our results represent the first time that such a comparison has been carried out for the invasion dynamics of a real species with strongly age-dependent demographic and dispersal parameters. Given the role of stochasticity in slowing spread rates (e.g., Lewis 2000), it is perhaps unsurprising that the difference between our projected wavespeeds from the IBM and the analytical model are slightly greater than those found in the theoretical study (Snyder 2003). It is also not surprising that we found the greatest proportional difference for the 60 age class version of the model as it is in this form that there is the greatest number of transitions and hence the greatest number of stochastic processes within the IBM.

Comparing our equilibrium wavespeed gained through an IBM with that obtained using the analytical methodology provides a useful cross-validation. While stochasticity in demography and dispersal clearly cause some reduction in wavespeed, the trends in wavespeed with different age classes shown in Fig. 2 are extremely similar. This provides a measure of reassurance that our IBM is coded correctly. Secondly, the other way around, the close match between the models provides reassurance that the analytical model, despite its simplifications, captures the equilibrium dynamics correctly. This cross-validation increases our confidence in using both the approaches to provide applied management guidance.

The number of age classes incorporated in the model has a major influence on the projected rate of range expansion. Originally, we compared the rate of range expansion obtained from our IBM with that projected by the matrix model using just four age classes. The analytical model projected a much greater rate of spread, which we thought to be the result of an implementation error. However, when reducing the number of age classes in the IBM to four, we found a similar increase in wavespeed (Fig. 2). The reason for this substantial difference in projected wavespeed is that a proportion of individuals rapidly progress through the age classes and start producing many seeds that disperse long distances sooner than would ever occur in reality. This result highlights that extreme caution needs to be exercised in using broad age classes in integrodifference models developed to estimate wavespeed. Ideally, the model should be set up to include an age class for every age up until the age at which there is no longer any change in life-history parameters. While creating the matrix for a 60 age class model for R. ponticum was much more time-consuming than for a model with 4, 8 or 20 age classes, it is required to obtain a reliable prediction for wavespeed, and it still represents a far more time-efficient method for providing elasticity analyses than the IBM. This argument applies not only to age-classified models, but also to those with size or stage classes – the accuracy in the modelling procedure must trade-off against effort in parameterising the model. The Neubert & Caswell (2000) formulation of the analytical modelling approach allows for such structured models, and our study supports their argument that this is more desirable than approaches with no structuring of the life cycle.

Elasticity analysis and planning management options

One of the great advantages of the analytical method is the rapidity with which it is possible to establish the extent to which changing each of the demographic or dispersal parameters influences spread rate. Not only is the analytical method faster, but also the results are much clearer. The elasticity analyses of equilibrium wavespeeds gained using the IBM show qualitatively the same results as those from the analytical model, but because they are so noisy it is impossible to draw any conclusions on possible quantitative differences. Even taking the mean of 100 replicates of the IBM provides noisy results, simply because the effect of making a small change to one of the matrix elements is submerged by all of the other stochasticities involved. The elasticity analyses from the analytical model provide important insights that can be used to recommend effective management targeted at impeding (in the case of invasive species) or facilitating (for species of conservation concern) range expansion (e.g., Buckley et al. 2005; Bullock, Pywell, & Coulson-Phillips 2008; Shea et al. 2010). For R. ponticum, the results suggest that the key transitions involve survival and transition of the early age classes. In the UK, where R. ponticum is invasive, this would suggest that preventing early age transitions, perhaps by removing young plants, might provide the best outcome rather than the more intuitive approach of targeting the large and highly fecund older plants (e.g., Edwards 2006). In Iberia, where the same plant is native and is of conservation concern (Mejias, Arroyo, & Ojeda 2002), efforts to recover its previous range might be best targeted at improving survival and growth of the younger plants.

However, caution has to be exercised in the interpretation of the elasticity of equilibrium wavespeeds in terms of providing management recommendations. The analytical method we have used provides information only on the equilibrium wavespeed. Thus, it tells us how much the wavespeed will eventually change if, year on year, we take some action that modifies one or more of our species’ life-history parameters. It does not provide information on the short-term effect of management, which results in a persistent change in one or more life-history parameters. Neither does it tell us how taking some action in a single year will affect the subsequent dynamics. Our IBM results demonstrate quite clearly that the greatest immediate impact on invasion rate is obtained by management actions targeted at middle-aged plants. This is in contrast to the finding that equilibrium wavespeed is impacted most by changing transition and fecundity probabilities of the youngest age classes. This difference makes intuitive sense for a long-lived species such as R. ponticum: changing the transition rates of very young plants has no immediate impact as it takes several years for them to start producing seeds that may lead to range expansion. The greatest immediate impact is gained by altering the transition rates for those plants about to enter age classes that are most likely to contribute to range expansion, and for R. ponticum it is these middle-aged plants that are key (this is very clearly illustrated in Fig. 5a).

In most real-world cases, management takes place within specific time frames, and recommendations based upon equilibrium rates of range expansion will not always be appropriate. Instead, we may want to identify the optimal management strategy for limiting or enhancing the spread of a given species over the next 10 years. Additionally, as modellers, we may need to provide advice relating to the optimal management when funding for intervention is limited in time; often funding is unlikely to be sustained and will instead be for one, three or perhaps 5 years duration (e.g., Dirnböck et al. 2008). In this case, we need a modelling approach that can be used to identify the strategy that minimises (or maximises) the range expansion over the next 10 years when intervention is only guaranteed over a shorter period. While it is possible to use the IBM for this, it is extremely time-consuming, and the results are likely to be noisy. Further development of analytical methods capable of providing transient elasticities (e.g., Caswell 2007; Ezard et al. 2010; Miller & Tenhumberg 2010) would be a major step forwards in terms of using models to inform the control or conservation of range-shifting species.

Where the IBM approach really comes into its own is in developing models to aid the management of invasive populations at particular locations where, for example, there may be spatial complexity in the landscape or spatial constraints on the form of control that is possible (as illustrated in Fig. 6). The flexibility of the IBM makes incorporation of these complexities relatively straightforward. Additionally, the visual output provided by the IBM (Fig. 6) is easily accessible by stakeholders, and this provides a major advantage in terms of demonstrating the likely impact of alternative management approaches. The flexibility of the IBM approach is also important where spatial control strategies such as ‘quarantine lines’ are being considered (e.g., Harris et al. 2009). With an IBM, it is possible to ask, for example, how long an invasion is likely to be contained by a quarantine line of a certain width. It would be impossible to use the currently available analytical approaches to provide equivalent information.

To conclude, rather than considering analytical approaches and IBMs as competing methodologies, we should view them as highly complementary. Each has major advantages as well as considerable drawbacks. The analytical approach is fast and precise, providing rapid and readily interpretable information in the form of, for example, elasticity analysis of equilibrium wavespeeds. It also has the advantage that the results can be readily verified. The IBM is, in comparison, much slower. It requires a greater investment of time to implement the model and runtime can be substantial, especially as many repetitions are required to characterise the behaviour of a stochastic model. This consequently makes verification of the results time-consuming and, in general, independent verification of IBMs does not occur. This is a major disadvantage especially because, despite calls for standardisation of IBMs (Grimm et al. 2006), their formulation varies widely in terms of which processes are included and how they are represented. Such variation is not an issue with analytical models because of their mathematical exactitude. We propose, therefore, that IBMs can be verified using the results of an analytical model. As we have discussed earlier, there are good reasons why we do not necessarily expect a perfect match, but nonetheless a comparison of the two provides a very useful check that the IBM is behaving sensibly. Obtaining similar simulated spread rates also provides reassurance that the many assumptions made in the analytical approach do not lead to inaccurate projections of wavespeed. The analytical approach is ideally suited to providing strategic management advice based on elasticity analyses, and the utility of this method could be greatly enhanced by increased availability of methods to consider transient effects. On the other hand, the flexibility of the IBM makes it ideal for tactical applications, for example providing site-specific management recommendations where the landscape is complex. There is also little doubt that the graphical output of an IBM can be extremely useful in engaging stakeholders with the modelling process and the importance of this benefit should not be overlooked.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References

Funding for this work was provided by The Carnegie Trust for the Universities of Scotland and by CEH project CO4166.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Study organism
  5. Modelling
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
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