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Present address and correspondence: Steven Higgins, UFZ-Centre for Environmental Research, Department of Ecological Modelling, Permoserstrasse 3, D-04318 Leipzig, Germany (fax 49 341235 3500; e-mail firstname.lastname@example.org).
David M. Richardson,
Institute for Plant Conservation, Department of Botany, University of Cape Town, Private Bag Rondebosch 7701, South Africa
1 Process-based models, and spatially explicit models in particular, will play an important role in predicting the impacts of future environmental change. Enthusiasm for the rich potential of these models, however, is tempered by the realization that their parameterization is often challenging and time consuming. Moreover, these models are seldom validated; this makes their predictive value in applied contexts uncertain.
2In this paper we describe the process of parameterizing and validating a spatial demographic model of a spreading alien plant population. The model, a spatially explicit individual-based simulation, has modest data requirements (for a spatial simulation model) in that it concentrates on simulating recruitment, dispersal, mortality and disturbance and ignores the environmental and biotic heterogeneity of the receiving environment.
3We tested the model using the invasion of Acacia cyclops and Pinus pinaster into fynbos, the mediterranean shrublands of South Africa, as a case study. Dispersal, recruitment and mortality data were collected for each species at six different sites. Aerial photographs from six independent sites (two sites for A. cyclops and four sites for P. pinaster) were used to reconstruct the invasion histories of the two species between 1938 and 1989. Demographic data were used to parameterize the model, and the 1938 distribution of alien plants, derived from aerial photography, was used to initialize the model.
4The empirically estimated indices of rate and pattern of invasion fell within the range of model predictions made at all six sites studied. The indices of rate and pattern of invasion predicted by the model did not differ significantly from the empirically estimated indices for 76% of the model data comparisons made. These analyses suggested that the model predictions are good, given the variance in parameter estimates.
5The proportion of grid locations where the model correctly predicted alien plant distribution was typically above 0·75 and always above 0·5 for both species. A permutation test showed that locations of invasive plants predicted by the model were significantly better than random for P. pinaster, but not always for A. cyclops; this may be because A. cyclops is bird dispersed, and its dispersal may be biased towards perch sites, whereas P. pinaster is wind dispersed.
6We conclude that, although spatial simulation models are often more difficult to parameterize and validate than statistical or analytical models, there are situations where such effort is warranted. In this case the validation process provides confidence to use the model as a tool for planning the control of invasive plants. In a more general sense we believe that the approach outlined here could be used for model parameterization and validation in situations where spatial simulation models seem appropriate.
Growing concern over our ability to manage biota in the face of global change is pressurizing ecologists to improve their capacity to make predictions. The rich potential of spatial simulation models to represent ecological processes has seen them emerge as the leading paradigm for predicting environmental change. Proponents of spatial simulation models argue that analytical models, although unquestionably valuable in a strategic context, are unlikely to provide context-specific predictions; and that statistical models (e.g. Peters 1992) will be of limited value given the novelty of many of the anticipated environmental changes. However, spatial simulation models are not a panacea, particularly because they can be difficult to parameterize (Doak & Mills 1994; Wennergren, Ruckelshaus & Kareiva 1995; Ruckelshaus, Hartway & Kareiva 1997). Generating confidence in process models is not straightforward: ecological systems are complex, and modelling them involves parameter estimation, assumptions, abstractions and aggregations (Loehle 1987). This means that a modeller could make a great number and many types of errors in constructing a process model. It follows that the validation stage in the modelling process is crucial for generating confidence in the model’s behaviour, especially if the model is to be used as a decision tool (Gentil & Blake 1981). Validation refers to the process of testing a model on its agreement with observations independent of the observations used to structure the model and estimate the parameters. The validation of models is, however, often complicated by the fact that suitable data for validation are not always available. Moreover, since criteria for validation must be defined in the context of the model’s purpose (Rykiel 1996), novel techniques must often be explored in the validation process. In this study we attempted to validate a spatial simulation model that aims to predict rates and patterns of alien plant spread. Our aim was to develop confidence in the behaviour of the model by exploring how variance in independent parameter estimates drives variance in the model predictions and how these model predictions compare with invasion histories that were independently reconstructed from aerial photographs.
We were interested in plant spread because rates of colonization and plant spread are of critical importance for maintaining biodiversity in landscapes (Tilman 1997); for determining the invasive success of alien organisms (Williamson 1996); and for determining the ability of organisms to shift their ranges in response to global climate change (Pitelka 1997). Despite the obvious and critical importance of making predictions about the colonization potential of plants, very few models of plant spread exist (Higgins & Richardson 1996). We have developed a spatially explicit individual-based simulation (SEIBS) model for predicting rates and patterns of alien plant spread (Higgins, Richardson & Cowling 1996; Higgins & Richardson 1998; Higgins & Richardson 1999). The philosophy of the SEIBS model was to develop a flexible modelling approach that could incorporate the key processes that determine the spatial population dynamics of invading plant species (Higgins & Richardson 1998). A fundamental component of this modelling philosophy was to operationalize these key processes in functions that could be easily parameterized from field data. The first objective of this study was to describe the methods for collecting the field data and show how these data can be used to parameterize the SEIBS model. The second, and more fundamental, objective was to use an independent data set reconstructed from aerial photographs to validate the SEIBS model. The process of validating the model will allow us objectively to define confidence in the model predictions so that these confidence levels can be used to qualify any predictions made by the model. Most previous attempts to validate spread models have compared model predictions to spread rates reconstructed from colonization of recently deglaciated landscapes (Skellam 1951; Collingham, Hill & Huntly 1996; Williamson 1996; Shigesada & Kawasaki 1997; Cain, Damman & Muir 1998; Clark 1998). Few contemporary invasion scenarios have been validated: Auld & Coote’s (1990) INVADE model is the only spread model that we are aware of that has been validated using data on contemporary invasions. The finer scale and resolution of contemporary invasion data available for this study meant that a detailed comparison of the predictions of the spread model and empirical data was possible.
The invasion of Pinus pinaster Aiton and Acacia cyclops A. Cunn. ex G. Don into unmodified fynbos (the mediterranean shrublands of South Africa) ecosystems was used as a case study. Both species are ‘declared invaders’ and are widespread within the fynbos biome, and they often form dense stands with interlocking crowns that suppress native plant species. Acacia cyclops shows a preference for lowland sites and P. pinaster has a preference for mountainous sites (Higgins et al. 1999). Pinus pinaster is a tree that reaches reproductive maturity at 6 years in fynbos; it has canopy-stored seeds and no resprouting ability. Acacia cyclops is a bird-dispersed shrub or tree that reaches reproductive maturity at 3 years in fynbos. It has a soil-stored seed bank and occasionally resprouts, but not enough to ensure persistence after fire. Richardson et al. (1992) provide more detailed life-history information.
The process of invasion (Richardson et al. 1992; Richardson & Cowling 1992) and the ecological and economic impacts of alien plants (Higgins et al. 1997b) in fynbos have been well documented. Fire drives both the natural dynamics of fynbos ecosystems and the invasion of alien plants into these systems. Fires cause widespread mortality of adult plants and provide opportunities for plant recruitment. The alien invader plants compete aggressively in the post-fire regeneration niche, ensuring that they dominate the post-fire community.
Model description and parameter estimation
The SEIBS model and its assumptions have been described elsewhere (Higgins, Richardson & Cowling 1996; Higgins & Richardson 1998; Higgins & Richardson 1999) and we describe here only the details that are relevant to this application. The model uses an annual time step, and a two-dimensional grid of sites represents space. Based on the potential sizes of P. pinaster and A. cyclops individuals, the model uses a grid cell size of 100 m2. During each simulation year the model simulates the processes of plant growth, senescence of recruits, fire ignition and spread, recruit production, dispersal, fire-induced mortality and establishment (Fig. 1).
A sensitivity analysis of the five factors hypothesized to influence the rate and pattern of spread showed that the model was most sensitive to the dispersal, fire frequency, age of reproductive maturity and recruitment potential parameters, and that the probability of surviving a fire was less important (Higgins, Richardson & Cowling 1996). These results meant that, as adequate data on fire frequency and age of reproductive maturity exist, data collection effort should be directed towards improving the parameterization of the recruitment potential and dispersal functions.
where f is a constant that defines the fire return interval and a is the vegetation age (years). In the model a fire can spread from a burning cell to adjacent cells in the landscape, if the vegetation is flammable. Older vegetation (> 5 years) is assumed to be more flammable (Higgins & Richardson 1998).
The model only allows seed production to occur if an individual stem at a site is reproductively mature. Acacia cyclops and P. pinaster reach reproductive maturity in fynbos after 3 and 6 years, respectively (Richardson et al. 1992). Recruitment can only occur if a site is unoccupied and recently burnt. Potential recruits that do not establish decay in numbers each iteration. We assumed that 0·75 of A. cyclops recruitment bank decays each year (Holmes 1989) and that the entire recruitment bank of P. pinaster decays each year (Richardson et al. 1992).
where s is the dispersion parameter; Rs is the number of recruits produced by a tree of standard size, ds (ds was arbitrarily set to 10 cm diameter for this study); and n is a normalizer that ensures that the area under the distribution equals 1 (Ribbens, Silander & Pacala 1994). The data for estimating the parameters of equation 2 were collected by mapping adult trees of reproductive size in belt transects that ranged in size from 150 × 50 m to 300 × 200 m and by mapping recruits in contiguous 1-m2 plots located in the centre of each belt transect. Stands that had been burnt in the previous season were mapped because alien plant recruitment in fynbos is confined to the immediate post-fire period (Higgins, Richardson & Cowling 1996). The stem diameter and the relative spatial coordinates of each adult tree were recorded. Recruits were defined as seedlings that had emerged and established after the fire. Six independent belt transects for each species were measured in a range of sites characterized by different fynbos communities (Table 1).
Table 1. Description of the study sites used for parameter estimation and aerial photograph interpretation for P. pinaster (P) and A. cyclops (A). More detailed descriptions of these sites can be found in the corresponding references
Working in recently burnt stands also allowed the estimation of the probability of fire-induced tree mortality. We recorded whether each mapped tree in the belt transects was dead or alive and assumed that the probability of surviving a fire increases with tree size (Higgins, Richardson & Cowling 1996; Higgins & Richardson 1998). These data were used to fit a sigmoidal function that described the probability (Ps) of fire survival as a function of stem diameter size (d):
where l is the probability of a tree less than diameter b (cm) surviving, and u is the probability of a tree greater than diameter b surviving; v is a constant that describes the slope between l and u.
The dispersal of recruits is simulated using a mixture of exponential distributions and the assumption that the dispersal direction is uniform (Higgins & Richardson 1999). We use a mixture of distributions because the dispersal parameter estimated using equation 2 is only likely to be adequate for describing local dispersal (Ribbens, Silander & Pacala 1994). Rare long-distance dispersal events are critically important in invasions and plant migration (Higgins, Richardson & Cowling 1996; Pitelka 1997; Clark et al. 1998; Higgins & Richardson 1999). For pine trees invading fynbos, gale force winds are likely to disperse a small proportion of pine seeds considerable distances. While most A. cyclops seeds are likely to be dispersed short distances by passive means, birds disperse some seeds much further (Glyphis, Milton & Siegfried 1981). A mixture of distributions can be used to describe the stratified nature of dispersal (Higgins & Richardson 1999). A mixture of two exponential distributions can be described as:
where p1 is the proportion of recruits in the first component and b1 and b2 are the means (meters) of the two exponential distributions. Higgins & Richardson (1999) describe the processes of fitting mixture models to dispersal data sets. Higgins & Richardson (1999) could accurately estimate the local dispersal components for P. pinaster, but they could only estimate probable parameterizations of the long-distance dispersal component. Conclusions drawn by Higgins & Richardson (1999) guided the definition of a dispersal function. First, data on rare long-distance dispersal will remain (by definition) hard to come by. Secondly, the rare long-distance dispersal component of the mixture model can, if sufficiently rare, be estimated independently of the local dispersal components. Thirdly, because the scale of long-distance dispersal is larger than the spatial extent of this study, the model may not be very sensitive to the exact parameter values of the long-distance dispersal component. For this study we use the parameter estimates from equation 2, the P. pinaster mixture modelling study (Higgins & Richardson 1999) and data on dispersal of A. cyclops seeds by birds (Glyphis, Milton & Siegfried 1981) to define a range of possible mixture models.
Both the mortality and recruitment functions use stem diameter. We collected data on the size–age relationship of A. cyclops by measuring the diameter of trees in even-aged stands of various known ages. Most stands of alien trees are even-aged in fynbos because almost all recruitment occurs after fires. Published data on the size–age relationship for P. radiata (von Gadow 1983) were used for P. pinaster. These data were used to estimate stem diameter (d) as a function of age (a):
where m cm is the maximum diameter and r is the growth rate cm year−1.
In summary (Fig. 1), each iteration (1 year) of the model increments the age of stems in every cell and calculates the diameter of each stem (equation 5). A proportion of recruits that did not establish in the previous year senesce. The model then evaluates whether a fire ignition (equation 1) will translate into fire spread; and fires spread if the vegetation is old enough. The model next simulates dispersal; the stem diameter is used calculate, for all reproductively mature stems, the number of recruits to be dispersed (equation 2); the recruits are dispersed using a mixture of exponential distributions (equation 4). If a site has been burnt then the model evaluates the chance of mortality (equation 3). The model then allows plants to establish; establishment requires that recruits are present and that the site is recently burnt and unoccupied.
Reconstruction of invasion histories
The spatial distributions of P. pinaster and A. cyclops were interpreted from historical aerial photographs (1938, 1961, 1973 and 1989) obtained from the Surveyor General (Chief Directorate: surveys and mapping, Private Bag X10, 7705 Mowbray, South Africa). We selected sites where cover was low in 1938 and where there was little or no evidence of human disturbance, such as ploughing and vegetation clearing. This proved more difficult for A. cyclops as it tends to invade the more transformed lowlands (Higgins et al. 1999); we found four sites for P. pinaster and two sites for A. cyclops (Table 1). The areas mapped ranged from 9 km2 to 16 km2. The scale of the available photographs varied from 1 : 30 000 to 1 : 50 000; all the photographs were enlarged to an approximate scale of 1 : 5000. This scale was suitable for the identification of individual adult trees. Each photograph was geo-referenced by matching features to orthophotos and projected. The interpreted photographs were digitized. The digitized images were converted into raster coverages, which were converted to ASCII files for analysis (see below). Arc/Info (1995) GIS software was used for these procedures.
Indices of rate and pattern of spread
We used four spatial indices to describe the rate and pattern of plant spread for both the empirical and simulated invasion data sets. (i) Plant density: the aerial cover of plants in the site. (ii) Box dimension: the slope of a log-log least-squares linear regression of the number of boxes with sides of size h needed to cover the plant distribution vs. h (Maurer 1994). The use of least-squares regression to estimate the slope was reliable (r2 values generally exceeded 0·97). The box dimension is small for complicated and dispersed patterns, but approaches 2 for solid patterns with smooth boundaries. (iii) Mean neighbour distance: the mean distance of the nearest neighbour from each plant. (iv) Aerial rate of spread: the spread rate was estimated using least-squares regression of the natural log of the area invaded against time (years).
The invasion histories derived from aerial photographs were independent of the data used to parameterize the model, hence the statistical comparisons used to validate the model were between two independent data sets. Two complementary approaches were used to validate the model. First, we evaluated how the data differed from the model’s predictions of the four response variables. Secondly, we tested the level of spatial agreement between the model and the data by doing a cell by cell comparison of the model and data. The level of spatial agreement was evaluated using a measure of the deviance between the model and data for each cell location, and the significance of the deviance was evaluated using a permutation test (described below).
Indices of rate and pattern of spread
Each sequence of historical aerial photographs used in this study is only one possible realization of how the invasion sequence could have progressed. Variations in climatic conditions, fire history and anthropogenic disturbance history mean that many possible invasion sequences could have developed at each site. Because of the observed variation in parameter estimates, a range of parameter estimates are consistent with the observed data. One way of validating the model is to test whether the observed data fall within the range of the model predictions. In addition it is possible to test whether the modelled data are from a different population than the observed data (assuming that the observed data are the true population mean) by using a Wilcoxon signed rank test. If the data fall outside the range of model predictions we will have low confidence in the model predictions, whereas if the Wilcoxon test indicates that the model predictions are consistent with the observed data, we would have higher confidence. These tests were performed for the each of the indices of invasive plant spread described above. Loehle (1997) advocates a similar approach but asked whether the model’s predictions fall within the confidence limits of the data. The danger with the approach we adopted is that if the range of model predictions is wide then the test has low power and will suggest agreement when there is none (Type II error).
The above tests enabled us to determine how well the model predicts the rate and pattern of spread but not whether the model can predict the explicit locations of alien plants. We developed a simple permutation test that allowed us to ask whether the spatial distribution of the model differs from that of the data, i.e. we tested the null hypothesis that the distributions of the populations are the same. A permutation test has the advantage that it makes no assumptions regarding the underlying distributions of the populations being compared, but does assume that the data sets being compared are independent (Manly 1991). Hence the fact that the plant distribution in both the predicted and the observed data are spatially autocorrelated should not bias the test. We use a test statistic A, which is the sum of the absolute differences between the model and the data, to describe the spatial agreement:
Here Dij and Mij are the plant covers at location (i,j) for the observed data (D) and model prediction (M), respectively, and N is the number of sites. By randomly permuting (1000 permutations were used for each site and for each time interval where a comparison between the observed data and model prediction was possible) the spatial locations of model predictions, it is possible to estimate a distribution for the test statistic and hence the significance of the test (Manly 1991). The spatial grain at which the data are compared to the model will influence the agreement (Costanza 1989). A fine-grained comparison is likely to indicate a poorer agreement between data and model than a coarser grained comparison. For this reason we used the concept of a multiple resolution procedure (Costanza 1989) and repeated the test for a range of spatial grains from 1 ha to 36 ha (10 cells to 60 cells).
The recruitment and dispersal parameter estimates are shown in Table 2. Estimates of recruitment potential (Rs) for P. pinaster ranged from 1 to 15 recruits per adult of 10-cm stem diameter. The lower and upper confidence limits of this range were 0·68 and 28·34, respectively (Table 2). Estimates of mean dispersal distance ranged from 4 to 30 m. The confidence limits of the Rs estimates for A. cyclops were 3–89 recruits, and the mean dispersal distance estimates ranged from 3 to 10 m (Table 2). The fitted sigmoidal function describing the probability of a stem surviving a fire as a function of stem diameter [d (cm), equation 3] was Ps = 0·0441 + (0·0819 − 0·0441)/1 + exp(10·784 − d)/1·060 (R2 = 0·69) for P. pinaster. No relationship between stem diameter and probability of mortality was found for A. cyclops. Consequently the frequency of tree survival was used to estimate the probability of fire survival (Ps = 0·017 for A. cyclops). The fitted parameters of the age (years)–size (stem diameter, cm) relationship (equation 5) for P. pinaster were m = 41·4 cm and r= 0·0669 (R2 = 0·98); for A. cyclops m = 20·4 cm and r= 0·093 (R2 = 0·75).
Table 2. Parameter estimates and confidence intervals (CI) for the RECRUITS model (Ribbens, Silander & Pacala 1994). Rs (standard total recruitment), s (dispersion) and normalizer (equation 1) for six sites invaded by P. pinaster and for six sites invaded by A. cyclops (Table 1). Mean dispersal distance (MDD) is presented as a more intuitive interpretation of s. Each site is an independent replicate and the correlation between data and the RECRUITS model prediction were significant (P < 0·05) for all sites
Low CI Rs
High CI Rs
Low CI MDD
High CI MDD
Empirical invasion pattern
The ranges of rates of spread estimated for P. pinaster and A. cyclops were very similar (Table 3). The annual rate of increase in the aerial cover (estimated using linear regression of ln area against time) ranged from 0·027 to 0·062 for the four sites invaded by P. pinaster and 0·027 to 0·061 for the two sites invaded by A. cyclops (Table 3 and Fig. 2). These parameters suggest that it will take 10–30 years for the area invaded to double. As typical fire return intervals are between 8 and 25 years, it will take one or two fires for the area invaded to double. The square root of the area invaded per year can be used to estimate the linear rate of spread (Table 3). These data are reported because rates of spread are often reported this way, although for this study these estimates are likely to be biased by boundary effects. The estimated linear rate of spread ranged from 17 to 31 m year−1 (Table 3).
Table 3. Estimated invasion rates and linear rates of spread for four sites invaded by P. pinaster and two sites (Gansbaai and Hawston) invaded by A. cyclops. Aerial rates of spread were estimated using a natural log of area vs. time linear regression; linear rates of spread were estimated using a square-root of area vs. time linear regression (n = 4)
Aerial rate of spread
Linear rate of spread (m year−1)
The pattern of invasion was also similar for the two species (Fig. 2). A comparison between the density and box dimension data showed that the box dimension increased with alien density. This means that the distribution of alien plants is scattered early on in the invasion but becomes more aggregated as the invasion progresses. Clumping only occurred later at two of the P. pinaster sites (Elim and Genadendal; Fig. 2). The mean nearest neighbour distance tended to decrease as the invasion progressed, again indicating that plant distribution aggregates as the invasion progresses. The correlations between the variables indicated that rate and pattern of spread were related (Table 4).
Table 4. Correlations between empirical measurements of invasion pattern over time (1938, 1961, 1971, 1989) at four sites invaded by P. pinaster (n = 16) and two sites invaded by A. cyclops (n = 8)
Mean neighbour distance
Mean neighbour distance
Mean neighbour distance
For the validation runs, the model was initiated with the 1938 spatial distribution of alien plants. The parameters that the model is most sensitive to (recruitment, dispersal and fire frequency parameters; Higgins, Richardson & Cowling 1996) were varied. The age at maturity (the other important parameter identified in Higgins, Richardson & Cowling 1996) was not varied as this is unlikely to change from site to site for the species considered here. We used our knowledge of the variance of these key parameters to define a mean and standard deviation for each parameter (Table 5). The range of fire return intervals (f) was selected to include fire management regimes that ranged from arson to fire prevention. The recruitment levels (Rs) represented the ranges recorded in the field (Table 2). The dispersal parameters (p1, b1, b2) were estimated from the parameters estimated in this study, the analysis of evidence for long-distance dispersal in P. pinaster dispersal data (Higgins & Richardson 1999), and data on dispersal of A. cyclops seeds by birds (Glyphis, Milton & Siegfried 1981). Assuming these parameters come from truncated normal distributions (Table 5), we randomly selected a set of parameter values for a simulation run (27 factor combinations were run for each site).
Table 5. Mean and standard deviations and constraints used to draw parameter values for the simulation runs for P. pinaster and A. cyclops. Parameters were drawn from truncated normal distributions (f= fire return interval*; Rs= standard total recruitment†; p1, b1, b1= parameters of a mixture of exponential distributions used to describe dispersal‡)
† equation 2,
‡ equation 4.
Simulations using these empirical parameter estimates showed good agreement between the model and the data (Fig. 2). The empirical estimates of the indices always fell within the range of values predicted by the model; moreover the indices predicted by the model agreed with the empirically estimated indices in 41 out of 54 comparisons (Fig. 2). This suggests that small errors characterizing the initial conditions, particularly errors in digitizing outlying individuals, and not knowing the exact fire history does not result in the propagation of errors. The range of index values did, however, become wider for the plant density and box dimension metrics as time progressed, hence we had less power to detect differences between the model and the data in 1989 than in 1961. It follows that the fact that the predicted and observed index values did not differ in 1989 may be misleading (Fig. 2). In addition, it may be that this increasing agreement over time is because stand density is approaching 1: Genadendal (P. pinaster) and Hawston (A. cyclops) have 1989 densities above 0·7; however, at the other sites the 1989 densities were less than 0·5. The predicted and empirical spread rates also agreed well in that the rates of spread predicted by the model did not differ from the empirically observed rates of spread for all six sites (Fig. 3). The correlation coefficients between the response variables generated by the simulation model were very similar to those reported empirically (compare Table 6 with Table 4).
Table 6. Correlations between simulated measurements of invasion pattern over time (1938, 1961, 1971, 1989) at four sites invaded by P. pinaster (n = 432) and two sites invaded by A. cyclops (n = 216)
Mean neighbour distance
Mean neighbour distance
Mean neighbour distance
The results of the tests of the spatial agreement between the model and the data are presented in Fig. 4. As expected, the general trend showed that, as the spatial grain of the analysis increased, the agreement between the data and the model improved. The average level of spatial agreement between the model and data was generally above 0·5 and often above 0·75 (Fig. 4). The level of spatial agreement may be regarded as somewhat biased in the early and late phases of invasion, as a model that predicts the right density at these phases is likely to have a high level of spatial agreement. However, the agreement remained high at intermediate densities (0·2–0·8; cf. Fig. 2), suggesting that the level of spatial agreement (Fig. 4) was not an artefact. The permutation test (see the Methods) aimed to assess whether the agreement between the model and data was better than would be predicted by a randomly permuted distribution of invaded sites. For P. pinaster the spatial agreement between the model and the data was generally better than would be expected for a random distribution of invaded sites. In the case of the Genadendal site, however, the spatial agreement was sometimes poorer than random in 1989 (Fig. 4); this could be due to either sporadic clearing of alien plants at this site or the rugged topography of the site, which may have caused digitizing errors. The spatial agreement between data and model was poorer for A. cyclops than for P. pinaster. At the Gansbaai site the model was not significantly better than random in 1973 or 1989. At the Hawston site the agreement was occasionally no better than random in 1973 and 1989. The poorer spatial agreement for the A. cyclops sites may be because this species is bird dispersed: i.e. dispersal may be directed towards perch sites. Interestingly, the permutation test appeared to have more power to detect weakness in the agreement between the model and the data at larger spatial grains.
There are many potential sources of error in ecological models and it is difficult to evaluate all of them explicitly. Important potential sources of error in this study included the assumptions of the size of an individual plant, an absorbing boundary and spatial homogeneity of the environment, no interfire recruitment, parameter estimation error and initial condition estimation error. In addition, the exact fire histories for the different sites were unknown. Despite these many potential sources of error, the model’s predictions agree well with the independently reconstructed invasion sequences at the stand (< 20 km2) scale; this agreement was robust given the observed variation in independent parameter estimates. We consider the model validated in that it makes predictions about the spread rate that do not differ significantly from independently observed rates (Fig. 3). In addition plant density, box dimension and nearest neighbour comparisons, made at intervals defined by the availability of aerial photograph records, show that the model and data are indistinguishable for 76% of the comparisons made (Fig. 2). The validated spread model gives us confidence to explore a range of management scenarios. For example, we could determine the most cost-effective alien-clearing strategy and the potential impact of alien plants on fynbos ecosystems (Higgins etal. 1997a,b; Higgins, Richardson & Cowling 2000; see Wadsworth etal. 2000a, for a similar approach).
The agreement between the model predictions and data implies that some of the fundamental assumptions the model makes regarding how invasions proceed in fynbos are acceptable. Importantly, it was assumed that the community composition of the receiving environment does not influence the rate and pattern of invasion at this scale. The sites invaded by A. cyclops ranged from densely vegetated dune asteraceous fynbos to a sparsely vegetated headland by-pass dune, while the sites invaded by P. pinaster ranged from proteoid fynbos (characterized by a proteoid overstorey and ericaceous understorey) to ericaceous fynbos (single stratum). In addition, high levels of alpha, beta and gamma diversity within these vegetation types (Cowling 1990; Simmons & Cowling 1996) imply that local species composition would have varied considerably across sites. While the high levels of functional redundancy characteristic of fynbos (Cowling et al. 1994) may explain the limited effect of species composition on invasion rates, the limited effect of structural and soil differences between the sites is intriguing. This circumstantial evidence contrasts with finer scaled experimental studies that have illustrated that the composition, diversity and nutrient availability of the receiving environment strongly influences invasive success and invasion patterns (Tilman 1997). The limited effect of the receiving environment in fynbos can be attributed to two factors. First, the extreme fecundity of the alien populations may make differences in the invasability of different fynbos communities insignificant (Milton 1980; Honig, Cowling & Richardson 1992). Secondly, the environmental tolerances of fynbos invaders are considerably wider than those of the native species and the range of environmental variance at the study sites (Richardson et al. 1992; Higgins et al. 1999).
The empirical data on the rates of invasion presented here contribute to a growing database on plant invasion rates. The rates of invasion (proportion of area invaded per year) estimated from historical aerial photographs were between 0·027 and 0·062. This is slower than the invasion rate of P. radiata in fynbos (0·079, calculated from Richardson & Brown 1986). Forcella (1985) reviewed the spread of 40 alien weed species in north-western United States; invasion rates calculated from his data ranged from 0·0262 to 0·0562 (mean = 0·0395). Other invasion rates found in the literature provide similar estimates: 0·059 for Ammophila arenaria invading dunes in California (Buell, Pickart & Stuart 1995), 0·042–0·110 for a range of species in riparian habitats in the Czech Republic (Pysek 1991), 0·031–0·056 for three Impatiens species in the British Isles (Perrins, Fitter & Williamson 1993), 0·131 for Eragrostis lehmanniana in southern Arizona (calculated from data in Anable, McClaran & Ruyle 1992) and 0·12 for Bromus tectorum in north-western United States (calculated from Mack 1981). However, extremely rapid invasion rates have also been reported: 0·59 for Mimosa pigra in northern Australia (Lonsdale 1993) and 0·701 for Baccharis pilularis in northern California (calculated from Williams, Hobbs & Hamburg 1987). The range of aerial invasion rates reviewed above is large (0·02–0·7); these rates provide valuable context for evaluating the match between the model’s predictions (0·01–0·076) and the range observed in this study (0·027–0·062).
Fewer studies report linear rates of spread, and the estimates reported here should be regarded as minimal estimates as long-distance dispersal events fell outside the scale of this study. Our study suggests that P. pinaster and A. cyclops spread at 21–31 m year−1 in fynbos; this is slightly slower than P. radiata invading fynbos (31 m year−1 calculated from Richardson & Brown 1986). Mimosa pigra spread at 76 m year−1 in northern Australia (Lonsdale 1993), and Ammophila arenaria spread at 14 m year−1 in California (Buell, Pickart & Stuart 1995). Linear invasion rates estimated at continental scales can be more spectacular: Mack’s data on Bromus tectorum invasion into north-western United States suggest rates of up to 5 km year−1 (estimated from Mack 1981); Plummer & Keever (1963) report spread rates of between 4 and 13 km year−1 for Heterotheca latifolia invading the Georgia piedmont region; while Williamson (1996) reports Impatiens at 38 km year−1. Clearly the scale of the invasion (and the invasion study) strongly influences the reported spread rate (Higgins & Richardson 1998, 1999). The estimates of P. pinaster invasion rates of up to 31 m year−1 (reported here) contrast strongly with the 200–500 m year−1 predicted by the same model at an unlimited spatial extent (Higgins & Richardson 1999). The latter rates are in the same order of magnitude as post-glacial migration rates of trees (c. 100–1000 m year−1; Delcourt & Delcourt 1987; Birks 1989).
The importance of the scale of investigation on the reported invasion rates suggests that we should be cautious in declaring a model validated; the model’s behaviour is not necessarily robust under all conditions and at all scales. While the coupling between parameter estimation error and model prediction error is likely to vary depending on the model’s sensitivity to that parameter (Ruckelshaus, Hartway & Kareiva 1997), this sensitivity may only be revealed at certain scales. For instance, the analysis presented here suggests that any of the dispersal parameterizations used are adequate; however, under severe levels of fragmentation and at more extensive spatial scales the inadequacies of some of these parameterizations would be revealed (Higgins & Richardson 1999). This observation further emphasizes the critical importance of dispersal and scale in spatial simulation models (Gardner et al. 1991; Higgins, Richardson & Cowling 1996; Ruckelshaus, Hartway & Kareiva 1997; Higgins & Richardson 1999).
Linking of spatial simulation models to pattern analysis provides a direct link between pattern and process (Cale, Henebry & Yeakley 1989). We found correlation between rates and patterns of spread in both the empirical data and in the simulation model. Similar relationships were found in a simulation study (Higgins, Richardson & Cowling 1996) and in studies of post-glacial migration of trees (Davis 1987). Because species that spread with a more diffuse pattern tend to spread faster, it may be possible to use pattern analysis of migrating species to separate species capable of rapid spread from those that are likely to be slow spreaders. Such an analysis would only be useful if there was evidence that environmental conditions do not limit the species at its range boundary. Achieving this necessitates linking spread models more tightly with models of environmental tolerance than they are at present.
Very few models of the spatial dynamics of plants have been validated (for an example see Auld & Coote 1990). Although empirical parameterizations can generate confidence in models, good parameter estimates do not guarantee that the key processes are included or correctly modelled. Although the process of parameterizing and validating a model can be long and arduous, it remains possible. Moreover, the insights of process-orientated models can be profound (Pacala & Deutchman 1995; Pacala et al. 1996). Assessing when these additional efforts are warranted remains difficult. We believe that this effort is warranted when statistical precedents for the phenomena of interest do not exist or when factors known to influence the phenomena cannot be readily included in an analytical model (Higgins & Richardson 1996). For instance, in invasions the importance of chance events, discrete disturbance events, long-distance dispersal and propagule pressure motivate against analytical models (Higgins, Richardson & Cowling 1996), while the importance of plant–environment interactions and the novelty of many invasion scenarios motivate against statistical models (Higgins & Richardson 1998). The challenge is to develop models that summarize the key processes in the most parsimonious way. In practice this results in the development of case-specific rather than general models. Our experience with biological invasions suggests that the importance of plant–environment interactions ensures that general models are of limited value when context-specific and quantitative predictions are needed (Higgins & Richardson 1998). Hence we anticipate that many of the functions and assumptions used in this model would have to be revised when applying the model to another system. However, we believe that the approach demonstrated here can be used to define, parameterize and validate simple context-specific spatial demographic models.
This work was funded by the Foundation for Research Development, the Institute for Plant Conservation, WWF South Africa and BP South Africa. Thanks to Jessica Kemper, Fernando Ojeda and Maria Ojeda for help with the field work. Thank you to the reviewers for their useful comments and suggestions.
Received 3 April 2000; revision received 29 December 2000