Yvonne M. Buckley, The Ecology Centre, University of Queensland, School of Integrative Biology, St Lucia, QLD 4072, Australia (e-mail firstname.lastname@example.org).
1Management decisions regarding invasive plants often have to be made quickly and in the face of fragmentary knowledge of their population dynamics. However, recommendations are commonly made on the basis of only a restricted set of parameters. Without addressing uncertainty and variability in model parameters we risk ineffective management, resulting in wasted resources and an escalating problem if early chances to control spread are missed.
2Using available data for Pinus nigra in ungrazed and grazed grassland and shrubland in New Zealand, we parameterized a stage-structured spread model to calculate invasion wave speed, population growth rate and their sensitivities and elasticities to population parameters. Uncertainty distributions of parameters were used with the model to generate confidence intervals (CI) about the model predictions.
3Ungrazed grassland environments were most vulnerable to invasion and the highest elasticities and sensitivities of invasion speed were to long-distance dispersal parameters. However, there was overlap between the elasticity and sensitivity CI on juvenile survival, seedling establishment and long-distance dispersal parameters, indicating overlap in their effects on invasion speed.
4While elasticity of invasion speed to long-distance dispersal was highest in shrubland environments, there was overlap with the CI of elasticity to juvenile survival. In shrubland invasion speed was most sensitive to the probability of establishment, especially when establishment was low. In the grazed environment elasticity and sensitivity of invasion speed to the severity of grazing were consistently highest. Management recommendations based on elasticities and sensitivities depend on the vulnerability of the habitat.
5Synthesis and applications. Despite considerable uncertainty in demography and dispersal, robust management recommendations emerged from the model. Proportional or absolute reductions in long-distance dispersal, juvenile survival and seedling establishment parameters have the potential to reduce wave speed substantially. Plantations of wind-dispersed invasive conifers should not be sited on exposed sites vulnerable to long-distance dispersal events, and trees in these sites should be removed. Invasion speed can also be reduced by removing seedlings, establishing competitive shrubs and grazing. Incorporating uncertainty into the modelling process increases our confidence in the wide applicability of the management strategies recommended here.
Management models of invasive plants have been developed with the aim of reducing population growth rate (Shea & Kelly 1998; Buckley, Briese & Rees 2003a) and area of occupancy (Rees & Paynter 1997; Buckley et al. 2004). These approaches are appropriate for management of invasive plants locally or where a population is well established; however, where management is required on a landscape scale, or invasion is in its early stages, it may be more appropriate to focus on containing or eradicating the invasion or reducing spread rates.
Spatial management models have focused on whether control can be best achieved by targeting satellite infestations or the main body of the invasion. Moody & Mack (1988) highlighted the importance of eradicating satellite infestations, while Hulme (2003) suggested targeting the main body of the population, rather than the satellites, might be appropriate for a source–sink meta-population. However, recent work has shown that both of these strategies might be appropriately applied, with the most effective strategy depending on characteristics of the population (Taylor & Hastings 2004; Travis & Park 2004). Another approach to spatial management includes Sharov & Liebhold's (1998) demonstration of the use of barrier zones to slow down spread.
In models for both endangered and invasive species we are often faced with the need for swift management action in the face of considerable uncertainty regarding the underlying processes. In order to make choices between management actions, we need to take into account estimates of the uncertainty attached to parameters in our models (Akcakaya & Raphael 1998; Caswell et al. 1998; Ellner & Fieberg 2003; Higgins et al. 2003). We can then ask whether there is a clear management strategy despite our uncertainty, or whether the uncertainty is so great that it precludes any recommendation for management.
At least 16 species of pine (Pinus) are invasive and pose serious threats to biodiversity, ecosystem function and landscape values in South Africa, Australia and New Zealand (Richardson, Williams & Hobbs 1994; Richardson & Higgins 1998). A wide range of conifers was originally introduced to New Zealand for production purposes, shelter-belts and erosion control. Spread outside of planted areas was noted around 100 years ago (Smith 1903) and the total area of New Zealand currently affected (at least 1 tree/ha) is estimated at 150 000 ha (Ledgard 2001). These invasions primarily threaten natural grasslands and shrublands and abandoned pastures (Allen & Lee 1989).
We have summarized the data available on the demography and dispersal of the invasive pine Pinus nigra Arn. ssp. lauricio (Poir.) Maire and parameterized a stage-structured spread model. The data come from a range of mostly unpublished sources and are typical of the kind of data often available to those making management decisions for invasive plants. Occasionally sufficient data exist for partitioning uncertainty into error distributions at multiple spatial and temporal scales (Buckley, Briese & Rees 2003b), or bootstrap methods are used where the original data are resampled to give uncertainty estimates. In this case, however, data are too sparse for either of these methods and, as in a similar problem analysed by Caswell et al. (1998), we used Monte Carlo sampling of a large number of random parameter values from an estimated distribution or range for each parameter, and presented model outputs with empirical confidence intervals (CI) from this large number of model runs (100 000).
Sensitivity and elasticity analyses of matrix models have been used to identify ecologically appropriate targets for reducing the population growth rate of invasive plants (Shea & Kelly 1998; Parker 2000). Neubert & Caswell's (2000) technique allows the rate of spread of a structured population to be predicted by combining matrix models with dispersal kernels, allowing sensitivity and elasticity of invasion wave speed as well as population growth rate to be calculated. Neubert & Parker (2004) have recently applied this approach to an invasive plant population. We show, using P. nigra, how, despite uncertainty in parameter estimates, it is possible to guide management by identifying the demographic and dispersal processes with the greatest potential impact on invasion wave speed in three habitat types: ungrazed grassland, shrubland and grazed grassland. A novel and important development of this method is the generation of CI on our measures of sensitivity and elasticity, allowing us to assess the robustness of management recommendations to uncertainty.
Parameter estimates for the model were obtained from published data and from fieldwork carried out in 1986–90, 1992–97 and 2001–03 on the South Island of New Zealand.
Pinus nigra sheds its cones annually. In March 2003 14 trees between the ages of 15 and 34 years in open parkland (Lake Ruataniwha, 170°03′E, 44°16′S) were used to determine the relationship between tree age and number of cones produced. Last season's cones on the ground within 1 m of the canopy margin were counted. Cones are relatively heavy and we assumed that few cones fall further away than 1 m. A regression of cone number against tree age was used to identify two adult fecundity classes; data were analysed using a generalized linear model with Poisson errors and log-link function.
Isolated trees in the same location (Lake Ruataniwha) were assessed for cone production for 6 years between 1992 and 1997. Ten ‘normal’ and 10 ‘low coning’ (categories based on cone status in 1992) trees 15–16 years old were chosen for monitoring, and were labelled to enable identification in subsequent years. Each year the number of cones per tree was counted. These data were used to estimate the variability in coning in order to calculate averages for fecundity over several years.
Samples of unopened cones from seven individual trees were taken in March 2003 and examined for seed viability by dissection. Viability was assessed as the percentage of fully formed seed from three lots of 100 seeds tree−1.
The probability of germination and survival to 1 year and yearly juvenile survival were determined from a seedling germination and establishment experiment undertaken in the Harper-Avoca catchment, 171°34′38″E, 43°09′41″S (grassland plots) and 171°33′36″E; 43°09′31″S (shrubland plots), in 1985–92. Seeds were sown by hand in October 1985 into two native vegetation types: native tussock grassland and open shrubland. Four 40 × 50-cm plots and two control plots were established in each vegetation type and 200 sound seeds plot−1 were sown. Seedling numbers were recorded four times during the first growing season, three times during the second growing season and then at 28, 59 and either 78 or 86 months. The progress of all seedlings was followed from one assessment to another to obtain survival probabilities. Establishment was calculated as the proportion of seedlings per plot still alive after 12·5 months. Survival of those seedlings was calculated as the proportion of established seedlings alive 78 months after sowing (grassland) and 86 months after sowing (shrubland). A per year average survival rate was calculated as: (no. survived 5 years/no. established)1/5·5 for grassland and (no. survived 6 years/no. established)1/6 for shrubland.
There were no data available on longevity or survivorship of mature trees (> age 10) in New Zealand. We assumed that all adult trees have a similar probability of survival from year to year. We assumed that the proportion of adult trees alive at some maximum age, u, was 0·05 and that the range of maximum ages was between 100 and 250 years, with a median value of 175 (using table 12.2 in Keeley & Zedler 1998 as a guide to maximum age for ecologically similar pine species). Therefore , where sa is the yearly adult survival probability and u is the maximum age. Yearly survival probability, sa, is then calculated as sa = 0·051/u. In the same way we calculated the probability of 5% of trees being retained in the adult 1 class for y1 years (y1 minimum 15 and maximum 30 years) as ; the uncertainty range for r was calculated with sa set to its longevity value of 175 years. The parameter range for y1 was determined from the regressions of cones produced against age of plant (see above).
We assumed that dispersal is biphasic, with most seeds falling near the parent plant as cones open and seeds drop out but with rare long-distance dispersal events also occurring (Horn, Nathan & Kaplan 2001). We fitted a Gaussian kernel to the short-distance dispersal data, allowing dispersal in two directions, and a negative exponential function to the long-distance data, allowing dispersal in one direction only. Conditions for the dispersal of seeds from P. nigra are thought to coincide with warm north-westerly winds causing most long-distance dispersal events to be in one direction only (as in Fig. 1). The Gaussian and negative exponential functions have been used previously in phenomenological models of short- and long-distance dispersal (Nathan & Muller-Landau 2000; Neubert & Caswell 2000; Levin et al. 2003).
Fieldwork was carried out in Glenbrook (170°02′E, 44°19′S) in March 2003 to quantify short-distance dispersal patterns from a known source. The source trees were in a shelter-belt planted in the 1960s and ungrazed land to one side had been colonized by a mixed-age stand of offspring. We ran transects perpendicular to the source trees and recorded position and age of each tree on the transect. We used the data from the oldest colonizers (15–18-year-old trees) to parameterize a short-distance dispersal kernel. We used a Gaussian kernel with µ= 0 and maximum likelihood to estimate the standard deviation of the data, which were truncated at 0; CI were calculated from the likelihood ratio (Aitkin et al. 1989) (n = 129 15–18-year-old trees).
We used data from aerial photographs taken in 1965 and 1980 for the Mount Barker site to estimate parameters for the long-distance dispersal function. The original source of P. nigra wildings was a shelter-belt containing trees planted in approximately 1910 (171°35′15″E, 43°21′30″S), marked by an S in Fig. 1. By 1945 the first outliers appeared to the right of the shelter-belt. In the late 1950s/early 1960s there was a major spread event over Mt Barker just to the right of the shelter-belt and down wind in a south-easterly direction for several kilometres (Langford 1984) (Fig. 1a). In both the 1965 and 1980 aerial photographs, an area of dense, closed canopy pines was visible closer to the shelter-belt, with a tail of scattered wildings stretching towards the right (south-east; Fig. 1a,b). It could clearly be seen that long-distance dispersal was primarily in one dimension only, south-east of the shelter-belt. We assumed that all dispersers were detected with the same error regardless of distance from the original source; a one-dimensional kernel could therefore be calculated directly from these data.
The black and white 1965 and 1980 aerial photographs were digitized and orthorectified to the New Zealand map grid (to a positional accuracy of ±15–20 m) and the scattered trees were identified in both. An image-processing routine was written for automated mapping of the trees in the digital images, using a template-matching method. This was successful for the majority of trees; manual identification was used for trees in the more complex terrain. One-thousand and seventy-six trees from the 1965 image and 7532 trees from the 1980 image were identified. Trees older than c. 10 years could be identified from aerial photographs in this way (N. Ledgard, personal observation). Field work in February 2004 was carried out to ground-truth and age some of the trees in the tail of the dispersal distribution; in total 10 trees were aged in the area marked with an ellipse in Fig. 1b.
Trees present in the 1980 photograph but not in the 1965 photograph formed the basis for our analysis of long-distance dispersal. We assumed that these trees were derived from a source present in 1965. In the 1965 photograph we identified 43 large trees in the area within 1000 m south-east of the shelter-belt; there were four large trees within 1000–2000 m and three large trees within 3000–4000 m. We estimated the negative exponential function parameter, α, from the 1980 data under four different scenarios: that the seed source was at the shelter-belt; or within 1000 m, 2000 m or 4000 m of the original shelter-belt. We used parametric survival analysis, substituting distance for the conventional failure time (McCallum 2000), to estimate the parameter for a negative exponential model of the distribution of distances. Upper and lower 95% CI across all four scenarios were calculated from the standard errors of the highest and lowest estimates of α obtained.
Trees in the tail of the 1980 distribution were at least 20 years old at that time (Langford 1984). We confirmed this estimate by locating and ageing trees in this area in February 2004; trees were aged using tree cores and by counting internodes. Ten trees were aged, with two older age classes found: one group approximately 30 years old and one group approximately 40–45 years old. There was little short-distance recruitment around these trees and we therefore assumed that data in the tail of the 1980 distribution were not overlaid with short-distance dispersal events. These data also indicated that trees present in the tail of the 1980 distribution were probably not short-distance dispersers from trees present at those locations in 1965 but too small to be picked up in that photograph, as these putative sources would currently be at least 50 years old.
We had no independent data on the proportion of seeds entering the long-distance dispersal phase (p). Estimates in the literature (Nathan et al. 2002a) give values of 1–5% for p. We set a wider uncertainty range of 0·001–0·2 for p, i.e. between 0·1% and 20% of seed going into the long-distance dispersal phase.
Age at reproductive maturity
We used data from the short-distance dispersal transects to estimate the minimum age at reproductive maturity. Trees on transects were aged and examined for coning; this gave an estimate for the minimum age at coning. A maximum age for coning was determined from the 1992–97 Ruataniwha data where one of the low-coning trees did not start coning until it was 19 years of age. We calculated q (the probability of remaining in a non-reproductive state) assuming that only 5% of trees remain non-reproductive for y2 years (y2 has a minimum of 1, maximum of 12 and median of 6): , where sj is the juvenile survival rate described above, and the uncertainty range for q was calculated with sj set to its value for retention in the non-reproductive stage for 6 years.
We estimated invasion speed using Neubert & Caswell's (2000) integrodifference equation approach for a stage-structured population. We developed code in the computer program R (R Core Development Team. 2004). A life-cycle diagram for P. nigra is shown in Fig. 2 and the corresponding stage-structured population projection matrix, A, is shown below (equation 1); parameter estimates and their sources are given in Table 1.
Table 1. Parameter values for the life-history matrix, A, and dispersal parameters for dispersal matrix, K
Uncertainty range, minimum–maximum or distribution
Fecundity for adult 1 trees
Fecundity for adult 2 trees
0, 0·5, 0·9
Grazing intensity, 0 = no grazing
Proportion of seeds in long-distance dispersal kernel
Probability of remaining in final juvenile class, delay in reproductive maturity
Probability of remaining in adult 1 class
SD = 0·043
Seedling establishment for grassland
SD = 0·035
Seedling establishment for shrubland
Seedling to juvenile, for grassland
Seedling to juvenile, for shrubland
Seed viability for adult 1 trees
Seed viability for adult 2 trees
Long-distance dispersal kernel shape parameter
Standard deviation of short-distance dispersal
( eqn 1)
The parameters are: z, intensity of grazing (0, no grazing, up to 1, where grazing completely prevents establishment); b, probability of a seed establishing; f1, fecundity of the adult 1 class; f2, fecundity of the adult 2 class; sj, survival of juveniles; q, reproductive delay parameter (0, immediate progression, up to 1, where all trees are retained in the juvenile 5 class); sa, adult survival; r, retention parameter for the adult 1 class (0, immediate progression, up to 1, where all trees are retained in the Adult 1 class) (Table 1). The matrix A describes life-stage transitions with a yearly time-step, including one juvenile stage class where residence is dependent on grazing, z, and three further classes where, if juveniles survive, they progress yearly. The final juvenile stage class has a reproductive delay parameter, q; individuals can be retained in that class before moving on to the first reproductive adult class (Fig. 2). Yearly progression through the juvenile classes is introduced to ensure that no trees reproduce at ages younger than an observed minimum. Parameters are assumed to be typical of low-density populations and both classes of adults have the same dispersal kernels. The dispersal kernels are contained in the matrix K of the same dimensions as A. If there is no dispersal the kernel is the Dirac delta function. Where dispersal does occur, the kernel is a mixture of short-distance (Gaussian) and long-distance (negative exponential) dispersal (see for a similar approach Clark 1998):
where P(d) is the probability of travelling a distance d, p is the probability of a seed entering the long-distance dispersal kernel, Pshort(d) is the Gaussian short-distance dispersal kernel with a mean of 0 and a standard deviation of σ:
and Plong(d) is the negative exponential long-distance dispersal kernel with a mean of β:
Moment-generating functions of the dispersal kernels are needed for the analysis of the Neubert & Caswell (2000) model and for P(d) are given by:
The moment-generating function of a kernel only exists for some finite interval around s= 0 and i= 7 or 8 (where the subscripts of m represent movement of seeds from either of the two adult stages to the seedling stage). The minimum value of the wave speed is calculated as c* = min0<s<ŝ[1/s ln ρ1(s)], where ρ1 is the dominant eigenvalue of the matrix H, which includes both the stage-structured demography and dispersal parameters. H =A°K, where ° is the element by element or Hadamard product (for details see Neubert & Caswell 2000).
Sensitivity of wave speed to model parameters measures how a small additive change in the parameter affects wave speed, whereas elasticity measures how a proportional change in the parameter affects wave speed. Some transitions were functions of underlying parameters (equation 1) and some underlying parameters occurred in more than one transition (e.g. sj). We therefore used the following formulae to decompose sensitivity of λ, population growth rate, and c*, wave speed, to particular stage transitions (ak.l) into sensitivity of λ and c* to the underlying demographic parameters (x), using methods in Caswell (2001):
where k,l give the subscripts of each transition, a. ∂λ/∂ak,l is the sensitivity of λ to the transition ak,l, which is the conventional sensitivity of population growth rate to changes in the transition value calculated as per Caswell (2001). Similarly elasticities of λ and c* to the underlying demographic parameters are given by:
and are easily worked out given terms for ∂λ/∂x or ∂c*/∂x (see Appendix S1 for ∂λ/∂x).
Sensitivities and elasticities of c* to demographic transitions ∂c*/∂ak,l are calculated according to equations 26 and 27 in Neubert & Caswell (2000), and sensitivity and elasticity of c* to dispersal parameters from equation 28 in Neubert & Caswell (2000) and equation 8 in Caswell, Lensink & Neubert (2003), which corrects the error in Neubert & Caswell (2000) (see Appendix S1 material for the sensitivity of the dispersal kernel moment generating functions, ∂m/∂x). Sensitivities and elasticities of c* to the underlying parameters in the transitions are calculated in the same way as outlined above and in Appendix S1 for λ.
Sensitivity and elasticity analyses indicate which parameters, if perturbed, lead to the greatest additive or proportional changes in λ or c*. Elasticities are commonly used to rank parameters in a management context to allow comparison of changes in parameters with very different scales. For parameters close to 0, however, such as p or b, a very large proportional change in the parameter is needed in order to achieve a change in the wave speed relative to another parameter slightly greater in magnitude. In reality, changes in different parameters are not equally easy or costly to achieve; sensitivity and elasticity analyses therefore only provide part of the answer to how best to manage populations (de Kroon, van Groenendael & Ehrlén 2000).
Our data were insufficient to partition the variance in parameters systematically according to measurable covariates. Therefore, we defined distributions for each parameter based on published and unpublished data and, where data were not available, distributions were estimated with high uncertainty from personal observations or indications in the literature. Distributions were obtained either by calculating a best-fit distribution from existing data or, if insufficient data were available to specify a best-fit distribution, a triangular distribution was specified with a median point estimate, minimum and maximum values, where probability decays towards the maximum and minimum.
Given a distribution of values for each parameter, we sampled a value at random from each distribution and used these random parameters to populate the demographic matrix A and the dispersal matrix K. Each matrix contained a random value for each parameter, there was no correlation between parameters and 100 000 different matrices were created. Empirical 95% CI were constructed around median values of λ, c* and their sensitivities and elasticities, ensuring that uncertainty in the parameters was translated into uncertainty in model outputs.
From the 2003 Ruataniwha data we found that the number of cones produced depended on age (P < 0·0005), with younger trees producing significantly fewer cones. Between 1992 and 1997 average cone production for the ‘normal’ population was 170 cones tree−1 year−1 and for the ‘low coners’ 20 cones tree−1 year−1. If we assumed that approximately 30% of the population comprised low coners (N. Ledgard, unpublished data), then the total average production for the Lake Ruataniwha population was 125 cones tree−1 year−1 for young trees (< 25 years old).
We had no data on cone production through time for older trees (> 25 years); the 2003 data from Ruataniwha were representative of a very good coning year. Our fecundity estimates in 2003 were therefore at the upper end of the coning range for this species. Miller & Knowles (1986) state that P. nigra produces large seed crops every 2–5 years, with some seed usually produced each year. For young trees, average cone production per tree over 5 years was 125 cones tree−1 year−1, which was 0·32× the maximum value observed from that time period. We assumed that this relationship was similar for older trees and estimated the average cone production for older trees from 0·32 × 2003 observations. Maximum cone production for the older trees was set to the 2003 value and minimum cone production was set to the point estimate for younger trees.
Seed viability and number of seeds per cone
Seed viability varied from 0·34 to 0·74, with younger trees tending to produce proportionally less viable seed than older trees; however, not enough trees were sampled to be able to confirm this relationship statistically. For all fecundity estimates we used a figure of 40 seeds per cone (N. Ledgard, unpublished data) and multiplied this by 0·75 for trees > 25 years and 0·5 for trees < 25 years, to account for the trend in seed viability.
From the Harper-Avoca data, establishment was higher in grassland (18·8% of sown seed germinated) than in shrubland (5·8%). Establishment from seed is not possible in improved pasture because of herbaceous sward competition (Davis, Grace & Horrell 1996). Pinus nigra has only a transient seed bank (L. Langer, unpublished data) and cones are dropped each year; we assumed that seedlings germinate within the same year that they are dispersed.
The exclusion of rabbits during a plantation establishment trial doubled establishment of P. nigra from seed (Davis, Grace & Horrell 1996). Using grassland establishment parameters we ran the model and uncertainty analysis with grazing set to three different levels, 0, 0·5 (half the normal establishment) and 0·9, to explore how grazing impacts on wave speed, sensitivities and elasticities.
The mean of the Gaussian short-distance dispersal kernel was assumed to be 0 and the standard deviation (σ) was estimated as 27·4 m ± 95% CI 24·4 m and 31·1 m.
Using data on tree locations from the 1980 and 1965 aerial photographs of the Mt Barker and Acheron site, we parameterized negative exponential functions for the four long-distance dispersal scenarios as follows: source at shelter-belt, β = 1845 ± 42 SE , n= 7532; source at 1000 m from shelter-belt, β = 1287 ± 18 SE, n= 5491; source at 2000 m from shelter-belt, β = 1326 ± 27 SE, n= 2491; source at 4000 m from shelter-belt, β = 1236 ± 47 SE, n= 718. Ninety-five per cent CI were estimated for each scenario and the maximum and minimum of these were used to define a triangular distribution for β in the uncertainty analysis, with an estimate of the median from scenario 2 as this was considered to be the most likely scenario because of the large number of potential source trees within 1000 m.
The median population growth rate λ and asymptotic spread speed c* were calculated from the uncertainty analysis and are shown with 95% empirical CI in Table 2. Sensitivity and elasticity of λ and c* to the demographic parameters were very similar; for brevity we present only sensitivity and elasticity of c*.
Table 2. Median values of λ and c* for each environment, 95% empirical CI are in parentheses
Median c* (m year−1)
2·5 (2·2, 2·8)
1500 (1190, 1957)
1·9 (1·0, 2·3)
1109 (677, 1559)
Grazed grassland (z = 0·5)
2·0 (1·8, 2·2)
1173 (918, 1542)
Grazed grassland (z = 0·9)
1·3 (1·3, 1·4)
527 (382, 717)
Median sensitivity of c* was highest for the proportion of seed going into the long-distance dispersal kernel p; however, the CI on sensitivity to establishment, b and juvenile survival sj overlapped substantially with those of p (Fig. 3a). The long-distance dispersal parameter, β, consistently had the largest elasticity (Fig. 3b; no overlap in CI), and the strong correlation between β and c*, despite uncertainty in the other parameters, is shown in Fig. 4. Juvenile survival, sj, had the next consistently highest elasticity value.
Wave speeds and population growth rates were reduced in shrubland relative to ungrazed grassland (Table 2). The highest median sensitivity was to the probability of establishment, b, with the CI on sensitivity to juvenile survival sj and the proportion of seeds in the long-distance dispersal phase, p, overlapping with those of b (Fig. 5a). At very low values of b wave speed was greatly reduced, as low probability of establishment acted as a bottleneck in the model. The pattern of sensitivity of c* to the other parameters was similar to that of the ungrazed grassland model, and the patterns of elasticities in the ungrazed grassland and shrubland models were very similar.
When the grazing parameter z was set to 0·5 (representing a 50% decrease in normal grassland establishment), λ and c* were reduced relative to the ungrazed environment, and the sensitivity and elasticity patterns were very similar to the ungrazed grassland model, but with the grazing parameter having a negative elasticity. When the grazing parameter was set to 0·9, λ and c* were reduced still further and the grazing parameter, z, had the highest sensitivity and elasticity (Fig. 6a,b).
This case study of the spread of the invasive P. nigra has given us a general insight into the importance of long-distance dispersal (LDD), juvenile survival and seedling establishment for determining invasion speed. We have shown that explicitly including uncertainty as 95% CI on sensitivities and elasticities allows us to assess the robustness of different management strategies to that uncertainty. Elasticity of invasion speed to the long-distance dispersal parameter β was consistently highest in the most vulnerable habitat, ungrazed grassland. This result agrees with the analysis of Neubert & Caswell (2000) and Caswell, Lensink & Neubert (2003) and, as they point out, this is not a new result. It is generally accepted that the modelling of long-distance dispersal kernels is of great importance in making predictions for how fast a species will spread (Clark et al. 2003; Levin et al. 2003). Our results have shown that this result is robust across a wide range of uncertainty in dispersal and demographic parameter values in vulnerable habitats.
For P. nigra, despite uncertainty in parameter estimates, spread is promoted by high LDD probabilities and high dispersal distances. It is therefore particularly important to prioritize removal of plants ‘upstream’ of vulnerable habitat and particularly from areas where dispersal of propagules is maximized. For P. nigra this means removal of trees where there is a high probability of diaspores being taken up into the air-stream and transported long distances, preferably before these trees become reproductive. Trees that are sources of LDD events can be removed from spread-prone positions (such as exposed ‘take-off sites’). Another suggestion is to plant shelter crops around plantations. For short-distance dispersal, Nathan et al. (2002b) showed that dispersal distances are lower in dense forests compared with open landscapes; their results, however, cannot be extended to long-distance dispersal as seed uplifting, a process thought to be important for long-distance wind dispersal (Horn, Nathan & Kaplan 2001; Nathan et al. 2002a; Tackenberg 2003), was not included in the model. Non-spread-prone trees planted as a barrier may not therefore be effective at preventing long-distance dispersal. More work on the conditions needed for long-distance dispersal to occur is necessary in order to characterize high-risk sites so that they are identifiable using digital elevation and land-use maps. We will then be in a position to determine which trees to remove in a landscape that is categorized according to spread risk.
There was considerable overlap between CI of sensitivities to juvenile survival, seedling establishment and probability of long-distance dispersal in the ungrazed grassland. Sensitivities are particularly revealing when a parameter value is low and limiting, as may be the case for these three parameters. Small absolute changes in these parameters will have a large impact on invasion speed and can be reduced by introducing grazing, encouraging the establishment of competitive shrubs or removal of seedlings through herbicide use or physical removal. Periodic high-intensity grazing has been successfully applied to prevent ‘in-fill’ of seedlings in an invaded area (J. Foster, personal communication). In habitats less vulnerable to invasion, demographic parameters can have equal or greater potential impacts on spread rates than long-distance dispersal parameters. In shrubland there was considerable overlap between CI on elasticities to the long-distance dispersal shape parameter and juvenile survival, indicating that proportional reductions in juvenile survival would be of similar benefit to similar reductions in long-distance dispersal. In grazed grassland it is the grazing parameter that has the highest sensitivity and elasticity.
In this model we used dispersal kernels whose tails decay at least exponentially fast (e.g. the Gaussian and Laplace functions). Fat-tailed kernels, with tails that decay slower than exponentially, have also been used to model long-distance dispersal events (Clark 1998). However, the use of fat-tailed kernels leads to accelerating wave speeds (Kot, Lewis & van den Driessche 1996), resulting in asymptotically infinite wave speeds. Clark, Lewis & Horvath (2001) and Clark et al. (2003) have developed methods for calculating wave speeds from samples of fat-tailed kernels using empirical moment-generating functions combined with the net reproductive rate R0. An advantage of the Neubert & Caswell (2000) model for invasive plant management purposes is that it is stage structured. Contributions to wave speed from any parameter in a life cycle can be assessed, rather than aggregating all population processes into a single value of R0. The value of the stage-structured approach is demonstrated here by the identification of particular demographic parameters, seedling establishment and juvenile survival, with the potential to alter growth rates and invasion speed.
Fitting long-distance dispersal kernels to available data is at the very least problematic. Pinus nigra can be picked up in aerial photographs, enabling us to detect outlying trees up to 10 km away from the original source. The patterns observed are not only a result of dispersal, however, but are affected by establishment and survival. This could result in distortion of the estimated dispersal distribution if probabilities of seedling establishment and survival vary with distance from the source (Nathan & Muller-Landau 2000). The pattern of long-distance dispersal is also obscured to a certain degree by short-distance dispersal events. This problem would have affected the fit of the long-distance dispersal kernel in all but scenario 4, where the source of trees in the tail of the distribution in 1980 was assumed to be the furthest ahead large tree in 1965.
Wave speeds estimated here are very high, > 1000 m year−1; the main reason for this is that the long-distance dispersal parameters were calculated from data in one take-off site, a hill exposed to warm north-westerly winds in the coning season, with vulnerable ungrazed grassland habitat down-wind of the source. The landscape over which the invasion wave moves will change, leading to variable dispersal kernels. Our estimates should therefore be taken as the upper limit of an invasion speed under good conditions for long-distance dispersal. Average rates of post-glacial migration of Pinus species, as determined from the pollen record, range from 81–400 m year−1 in eastern North America, < 100–700 m year−1 in the British Isles, to 1500 m year−1 in Europe (MacDonald 2004). Our estimates are not inconsistent with these spread rates; however, for management purposes it is the contributions of parameters to c* that are important.
The Neubert & Caswell (2000) model also assumes that Allee effects are unimportant; Allee effects occur when population growth rate is depressed at low densities, an assumption for which we have no data for P. nigra. Data for other pine species suggest that isolated trees that have to self-pollinate produce fewer sound seed (Lanner 1998). If Allee effects do occur then the invasion will be slower (Taylor et al. 2004). For an alternative way of using integrodifference equations to estimate invasion speeds with Allee effects (but without stage structure), see Wang, Kot & Neubert (2002).
Pinus nigra and other wind-dispersed invasive conifers are of major conservation concern because of their conversion of grassland into exotic monocultures, excluding native species and reducing land-use options for managers. By identifying robust management options in the face of considerable parameter uncertainty we hope to narrow down the range of control options open to managers to those we think will have most impact. Challenges remain, however; we need to use mechanistic wind-dispersal models to determine the impact of landscape on the nature of the long-distance dispersal kernel and to integrate economics into the decision-making framework together with elasticity and sensitivity analyses.
Funding: Leverhulme Trust, New Zealand Forest Research Institute, Environment Southland, Environment Canterbury and New Zealand Department of Conservation, New Zealand Foundation for Research, Science and Technology, contract number C09X0010 to Landcare Research. Thanks to Hal Caswell for helpful comments, Dave Henley for data collection and John Foster, Euan Mason and members of the Mathematics-in-Industry Study Group for valuable discussion.