1. A mechanistic understanding of seed dispersal is important for understanding and predicting dispersal patterns and spatial population dynamics. We analysed a mechanistic model for the ballistic dispersal of the widespread invasive plant Impatiens glandulifera. The model deterministically simulates individual seed trajectories, including the interaction between seeds and atmospheric winds. It is fully specified using independently measured plant traits and wind velocities.
2. To parameterise the model, we conducted field surveys and analysed high-speed video footage of seed release. We explicitly incorporated intraspecific variation in key biological parameters (seed size and mass, release height, velocity and direction), as well as variation in wind velocities. This parameter variation introduces process-based stochasticity to the deterministic model, allowing the derivation of population-level dispersal kernels.
3. Analysis of the parameterised model showed complex interactions between plant dispersal parameters and wind conditions. For example, the optimal seed launch angle in still wind is 32° above horizontal, but this rises in stronger tailwinds. In a headwind, it can be optimal to either launch the seed straight upwards or slightly below the horizontal. Headwinds also blow projected seeds back towards the source, initially decreasing but then increasing net dispersal distance, which can reverse the positive effect of seed release height on dispersal distance.
4. To validate the model, we performed a field experiment in which seeds were trapped up to 6 m from potted I. glandulifera plants. Seed dispersal was biased in the direction of the prevailing wind, which was only reproduced by simulations of the model that included the wind modification of ballistic trajectories. The correspondence between the model and the data was good (R2 = 0.882), but improved if seed release was biased to occur in more windy conditions (R2 = 0.945) as might occur through mechanical disturbance to the seed pods.
5.Synthesis. Our validated model provides a functional understanding of seed dispersal in I. glandulifera. The model reveals surprisingly complex interactions between the ballistic dispersal mechanism and the wind and shows how population-level dispersal patterns emerge from intraspecific variation in dispersal traits.
Seed dispersal is a fundamental process in plant ecology, impacting individual fitness, population dynamics, community structure, gene flow and evolutionary processes (Levin et al. 2003; Cousens, Dytham & Law 2008). As such, it has wide-ranging consequences for invasive species distributions and conservation ecology. Given its significance, much effort has been applied to the study of seed dispersal (Bullock, Shea & Skarpaas 2006). This includes measurement of seed deposition patterns at varying distance from parent plants to provide data on how far seeds are actually dispersed. These data are often then modelled using phenomenological dispersal kernels (e.g. Dalling et al. 2002; Clark et al. 2005) or mechanistic models that incorporate physical factors affecting seed movements and predict dispersal patterns based on measured seed parameters and environmental conditions (Nathan, Safriel & Noy-Meir 2001; Katul et al. 2005; Kuparinen 2006). Mechanistic models are often considered superior to phenomenological approaches as they provide a generalised understanding of the dispersal process in which spatio-temporal variation in dispersal patterns is implicit (Kuparinen 2006). Although labour-intensive dispersal data are needed to adequately parameterise and validate such models, they can predict dispersal in novel settings more readily than phenomenological models.
Ballistic dispersal, where seeds are projected from the parent plant via explosive release mechanisms, involves both physical and biological parameters that may be well suited to a mechanistic approach (Beer & Swaine 1977; Stamp & Lucas 1983). Using seed dispersal models based on the physical laws of projectile motion, it should in theory be possible to predict dispersal distance and direction from information on the size and shape of the seed, along with its initial projection velocity, angle and height (Beer & Swaine 1977; Garrison, Miller & Raspet 2000, Vogel 2005; Robinson & Geils 2006; Hayashi, Feilich & Ellerby 2009). However, despite the relative tractability of ballistic dispersal, we know of only one study that has attempted to validate such models by comparing observed dispersal patterns to those predicted by a ballistic model. Swaine & Beer (1977) recorded the pattern of seed rain around a single Hura crepitans L. tree. This exhibited a bimodal pattern with one peak centred roughly at the modal distance predicted by their parameterised model (30 m) and a second peak about the parent tree (interpreted as seeds rebounding from collisions with the tree crown).
One complicating factor in ballistic seed dispersal models, which has yet to be addressed, is that seeds do not disperse through still air conditions, but into winds. This has been extensively studied for conventionally wind-dispersed seeds (Okubo & Levin 1989; Horn, Nathan & Kaplan 2001; Katul et al. 2005; Skarpaas, Auhl & Shea 2006; Schurr, Steinitz & Nathan 2008), suggesting that wind conditions can often be more important than plant or propagule characteristics in determining seed dispersal distances. Ballistically dispersed seeds are generally heavier and suffer lower drag than wind-dispersed seeds, but it is still highly likely that the wind will interact with their trajectory during dispersal. Indeed, during windy conditions, ballistically dispersed Arceuthobium tsugense (Rosend.) G.N. Jones seeds exhibited an anisotropic dispersal pattern more concentrated towards the prevailing downwind direction than was observed during calm conditions (Smith 1973).
This study considers ballistic dispersal of Impatiens glandulifera Royle, which is a widespread and extensive invader of riparian habitats in the UK and much of Europe (Beerling & Perrins 1993). The I. glandulifera seed pod is 1.5–3.5 cm long with five outer valves that rapidly coil upwards during dehiscence, ejecting the seeds. Most seeds land within 3 m of the parent plant, although new seedlings may be found up to 10 m from the nearest potential parent (Beerling & Perrins 1993). At a national scale, the rapid spread of I. glandulifera outstrips the range of ballistic dispersal and is more likely due to human-aided movement and dispersal within watercourses (Pysek & Prach 1995). However, understanding its primary ballistic dispersal mechanism remains important for predicting local invasion dynamics and is a mechanism by which seeds are deposited into watercourses to become long-distance dispersers. It is therefore likely to be an important consideration when devising effective spatially explicit control measures.
Our aim was to analyse the interactions between ballistic dispersal and wind and examine how population-level dispersal patterns may emerge from intraspecific variation in dispersal traits. Specifically, we: (i) adapt ballistic equations for projectile motion to incorporate wind effects; (ii) parameterise the model through quantification of key dispersal traits and wind conditions; (iii) analyse the behaviour of the model to examine interactions between seed traits, the ballistic dispersal mechanism and the wind; and (iv) conduct a seed trapping experiment to validate predictions of the model.
Ballistic dispersal model
We simulated seed dispersal by adapting the equations of Hayashi, Feilich & Ellerby (2009) previously used for modelling seed dispersal in Impatiens capensis Meerb. In overview, seeds are represented as spheres in three-dimensional space and are launched from the parent plant at a particular initial velocity, pitch (angle from horizontal), direction (horizontal angle from north) and height. Subsequently, the seed is accelerated by drag and gravity forces (Fig. 1), causing deviation from its initial trajectory that allows us to project its flight through space. We assume that acceleration due to the buoyancy or lift forces are negligible. Once the seed reaches the height of the surrounding vegetation, we assume it is intercepted, terminating further movement in the horizontal plane and ending primary dispersal.
Consider a spherical seed of mass m (kg) and diameter d (m) travelling through the air with an instantaneous three-dimensional velocity vector v (m s−1). At seed height, z (m), the wind has a velocity vector u(z) (m s−1). The drag force acting on the seed in all three dimensions, fD (N), is:
where ρ is the air density (set at 1.204 kg m−3, equivalent to 20 °C and 100 kPa pressure), A is the seed’s frontal area (πd2/4 m2) and vR is the resultant velocity relative to the wind, v − u(z) (ms−1) (Vogel 1988). CD is the coefficient of drag, which we estimate as a function of the Reynolds number Re as:
where μ is the viscosity of the air (set at 1.8 × 10−5 Ns m−2, equivalent to 20 °C) (Vogel 2005). Varying the air density and viscosity over values for temperatures between 0 and 35 °C causes <10 cm of total variation in modelled seed dispersal distances, justifying our use of constants for these variables.
The gravity force fG accelerates the seed downwards in the vertical dimension with a force of mg (N) (g = 9.81 ms−2) and takes a value of 0 in both horizontal dimensions. Thus, the instantaneous change in the seed’s velocity is:
The minus sign for fG signifies its downwards action in the vertical dimension. The trajectory of the seed is simulated from this expression, by numerical integration using Euler’s method and a 0.002 s time step. We investigated alternative time steps and found that below 0.002 s, there was almost no difference in the resulting dispersal distances, while distances were underestimated for longer intervals. Integration is terminated when the seed reaches the height of the surrounding vegetation (or bare ground), allowing us to calculate the dispersal distance of the seed in the horizontal plane.
Effects of the wind are incorporated via the drag force, as drag accelerates the seed in the opposite direction to its resultant velocity relative to the wind (Fig. 1). For simplicity, we do not include temporal changes in the wind vector during the dispersal of any one seed and assume that wind flow is non-turbulent. We also assume that wind speed increases with height (z) following a logarithmic profile typical of open habitats (Oke 1987) as:
where z0 is the roughness length of the vegetation or ground surface (m), scaling the drag exerted by the surface on the wind. In the model, we use an empirical approximation for natural vegetation to estimate z0 = max(zveg/10, 0.003), where zveg is the height of the surrounding vegetation (m) (Oke 1987). Parameter zd is the displacement height of the vegetation and represents the height below which U = 0 m s−1. The empirical approximation of zd = ⅔zveg is used in the model (Oke 1987). κ is the Von Kármán constant, which takes a value of 0.41, and u* is the friction velocity of the air. Friction velocity characterises the shear stress of the air as it passes over the surface and is calculated by rearranging the above equation so that u* is on the left hand side and then entering a value of u at known z. We had wind speed data for a height of 2.5 m (see below), and so this was used as the reference height.
To acquire information on seed mass and shape, 60 balsam plants from a naturalised population on the River Esk (Musselburgh, Scotland, 3°3′W 55°55′N) were selected in August 2009. These plants were cut near their base, stood in water and brought back to the laboratory. A ripe seed pod from each of these was opened and three mature seeds randomly sampled (n = 180). Seeds were immediately weighed to the nearest 0.1 mg, and the side and top profiles were photographed under ×10 magnification. From this, we estimated the minimum and maximum frontal areas presented by each seed. In the model, we set spherical seed diameters to the mean of these two frontal areas, as I. glandulifera seeds rotate in the air similarly to I. capensis seeds (Hayashi, Feilich & Ellerby 2009). Seed release heights were estimated by measuring the height of 2819 dehisced pods to the nearest 5 cm on a further 60 plants used in a seed trapping experiment (see below).
Seed release parameters were obtained by analysing slow motion video footage of the dehiscence of 170 seed pods from different plants to those described above. Digital videos were taken using a Phantom Miro 4 camera (Vision Research, Bedford, UK) with a frame resolution of 800 × 600 pixels, frame rate of 1000 Hz and exposure of 40 μs. The camera was mounted directly above the seed pod and faced downwards to view the horizontal plane. One 500 W and two 150 W lamps were used to illuminate the arena, and a mirror was mounted at 45° to the horizontal, allowing a simultaneous view of the vertical plane in the camera’s field of view. A marked ruler was positioned adjacent to the seed pod as a scale marker.
Impatiens glandulifera plants were brought into the laboratory from the field site, and all leaves and pods except for the ripest pod were removed. Plants were then stood in water for a maximum of 5 h before use in the experiment, which did not cause any visible wilting. To record pod dehiscence, the plant stem was cut approximately 3 cm below its terminal node, and the remaining 3 cm of stem was mounted vertically in a retort stand at bench level under the camera. In this way, the pod’s typical attitude relative to the stem was maintained, and the natural recoil ability of the peduncle (flower stalk) was also retained. A pair of fine forceps was used to apply light pressure to a single pod outer valve at its proximal end (i.e. near the peduncle) to trigger dehiscence. Any pods not dehiscing under the light pressure were rejected, ensuring that only fully ripe pods were included in the study. The proximal end of the pod was manipulated as I. glandulifera’s dispersal mechanism is for the turgid outer pod valves to coil upwards and eject the seeds (Hayashi, Feilich & Ellerby 2009 describe a similar mechanism). In preliminary trials, we found that manipulating the distal end of the pod interfered with this coiling mechanism and slowed the seed release. Video frames were imported into ImageJ v.1.42q (Rasband 1997–2011), and a single seed was randomly selected from each pod, excluding any that collided with the forceps. The change in position of the seed over the first two frames in which it was clearly separated from the pod was used to estimate its launch velocity and pitch. The pod length and orientation prior to dehiscence were also estimated from the footage.
To characterise intraspecific variation in these parameters, we fitted flexible statistical distributions to the data, by a maximum-likelihood simplex algorithm (Nelder & Mead 1965). Seed masses, release heights and launch velocities were characterised with Gamma distributions as they cannot take negative values. Seed pitches were characterised with a Cauchy distribution, truncated between −90° and 90° from the horizontal. This distribution was used as it is flexible enough to represent a range of unimodal distributions constrained over a fixed range (Nadarajah & Kotz 2007). Seed diameters were found to exhibit a strong correlation with mass, and so a nonlinear power relationship was fitted by least squares and the residuals modelled as a Gaussian distribution.
A seed trapping experiment was conducted to validate model predictions. In April 2010, seedlings were collected from the River Esk site and transplanted into 12 pots (26 cm diameter) filled with sand and compost (1 : 1 by volume) at an evenly spaced density of five seedlings per pot. These plants were grown in an unheated glass house with ad hoc, but equitable watering and fertilisation. On 15 August 2010, when plants were beginning to produce mature seed pods, the pots were moved to a mown grass field (Midlothian, Scotland, 3°12′13″W 55°51′43″N). The 12 pots were dug into the ground at 50 cm intervals in a line perpendicular to the direction of the prevailing wind (blowing from 210° over the previous 10 years, see below). To prevent the I. glandulifera stems breaking under high wind conditions, a bamboo cane was planted in each pot, and the plants were loosely bound with string.
The seed traps were rectangular plastic trays (30 × 40 × 10 cm) filled with water to a depth of at least 5 cm throughout the experiment. Thirty traps were arranged in a line running perpendicular to the row of balsam pots (i.e. aligned with the prevailing wind) and through their centre. There were 15 traps on each side of the pots evenly spaced at 40 cm intervals so that trap centres were between 0.25 and 5.95 m from the line of pots. Preliminary simulations of the dispersal model, performed prior to the experiment as recommended by Skarpaas, Shea & Bullock (2005), suggested that very few seeds would be trapped at greater distances than this. Between 16 August and 28 September 2010, the traps were checked daily (excluding weekends), and all seeds were collected. By the end of the experiment, there were very few intact seed pods remaining. The number and heights of all dehisced seed pods were determined from the stalks remaining after dehiscence (see above). These were recorded to replicate the starting positions of seeds in the experiment during model simulations.
Raw data from seed trapping experiments do not give dispersal kernels (i.e. frequency or probability distributions of dispersal distances) because of the observational bias introduced by the trapping design (see Chapter 5 in Cousens, Dytham & Law 2008). To correct this bias and validate the model predictions, simulations were designed to mimic the seed trapping experiment. In each simulation run, a seed with random size and mass (drawn from the fitted distributions described above) was assigned to a randomly selected pod dehiscing over the course of the experiment. The pod height was known, but its exact horizontal position was not, and so we set this to a randomly generated coordinate within the relevant pot. The seed was then launched with a randomly drawn velocity, pitch and bearing and a randomly selected wind vector (see below) and if it landed within the area covered by one of the seed traps it was retained. We repeated this until the number of seeds trapped in the simulation equalled the number trapped in the experiment. The whole procedure was repeated 10 000 times to obtain 95th percentile ranges for the numbers of seeds collected from each trap. These were then compared with the observed data. Because this process required the modelling of millions of seed dispersal events, we increased the model’s computational efficiency by training neural networks to accurately predict the dispersal distance and bearing of seeds from the model parameters (see Appendix S1 of the Supporting Information).
Horizontal and vertical wind conditions during the simulations were randomly drawn from observed wind vectors recorded during the experimental period. We randomly sampled wind vectors rather than statistically model them to preserve complex correlation structures between wind velocities and directions that would be difficult to reproduce. Wind vectors were obtained from a Metek USA-1 Sonic anemometer mounted 2.5 m above the ground at a weather station approximately 400 m from the experimental set up. Measurements were recorded at 30 min intervals, but the data contained a gap between 25 August and 16 September when the sensor was non-operational. We therefore assume that wind conditions during the operational period were representative of those in the non-operational one.
We also performed a simulation where seeds were dispersed preferentially into strong winds. This was performed as I. glandulifera seed pod dehiscence may be triggered by physical disturbance during windy conditions. As such, a random sample of the wind vectors may not be representative of those experienced during seed dispersal. In this simulation, wind vectors were sampled with an arbitrarily chosen weighting proportional to their absolute magnitude, that is the probability of selecting a particular vector was its magnitude divided by the sum of all wind vector magnitudes.
Summaries of measured seed masses, diameters and release heights are given in Table 1. There was a strong correlation between mass and diameter (r = 0.972, P < 0.001), with the relationship well captured by the fitted power function (R2 = 0.959, see Table 1). Seeds were released with a median velocity of 6.2 m s−1 and a median pitch of 48° above horizontal (Table 1). The maximum observed velocity was 12.4 m s−1, and there was a tendency for greater launch speeds from longer seed pods (r = 0.454, P < 0.001). Seed launch pitches were also strongly correlated to the pitch of the pod prior to dehiscence (r = 0.736, P < 0.001), and we observed a weak negative correlation between launch pitch and velocity (r = −0.159, P < 0.039). Intraspecific variation in these parameters was well captured by the fitted distribution functions (Table 1), as evidenced by non-significant Kolmogorov–Smirnov tests against the theoretical distributions (all with P > 0.393). The one exception to this was for seed release heights (P < 0.001) for which the quantiles were highly correlated (every 0.01 from 0.01 to 0.99, r = 0.984), but there was a very large sample size (n = 2819).
Table 1. Intraspecific variation in biological parameters of the model, including their median values, variances and statistical distributions, used in model simulations. In the case of seed mass, release height, velocity and pitch parameter distributions were fitted to the data by the maximum-likelihood simplex algorithm (Nelder & Mead 1965). The relationship between seed diameter and mass was fitted by nonlinear least squares (R2 = 0.959) and the standard deviation of the residuals used to add stochasticity to the output. Release angles in the horizontal plane were assumed to be uniformly distributed, and the vegetation height was set to that during the seed trapping experiment
Median (95th percentile range)
Mean within-sample unit variance
Seed mass (mg)
Three seeds from each of 60 pods on different plants (n = 180)
Truncated Cauchy (location = 48.47, scale = 11.85, min = −90, max = 90)
Seed release direction in horizontal plane (° from north)
None (assumed uniform)
Uniform (min = 0, max = 360)
Analysis of the parameterised model
In still air, dispersal distances calculated by the model increased with seed mass, launch height and launch velocity (Fig. 2a–c), while the optimal launch angle was around 35° above horizontal (Fig. 2c). The dispersal of light seeds was more strongly influenced by the wind than that of heavy seeds (Fig. 2a,e), and a tailwind reversed the relationship between release height and dispersal distance from positive to negative (Fig. 2b). Dispersal distance in the model increased with increasing horizontal tailwind speed (direction = 0° in Fig. 2d). However, with a horizontal headwind (direction = −180° or 180° in Fig. 2d), dispersal distances were curtailed up until a point where the wind blew the seed in the opposite direction in which it was launched. Higher vertical wind speed increased dispersal distance in a tailwind, especially for light seeds, but had little effect in a headwind other than for the lightest seeds (Fig. 2e). These relationships illustrate the complex interactions between ballistic seed release and the wind that are likely to determine patterns of I. glandulifera dispersal (Fig. 3).
During the seed trapping experiment (when the anemometer was operational), the median horizontal wind velocity was 2.92 m s−1 (95th percentile range 0.34–6.17 ms−1), and the median vertical wind velocity was −0.037 m s−1 (95th percentile range −0.107 to 0.107 m s−1). The circular mean wind direction was 245° (circular variance = 32.0°), although the distribution of wind directions was actually bimodal with a major peak around 210° and a minor peak at approximately 25°. Thus, as expected, seed traps were aligned with the prevailing wind.
Over the experimental period, 507 seeds were collected, mainly in traps close to source plants (Fig. 4). Significantly, more seeds were found in traps downwind of the prevailing wind than those upwind of it (369 vs. 138, = 105, P < 0.001). Simulations of the model that did not include any interaction between the wind and seed trajectories did not reproduce this asymmetric pattern and gave a poor fit to the observed data (Fig. 3). By contrast, simulations of the dispersal model that included the wind effect gave a much better fit, with 24 of the 30 observed seed trap counts lying within the 95th percentile range of model predictions (Fig. 4b). However, there appeared to be a slight tendency for greater downwind dispersal in the data than was produced by the model. It is possible that this is caused by a higher likelihood of pod dehiscence with increased physical disturbance during windy conditions. Indeed, biasing seed release to occur more frequently during high winds improved model fit (Table 2, Fig. 4c). In this case, 28 of the 30 trap counts were within the 95th percentile range of the simulated data.
Table 2. Approximate model fit statistics derived by comparing the observed number of seeds per trap to the mean number predicted over 10 000 model simulations. R2 is the squared correlation between observed and expected. An approximate log-likelihood was calculated assuming the observed data are drawn from Poisson distributions with means per trap equal to those simulated. Log-likelihoods were used to estimate the ΔAIC statistic to compare the models (ΔAIC = 0 indicates the most parsimonious model). Finally, the χ2 goodness of fit statistic was calculated, with P values estimated through a Monte Carlo test with 10 000 replicates
χ2 G.O.F. (P)
No wind effect
Random wind effect
Weighted wind effect
Our model for the primary dispersal of I. glandulifera seeds through ballistic release, and experimental results, suggest that I. glandulifera seeds are rarely dispersed further than 6 m. This supports the prevailing view that ballistic dispersal is an efficient but relatively short-distance dispersal mechanism (Stamp & Lucas 1983; Cousens, Dytham & Law 2008). However, unlike other similar models (Garrison, Miller & Raspet 2000; Hayashi, Feilich & Ellerby 2009), we explicitly incorporated the effects of wind flow on modelled seed trajectories. Our results suggest that failing to include this seed-atmosphere interaction means that models will be unable to reproduce the clear dispersal bias in the direction of the prevailing wind (see seed trapping experimental results). By contrast, simulated seed trapping patterns produced with wind vectors recorded at the experimental site closely matched the observed pattern, demonstrating the important role of wind in the primary dispersal of I. glandulifera.
Analysis of the dispersal model revealed a surprising array of complex interactions between plant dispersal traits and the wind (see Figs 2 and 5). In the main, modelled relationships between traits and dispersal distances produced in still air were qualitatively similar to those produced with tailwinds, but with increased dispersal distance. For example, heavier seeds were dispersed further in both situations because of lower relative drag compared with light seeds. This relationship could only be reversed by strong updraught, in which the higher drag of light seeds retarded their rate of descent under gravity allowing further dispersal. However, the most complex behaviour of the model was observed when the orientation of the wind was varied from the direction of the seed. Tailwinds blew seeds back towards the point they had been launched from. Ignoring any interception by the parent plant, this caused a nonlinear effect of tailwind velocity on net dispersal distance, characterised by an initial decrease and then increase with progressively stronger winds. Tailwind was also able to reverse the otherwise positive effect of seed release height on dispersal distance. The reason for this is that the increase in wind velocity with height above the ground (Oke 1987, Horn, Nathan & Kaplan 2001) meant that taller plants launched their seeds into stronger tailwinds. This resulted in a slightly lower dispersal distance for a typical seed, although it may be that very light seeds could be carried further in the downwind direction when released at height.
Another consequence of the height variation in wind velocity was to alter the ‘optimal’ seed launch pitch, that is the initial launch angle resulting in the greatest dispersal distance. In still air, the optimal pitch was c. 35° above the horizontal, which is similar to values estimated for other ballistically dispersing seeds (Garrison, Miller & Raspet 2000; Vogel 2005; Hayashi, Feilich & Ellerby 2009), but shallower than the median pitch of 48° observed in the seed release high-speed videos. However, our model shows how such estimates of optimal pitches are overly simplistic, as the optimal pitch varies dramatically with the strength and orientation of the wind (see Fig. 5). As headwinds increase, there is an advantage in exposing the seed to the greater wind velocities that occur higher above ground. Therefore, the optimal pitch steepens. Under a pure tailwind, the opposite is true so that optimal pitch decreases with velocity, eventually falling below the horizontal. When there is a strong crosswind, it becomes optimal to launch the seed vertically, maximising its exposure to the wind, but imparting no horizontal movement. In this case, all horizontal dispersal achieved by the seed will result from it being carried on the wind. Launch pitch may be a readily adaptable trait because of the strong correlation we found between pitch and seed pod orientation, which may be under heritable genetic control. Therefore, we suggest that I. glandulifera has evolved a steeper pitch than is optimal under still air conditions to increase the chance that seeds achieve high trajectories and can increase their dispersal distance if encountering favourable winds.
Although our model simulates the trajectory of individual seeds deterministically, we incorporated process-based stochasticity via intraspecific variation in seed dispersal traits and by randomly sampling wind velocity vectors. This meant that dispersal kernels could be derived by simulating large numbers of dispersal events (see Fig. 3). We have produced population-level kernels, but individual-level kernels could also be derived (Thiede & Augspurger 1996). These kernels vary with local meteorological conditions and the distribution of dispersal traits among the focal population or individual. For example, the observed unimodal distribution of I. glandulifera launch pitches (angles relative to horizontal) generated unimodal dispersal kernels, rather than the bimodal kernels imparted by uniform pitch distributions (Beer & Swaine 1977; Neubert & Parker 2004; Rezvani et al. 2010). Spatio-temporal variation in meteorological conditions and plant traits, either through phenotypic plasticity or local adaptation, will therefore cause dispersal kernels to vary greatly in space and time (Horn, Nathan & Kaplan 2001; Bullock et al. 2003). For example, I. glandulifera adapts to northerly latitudes by growing to a shorter height (Kollmann & Bañuelos 2004) and produces heavier seeds in cooler temperatures (Willis & Hulme 2004). We have also observed that shaded plants grow to be taller, but produce shorter seed pods (D. Chapman unpublished data), which will impart lower launch velocity. This intraspecific variability in response to environmental gradients highlights the importance of the mechanistic approach to seed dispersal modelling (Katul et al. 2005; Kuparinen 2006). In principle, our model provides a functional understanding of dispersal, which can accommodate between-individual and between-population variation in dispersal kernels caused by both trait variation and abiotic gradients. However, a limitation of our study is that we did not model the hierarchical or nested structure of variation in seed traits (within-pod) or pod traits (within-plant, within-population), other than for the distribution of seed release heights (where it is implicit that there is no within-pod variability). This was because heights were measured for every exploded seed pod on all our experimental plants, whereas the other traits were measured with sampling designs that did not permit such a representation (Table 1).
Despite this simplicity, the model gave a good prediction of observed seed trapping patterns. Importantly, the field validation of the dispersal model was totally independent of the theoretical model formulation or its parameterisation. As such, the adequate reproduction of observed dispersal distributions suggests that the model captured salient features of the dispersal process. However, we also found that if the model was adjusted so that seed dispersal preferentially occurred during high winds, then its fit was improved. Although we have no data to support it, there is good reason to suspect that windy conditions promote pod dehiscence through increased mechanical disturbance that breaks the joins between pod outer valves (Beerling & Perrins 1993; Hayashi, Feilich & Ellerby 2009). There may also be a link between dehiscence probability and rainfall or humidity not considered in our modelling. This is because pod valve turgor maintained by plant hydration promotes dehiscence (Hayashi, Feilich & Ellerby 2009). In Pinus halepensis, seeds are disproportionately released when upwards wind movement is highest, which promotes long-distance dispersal (Nathan et al. 1999). However, we would not expect a similar pattern in I. glandulifera as our model suggests that vertical wind has only a moderate effect on dispersal distance compared with horizontal wind (Fig. 2). Nevertheless, our results add to existing evidence that non-random seed release, with respect to meteorological parameters, is an important but under-studied aspect of seed dispersal (Nathan et al. 1999; Narbona, Arista & Ortiz 2005; Soons & Bullock 2008).
At the field site where plants used to parameterise, the models were collected, pods contain a mean of 6.8 seeds (n = 301, standard deviation = 3.0; D. Chapman, unpublished data). This suggests that given the number of dehisced pods counted on our experimental source plants, around 19 300 seeds would have been produced. If this number of seeds were dispersed according to our model, then we would have expected 1.8–2.1 times as many seeds to have been trapped as were recovered (with the least difference for the wind-weighted model). There are probably two explanations for this. First, seeds in the traps may have been subjected to predation, for example by birds. Second, some seeds may have collided with the adult balsam plants, fallen to the ground and thus not reached the traps. The experimental set-up consisted of a row of 12 pots, each with five I. glandulifera plants arranged like the dots on the five-face of a dice. This formed a more-or-less continuous ‘hedge’ of source plants in which many seeds would have been launched directly towards the parent or another plant. The central plants in each pot were surrounded by neighbours on all sides, while most outer plants had close neighbours on half of their perimeter. Collisions with the parent have also been suggested as important determinants of ballistic dispersal patterns in Hura crepitans (Swaine & Beer 1977), Oxalis corniculata L. (Rezvani et al. 2010) and Cytisus scoparius L. (Malo 2004), suggesting that such effects require consideration before mechanistic models are used to predict population dynamics or spread.
In conclusion, our validated model provides a functional understanding of primary seed dispersal in I. glandulifera that should also apply to other ballistically dispersed species and reveals a complex array of previously unexplored interactions between ballistic dispersal and the wind. Our results also have a wider relevance for invasive species ecology. For I. glandulifera, which is an annual with a transient seed bank and strong local density dependence (Beerling & Perrins 1993), local population and invasion dynamics must depend critically on dispersal. As such, our results on primary seed dispersal should be informative for planning management strategies at a local scale, with the strong caveat that secondary dispersal mechanisms including water transport also require consideration (Wadsworth et al. 2000). For example, we predict that within-site invasion speeds should be greatest in the downwind direction relative to the prevailing wind, where the river is aligned to the prevailing wind direction. Under this scenario, the individuals furthest downwind will have the greatest contribution to population growth, and so control should target these. Likewise, our modelling shows that plants further than 6 m from the river will deposit very few seeds into the river directly. Thus, they may be relatively unimportant for catchment-scale spread, depending on the extent of secondary dispersal due to flooding, animals or surface winds. This illustrates the potential for validated mechanistic dispersal models to be used in planning control strategies for problem invasive species and emphasises that considering interactions between dispersal traits and the wind will be a necessary step towards this goal.
This research was funded through the Natural Environment Research Council (CEH project NEC03681). We thank Mhairi Coyle for providing wind speed data.