A mechanistic model for secondary seed dispersal by wind and its experimental validation

Authors

  • FRANK M. SCHURR,

    Corresponding author
    1. Department of Ecological Modelling, UFZ – Centre for Environmental Research Leipzig-Halle, PO Box 500135, 04301 Leipzig, Germany,
      *Present address and correspondence: F. M. Schurr, Plant Ecology and Nature Conservation, Institute of Biochemistry and Biology, University of Potsdam, Maulbeerallee 2, 14469 Potsdam, Germany (fax + 49 331 9771948; e-mail frank.schurr@ufz.de).
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  • WILLIAM J. BOND,

    1. Botany Department, University of Cape Town, Rondebosch 7700, South Africa,
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  • GUY F. MIDGLEY,

    1. Climate Change Research Group, South African National Biodiversity Institute, Private Bag x7, Claremont 7735, South Africa, and
    2. Center for Applied Biodiversity Science, Conservation International, 1919 M Street, Washington, DC 20036, USA
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  • STEVEN I. HIGGINS

    1. Department of Ecological Modelling, UFZ – Centre for Environmental Research Leipzig-Halle, PO Box 500135, 04301 Leipzig, Germany,
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    • Present address: Chair of Vegetation Ecology, Technical University of Munich, 85350 Freising-Weihenstephan, Germany.


*Present address and correspondence: F. M. Schurr, Plant Ecology and Nature Conservation, Institute of Biochemistry and Biology, University of Potsdam, Maulbeerallee 2, 14469 Potsdam, Germany (fax + 49 331 9771948; e-mail frank.schurr@ufz.de).

Summary

  • 1Secondary seed dispersal by wind, the wind-driven movement of seeds along the ground surface, is an important dispersal mechanism for plant species in a range of environments.
  • 2We formulate a mechanistic model that describes how secondary dispersal by wind is affected by seed traits, wind conditions and obstacles to seed movement. The model simulates the movement paths of individual seeds and can be fully specified using independently measured parameters.
  • 3We develop an explicit version of the model that uses a spatially explicit representation of obstacle patterns, and also an aggregated version that uses probability distributions to model seed retention at obstacles and seed movement between obstacles. The aggregated version is computationally efficient and therefore suited to large-scale simulations. It provides a very good approximation of the explicit version (R2 > 0.99) if initial seed positions vary randomly relative to the obstacle pattern.
  • 4To validate the model, we conducted a field experiment in which we released seeds of seven South African Proteaceae species that differ in seed size and morphology into an arena in which we systematically varied obstacle patterns. When parameterized with maximum likelihood estimates obtained from independent measurements, the explicit model version explained 70–77% of the observed variation in the proportion of seeds dispersed over 25 m and 67–69% of the observed variation in the direction of seed dispersal.
  • 5The model tended to underestimate dispersal rates, possibly due to the omission of turbulence from the model, although this could also be explained by imprecise estimation of one model parameter (the aerodynamic roughness length).
  • 6Our analysis of the aggregated model predicts a unimodal relationship between the distance of secondary dispersal by wind and seed size. The model can also be used to identify species with the potential for long-distance seed transport by secondary wind dispersal.
  • 7The validated model expands the domain of mechanistic dispersal models, contributes to a functional understanding of seed dispersal, and provides a tool for predicting the distances that seeds move.

Introduction

Seed dispersal by wind consists of two phases (Watkinson 1978). Following primary seed dispersal (the airborne movement of seeds from the mother plant to the ground surface), a seed may be blown along the surface until it germinates, until it is permanently entrapped, or until its dispersal structure has deteriorated (Johnson & Fryer 1992; Greene & Johnson 1997). This wind-driven movement along the ground surface is often termed secondary wind dispersal (e.g. Greene & Johnson 1997) but it has also been referred to as phase II dispersal (Watkinson 1978) or tumble dispersal (e.g. Bond 1988).

Secondary dispersal by wind is effective when seeds remain mobile for sufficient periods of time, when the ground surface is smooth, when few obstacles impede seed movement, and when the vertical wind velocity profile (Monteith & Unsworth 1990) results in high wind velocities close to the ground. Such conditions are found in various environments (Chambers & MacMahon 1994). Indeed, wind-driven seed dispersal along the ground has been observed in temperate grasslands (van Tooren 1988), coastal environments (Watkinson 1978; Redbo-Torstensson & Telenius 1995), tundra (Glaser 1981), alpine habitats (Chambers et al. 1991), arid and semi-arid environments (Reichman 1984; Milton 1995; Aguiar & Sala 1997), on snow (Matlack 1989; Greene & Johnson 1997), as well as in environments disturbed by fire (Bond 1988; Hammill et al. 1998), human activities (Chambers et al. 1991; Campbell et al. 2003) or volcanic eruptions (e.g. Fuller & del Moral 2003).

Secondary wind dispersal can markedly alter the seed shadows resulting from primary dispersal (Harper 1977; Chambers & MacMahon 1994; Nathan & Muller-Landau 2000) and may be more important than primary dispersal with respect to the spatial patterning of plant populations (Chambers & MacMahon 1994) and the long-distance movement of seeds (Higgins et al. 2003b). Consequently, secondary wind dispersal is considered important for processes as diverse as plant migration in response to climate change (Midgley et al. 2002), species persistence in fragmented landscapes (Bond 1988), succession after natural or anthropogenic disturbances (Chambers et al. 1991; Campbell et al. 2003; Fuller & del Moral 2003), and ecosystem functioning (Wiegand et al. 1995; Aguiar & Sala 1997).

To understand and forecast these processes, we need a quantitative description of secondary wind dispersal (Chambers & MacMahon 1994). The quantitative study of other seed dispersal processes has been advanced by the development of mechanistic models (Nathan & Muller-Landau 2000). Such models describe the mechanisms underlying seed movement and predict seed dispersal from properties of species and their dispersal agents. Many mechanistic models have been developed to describe primary wind dispersal (e.g. Greene & Johnson 1989, 1996; Okubo & Levin 1989; Andersen 1991), and some of them (Nathan et al. 2002; Tackenberg 2003) reliably predict the airborne long-distance dispersal of seeds. In contrast, few authors have studied the mechanisms determining secondary wind dispersal, although Greene & Johnson (1997) developed a model for secondary wind dispersal over snow, based on Johnson & Fryer's (1992) detailed treatment of the physics of secondary seed movement by wind. However, this model has to be calibrated with data from experimental seed releases.

Here we develop a mechanistic model for secondary wind dispersal that builds on the principles outlined by Johnson & Fryer (1992) but, unlike Greene & Johnson (1997), describes the effects of both obstacles and the vertical wind velocity profile and can be fully specified with independently measured parameters. We formulate the model, derive an aggregated model version suitable for large-scale simulations and describe a protocol for estimating model parameters. In addition, we show that the model reliably describes seed movement in field experiments and explore model behaviour through extensive parameter variation. Finally, we explore the implications of the model for long-distance seed dispersal and for the relationship between seed size and dispersal distance.

Model description

physical background

The physical forces considered in the model are drag and friction, with friction being a function of lift and gravity (Johnson & Fryer 1992). The wind drag on a seed (inline image) acts in the direction of the horizontal wind vector experienced by the seed (inline image ). The strength of this drag force is

image( eqn 1 )

where CD is the seed's coefficient of drag, ρ is air density, A is the planform area of the seed, and inline image is the seed velocity vector (Monteith & Unsworth 1990; Johnson & Fryer 1992). Opposed to the drag force is friction, inline image, whose maximum absolute value depends on the balance of seed weight (inline image) and lift (inline image)

image( eqn 2 )

where µ is the seed's coefficient of friction on the surface (Johnson & Fryer 1992). When the seed is stationary, µ = µs (coefficient of static friction), and when it is moving µ = µk (coefficient of kinetic friction). The strength of the weight force acting on the seed is

image( eqn 3 )

where m is seed mass and g is gravitational acceleration. The strength of the lift force experienced by the seed is

image( eqn 4 )

where CL is the seed's coefficient of lift (Landau & Lifschitz 1991; Johnson & Fryer 1992).

The wind vector experienced by a seed, inline image, depends on the vertical wind velocity profile. This profile describes how wind velocity decreases with the height above ground. On open ground, the horizontal wind velocity |inline image(z) | at height z above the ground typically follows a logarithmic profile

image

where z0 is the aerodynamic roughness length, U* is the friction velocity and K the von Karman constant (Monteith & Unsworth 1990). This logarithmic wind velocity profile can be expressed as a function of |inline imageref |, the wind velocity measured at a reference height zref (Monteith & Unsworth 1990)

image( eqn 5 )

We calculate the wind velocity experienced by a seed, |inline image | , as |(inline imagez) | averaged over the vertical seed projection, h

image( eqn 6 )

where p is the ‘wind interception parameter’, a dimensionless ratio between the wind velocity experienced by the seed and the wind velocity at reference height. p summarizes the effects of the vertical seed projection and the wind velocity profile. For a logarithmic profile, the interception parameter is

image( eqn 7 )

explicit model version

We use the above equations to formulate a mechanistic model for secondary wind dispersal that represents obstacles in a spatially explicit fashion. This explicit version was implemented in Pascal (using Borland Delphi 5, Borland Software Co., Scotts Valley, USA). For a given time t, the model first determines whether seed movement is possible. If it is, the seed position at time t+ Δt is calculated from

image

where inline image(t) and inline image(t) are the position and the velocity of the seed's centre at time t, respectively. If T is the period of secondary wind dispersal (the amount of time for which a seed remains mobile) the seed's post-dispersal location is inline image(T).

conditions for seed movement

A stationary seed (inline image = inline image) that is not retained by an obstacle starts moving if drag overcomes friction, that is if | inline image | > | inline image |. Using equations 1–4 and 6 we can write down the condition for the start of seed movement in the absence of obstacles in terms of a threshold lift-off velocity, Ulift (Johnson & Fryer 1992)

image( eqn 8 )

Ulift is thus an aggregated parameter that summarizes seed properties (the wing loading m/A, Norberg 1973), seed–surface interactions (µs, CD, CL), and physical constants (g, ρ). Note that we define Ulift in terms of the wind velocity experienced by the seed (| inline image |), whereas Johnson & Fryer (1992) express it in terms of a wind velocity at some arbitrary reference height (| inline imageref |). Our definition allows a separation of the effects of a seed's lift-off velocity from the effects of its wind interception, p. We assume that Ulift is constant for a given seed on a given surface. Ignoring seed momentum, we furthermore assume that seed movement stops as soon as the condition for the start of movement is no longer fulfilled.

speed of seed movement

Kinetic friction experienced by a moving seed is assumed to be small and we therefore ignore it in the model (inline image = inline image for a moving seed). Moreover, we assume that seeds accelerate and decelerate instantaneously. Under these assumptions, it follows from equation 1 that at any time a moving seed has the same speed as the wind it experiences,

image( eqn 9 )

This assumption is in agreement with field observations: 25 Protea repens seeds released on a 50-m long section of a sandy, obstacle-free beach moved at 96% (standard deviation 15%) of the estimated wind velocity they experienced (F. Schurr, unpublished data).

interaction with obstacles

We assume that the horizontal cross-sections of seeds are circular (with diameter s) and that the horizontal cross-sections of obstacles are elliptical (with diameters a and b). The centre (inline image)of a seed situated at an obstacle then approximately lies on an ellipse E with diameters a+s and b+s (Fig. 1a). If E intersects the movement vector of a seed, the seed is either stopped or it changes its direction of movement. In our model, the outcome of this seed–obstacle interaction depends on the effects of the obstacle on wind conditions in its neighbourhood. While such effects are complex, we describe them with a simple rule: a seed situated at an obstacle experiences a wind vector that is the projection of inline image on the tangent on ellipse E in seed location inline image (Fig. 1a). If ω is the angle between inline image and this obstacle tangent, the wind velocity experienced by the seed is |inline image | cos ω. At an obstacle, the condition for seed movement is thus

Figure 1.

Representation of seed–obstacle interactions in the model. (a) The centre, inline image, of a circular seed (diameter s), that is situated at an elliptical obstacle (diameters a and b), lies approximately on an ellipse E with diameters a+s and bs. The wind vector experienced by the seed is calculated as the projection of the ambient wind vector inline image on the tangent on E in inline image. This projection depends on the angle ω between inline image and the tangent. (b) The seed moves along this tangent until it reaches the tangent of inline image on E. Thereafter the seed is assumed to have moved ‘past’ the obstacle and its movement is again determined directly by inline image.

image( eqn 10 )

If this condition is met, the seed moves along the obstacle tangent with velocity

image( eqn 11 )

Once the seed has moved ‘past’ the obstacle (see Fig. 1b), seed velocity is again calculated from equation 9.

aggregated model version

The model version described above is not suitable for simulating the dispersal of many seeds over extended periods because the explicit representation of obstacles makes simulations very time-consuming. Thus, we derived an aggregated version of the mechanistic model that enables large-scale simulations of secondary wind dispersal (see Appendix S1 in Supplementary Material). The aggregated model version was implemented in R 1.8.1 (R Development Core Team 2004) with computer-intensive subroutines coded in C.

The model aggregation is based on the idea that secondary wind dispersal consists of an alternating series of periods in which seeds move between obstacles, and periods of seed retention at obstacles. The final position of a seed is then a function of the sum of individual movement periods within the dispersal period T, and of the wind the seed experiences while moving. In Appendix S1, we derive probability distributions for retention and movement times that can be used in dispersal simulations. We show how empirical distributions of retention time can be calculated under the assumption that obstacles have a circular basal area. For a wide range of wind measurements, Ulift and p-values, we found these empirical retention time distributions to be well approximated by Gamma distributions. Movement times follow an exponential distribution if (i) seeds moving between obstacles follow a straight line, (ii) the spatial distribution of obstacle centres is completely random, and (iii) obstacle diameters are substantially smaller than typical distances between obstacles. The parameter of this exponential distribution is the obstacle encounter rate λ, which specifies the mean number of obstacles a seed encounters per unit distance moved (1/λ is the mean free path between two obstacles). The obstacle encounter rate can be calculated as

image( eqn 12 )

where d is the density of obstacle centres and inline image is the mean diameter of obstacles (see Appendix S1).

Model parameterization and validation

To validate the model, we conducted a field experiment with seeds of seven species of Proteaceae native to the Cape Floristic Region, South Africa. The study species cover the range of seed sizes and the main seed morphologies found in wind-dispersed fynbos Proteaceae (Fig. 2, Table 1, Rebelo 2001): plumed seeds (Protea repens, P. lorifolia, P. neriifolia), winged seeds (Leucadendron laureolum, L. xanthoconus, L. salignum), and parachute seeds (L. rubrum, nomenclature follows Rebelo 2001). All study species are serotinous, that is they store their seeds in cones that open after the mother plant has burnt. The high intensity fires in fynbos create a vegetation-free environment in which secondary wind dispersal is promoted (Bond 1988).

Figure 2.

The three basic morphologies of Proteaceae seeds that were used for experimental model validation: (a) plumed seed, Protea repens, (b) winged seed, Leucadendron salignum, (c) parachute seed, L. rubrum. Scale bars mark a length of approximately 1 cm.

Table 1.  Seed traits of seven species of Proteaceae and summary statistics of their seed movement in the field experiment. The table gives means and standard deviations (in brackets) of quantitative seed trait estimates. h and s, respectively, are the mean vertical projection and the mean horizontal diameter of a seed. Lognormal distributions of lift-off velocity, Ulift, are characterized by the mean and the standard deviation (the standard deviations associated with estimates of these two parameters were determined by non-parametric bootstrapping). Experimental results are summarized as the mean (and range) of the proportion of seeds collected at the first trap check after seed release (Fig. 3)
SpeciesSeed typeh (cm)s (cm)Ulift (m/s) Proportion trapped (%)
MeanSD
Protea repensPlumed2.1 (0.3)4.6 (0.8)2.7 (0.1)0.9 (0.1)72.8 (37–100)
P. lorifoliaPlumed0.5 (0.1)3.1 (0.2)3.0 (0.1)1.0 (0.1)47.4 (0–88)
P. neriifoliaPlumed0.9 (0.2)3.9 (0.4)3.7 (0.1)1.2 (0.1)32.8 (0–71)
Leucadendron laureolumWinged0.3 (0.1)0.9 (0.1)4.0 (0.3)1.8 (0.3)10.1 (0–32)
L. xanthoconusWinged0.2 (0.1)0.5 (0.03)3.4 (0.2)1.6 (0.2) 9.4 (0–26)
L. salignumWinged0.3 (0.1)0.8 (0.1)3.1 (0.2)1.2 (0.1)13.8 (0–45)
L. rubrumParachute1.9 (0.4)3.1 (0.3)2.6 (0.4)1.1 (0.6)64.8 (9–94)

field validation experiment

For the validation experiment, we set up a semicircular arena of 25 m radius (Fig. 3a) on a sandy, level and obstacle-free section of Noordhoek Beach, Cape Peninsula, South Africa (34°8′ S, 18°21′ E). The arena was delimited by a seed trap, a 50 cm high strip of 40% shade cloth that was attached to the surface so that seeds moving along the ground could not slip underneath. We repeatedly released batches of 100 seeds at the arena centre and subsequently determined the number of seeds caught in the seed trap. Seeds released at different times were stained with fluorescent powder of different colours (Magruder Color Company, Elizabeth, New Jersey, USA). To quantify the direction of seed movement, we divided the trap into four sectors of equal length (Fig. 3a). As artificial obstacles we used sand-filled paper bags that had an elliptical basal area (diameters 64 cm and 32.5 cm). These obstacles were introduced into the arena at four different densities (0, 60, 120 and 180 obstacles, resulting in densities from 0 to 0.18 obstacles m−2). They were arranged in spatially completely random patterns with their larger diameter parallel to the base line of the arena (i.e. the diameter of the semicircle). The randomized co-ordinates of the obstacles were simulated prior to the experiment. At each obstacle density, seeds of each study species were released at two different times (Fig. 3b). To validate the model, we used data from the trap checks immediately following each of the 56 seed releases (four obstacle densities × two releases × seven study species). The dispersal period T up to this first check ranged from 22 to 58 minutes. Additionally, we considered data from the second trap check after seed release if the obstacle density had not changed since the release (Fig. 3b). This was the case for 28 seed releases (four obstacle densities × one release × seven species). Dispersal period T up to the second check ranged from 52 to 93 minutes.

Figure 3.

Set-up of the field experiment used to validate the seed dispersal model. (a) Set-up of the experimental arena on a sandy beach. Seeds were released at the release point and the number of seeds collected in different sectors of the semicircular trap was determined. Wind conditions during the experiment were measured with a sonic anemometer. (b) Temporal sequence of seed releases (R) and trap checks (C). At each of the four obstacle densities (0, 60, 120 and 180 obstacles within the experimental arena), seeds of each study species were released at two different times. For model validation we considered all seeds that were found at the first trap check after their release. Seeds found at the second trap check after release were only considered if the obstacle density had not changed since their release.

wind measurements

During the experiment, time series of horizontal wind velocity components (at reference height zref = 145 cm) were recorded with a triaxial sonic anemometer (Model USA-1, Metek GmbH, Elmshorn, Germany) at a frequency of 10 Hz (therefore Δt = 0.1 seconds in the model simulations). Summary statistics of the wind conditions are given in Table 2. To determine the vertical wind velocity profile we took additional cup anemometer measurements (WatchDog 700, Spectrum Technologies, Inc., Plainfield, Illinois, USA) at 33 cm, 47 cm and 103 cm above ground. By fitting equation 5 to the mean velocities per minute interval with a non-linear least squares model (R package NLS) we estimated the roughness length as z0 = 0.018 cm. The 95% confidence interval around this estimate is broad (0.005–0.052 cm) but falls within the range of values reported for a similar beach (Jackson 1996).

Table 2.  Summary statistics of wind conditions (Uref) during the field experiment at Noordhoek Beach. The table shows the ranges observed for each statistic in the eight periods between seed release and the first trap check (Fig. 3b). Wind measurements were taken 145 cm above the ground with a sonic anemometer at a temporal resolution of 0.1 s
VariableMeanMedianVarianceMinimumMaximum
Wind velocity (m second−1)4.20–7.374.18–7.351.08–1.721.33–3.058.69–12.69
Wind direction (radians from E)1.10–1.531.11–1.530.03–0.060.37–0.531.84–2.68

estimation of seed parameters

The vertical seed projection of a species, h, was calculated by averaging calliper measurements of 100 seeds placed randomly on a smooth board (Table 1). To determine horizontal seed extent, we scanned > 50 seeds of each species with a digital scanner, and measured maximum and minimum seed extent with the image processing software KS 300 3.0 (Carl Zeiss Vision GmbH 1999). The horizontal seed diameter, s, was then calculated as the mean diameter of an ellipse with diameters equal to the maximum and minimum seed extent (Table 1).

Lift-off velocities, Ulift, were measured with the methodology of Johnson & Fryer (1992) in a wind tunnel of the open jet return circuit type at the Department of Mechanical Engineering, University of Cape Town. In the wind tunnel, we placed seeds on sandpaper (mesh 40, average grain size c. 400 µm), a surface similar to the surface of sandy fynbos soils. Starting at 2 m second−1, we then increased the free stream velocity of the wind tunnel in steps of 1 m second−1 and recorded the velocity at which each seed started moving (n = 72 seeds per species). This free stream velocity was translated into Ulift, the threshold wind velocity experienced by the seed, by assuming that the velocity profile in the wind tunnel is logarithmic (equation 5). From the free stream velocity and Pitot tube measurements at four different heights (0.1 cm, 0.6 cm, 1.1 cm and 2.1 cm), we estimated the roughness length of this velocity profile as z0 = 0.0024 cm (R2 = 0.98). To estimate the probability distribution of Ulift for each species, we fitted lognormal density functions to the distributions of measured Ulift values (R-function fitdistr, Venables & Ripley 2002; Table 1).

model validation

We used the explicit model to simulate the dispersal of 10 000 seeds for each of the experimental seed releases. In these simulations, the positions and orientations of obstacles were identical to those in the experiment. The release time of individual seeds was selected randomly within the first minute of the respective dispersal period, and initial seed positions were distributed randomly within a square metre centred at the release point. We compared model predictions with the experimental results both in terms of the overall proportion of seeds trapped, and in terms of the proportion of seeds trapped in each trap sector. The amount of variation in the observed data that is explained by the model was calculated as the generalized coefficient of determination (adjusted R2, Nagelkerke 1991). In the calculation of this adjusted R2 we assumed binomial errors for the overall number of trapped seeds, and multinomial errors for the number of seeds per trap sector. In cases where no seeds were simulated to reach a trap sector, the predicted proportion of trapped seeds was set to 7.5 × 10−5. This is the per-sector trapping probability of a multinomial distribution for which the probability of trapping none of 10 000 seeds is 5%.

Results

model validation against experimental data

The proportion of seeds trapped in the field experiment ranged from 0 to 100% per seed batch. Seed distribution was variable in space, with seeds being trapped in three of the four trap sectors. Additionally, seed distribution varied in time; in the seed batches undergoing two checks, 0–100% of the trapped seeds were found in the second check. Moreover, the mean proportion of seeds trapped up to the first check differed between obstacle densities (ranging from 11.9% at 180 obstacles to 61.7% at 0 obstacles) as well as between species (ranging from 9.4% for L. xanthoconus to 72.8% for P. repens). Plumed and parachute seeds were markedly more mobile than the smaller winged seeds (Table 1).

The explicit version of the mechanistic model was able to explain most of the variation in the experimental data from independently measured parameters. The model provided a good explanation both of the overall proportion of seeds that covered 25 m up to the first check (Fig. 4a, adjusted R2 = 0.77, n = 56 releases) and of the distribution of these seeds to the different trap sectors (Fig. 4b, adjusted R2 = 0.69). The model also performed well at explaining the overall proportion of seeds trapped up to the second check (Fig. 4c, adjusted R2 = 0.70, n = 28 releases) and the spatial distribution of these seeds (Fig. 4d, adjusted R2 = 0.67).

Figure 4.

Observed vs. predicted proportions of seeds dispersed over a distance of 25 m in seed release experiments on Noordhoek beach. Figures show the overall proportion of seeds found in the seed trap (left column) and the proportion of seeds found in each trap sector (right column) for the first trap check (top row) and the second trap check (bottom row) after seed release. Symbols represent different species (filled symbols, plumed seeds: square = Protea repens, diamond = P. lorifolia, triangle = P. neriifolia; empty symbols, winged seeds: square = Leucadendron laureolum, diamond = L. xanthoconus, triangle = L. salignum; stars, parachute seeds = L. rubrum). Also shown are lines of agreement between predictions and observations.

Model bias, the mean difference between model prediction and experimental observation, was −7.7% for the proportion of seeds trapped up to the first check. The model thus tended to underestimate seed dispersal at the maximum likelihood parameter estimates. As our estimate of the aerodynamic roughness length z0 was uncertain, we varied this parameter within the 95% confidence interval of the estimate. This parameter variation showed that the model is sensitive to z0 and that the underestimation of seed dispersal disappears as z0 becomes smaller (Fig. 5). For instance, at z0 = 0.012 cm (a value within the 50% confidence interval of the z0 estimate), model bias is reduced to −2.2% and the adjusted R2 is 0.86. Similar results were found when analysing data from the second check (results not shown).

Figure 5.

The effect of the aerodynamic roughness length (z0) on the bias of the explicit model version. Model bias is the mean difference between the proportion of seeds predicted to disperse over 25 m and the respective experimental data (only data from the first trap check were considered). The solid line indicates the maximum likelihood estimate of z0. Bold hatched lines show the 50% confidence interval of this estimate and thin hatched lines the 95% confidence interval.

comparison of explicit and aggregated model version

Predictions of the aggregated and the explicit model showed some deviation if the explicit model was run with the exact distributions and orientations of obstacles used in the beach experiment (Fig. 6, R2 = 0.96, bias = 3.7%). This difference may arise because the aggregated model assumes that seeds start at random locations relative to a random obstacle pattern. In contrast, in simulations of the experimental setting with the explicit model, the initial position of seeds relative to the obstacle pattern varied little. In these simulations, seed trajectories were thus more strongly correlated than in the aggregated model. To introduce variation in the relative location of initial seed positions and obstacles, we generated 100 random obstacle patterns for each obstacle density. These patterns were created by assigning random positions and orientations to individual obstacles. For each obstacle pattern we simulated the dispersal of 100 seeds with the explicit model version. The aggregated model provided an excellent prediction of the proportion of seeds dispersed over 25 m in these simulations (Fig. 6, R2 > 0.99, bias = −0.7%).

Figure 6.

Comparison of explicit and aggregated model version in terms of the proportion of seeds predicted to disperse over 25 m. Empty symbols represent simulations of the explicit model for the configuration of initial seed positions and obstacles in the field experiment (R2 = 0.96, bias = 3.7%, explicit = 1.05 × aggregated + 2%). Filled symbols represent simulations of the explicit model for 100 random configurations of initial seed positions and obstacles (R2 > 0.99, bias =−0.7%, explicit = 0.99 × aggregated − 0.4%).

sensitivity analysis

To explore the behaviour of the mechanistic model we performed an extensive sensitivity analysis. We used the aggregated model version for this analysis as we were not interested in analysing the dispersal distance of seeds released in a particular location relative to a specific obstacle pattern. In the sensitivity analysis we independently varied the lift-off velocity Ulift, the obstacle encounter rate λ, and the wind interception parameter p. For each combination of these parameters, we simulated the dispersal of 100 seeds using wind data measured during 1 hour of the field experiment and calculated the median dispersal distance.

In the absence of obstacles (λ = 0 cm−1), we find a sigmoidal relation between Ulift and dispersal distance (Fig. 7a): seeds with high Ulift are not dispersed along the surface whereas for low Ulift dispersal distance approaches an upper bound. This upper bound reflects the fact that no seed can travel further than the wind it experiences within the dispersal period. Note that in this respect our model differs from that of Greene & Johnson (1997); the latter predicts an inverse relationship between distance of secondary wind dispersal and wing loading, implying that dispersal distance becomes infinitely large as wing loading (and thus Ulift, see equation 8) approaches 0. Our model furthermore reveals an interaction between lift-off velocity and obstacle encounter rate: higher obstacle encounter rates (e.g. λ = 0.1 cm−1) keep all seeds from realizing the maximum dispersal distance and allow substantial dispersal only for seeds with low Ulift (Fig. 7a).

Figure 7.

The joint effects of a seed's lift-off velocity (Ulift), the wind interception parameter (p) and the obstacle encounter rate (λ) on the median distance of secondary wind dispersal predicted by the aggregated model version. For model simulations we used 1 hour of wind data measured during the field experiment.

The sensitivity analysis also demonstrates the importance of the wind interception parameter p. The three curves in Fig. 7(b) show that differences in p can result in different median dispersal distances even if the lift-off velocity measured at reference height (Ulift/p) is identical. This is because higher values of p increase seed velocity (equation 9), thereby decreasing the travel time between obstacles and promoting dispersal distance. Ulift and p thus affect secondary wind dispersal independently and cannot be aggregated into a single parameter.

Figure 7(c) shows the joint effect of obstacles and wind interception on the distance of secondary wind dispersal. For λ = 0 cm−1, dispersal distance attains an upper bound that is defined by p and Ulift (see Fig. 7a, b). At low values of p, dispersal distance decreases steeply with λ, whereas the rate of decrease is less pronounced for larger p. This effect arises because an increase in p reduces the importance of individual obstacles by decreasing the travel time between obstacles and the retention time at obstacles. Finally, for large values of λ (when obstacle encounters are frequent), median dispersal distance is approximately correlated with the inverse of λ (that is with the mean obstacle-free path).

Discussion

model evaluation

In our field experiment, we observed large variation in the wind-driven movement of seeds along the ground. The explicit version of our mechanistic model, when parameterized with independent measurements, explains most of this variation (Fig. 4). The aggregated model version seems to be an excellent approximation of the explicit version if initial seed positions vary randomly relative to the obstacle pattern (Fig. 6). This condition will be met in most cases where the large-scale simulation of secondary wind dispersal is of interest.

The good agreement between model predictions and experimental data is remarkable as the model makes a number of simplifying assumptions: it (i) ignores turbulence, (ii) assumes spatially homogeneous wind conditions away from obstacles, (iii) ignores seed momentum, (iv) uses a simple rule for seed–obstacle interactions (Fig. 1), and (v) describes complex seed morphologies (Fig. 2) using three easy to measure parameters (lift-off velocity, vertical seed projection and horizontal seed diameter). Of these assumptions, the omission of turbulence seems particularly important. Turbulence may significantly increase distances of primary seed dispersal by wind (Nathan et al. 2002; Tackenberg 2003), and it evidently can also affect secondary dispersal by wind. First, turbulent wind fluctuations close to the ground may move seeds smaller than the roughness length z0. Second, the turbulence created by obstacles may affect seed retention at obstacles. Finally, turbulent eddies may pick up seeds, thereby terminating secondary seed movement and initiating a new (tertiary) phase of airborne seed dispersal. The importance of turbulence for secondary dispersal by wind will thus increase with decreasing seed size and increasing roughness length. Moreover, it will vary with the size and shape of obstacles.

Although our model ignores turbulence, it reasonably described seed movement in the field experiment, in which we varied seed size by more than one order of magnitude (Table 1). While the model's slight underestimation of seed movement might be due to the omission of turbulence, it can probably be attributed to the imprecise estimation of roughness length, z0 (Fig. 5). In summary, our model seems to capture the essential processes that determined secondary seed movement in the field experiment. The simplicity of our model is in fact one of its advantages: in comparison with a more complex model, it is easier to parameterize and less sensitive to parameter uncertainty (Burnham & Anderson 1998; Clark et al. 2003; Higgins et al. 2003a). Nevertheless, the domain for which our model makes valid predictions is obviously limited. It will be interesting to assess these limits by releasing an even wider range of seed sizes into environments that have differences in obstacle shape and the amount of turbulence.

The measurement of seed movement in our experimental arena cannot directly validate model predictions with respect to long-distance dispersal. In fact, dispersal models can hardly be validated at large spatial scales, because the empirical quantification of long-distance dispersal requires a prohibitive sampling effort (Greene & Calogeropoulos 2002; Nathan et al. 2003). Instead, the validation of mechanistic dispersal models has to focus on aspects of seed movement that are both measurable and relevant for long-distance dispersal, such as (for primary wind dispersal) the proportion of seeds that are uplifted and therefore likely to be dispersed over long distances (Nathan et al. 2002; Tackenberg 2003). Our field experiment similarly quantified the proportion of ‘fast’ seeds: a seed that travelled the 25-m extent of our experimental arena within 22 minutes (the minimum time up to the first check, Fig. 3b) may disperse over 1.6 km in a 24-hour period. The field experiment thus validated model predictions at scales relevant for long-distance dispersal.

The experimental validation gives us confidence in applying the model to a range of species and environmental settings (Nathan & Muller-Landau 2000).

long-distance dispersal by secondary seed movement

The distance of secondary wind dispersal depends both on the way in which seeds move along the ground, and on the length of the dispersal period (T) during which seeds can move. While our model provides, for a given T, a good description of wind-driven seed movement along the ground (Fig. 4), it is not immediately clear which factors influence T.

Secondary wind dispersal can be terminated by the germination of seeds, by their permanent entrapment (e.g. through burial under litter) or by the deterioration of the seed's dispersal structure. As the factors determining these processes vary substantially between environments (Chambers & MacMahon 1994), it is not surprising that literature estimates of T range from a few hours (Greene & Johnson 1997) to several months (Watkinson 1978). In South African fynbos, rainfall appears to trigger all three causes of termination of secondary wind dispersal (A. Rebelo, P. Holmes, J. Vlok, D. LeMaitre, B. van Wilgen, personal communication). As serotinous Proteaceae release their seeds after the mother plant has burnt, T can be estimated as the time between a fire and the first significant rainfall event. By combining seasonal fire frequencies for fynbos (Brown et al. 1991; Richardson et al. 1994) with seasonal rainfall distributions (Zucchini et al. 1992), one obtains values of T that range from a few days to 1 year, with a median of 73 days (F. Schurr, unpublished data).

If T is long, interspecific differences in seed traits can strongly affect the distance of secondary wind dispersal. This is illustrated by model simulations for L. salignum and P. repens, in which we used long-term wind measurements and parameter values typical of fynbos conditions (T = 73 days, z0 = 0.1 cm, λ = 0.098 cm−1 for P. repens and λ = 0.060 cm−1 for L. salignum, F. Schurr, unpublished data). Although roughness length and obstacle encounter rates were higher than in the beach experiment (z0 = 0.018 cm and λc. 0.001 cm−1 for both species), the maximum of 10 000 simulated seed dispersal distances was 59 km for P. repens, but only 3 m for L. salignum. These results support the notion that secondary wind dispersal is a mechanism by which seeds can move long distances (Bond 1988; Higgins et al. 2003b). They also agree with the empirical finding that serotinous Proteaceae differ substantially in their potential for long-distance dispersal by secondary seed movement (Bond 1988).

It should be noted that our simulations probably overestimate the long-distance dispersal of P. repens because they assume that environmental conditions (vertical wind profile and obstacle pattern) are spatially homogeneous. In reality, however, a seed dispersing over several kilometres will enter areas where high obstacle encounter rates and/or low wind interception effectively prevent secondary wind dispersal. Such impermeable areas may be boulder fields, steep slopes, rivers and roads, but the ultimate spatial limit to secondary wind dispersal in fynbos is the dense vegetation characteristic of unburnt patches (Bond 1988). Typical fynbos fires burn areas greater than 1 km2 (Horne 1981). Hence, fire extent may well limit the secondary wind dispersal of P. repens, whereas it is less likely to do so in L. salignum. The finding that large fires promote the spread of good secondary wind dispersers, whereas fire size may be irrelevant for poorer dispersers, has important implications for conservation management in fynbos.

seed size and the distance of secondary wind dispersal

Secondary wind dispersal in a given environment is promoted by a decrease in lift-off velocity (Ulift), a decrease in the obstacle encounter rate (λ) or an increase in wind interception (p) (Fig. 7). λ and p increase with horizontal seed diameter (equation 12) and vertical seed projection (equation 7), respectively. Because Ulift scales with the square root of wing loading (equation 8), allometric considerations suggest that Ulift increases with the square root of a linear measure of seed size (Johnson & Fryer 1992). For a given seed morphology, a change in seed size should thus affect Ulift, p and λ simultaneously.

The relationship between seed size and the distance of secondary wind dispersal has been discussed by Greene & Johnson (1997), who suggested that dispersal distance is maximized for small seeds because Ulift decreases with seed size. Small seeds also have lower obstacle encounter rates, which should further promote their dispersal (Fig. 7). According to our model, however, there is an overwhelming disadvantage of small seed size: seeds with a vertical projection below the roughness length z0 will not be moved at all (because for them p = 0, equation 7). Therefore, the distance of secondary wind dispersal is predicted to be maximal for some intermediate seed size.

Turbulence may promote the secondary wind dispersal of small seeds, and thus decrease the difference in dispersal distance between small and intermediate seed sizes. However, seed size did increase dispersability in our field experiment: for example, the seeds of P. neriifolia are larger and better dispersed than those of L. salignum, although their lift-off velocity is higher (Table 1). In addition, Bond (1988) tracked seeds of six Proteaceae species (four of which were not included in our study) and observed that the distance of secondary wind dispersal increased with seed size, and Chambers et al. (1991) showed that for eight species in a disturbed alpine environment, the probability of horizontal seed movement increased with seed size on each of five surface types. Thus, for a range of seed sizes (vertical projection between c. 1 mm and several cm) and environmental conditions, the secondary dispersal distance does appear to increase with seed size.

The unimodal relationship between dispersal distance and seed size predicted by our model differs from the negative correlation predicted by mechanistic models for primary wind dispersal (e.g. Greene & Johnson 1996). Such a negative correlation is assumed in many ecological and evolutionary models (Ezoe 1998; Levin et al. 2003), and is often regarded as mediating a competition-colonization trade-off (e.g. Crawley 1997). However, this negative correlation does not seem to hold for animal-dispersed plants (e.g. Coomes & Grubb 2003). Our model predicts that it also will not hold for wind-dispersed plants if seed movement along the ground is important, and the effect of turbulence is small.

Conclusions

Mechanistic models present the only realistic option for predicting the dispersal of large groups of species. This is for the simple reason that we cannot hope to empirically measure dispersal distances for large numbers of species. Currently, primary seed dispersal by wind is probably the only dispersal syndrome where mechanistic models can be applied with any confidence (Nathan et al. 2002; Tackenberg 2003). By developing and validating a model for secondary wind dispersal, we expand the domain of mechanistic seed dispersal models.

Acknowledgements

We are grateful to Anthony Sayers for permission to use the wind tunnel. Permission to conduct fieldwork was granted by the Cape Peninsula National Park and Western Cape Nature Conservation. Robert Taylor, Erna Davidse, Claude Midgley and Carolyn Holness assisted with field and laboratory work. We thank Björn Reineking, Jürgen Groeneveld, Alexander Singer, Oliver Tackenberg, Peter Poschlod, Christian Wissel, Tim Myers, Anthony Sayers, Lindsay Haddon, Michael Hutchings and two anonymous reviewers for many helpful comments and suggestions. Parts of this work were funded by the A.F.W. Schimper Foundation for Ecological Research and by the Center for Applied Biodiversity Science, Conservation International.

Supplementary material

The following material is available from http://www.blackwellpublishing.com/products/journals/suppmat/JEC/JEC1018/JEC1018sm.htm

Appendix S1 Aggregation of the mechanistic model for secondary seed dispersal by wind.

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