Errata: Erratum Volume 44, Issue 6, 1263, Article first published online: 25 October 2007
C.F.G. Thomas, Department of Agriculture and Food Sciences, University of Plymouth, Seale-Hayne, Newton Abbot, Devon TQ12 6NQ, UK (fax +44 1626 325605; e-mail email@example.com).
1Dispersal parameters are critical for modelling spatially dynamic populations yet remain among the most difficult to quantify. Linyphiid spider aerial dispersal by ‘ballooning’ on silk threads is dependent on meteorological factors and amenable to analysis and quantification.
2Spider aerial activity was measured during consecutive 10-min periods for up to 11 h a day. Aerial density was measured at four heights using sticky traps. The time intervals between successive flights were measured by observing spiders landing and taking-off.
3Meteorological measurements (wind speeds and temperatures) were taken simultaneously with the collection of airborne spiders, and used to calculate Richardson numbers to estimate atmospheric turbulence.
4Numbers of airborne spiders on any given day, and their vertical density profile on different days, were significantly correlated with Richardson numbers.
5Single flight distances were modelled using estimates of ascent and descent rates, the vertical density profile of airborne spiders and the wind speeds they experience aloft. The distributions of single flight distances and the time spent between successive flights were combined in an alternating renewal process to model the number of flights and the total daily dispersal distances of ballooning spiders as a function of available dispersal time.
6On a day with, for example, 6 h of suitable weather, linyphiid spiders can potentially disperse a mean distance of approximately 30 km downwind.
7Synthesis and applications. We have developed a dispersal model of linyphiid spiders that is central to the further development of spatially dynamic population models of these spiders in agricultural landscapes. It could also be adapted for application to other wind-borne organisms. Such models have a key role in the future management of sustainable agricultural systems where natural predators are seen as major components of pest control.
The aerial dispersal of linyphiid spiders (Araneae, Linyphiidae) by ‘ballooning’ on silk threads is well-documented (Blackwall 1827; Duffey 1998). They are known to be able to travel considerable distances as linyphiid spiders are among the first colonists of newly available habitats, including reclaimed land (Meijer 1977) and volcanic islands (Thornton et al. 1988). They have also been observed landing on ships some distance from the nearest land (Darwin 1839).
In order to predict the effects of changing land use and landscape structure on population persistence and extinction of vagile species, it is necessary to develop spatially dynamic models that incorporate the essential parameters of their dispersal (Topping & Sunderland 1994; Halley, Thomas & Jepson 1996). For linyphiid spiders, these include the frequency and timing of dispersal episodes in relation to habitat disturbances, the proportion of a population that disperses on any occasion, and total dispersal distance and pattern of redistribution among available habitats.
Dispersal distance is a critical parameter in spatially dynamic models, as it affects the success with which an organism locates a suitable new patch (Fahrig & Paloheimo 1988; Moloney et al. 1992). In theoretical metapopulation models, the relationship between dispersal distance and the scale of spatial fragmentation of the environment can be explored by describing dispersal distance as a function of spatial scale (Doak, Marino & Kareiva 1992; Moloney et al. 1992). However, in real systems spatial scale is an environmental constant, and to model the spatial dynamics of a particular species or group actual dispersal distances need to be known. Linyphiid spiders engage in a relatively simple system of wind-borne dispersal, which, because of its passive nature and its dependence on meteorological conditions, is amenable to analysis by simulation modelling.
The role of meteorological factors in the aerial dispersal of spiders has been studied in the field (Vugts & van Wingerden 1976; Greenstone 1990; Suter 1999) and certain aspects have been analysed in fluid mechanic models (Humphrey 1987) and laboratory studies (Suter 1991, 1992). It has been established that the principal meteorological conditions necessary for spiders to balloon are wind speeds below 3 m s−1 and some turbulence of the air to provide the forces producing lift on an otherwise passive particle (Vugts & van Wingerden 1976). However, no work has related meteorological conditions to the vertical distribution of airborne spiders in the turbulent boundary layer or to estimate the distances that can be travelled.
Little is known of the precise stimuli that initiate and terminate spider ballooning behaviour (Weyman 1993). An ability to sense favourable meteorological factors is undoubtedly involved, and some evidence suggests that a spider's physiological state is important (Legel & van Wingerden 1980; Weyman & Jepson 1995).
Ballooning is typically, but not exclusively, associated with anti-cyclonic weather characterized by light winds and thermals of rising air. When suitably stimulated, a spider climbs a launch site (e.g. grass blade), hangs suspended or stands facing into the wind with legs extended and abdomen raised, and emits a silk thread of sufficient length for drag to enable take-off. Once airborne, the rate of ascent in a thermal is dependent on the vertical wind speed and the terminal velocity of the spider. The vertical density distribution is governed by turbulent diffusion. When the thermal dissipates, the spider descends in the surrounding subsiding or stable air. The downwind displacement of a spider during this process is therefore dependent on horizontal wind speeds experienced during ascent and descent. In simple terms, tractable within a simulation model, we assume wind speed increases linearly with ln(height) up to the top of the surface layer at an altitude of approximately 100 m. Above this, the wind speed remains relatively constant up to the top of the boundary layer at approximately 1000 m. The top of the boundary layer caps entrained material such as spiders and pollutants (Stull 1988).
During a dispersal episode (defined as a period of a day or less when suitable weather prevails), a linyphiid spider alternates between the airborne and grounded phases and makes several flights until behavioural or meteorological factors prevent it ballooning again. This may be when wind speeds are no longer favourable or the turbulent structure of the boundary layer collapses in the late afternoon or early evening. During our studies, ballooning has not been observed to exceed 12 h in mid-summer, usually between 06.00 and 18.00 GMT.
The work presented here focused on meteorological factors that relate to successful launches and the distances travelled. We describe a series of field experiments designed to (i) quantify the relationship between atmospheric turbulence and the numbers of spiders becoming airborne; (ii) describe the vertical distribution of spiders in the surface boundary layer in relation to meteorological conditions; (iii) quantify the time intervals between successive flights of ballooning spiders.
Estimates of a range of ascent and descent rates were used with field observations of the vertical distribution of spiders in the air, wind speeds and spider behaviour. From these data, we present a model that describes the duration and lengths of single flights. The single-flight distances (airborne phase) and interflight intervals (grounded phase) are then modelled together as an alternating renewal process (Cox 1967) to simulate the number of flights possible in a given time and hence the total dispersal distances achievable by linyphiid spiders within a day.
Materials and methods
All field observations were made at the Leckford Estates, Hampshire, UK, in a grass field in open farmland with slightly rolling terrain (grid reference SU 340387). There was a minimum fetch of unobstructed wind flow of 300 m. Sampling was performed on 5 days in July 1991 (10, 17, 22, 27 and 28), when a high probability of spider aerial dispersal was predicted from broadcast Meteorological Office weather reports forecasting low wind speeds.
meteorological measurements and field sampling
Wind speed and temperature were recorded from meteorological instruments attached to a 7-m instrument mast erected in the centre of the field. Wind speed was measured at six heights with sensitive cup anemometers (Type AN1; Vector Instruments, Rhyl, UK; threshold 0·3 m s−1, accuracy 1% ± 0·1 m s−1). The lowest anemometer was positioned just above crop height and adjusted for each experiment according to the height of the crop, in order to maintain a constant height above the zero plane displacement (d) (Stull 1988), although in practice the roughness length (Z0) (Stull 1988) of the crop changed little between sample dates. Successively higher anemometers were attached 0·2, 0·6, 1·4, 3·0 and 6·2 m above the lowest anemometer. Air temperature was measured at two heights with aspirated platinum resistance thermometers (Type T302; Vector Instruments; tolerance ±0·03 °C), shielded from direct solar radiation and positioned at the same heights as the second and fourth anemometers.
The use of cup anemometers averaging wind speeds over 10-min intervals necessarily ignores the fine detail of atmospheric turbulence and some of the true variance in wind speed. While this detail may influence the behaviour of spiders from second to second, we feel it is legitimate to ignore it for practical purposes and with respect to the larger-scale processes under consideration.
All instruments were attached to a 64-channel data logger (DeltaT, Cambridge, UK) configured to record wind speeds and temperatures at 10-min intervals. Wind speed was averaged over 10-min periods; temperatures were sampled every 15 s and averaged over 10-min intervals. Wind direction was continuously recorded on a paper chart by a weather vane (Lambrecht, Germany). Logged data were down-loaded into spreadsheets for processing.
Turbulence was quantified by the Richardson number (Ri) derived from a ratio of buoyancy and shear forces:
where g = gravitational acceleration, T = mean air temperature, ΔT = temperature difference, ΔU = difference in wind speed and ΔZ = height difference between measurements (Stull 1988). Negative values of Ri indicate turbulence or unstable air.
Aerial activity of spiders
Airborne spiders were sampled at intervals throughout each day, approximately 25 m from the instrument mast, using methods similar to those of Vugts & van Wingerden (1976). A 15-m length of string was suspended above the crop from three telescopic poles to form an L-shape with sides of equal length, so that the height of the string was level with the lowest anemometer. The tops of the support poles and a short length of adjacent string were smeared with insect-trapping gum (Biological Control Systems, Mid-Glamorgan, UK) to isolate the string from spiders climbing from the ground. Spiders could therefore reach the string only through the air.
Spiders were collected by aspirator during continuous patrols of the string, back and forth, for periods of 10 min separated by intervals of 10 min, for up to 11 h on each day. Each sampling period was synchronized with the logging of meteorological data. On each initial pass of the string, spiders that had accumulated during the previous 10-min interval were counted separately and designated set 1. Spiders collected during the remaining passes of the patrol were designated set 2. During the 10-min interval between patrols, spiders were transferred to 70% alcohol and labelled with sample number and time of collection.
Spiders were sampled from the air at four heights using sticky traps made of 0·29 × 0·91-m sheets of 13-mm mesh chicken wire, coated with insect trapping gum (Biological Control Systems Ltd) after Greenstone, Morgan & Hultsch (1985). Three 7-m instrument masts were erected in the centre of the field, positioned at the ends and in the right angle of an L-shape with 25 m between the pair of masts on each leg of the L. Each mast had a system of ropes and pulleys attached at four heights such that four horizontal ropes could be raised to any height between each pair of masts. Bulldog clips on the horizontal ropes allowed the attachment of three sticky traps, 1 m apart, at the centre of each horizontal span, giving a total of six traps at each height. The traps were raised as early in the morning as possible. At the end of the day, when spider ballooning had finished (determined from the continuous string captures), the traps were removed and placed individually between sheets of brown parcel paper and returned to the laboratory.
Spiders were removed from the traps by overnight immersion in solvent (white spirit). Spiders that transferred from the wire traps to the paper during transport were carefully removed with fine forceps and added to those in the solvent. The spiders were preserved in 70% alcohol for later quantification, after clearing the solvent in ethyl acetate.
Aerial density at four heights
The aerial densities of spiders at each height were calculated as the number of spiders per 1000 m3 of air (Nh/Vh), where Nh = the number of spiders from the six traps at height h and Vh = air volume sampled at height h. The volume of air sampled by the traps is a product of the cross-sectional area (a) of the traps in relation to the mean angle of the wind (θ) incident upon them, the wind speed at height h (wsh) and the total trapping time (t), such that Vh= a × wsh × t, where the total effective trap area a = (cosθ × 0·91 × 0·29 × 3) + (sinθ × 0·91 × 0·29 × 3).
Intervals between flights
To estimate the time spent in the grounded phase between successive flights, 87 spiders were observed landing in a grass field (Lolium perenne; vegetation height approximately 0·5 m), repositioning at the tip of a grass stem in preparation for a subsequent flight and taking-off. When a spider was seen to land, the time was noted and the spider continuously observed until it became airborne again, when the time was again recorded. Only one or two spiders failed to become airborne again, invariably in the late afternoon at the end of the observation period.
The turbulent boundary layer is too complicated to model in detail. However, the essential features relevant to spider ballooning may be contained in a simplified model of the process in which spiders are transported downwind during ascent to, and descent from, a given height. The airborne phase alternates with the grounded phase between flights to determine the total time spent airborne and hence the total distance travelled downwind as a function of time.
To simplify the model it is necessary to make certain assumptions as follows.
A spider becomes airborne in a parcel of rising air and is transported to an altitude HI at a rate A m s−1, where HI is a random variable with a probability distribution function (p.d.f.) f(h) derived from the observed vertical profile of aerial density. It was not possible to observe the maximum height attained by an individual spider directly. The maximum height that might have been attained by a spider could have been higher than the observed height h at which the spider was caught. However, it is possible to estimate the distribution of HI from the observed vertical distribution (see the Appendix for derivation) when the estimated f(h) is given by:
b is a parameter describing the change of aerial density with height derived from field data and c is the maximum altitude possible at the top of the boundary layer, which is assumed to be 1000 m.
On reaching this height the spider begins its descent at its terminal velocity of D, which approximates to 1 m s−1 in stable air with a silk filament length of 1 m (Humphrey 1987).
During both ascent and descent, the spider is transported downwind a distance L at the speed of the wind, which varies with height. Field measurements of wind speeds during ballooning were linearly regressed against ln(height) (Table 1) to give the intercept or wind speed at 1 m (the height of the crop acting as a launch platform for a spider at the start of the ballooning process), denoted ha, and slope of the regression or wind speed gradient, denoted hb. A general description of wind speed (m s−1) at height h above this initial ballooning height of 1 m, denoted by g(h), is given by:
Table 1. Mean wind speeds at six heights taken on days when the vertical distributions of ballooning spiders were sampled. Regression statistics of wind speed against ln(height) used to determine the parameters ha and hb are given together with an extrapolation to the wind speed at heights > 100 m (WS(>100))
10.7.91 Wind speed m s−1
17.7.91 Wind speed m s−1
22.7.91 Wind speed m s−1
27.7.91 Wind speed m s−1
28.7.91 Wind speed m s−1
Wind speed at H 101 m
Wind speed was assumed to have a constant value for h > 100 m, where the frictional effects of the surface become negligible. Wind speed above 100 m was estimated to be the wind speed at 100 m, and is denoted by WS(>100m), where:
On landing, the spider prepares for another flight and becomes airborne after a certain period, according to our observations of intervals between flights (Fig. 3). There was a minimum period of 1 min at the start of the grounded phase while a spider re-orientated itself on a new launch platform, after which the time to the next flight is exponentially distributed. Thus if the length of the grounded phase is R (a random variable) then the p.d.f., R(t)m, is given by:
For our data the mean interval between flights was 12·18 min, thus λ was taken to be 1/(12·18 – 1) = 0·089.
We assume the mean wind speed, the vertical wind speed gradient, and the wind direction are constant throughout a dispersal episode during a single day.
The influence of assumptions on model output is assessed during sensitivity analysis (see Using simulation to obtain the mean number of flights and distance moved).
Developing models of the process
Because this process is random, it is not possible to predict the behaviour of a single spider through time. However, for large populations it is possible to predict the population properties and, in particular, the mean and standard deviation of the number of complete flights during a given time, and the mean and standard deviation of the horizontal distance moved during that time as described below.
Modelling the number of flights
The cycle of airborne/grounded/airborne is an alternating renewal process (Cox 1967) so asymptotic properties hold for the number of flights. In particular, if the mean length of the (grounded/airborne) cycle is denoted by µ then the mean number of flights at time t after the start of the process (N(t)) is given asymptotically by:
The mean length of the cycle is thus the sum of the mean time in the grounded phase and the mean time in the airborne phase.
The properties of the alternating renewal process are well established for the case where both the flight and the interflight times have an exponential distribution, in which case explicit equations can be obtained for the mean and variance of number of flights. However, when either the distribution of flight or interflight times are non-exponential, as in this case, it is not possible to obtain explicit equations for the mean or variance of the number of flights, which were therefore obtained by Monte Carlo simulation.
Modelling the horizontal distance moved
If the maximum height above the initial launch platform at height 1 m during a flight is h, then the duration of the flight, tf, is given by tf = h/E, where 1/E = 1/A + 1/D, A= rate of ascent and D = rate of descent. The total horizontal distance moved during a single flight, l, is given by:
Using the equation for g(h) given above, it can be shown that for h < 100:
and for 100 < h < 1000:
Thus, if the distributional properties of the initial height are known, the corresponding properties of the distance moved in a single flight can also be estimated. The total distance moved in a given time is the sum of the distances moved in individual flights, and is known as a cumulative process (Cox 1967). Again, it was not possible to obtain analytic solutions for the mean distance moved and its variance, so Monte Carlo simulation was used.
Using simulation to obtain the mean number of flights and distance moved
The alternating airborne/grounded process was simulated for 10 000 spiders in each run using Genstat 5 (Genstat Committee 1993). The maximum height achieved during each flight (and hence the distance moved during each flight) and the duration of each grounded phase between flights were simulated using appropriate random number generators.
The model output, in terms of mean single-flight distances, mean number of flights and mean total dispersal distances, was tested for sensitivity to variation in parameter values used as inputs. These were as follows.
The parameters defining wind speeds and the change of wind speed with height [ha = intercept, i.e. the wind speed at an altitude of 1 m, and hb = slope of the linear regression of wind speed against ln(height)]. These values were representative of those expected within the weather window of suitable ballooning conditions. They were derived from the highest, middle and lowest values observed in the field during sampling (Table 1).
The parameter defining ascent and descent rates (E). This was calculated for the range of ascent rates of 0·5, 1·0, 1·5, 2·0, 3·0, 4·0 and 5·0 m s−1, with a range of corresponding descent rates set at 20%, 40%, 80% and 100% of the ascent rate. This generated a range of values for E between 0·083 and 2·5. These values cover a realistic range of vertical wind speeds expected in rising thermals and surrounding subsiding air during suitable ballooning weather, e.g. under fair weather cumulus (Meteorological Office, personal communication). They also cover the values calculated by Humphrey (1987) on fluid mechanics principles for the terminal velocity of a spider-silk filament system in stable and unstable air.
The parameter b defining the vertical density profile of ballooning spiders derived from the field observations. This was set at a mean value of 0·92 while other parameters were varied, and sensitivity tested over the range 0·5–1·5 in increments of 0·1 while ha, hb and E were held at 2·93, 1·03 and 0·25, respectively.
The effect of available dispersal time on dispersal distances was also generated during each run of the model, with output in the form of ‘snapshots’ of the downwind distribution of ballooning spiders at half-hour intervals up to 8 h.
composition of the ballooning population
The species composition and sex ratio (m : f) of the ballooning adult spider population (sample size n = 798) during July was as follows, [percentage (sex ratio)]: Erigone atra (Blackwall) [30(0·9)]; Erigone dentipalpis (Wider) [5(0·85)]; Meioneta rurestris (C.L. Koch) [26(1·09)]; Lepthyphantes tenuis (Blackwall) [9(1·32)]; Bathyphantes gracilis (Blackwall) [9(1·23)]; Savignya frontata (Blackwall) [1(0)]; Milleriana inerrans (O.P.-Cambridge) [2(0·3)]; Oedothorax spp. [1(0·83)]; others [17(NA)].
atmospheric turbulence and numbers of airborne spiders
For each 10-min period during which spiders were sampled from the string, Richardson numbers (Ri) were calculated from the corresponding meteorological data. The original data are given in Thomas (1992). To correct for differences in the wind run during each sampling period, the numbers of spiders captured on the string were divided by the average wind speeds measured during each 10-min period. For each day and each sample set, the numbers of spiders collected in each 10-min period were regressed against the corresponding Ri values (Fig. 1a–j).
A combined, weighted regression of spider counts against Ri, with 1/(count + 0·5) as the weight to stabilize the variance, gave a highly significant regression (P < 0·001) that accounted for 59·5% of the variance. The slopes and intercepts varied with day because of overall differences in atmospheric conditions and changes in the size of the spider population on the ground or their propensity to disperse. However, the slope for set 2 was consistently 0·709 (± 0·258) higher than set 1 independent of day. This reflected the net difference between the numbers of spiders landing on the string and taking-off again before being sampled. Set 1 comprised spider arrivals minus departures over the previous 10 min. Set 2 comprised the sum of spider arrivals minus departures during each pass of the string (approximately 2 min) during the continuous 10-min patrol. Regression models and parameter estimates are given in Fig. 1.
In common with other materials subject to vertical distribution in the atmosphere by turbulent diffusion (Johnson 1957; Stull 1988), ln(aerial density) of spiders declined linearly with ln(height) (Fig. 2a). Differences in intercepts between days were detected. The intercepts represent the aerial density of spiders at an altitude of 1 m and are dependent on the size of the population of spiders attempting to become airborne.
The values for the slopes of the relationships between ln(aerial density) vs. ln(height) were plotted against the corresponding average turbulence (Ri) for each day (Fig. 2b). The consequent bivariate relationship between aerial density, height above the launch platform (h) and Ri:
Aerial Density = (Intercept) × (h + 1)−a×exp(exp(k×Ri))(eqn 1)
where a and k are constants and 0 < h, was fitted using least squares and a ‘log transform both sides’ approach. The estimated parameters are given in Fig. 2.
Aerial densities of spiders caught in a net attached to a balloon at higher altitudes (approximately 200–250 m) on a number of dates in July 1999 over Cardington, UK (52°06′N, 0°25′W) (J. Chapman, unpublished data) are also plotted on Fig. 2a. These data, with the exception of one outlier (13 July 1999), fall in the range predicted from the sticky traps suspended at different heights from the ground-based instrument masts.
time intervals between flights
Upon landing in the grass field, a spider would take a minimum of 1 min to orientate itself at the tip of a grass blade and adopt an attitude in preparation for a fresh flight. The mean interval between successive flights (Fig. 3) was 12·18 min (SE = 1·45 on 64 d.f.), and was estimated by assuming that the time interval between flights followed an exponential distribution with an unknown mean time, M. Under this assumption the number of spiders having an interflight time interval of between t – 1 and t min for t = 1, 2, …tmax has a multinomial distribution, with category probabilities being determined as the differences of the cumulative distribution function (c.d.f.). The mean time spent grounded between flights, M, can then be estimated using maximum likelihood.
The simulated distributions for flight times (Fig. 4a), the time spent between flights (Fig. 4b) and the distance moved in a single complete flight under three different wind speed scenarios (Fig. 4c) gave a sequence of times of landing and starting the next flight for each spider. This was used to obtain the cumulative number of flights and the cumulative distance moved. The mean and variance were then obtained. This gave the time–course for the means and variances and allowed snapshots of the distributions for various dispersal times.
Almost 60% of spiders were airborne for less than 2 min during single flights, with about 16% achieving flight times of 8 min or longer (Fig. 4a). The mean duration of a flight was 3·3 min. Similarly, most spiders spent less than 15 min between flights (Fig. 4b), with less than 1% remaining grounded for more than 1 h between flights. The flight times (Fig. 4a) translated into various flight distances under different wind speed scenarios (Fig. 4c). With ha at 1·75, 2·2 and 2·93 m s−1, approximately 69%, 64% and 60%, respectively, of ballooning spiders travelled less than 600 m in a single flight. The maximum single flight distances achieved by only 0·4% of ballooning spiders were approximately 3·8, 4·9 and 6·5 km, respectively.
The model generates mean numbers of flights that increase at a rate of approximately five flights per hour (Fig. 5). Because suitable wind conditions for aerial dispersal persist for various amounts of time on a given day, the key results are presented as snapshots in Fig. 6. Even with only 30 min of suitable ballooning conditions, spiders can potentially make between 1 and 12 flights (mean 5·5, SD 1·5). During a day with wind speeds below 3 m s−1 for 8 h, ballooning spiders will have the potential to make between 18 and 49 flights (mean 33·9, SD 4·3). Figure 6 gives an example of how these flights translate into dispersal distances under one wind speed scenario (ha 2·2, hb 0·8) and two ascent/descent rates of E = 0·167 (Fig. 6a) and E = 2·5 (Fig. 6b), which are at the lower and upper end, respectively, of the range tested. Snapshot views of the distributions of total dispersal distances are given for a range of dispersal times up to 8 h. As ballooning time increases, the standard deviations of the mean dispersal distances increase. Thus, spiders dispersing from a point source (a single field) are spread over an increasingly wide area (number of fields) at increasing distances downwind.
As passive wind-borne organisms, it is self-evident that the dispersal distance of spiders is dependent on wind speed. The influence of wind speed parameters on mean distance per flight is shown in Fig. 7a for three wind speed scenarios, combined with the sensitivity to the ascent/descent rate parameter E.
Clearly, as wind speed (ha) increases, the mean distance per flight increases (Fig. 7a). However, the ascent/descent rate parameter (E) influences mean flight distance most markedly at lower values. If E is small, small increases in E result in relatively large increases (a few 10s of metres) in mean flight distance, and this effect is larger (several 10s of metres) at higher wind speeds. At values of E greater than approximately 1, as E increases so the mean flight distances increase by only a few metres.
The way in which these effects scale up as flights cumulate is shown in Fig. 7b, which gives mean dispersal distances after 6 h (± SD). For clarity only the highest and lowest wind speed scenarios are shown. Here, as ha increases the standard deviations of the dispersal distances also increase, indicating the spread of the dispersing population over larger areas further downwind. Similarly, standard deviations increase as E increases within each wind speed scenario. Thus, the distributions of dispersal distances generated by the lowest and highest wind speed scenarios overlap. However, throughout the range of varying E and ha, standard deviations remain approximately the same at 26% to 28% of the mean dispersal distance.
Within the realistic range of horizontal and vertical movements experienced by ballooning spiders and represented by E and ha, neither parameter has a large effect on dispersal distance. An approximate doubling of wind speed ha results in slightly less than a doubling of the mean dispersal distance. Similarly, increasing E by an order of magnitude from 0·25 to 2·5, equivalent to increasing the ascent and descent rates from 0·5 m s−1 to 5 m s−1, increases mean dispersal distance by only approximately 25%.
The effect of variation in the vertical density profile of airborne spiders (b) on mean flight time is given in Fig. 8a and the consequent effects on the mean number of flights possible (± SD) in 6 h in Fig. 8b. Similarly, mean flight distance as a function of b is shown in Fig. 9a and the consequent mean dispersal distance (± SD) as a function of b in Fig. 9b. Because b is negative by definition (density decreases with height), as the absolute value of the slope b increases a relatively larger proportion of ballooning spiders occurs at lower altitudes. Thus, as b increases by a factor of three so mean flight time decreases by a factor of seven (Fig. 8a). Shorter flight times enable a larger number of flights to be made; thus as b increases from 0·5 to 1·5 so the mean number of flights possible in a 6-h period increases from approximately 21 ± 3 to 29 ± 5 (Fig. 8b). Similarly, as flight times decrease with a threefold increase in the value of b, so mean flight distances (Fig. 9a) and the cumulative dispersal distance decrease by a factor of five (Fig. 9b).
The fieldwork described in the present paper corroborates and extends a previous study of spider ballooning. Vugts & van Wingerden (1976) correlated the numbers of linyphiid spiders caught on wires just above the vegetation height with a form of the Richardson number, and derived an ‘aeronautic index’ as a function of wind speed and temperature difference with height. This can be used to predict days when ballooning is likely to occur, the principal condition being wind speeds below 3 m s−1 when measured 2 m above the ground.
Our data corroborate the finding that, on a time-scale of 10s of minutes, the numbers of spiders becoming airborne is correlated with the local air turbulence. We further show a relationship between the rate of change of the aerial density of spiders with height and mean daily atmospheric turbulence. This enables the aerial density of spiders at any height to be estimated from the density at one height and some simple meteorological data. Thus, turbulence influences both the successful launching of spiders attempting to become airborne and their subsequent diffusion to higher altitudes in the boundary layer.
Few studies have reported the vertical density profiles of airborne spiders, but where this has been studied the same linear relationships of ln(density) with ln(height) have generally been found (Glick 1939). Thorbek, Topping & Sunderland (2002) found no seasonal difference in the proportion of spiders caught at 1·4 m and 12·2 m. However, Bishop (1990) found vertical distributions of spiders to differ in spring and autumn. Seasonal differences might be expected because variation in solar radiation affects surface layer turbulence. Further work to examine seasonal variation in the vertical distribution of airborne spiders is justified, as populations of linyphiid spiders can be large in the autumn, when intensive ballooning is often observed (Duffey 1956).
There were several assumptions made in the construction of the model, of which the most important was the simplification of dispersal to a single period of ascent followed by a single period of descent. This is reasonable, however, given that the actual trajectories of passive, airborne particles can be extremely complex (Burrows 1975) and to model them explicitly would mean modelling turbulence itself (an intractable task) rather than its gross effects.
Other assumptions include the maximum altitude attainable by ballooning spiders, constant wind direction and an invariant wind speed gradient during a single day. These assumptions are realistic given the narrow weather window of low wind speeds within which spiders can balloon.
The constraint of ballooning only in low wind speeds renders the model output, in terms of total dispersal distances, relatively insensitive to absolute wind speed and rates of ascent and descent. The vertical density profile of spiders determines the proportion of the ballooning population that is exposed to the higher wind speeds found at altitude, which renders the model moderately sensitive to the parameter b. However, the largest effects on dispersal distances are due to the duration of ballooning, which may be constrained by the duration of suitable weather conditions, the number of flights or the time spent ballooning. Further work is necessary to determine whether there are behavioural constraints on these two latter parameters. As the present model is configured for a generic linyphiid spider, further research is also required to quantify the propensity of individual species to balloon. Such behaviour may form a component of a specific dispersal strategy appropriate to the risks and resources associated with certain habitats or landscapes.
In spite of certain assumptions, we have shown that the dispersal distances of ballooning spiders are considerable. With some scenarios the maximum dispersal distances after 8 h approach 90 km. Many of the scenarios tested show that during a 6-h dispersal episode mean downwind dispersal distances are in the order of 30 km. In the context of the farmland landscape, this scale of dispersal allows a considerable redistribution of spiders from one field over a range of new habitats or field types.
Darwin (1839) reported large numbers of small, gossamer spiders landing on board The Beagle some 60 miles off-shore in the mouth of the River Plate. Although Darwin observed that the spiders could ‘run easily over water’, cold seas are considered barriers to dispersal because of a lack of thermal updraft and the predominance of sea breezes over land breezes in coastal areas. However, a large body of warm water may generate a widespread area of updraft. If the vertical wind speed in such an air mass equalled the terminal velocity of the spider and silk filament, spiders could sustain flight at a constant height, drifting with the horizontal component of the wind, as observed by Darwin.
At this stage we have only modelled spider dispersal distances achievable during a single day. Further development of the model presented here will include meteorological data of wind speeds and directions on different days in the year to estimate the two-dimensional area over which linyphiid spiders can range. Further parameterization with species-specific data on other aspects of dispersal behaviour and population dynamics, together with landscape structure, will enable an evaluation of the different levels of risk individual species experience in agricultural landscapes (Thomas et al. 2003). Such a model will have important applications in the management of sustainable agriculture and the survival, persistence and diversity of this important guild of natural predators.
C. F. G. Thomas is funded by BBSRC grant 321/D14032. Fieldwork (University of Southampton) was funded under NERC(JAEP) grant GST/02/478. IACR-Long Ashton receives grant-aided support from BBSRC, UK. Thanks to: Alan Ibbetson and Neil Wells for advice on design and analysis of meteorological aspects; Jason Chapman for unpublished data of spiders airborne over Cardington; Ben Gibbons, of Leckford Estates, Hampshire, for permission to carry out the fieldwork; and several anonymous referees for comments on earlier drafts.
estimating the initial vertical distribution of the spiders from the observed vertical distribution
The p.d.f. of the observed vertical distribution (averaged over time), H, is denoted by g(h). This is estimated using the proportion (adjusted for wind speed) caught at each height. The p.d.f. of the maximum height during a flight, HI, is denoted by f(h), and is required for the model.
A given spider is assumed to ascend to its maximum height, HI, at a rate A m s−1 and to descend at a constant rate D m s−1, so that a spider whose maximum height is HI m will have a flight of total length T given by:
The maximum possible height is denoted by c m; in the models presented here c = 1000 m. The minimum height is 1 m.
Modelling the distribution of the maximum height given the observed vertical distribution
The c.d.f. of the observed height of a spider, S(h), is given by:
The spider is equally likely to be caught at any height below its initial height, so that H | HI has the U(1, HI) distribution. Then: