Modelling spread of British wind-dispersed plants under future wind speeds in a changing climate

Authors


Correspondence author. E-mail: jmbul@ceh.ac.uk

Summary

1. Climate change impacts on habitat suitability and demography are often studied, but direct effects on plant dispersal are rarely considered. To address this we analysed climate model projections of future wind speeds and modelled their possible impacts on dispersal and spread of wind-dispersed plants.

2. Projections for 17 Global Climate Models and three emission scenarios suggested great uncertainty about wind speeds in southern England by the period 2070–99. Projections ranged from −90% to +100% change in the mean wind speed, although the average projection was for large falls in both summer and winter wind speeds.

3. Using a novel method for converting projected changes in mean wind speed to new seasonal wind speed distributions, we parameterized a mechanistic model of seed dispersal by wind using baseline and changes in mean wind speed from −80% to +80%.

4. The mechanistic seed dispersal model was combined with demographic data in an analytical model of plant spread. This was carried out for three British native and three non-native species, which represented a range of life-forms.

5. Dispersal kernels and population spread rates were affected disproportionately by changes in wind speed, demonstrating nonlinear propagation of uncertainty in wind speed projections through to modelled plant spread rates.

6. Sensitivity analyses showed differences among the plant species in which demographic transitions were most important in determining spread rates. By contrast, sensitivity of spread rates to dispersal parameters showed great consistency among species, with seed release height being more important than seed terminal velocity.

7.Synthesis. Plant populations will need to shift their geographic ranges to keep pace with climate change-driven habitat loss. This study shows that climate change may affect that ability by decreasing the dispersal distances of wind-dispersed plants and thus their potential spread rates. However, the modelling approach presented here illustrates that uncertainty in climate models leads to an even greater uncertainty about how dispersal and spread will change in future climates. Caution should therefore be exercised in making predictions as to how fast plant species may spread in response to climate change.

Introduction

Anthropogenic climate change is likely to necessitate range shifting by many plant species, as local climatic conditions become unsuitable (Thuiller 2004; Dawson et al. 2011). Recent estimates, using climate model projections, of the velocity of climate change across the world’s surface in the 21st century suggest minimum rates of spread needed for species to track climate change, with a global average of 0.42 km year−1 (Loarie et al. 2009). There is, therefore, great interest in modelling the rate at which different plant species may be able to spread, to determine whether they might keep pace with climate change (Thuiller et al. 2008). Such information is critical to understanding the broader aspects of species’ resilience to a changing climate (Dawson et al. 2011). A range of modelling approaches is used, which incorporate a number of processes depending on model complexity (Thuiller et al. 2008), but all include dispersal by necessity. Dispersal is generally characterized in these models using knowledge of current or past dispersal patterns (e.g. Clark et al. 2003; Higgins et al. 2003).

The assumption of unchanging dispersal is undermined by the increasing recognition that climate change may affect not only the demography and habitat suitability of plants, but also the dispersal process itself. This finding has been made possible by the development of mechanistic models that use plant and wind characteristics to derive dispersal kernels for seeds dispersed by wind (see Nathan et al. 2011b). Kuparinen et al. (2009) demonstrated that warmer conditions in a boreal forest led to greater turbulence in the airflow and, using this finding in a mechanistic wind dispersal model, predicted that dispersal distances could increase under climate warming. Nathan et al. (2011a) showed that modelled spread of North American trees was sensitive to a number of demographic and dispersal parameters, including wind speed. A climate model projected a slight decline, on average, in future wind speeds for that region, which could therefore lead to decreases in the rate of tree spread. An experimental manipulation of temperature and rainfall suggested that plant height of an invasive thistle might increase in future climates (Zhang, Jongejans & Shea 2011). By increasing seed release height, this process increased modelled dispersal distances, and the consequent impact on modelled spread rates was greater than any climatic effects on demographic variables.

These studies suggest that climate change could affect dispersal directly, at least for wind-dispersed seeds. This raises the question whether it will be possible to project the effects of future climates on seed dispersal by wind and thus on spread rates. A major uncertainty for such an attempt will be the projections by climate models of future wind speeds themselves.

Wind speed distributions simulated by global climate models (GCMs) have been analysed rarely, but there is evidence that GCMs are not yet able to replicate the magnitude and spatial variability of observed wind speeds (e.g. in Scandinavia; Pryor, Schoof & Barthelmie 2006), possibly because of the large variation in wind speed at spatial scales much lower than that of GCMs (typically using grid squares of several degrees). However, dynamical down-scaling using regional climate models retains these biases (Pryor, Barthelmie & Kjellstrom 2005a), which seem to be influenced by the simulated location of the pressure gradient. There is also low confidence in the signal of future changes in wind speed, which is linked to the uncertainty about the changes in large-scale atmospheric circulation (IPCC, 2007). Apart from these individual inaccuracies, projected changes in wind speeds vary among both the various available GCMs, even when empirically down-scaled (Pryor, Schoof & Barthelmie 2005b; Najac, Boe & Terray 2009; Kjellström et al. 2011), and the modelled outcomes of different emission scenarios (Pryor & Schoof 2010).

Given such uncertainty, although it is tempting to use plant dispersal and population models to make predictions about future spread rates, it is more useful to use these models to begin to understand the ramifications of this uncertainty. For example, although climate models vary, they show a trend in projecting future decreases in average wind speed for Europe (Kjellström et al. 2011). Therefore, we can ask: how sensitive are projections of dispersal and spread to the exact size of the wind speed change?

In exploring uncertainties in plant dispersal and spread, it is useful to begin with relatively simple analytical models. While they are in some ways unrealistic, the relatively low number of parameters and mathematical precision of such models facilitates a good and general understanding of the impacts of variation in certain parameters. Complexity can be added where necessary, as in this study in which we model seed dispersal integrated over realistic distributions of wind speeds.

We have three aims in this study: (i) we examine 17 GCMs across three emission scenarios and demonstrate great variation in projected future wind speeds of Great Britain (GB). (ii) We then combine well-tested analytical models of seed dispersal by wind and population spread to evaluate the consequences of this variation in climate projections for the dispersal and spread of six wind-dispersed, climate-limited species. These species represent a range of life-forms and comprise rare British natives and invasive non-natives. (iii) We analyse sensitivity of these spread rates to changes in a range of demographic and dispersal variables and examine to what extent these sensitivities are altered by changes in wind speed.

Materials and methods

Modelled Species

Six species were selected for this study, using the following criteria. They all have a southerly distribution in GB (See Appendix S1 in Supporting Information), which has been ascribed to their climatic tolerances (Preston, Pearman & Dines 2002). They are all primarily wind-dispersed. Finally, each has a published demographic matrix or data allowing construction of a matrix. We used the most appropriate available data to give a representation of the demography of each species at low density (Appendix S2), although the lack of data often meant that we used sources from outside GB or (in one case) for a closely related species. To encompass a variety of plant types, we selected three native and three non-native species, which comprised annual and perennial herbs, a shrub and a tree (Table 1). The species also differ in their dispersal periods, being dispersed in summer or early winter, which allowed a consideration of wind speed projections for different seasons.

Table 1.   Descriptions of the modelled species, included values of key parameters and model outputs
SpeciesHimantoglossum hircinumCirsium acauleErica ciliarisConyza canadensisLactuca serriolaAilanthus altissima
  1. 1Calculated using the matrix models in this study.

  2. 2From the LEDA traitbase http://www.leda-traitbase.org/LEDAportal.

  3. 3There are no data for this species, so we used the value for Dactylorhiza species, which has similarly sized seeds (Arditti & Ghani 2000).

  4. 4From Tremlová & Mūnzbergová (2007).

  5. 5J. M. Bullock & D. A. P Hooftman, pers. obs.

  6. 6Calculated using the WALD model for the baseline wind speed distribution.

  7. 7Derived for baseline wind speeds.

Native status in Great Britain (GB)NativeNativeNativeNon-nativeNon-nativeNon-native
Life-formPerennial herbPerennial herbDwarf shrubAnnualAnnualTree
Habitat in GBCalcareous grasslandCalcareous grasslandLowland heathOpen and cultivated landOpen and cultivated landUrban (planted)
Population growth rate (year−1) λ11.321.052.157.087.741.92
Dispersal period in GBAugustAugustNovember–DecemberJuly–AugustAugustOctober–December
Seed terminal velocity (m s−1)2F0.2830.391.090.260.850.56
Seed release height (m) H0.4520.140.650.5520.6221.252
Vegetation height (m) h0.0550.0540.150.0520.0550.15
Median dispersal distance (m)61.530.100.682.080.9899.2
Median wave speed (m year−1)75.340.032.1856.47.51529

Himantoglossum hircinum is a long-lived perennial, native orchid. Its distribution is restricted to the south of England (Carey & Farrell 2002). As with many other orchids (Arditti & Ghani 2000), its tiny seeds indicate that wind is the primary dispersal vector (Carey 1998). We used a published five-stage demographic matrix (Appendix S2) for a population near the range limit of this species in Germany (Pfeifer et al. 2006), rather than those for an English population (Carey 1998), because the former explicitly included the dispersing seed stage.

Cirsium acaule is a native perennial herb that reproduces by seed and clonally (we consider dispersal only by seed). It is limited to southern and central England, which is explained by reproductive failure at the northern range edge (Jump & Woodward 2003). It has plumed seeds. No demographic data are available for GB, so we constructed a four-stage matrix using data for the Czech Republic, in the centre of the species’ range (Mūnzbergová 2005).

Erica ciliaris is a rare, native, dwarf shrub with an extremely restricted distribution in the far south of England, which may be linked to specific climatic tolerances (Rose, Bannister & Chapman 1996). The seeds are simple and small and are similar to those of the related and co-habiting Erica cinerea and Calluna vulgaris, which are dispersed primarily by wind (Bullock & Clarke 2000). We used an 11-stage demographic matrix for the related E. cinerea– which has a very similar long-lived, perennial life cycle as E. ciliaris– measured from populations growing in the same heaths as E. ciliaris in Dorset, England (Soons & Bullock 2008).

Conyza canadensis is a non-native annual that has been established in GB since the 17th century, but has been expanding its range from the south of England over the last century (Preston, Pearman & Dines 2002). It has plumed seeds. No complete demography has been published, so we constructed a two-stage matrix using data sources from Europe and the USA.

Lactuca serriola is another long-established non-native annual with plumed seeds, which expanded northwards through the 20th century (Preston, Pearman & Dines 2002). Like C. canadensis, it remains absent from Scotland and the far north of England. A two-stage matrix was constructed from data gathered in the south-eastern Netherlands (D. A. P. Hooftman, unpubl. data).

Ailanthus altissima is a non-native tree that has been planted in urban areas since the 18th century. It is expanding its range in southern and central Europe, probably as a result of climate change (Kowarik & Säumel 2007). Ailanthus altissima is not yet regarded as a problem species in GB, where it is constrained mostly to south-eastern England (Preston, Pearman & Dines 2002). It reproduces by seed and clonally (we consider dispersal only by seed), and the seeds are winged (Kowarik & Säumel 2007). The species is dioecious, but we took account of this in the modelling only by restricting half of the trees to produce seed. No complete demography has been published, so we constructed a six-stage matrix using data sources from Europe and the USA.

Baseline Wind Speeds

Most climate change impact studies compare projections for a future time slice against a baseline, and change factors (that express the difference in the climatology of the future and the baseline) are usually calculated based on the period 1961–90 (e.g. Murphy et al. 2009). Baseline distributions of observed wind speeds were obtained using hourly mean wind speed data from 12 weather stations across England (Appendix S3) that are part of the Meteorological Office Integrated Data Archive System. These stations covered, roughly, the range of the most widely distributed of our six species, and were chosen because they had the most continuous wind data of the available stations. Even so, data were sparse for the early part of the baseline period, so we used data for the period 1971–90.

Wind speed data were extracted to represent the two dispersal seasons exhibited by our six species: the summer months of June, July, August and September for H. hircinum, C. acaule, C. canadensis and L. serriola; and the autumn-winter (which we will call ‘winter’ subsequently) months of October, November and December for E. ciliaris and A. altissima. The complete sets of summer and winter hourly mean wind speeds (u) over 20 years were summarized by fitting a two-parameter Weibull distribution (Fig. 1), for which the probability density function (PDF) is

image(eqn 1)

where β is the shape parameter and η is the scale parameter. This distribution is commonly used to describe wind speed data (e.g. Kiss & Jánosi 2008; Nathan et al. 2011a). For simplicity, we ignored wind direction in these data and assumed that baseline and future winds, and thus dispersal, were isotropic.

Figure 1.

 Weibull probability density functions describing observed (a) summer (June–September) and (b) winter (October–December) wind speed distributions for the period 1971–90 for 12 weather stations concentrated in southern England. Also plotted are these Weibull distributions modified (using the scale parameter) to match changes in the mean wind speed of −80% and +80%.

Future Wind Speeds

The World Climate Research Programme’s (WCRP) Coupled Model Intercomparison Project phase 3 (CMIP3) multi-model data set provides output data from a wide range of climate models (Covey et al. 2003; Meehl et al. 2007). Projections of percentage changes in winter (October, November and December) and summer (June, July, August and September) mean wind speeds for the years 2070–99 compared to the baseline were collated from 17 GCMs run for up to three emission scenarios (not all scenarios were available for all GCMs): SRES A1B ‘medium emissions’, SRES A2 ‘low-medium emissions’, and SRES B1 ‘low emissions’ (IPCC 2000). Details of the models can be found in Appendix S4. The spatial resolutions of the GCMs range from c. 20 000 to 200 000 km2, which provides the general projections for a large geographic area required for this study. Change factors were obtained for the grid square encompassing the Heathrow weather station near London (51°28′39″ N, 0°27′41″ W) to represent south-eastern England. The change factors were calculated from monthly time series of mean wind speeds representative of the control run (baseline) and runs assuming different emission scenarios (Fig. 2). These change factors were expressed as percentages to allow conversion of the observed baseline wind speeds to projected future wind speeds. All time series were obtained from the Climate and Environmental Retrieving and Archiving portal (http://cera-www.dkrz.de/CERA/index.html) and from the Program for Climate Model Diagnosis and Intercomparison website (http://www-pcmdi.llnl.gov).

Figure 2.

 Projected percentage changes in summer and winter mean wind speeds in south-eastern England for 2070–99. The projection are taken from 17 Global Climate Models run under three emission scenarios: SRES A1B ‘medium emissions’, SRES A2 ‘low-medium emissions’, and SRES B1 ‘low emissions’.

Modelling Wind Dispersal

The range of change factors suggested by the different models was used to inform an analysis of the impact of wind speed change on plant dispersal kernels. The projected percentage changes in the mean wind speed, from −90% to +100%, were used to inform our modification of the observed baseline wind speed distributions. The mean of a Weibull distribution is

image(eqn 2)

where Γ is the gamma function. We assumed that climate change effects on ū would affect η only, and not β. That is, the range of the wind speed distribution would be affected much more than the shape. Some support for this simplifying assumption is given by Kiss & Jánosi (2008), who showed that variation in the mean wind speed across Europe was strongly matched by variation in η, but not by the pattern of β. Therefore, we derived a set of summer and winter wind speed distributions for changes in ū of −80% to +80% at steps of 10%, by using eqn 2 to calculate a new value of η, while keeping β fixed.

We used the WALD model (Katul et al. 2005) to derive dispersal kernels for each species using the wind speed data. The WALD model is mechanistic, in the sense that it uses measured wind, plant and vegetation variables to derive a PDF of dispersal distances (i.e. a dispersal kernel), which is independent of measured dispersal data. The WALD model is based on theory describing three-dimensional movement in a turbulent airflow. Nathan et al. (2011b) give a detailed background to mechanistic wind dispersal models, and the WALD model in particular; while Bullock et al. (2006) and Jongejans, Skarpaas & Shea (2008b) provide more general overviews of the measurement and modelling of plant dispersal. Simply, the basis of the WALD model is that each seed is released from the plant at a certain height and in still air would fall to the ground at a rate determined by its terminal velocity. In a laminar (i.e. non-turbulent) airflow, the seed falls to the ground while travelling away from the plant at a rate determined by the horizontal wind speed. Once the seed enters the vegetation canopy, its movement is severely curtailed, especially in the situations used for this paper, in which the vegetation is short and dense. Wind turbulence during the seed’s trajectory causes it to deviate from the smooth trajectory, which may increase or decrease the distance dispersed. Vegetation height and structure (‘roughness’) are important in determining the degree of turbulence. The WALD model has been generally shown to match observed dispersal kernels (Katul et al. 2005; Skarpaas & Shea 2007), although Stephenson et al. (2007) showed it had a tendency to over-predict dispersal distances compared to their observations.

With the notation slightly modified from Katul et al. (2005) for clarity, the PDF of the WALD model is

image(eqn 3)

where r is distance, μ′(u) is the location parameter and λ′(u) is the scale parameter. The latter two parameters are dependent on wind speed, u, and related to measurable parameters as follows:

image(eqn 4)
image(eqn 5)

.

The required parameters are therefore: H, seed release height; F, seed terminal velocity; U(u), mean wind speed at the height of seed release; and σ(u), a turbulent flow parameter. Wind speeds were measured at 10 m height, and had to be corrected to give the wind speed at the height of seed release. We used the procedure described by Skarpaas & Shea (2007), who derived U(u) by integrating wind speed over a logarithmic wind profile,

image(eqn 6)

where U*(u) is the friction velocity, K is the von Karman constant (0.4), z is the height above ground and d and z0 are surface roughness parameters. The lower limit of the integral is l, where z0, which corrects a typographical error (which set 0 as the lower limit) in eqn A1 of Skarpaas & Shea (2007; O. Skarpaas, pers. comm.). Following Skarpaas & Shea (2007), and so assuming the species are dispersing over short herbaceous vegetation, the surface roughness parameters are related to h by d ≈ 0.7h and z0 ≈ 0.1h; U*(u) and σ(u) were calculated from their eqns A2 and A4, respectively, which are both functions of the measured wind speed and vegetation height. The values of H, F, and h for each species were taken from the LEDA data base of life-history trait values for plants of the north-western Europe (http://www.leda-traitbase.org/LEDAportal), other published sources or from our own observations (Table 1). We assumed each species had a single seed release height. This is a fair assumption as all the species, with one exception, disperse seed at a single demographic stage (Appendix S2). Erica ciliaris disperses seeds from multiple stages. It is possible to assign different dispersal kernels to each demographic stage (e.g. Travis et al. 2011), but we considered this an unnecessary complication here as E. ciliaris adults increase in size more by lateral expansion than by height increments (J. Bullock, pers. obs.)

The WALD model derives a dispersal kernel using a single wind speed, but our data comprise distributions of hourly mean wind speeds over whole summer or winter seasons and over 20 years. Nathan et al. (2011a) approached this issue by dividing the wind speed distribution into equal probability bins, calculating a WALD kernel and the resulting wavespeed for each wind speed bin mid-point, and then taking the mean of these wavespeeds. We prefer to derive a seasonal dispersal kernel by integrating the WALD model over a distribution of wind speeds (see also Skarpaas & Shea 2007), whereby the PDF is given by

image(eqn 7)

where p1(u) is the Weibull PDF given by eqn 1, p2(r,u) is the PDF of the WALD model given by eqn 3, and C is the normalization constant given by

image(eqn 8)

We limit the integral in eqn 7 to a maximum wind speed of 25 m s−1, matching the maxima of the observed wind speeds, to avoid unrealistically high wind speeds and, thus, dispersal distances.

Modelling Population Spread

The analytical wavespeed model of Neubert & Caswell (2000) combines a demographic matrix model with integrodifference equations describing dispersal. This represents a population spreading in one dimension from a starting location in discrete time steps and so is a general, if simplified, approach to modelling the expansion of a population into unoccupied habitat (e.g. Skarpaas & Shea 2007; Bullock, Pywell & Coulson-Phillips 2008; Soons & Bullock 2008). This approach allows stage-structured demography and realistically complex dispersal kernels, but includes simplifying assumptions such as no temporal variation or Allee effects. In particular, the environment is treated as spatially homogeneous. Lewis et al. (2006) present an accessible introduction to wavespeed modelling. Using the Neubert & Caswell (2000) approach population density at location x at time t + 1 is described by

image(eqn 9)

where inline image is the Hadamard product operator. Bn is a stage-structured population projection matrix which describes density-dependent population growth at location y. K(− y) is a matrix of dispersal kernels which describe the set of probabilities of the relocation from y to x of individuals undergoing each demographic transition, with the assumption that dispersal from y to x depends only on the relative locations of the two points. Thus, over a time step the population grows at each location y and individuals are dispersed. The population at location x is given by integrating this process over all locations y.

Calculation of the wavespeed requires a population projection matrix for demography at low density (i.e. at the forefront of the spreading population; A = B0), and the species’ matrices described earlier were constructed or chosen following this criterion. Also required is a matrix M(s), which describes the dispersal kernel for each demographic transition in terms of a moment generating function (MGF). The WALD model has an analytical MGF, but the PDFs resulting from integration over the wind speed distributions necessitated an empirical estimation of the MGF, using the method described by Clark, Horvath & Lewis (2001; see also Lewis et al. 2006). Taking an integrated WALD kernel we sampled 106 (N) dispersal distances r1,..., rN and used

image(eqn 10)

(where I0 is the modified Bessel function of the first kind) to calculate each entry for the matrix M(s) in which seed dispersal takes place (for non-dispersing transitions mij(s) = 1). The parameter s describes the shape of the wave (Neubert & Caswell 2000). This method marginalizes the two-dimensional dispersal kernel into the single dimension necessary for the wavespeed model.

Under this model a population forms a wave of constant shape that advances at constant speed c* (the wavespeed), which can be derived analytically by

image(eqn 11)

where ρ is the dominant eigenvalue of inline image. This method was used to calculate c* for each species at the baseline wind speed distribution, and for changes in mean wind speed from −80% to +80% at steps of 10%. The calculated wavespeed is sensitive to long distance dispersal events (Caswell, Lensink & Neubert 2003; Bullock, Pywell & Coulson-Phillips 2008), and so to the exact set of sampled dispersal distances r1,..., rN used for eqn 10. Therefore, we re-sampled the integrated WALD PDF 25 times for each wind speed scenario, and present the median value of c*. More detail about the mathematical methods is given in Appendix S5.

Sensitivity Analyses

Demographic elasticities (a measure of how different life-history stage transitions affect the wavespeed dynamics) can be calculated for this matrix-based wavespeed model, in the same way as they are performed for standard population growth matrix models (Neubert & Caswell 2000). The elasticity of c* to each transition aij of the demographic matrix A was calculated as

image(eqn 12)

where wij (which Neubert & Caswell 2000 call hij) is the respective matrix element of inline image, where s* is the value of s which gives the minimal value c* in eqn 11.

Because we used the integrated WALD model, dispersal was not characterized by a parametric function, so dispersal elasticities could not be determined using the approach just described (Caswell, Lensink & Neubert 2003). Instead, we determined the sensitivity of c* to the key parameters H, F and h using the method of Hooftman et al. (2008). The proportional changes of c* in response to a 50% change in each parameter value were calculated and were normalized across the three parameters to derive ‘relative sensitivities’, which are comparable to elasticities. A 50% change was chosen as a large, but not unrealistic change for each parameter (Hooftman et al. 2008). Note that seed release height (H) remained above the vegetation for all species even when the vegetation height (h) was increased by 50%. Dispersal would be severely curtailed if h > H.

Results

Baseline and Future Wind Speed Distributions

The observed baseline wind speeds were slightly higher for the winter (median = 4.1 m s−1; max = 23.7 m s−1) than for the summer (median = 3.6 m s−1; max = 20.1 m s−1) seasons (Fig. 1). The fitted Weibull distribution provided a good description of both the summer (r2 > 0.99; n = 698 359) and winter (r2 > 0.99; n = 527 509) wind speed data sets. The GCMs gave a wide variety of projections for change in mean wind speed in both seasons, which ranged from about −90% to +100% (Fig. 2). However, the median change was strongly negative in both the summer (−49%) and winter (−81%) periods. There were no systematic differences among the three emission scenarios in the percentage change projected for the summer (Kruskal–Wallis statistic = 2.4, = 0.3) or winter (1.8, = 0.4).

Modelling Dispersal and Wavespeeds under Different Wind Speed Scenarios

Under baseline wind speed distributions, the six species showed very different WALD-generated dispersal kernels (Fig. 3). Median dispersal distances ranged from 0.10 m for the short-stemmed C. acaule to over 99 m for the tree A. altissima (Table 1). The population growth rates λ derived from the demographic matrices also differed among species, from 1.05 for C. acaule to 7.74 for L. serriola (note λ is the population growth rate at low density). The wavespeeds derived from these dispersal and demographic matrices followed the pattern of species differences in the dispersal median (Spearman rank correlation, = 0.94, < 0.01) rather than those in λ (= 0.49, > 0.05), and varied over four orders of magnitude between 0.03 m year−1 (C. acaule) and 529 m year−1 (A. altissima).

Figure 3.

 Dispersal kernels derived for six species by integrating the WALD model over wind speed distributions for baseline conditions and for changes in the mean wind speed of −80% and +80%. The scales differ among the graphs. The insets show the tails of each kernel.

Although the species showed great differences in demography, dispersal and projected wavespeeds under baseline conditions, their modelled responses to changes in mean wind speeds were similar. Following the changes in wind speed distributions (Fig. 1), the modelled dispersal kernels for each species became more peaked with a lower mode and thinner tail when the mean wind speed was decreased and became flatter with a higher mode and fatter tail with an increase in the mean wind speed (Fig. 3). For example, the 95th percentile dispersal distances under baseline conditions were 0.45 m for C. acaule and 518 m for A. altissima, but these were altered to 0.06 and 85 m, respectively, with a −80% change in mean wind speed, and to 0.81 and 1013 m with a +80% change.

These effects of wind speed changes on the dispersal kernels were translated into nonlinear effects on wavespeeds, and the proportional effects were identical among the six species (Fig. 4). The percentage increase or decrease in the wavespeed was always greater than the corresponding percentage change in the mean wind speed. This deviation from a 1 : 1 relationship became greater for larger changes in the mean wind speed, except for the largest decreases in wind speed where the relationship would converge at a −100% change in wind speed. The larger differences among species’ responses at higher wind speeds was due to a greater probability of individual long distance dispersal events, which increased the variation among projected wavespeeds.

Figure 4.

 The percentage change in projected wavespeeds of six species from those under baseline wind conditions, plotted against the percentage change in mean wind speed used to derive the dispersal kernel. The value plotted is the median of 25 wavespeeds calculated from separate realizations of the dispersal kernel.

Sensitivity Analyses

Sensitivities of the modelled wavespeed were largely consistent over the range of wind speed change scenarios. For ease of interpretation, demographic transitions were grouped into stasis (staying in the same, or regressing to a smaller, stage), growth (transition to a larger stage) and fecundity, following Silvertown et al. (1993). The elasticities of these three classes varied greatly among species, but within each species hardly varied from the −80% to the +80% wind speed scenarios (Fig. 5). Very small changes were apparent in some species under the biggest decreases in wind speed, but these did not indicate a consistent pattern.

Figure 5.

 The elasticities of wavespeeds to demographic transitions for six species, calculated for models with altered dispersal kernels as a result of changes to the baseline wind speed from −80% to +80%. The transitions are grouped into stasis, growth and fecundity.

The relative sensitivities of the wavespeed to the WALD model parameters showed strong consistency among species as well as among the wind speed scenarios (Fig. 6). In almost every case, a wavespeed was by far the most sensitive to seed release height H. Seed terminal velocity F had moderate (e.g. A. altissima) to minimal (e.g. C. acaule) effects on the wavespeed, and vegetation height h had a consistently minimal effect. The result for C. acaule under −80% change in mean wind speed is an exception to this general pattern, probably because the projected wavespeed was so low under this scenario (0.003 m year−1) that sensitivities became rather stochastic.

Figure 6.

 The relative sensitivities of wavespeeds of six species to changes in the values of parameters of the WALD dispersal kernel. Sensitivities are examined for models of spread under baseline wind speed distributions (0%), and with changes in mean wind speed of −80% and +80%. H height of seed release, F seed terminal velocity, h vegetation height.

Discussion

Climate models show great variation in their projections of how wind speed will be altered by anthropogenic climate change over the next century. By combining a mechanistic model of seed dispersal by wind with an analytical wavespeed model, we have shown that this variation leads to great uncertainty in projections of how fast wind-dispersed plant populations may spread in the future. In this paper, we have analysed this uncertainty – which we discuss in detail later – and through this we can draw some conclusions about the key determinants of spread and plant responses to climate change.

Over southern England, the projections of 17 GCMs across three emission scenarios taken from CMIP3 generally suggest decreases in winter and summer mean monthly wind speed of up to −80% by the 2080s, while some models (GIER and CNCM3) suggest large increases in winter and summer, respectively. These findings confirm the large uncertainty in wind speed projections from climate models (e.g. Pryor, Schoof & Barthelmie 2005b; Najac, Boe & Terray 2009; Kjellström et al. 2011), and that this is due more to differences among GCMs than among emission scenarios (e.g. Pryor & Schoof 2010). The UK Climate Impact Programme (UKCIP) has recently developed a set of probabilistic scenarios of changes in monthly wind speed at a 25 km grid using a Bayesian methodology and simulations from the Met Office RCM ensemble HadRM3-PPE driven by HadCM3-PPE (Sexton & Murphy 2010). In contrast to the CMIP3 projections, the 50th percentile change from UKCIP suggests a small reduction in summer mean wind speed across most of the UK for the 2050s. For winter, the 10–90% probability change is typically ± 0.5 m s−1 (Sexton & Murphy 2010). This discrepancy with our findings might be due to the small changes in large-scale atmospheric circulation simulated by HadCM3 compared with other GCMs (Kjellström et al. 2011).

Data from these climate models is generally available as mean changes. We have described a method to translate such data into wind speed distributions, by changing the parameter values of a Weibull distribution of observed wind speeds. We changed only the scale parameter η because observed spatial and temporal variations in wind speed distributions are governed by variation in η rather than in β (Akpinar & Akpinar 2005; Kiss & Jánosi 2008) (We also note that it is not straightforward to alter the shape parameter β to match a changed mean.). This method is important because it allowed us to calculate projected changes in wind speed distributions for use with the WALD model to derive seed dispersal kernels. Using a Weibull description of wind speeds, Nathan et al. (2011a) projected seed dispersal and population spread by calculating dispersal and spread for a single wind speed at a time, and then averaging the wavespeed c* over the Weibull distribution. We recommend our approach, which is more true to the actual process in deriving a single WALD-based kernel by integrating over the distribution of wind speeds through the dispersal season, from which one then calculates c* (see also Skarpaas & Shea 2007).

These methods showed that the wavespeed was affected disproportionately by changes in mean wind speed, such that the percentage difference in wavespeed compared to the baseline was always greater than the percentage change in mean wind speed. This is probably because the wavespeed is sensitive to the tail of the dispersal kernel in such wavespeed models (Caswell, Lensink & Neubert 2003; Bullock, Pywell & Coulson-Phillips 2008). An alteration in the mean of the wind speed distributions caused changes in the tail of the distribution (Fig. 1), which translated directly into the tail of the dispersal kernels (Fig. 2). This is a consequence of altering η to match the changed mean, and this pattern is supported by observational data which show a positive correlation between mean and maximum (i.e. the tail) wind speeds (Graybeal 2006).

The great variation in climate model projections, combined with the nonlinear sensitivity of the wavespeed to wind speed change, results in a very large uncertainty about plant spread rates in future climates. The use of a relatively simple model of population spread allowed us to isolate and quantify this effect. Wavespeed models can be extended to examine the impact on population spread of other factors, including: variation in demography (Bullock, Pywell & Coulson-Phillips 2008; Jongejans et al. 2008a); the impacts of seed abscission mechanics on dispersal (Soons & Bullock 2008; Nathan et al. 2011a); and changes in dispersal parameters (Kuparinen et al. 2009; Nathan et al. 2011a; Zhang, Jongejans & Shea 2011). This modelling approach is less able to incorporate spatial environmental structuring, although Nathan et al. (2011a) did so in a simplified way by modelling Janzen–Connell processes and Dewhirst & Lutscher (2009) provided a more generic approach, albeit using numerical approximations. Analytical wavespeed models can deliver predictions which match observed population spread (Bullock, Pywell & Coulson-Phillips 2008), but in general these models are useful for general projection and exploration of the importance of particular processes, such as changes in wind speed. More specific and realistic predictions might come from spatially explicit simulations, but the utility of these is limited by their complexity, data requirements, and the difficulty in performing sensitivity analyses. Indeed, recent work is suggesting how analytical and simulation approaches to modelling of population spread might be combined so each can be informed by the other (Travis et al. 2011).

To analyse the complexities of the effects of changing wind speed on seed dispersal kernels and thus on plant spread, we have assumed other aspects of dispersal and demography remain unchanged. Other determinants of spread might also be altered in future climates. Dispersal might be enhanced by increased turbulence as the climate warms (Kuparinen et al. 2009), while there is some evidence for trees that increased CO2 could increase fecundity and speed maturation, and that these changes could have large effects on spread rates (Nathan et al. 2011a). One could speculate about other possible changes, but without specific predictions about future changes, a comprehensive approach is to analyse sensitivities to demographic and dispersal, as we have performed (see also Nathan et al. 2011a).

The approach to modelling wavespeed used here, which combines an integrodifference model for dispersal with a stage-structured matrix model for demography, has well-developed theory for sensitivity analysis (Neubert & Caswell 2000; Caswell, Lensink & Neubert 2003). In this study, sensitivity (elasticity) of the wavespeed to demographic transitions hardly varied as the dispersal kernel changed with wavespeed. This is not surprising as demography and dispersal make separate contributions to wavespeed (Neubert & Caswell 2000). This analysis does illustrate however, that the most important demographic contribution to the wavespeed varies greatly among species. As shown by Silvertown et al. (1993) for population growth rate, we found that the highest wavespeed elasticities may reside among the transitions relating to growth, stasis, or fecundity, depending on the species considered. Therefore, it is not possible to generalize about the most important demographic transitions in plant population spread.

In contrast to those for demography, the sensitivities of the wavespeed to dispersal parameters varied little among species or wind speeds. Seed release height was very important, terminal velocity less so, and vegetation height hardly at all. It is difficult to compare this finding against those of Skarpaas & Shea (2007) and Nathan et al. (2011a), who also used WALD kernels, but different analytical methods to ours. However, both found the wavespeed was sensitive to seed release height and seed terminal velocity. Our finding of the importance of seed release height rather than terminal velocity in governing dispersal distances is also becoming clear from a variety of studies. A cross-species analysis by Thomson et al. (2012) showed that dispersal distances are related more to plant height than seed size, and plant height responses of Carduus nutans to climate manipulations had a large effect on modelled dispersal distances (Zhang, Jongejans & Shea 2011).

The study presented here increases our understanding of the determinants of seed dispersal by wind and the consequent spread of wind-dispersed plants, and illustrates that uncertainty in climate models leads to an even greater uncertainty about how dispersal and spread will change in future climates. One should therefore be wary about making predictions as to how fast plant species may spread in response to climate change. This is not only because of the uncertainty we have just described, but also because these models do not reflect the complexity of the real world, as we discussed earlier. A further issue is that parameterization of these models is imperfect. The matrices we constructed cannot be regarded as precise representations of the demography of each species in southern England, although the matrix structures we have used are probably accurate and the vital rates are adequate approximations. The dispersal kernels used are probably incomplete as well. The WALD model seems to give a good representation of seed dispersal by wind (e.g. Katul et al. 2005; Skarpaas & Shea 2007), although reported discrepancies (Stephenson et al. 2007) suggest field testing should be continued. However, each species may also be dispersed by other vectors than wind. For example, A. altissima can be dispersed in flowing water (Säumel & Kowarik 2010) and is still sold in the UK as a garden plant. Indeed, humans can disperse seeds orders of magnitude further than can wind (Wichmann et al. 2009).

While exact predictions should not be made, the modelling presented in this paper does lead to clear conclusions. The spread rates under baseline wind speeds of all species except A. altissima (Table 1) are much lower than the velocity of climate change for the 21st century, which has been projected (using an ensemble of GCMs) to be 420 m year−1 as a global average and 350 m year−1 in southern Britain (although this is classified as temperate forest by Loarie et al. 2009). If wind speeds do decline in Britain, as is the usual projection, then species such as these will have to contend not only with decreasing suitability of local habitat, but also an increasing inability to disperse far enough to keep pace with climate change.

Acknowledgements

This research was supported by the projects SCALES EU–FP7–226852 and CEH C04166. We acknowledge the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP’s Working Group on Coupled Modelling (WGCM) for their roles in making available the WCRP CMIP3 multi-model data set. Support of this data set is provided by the Office of Science, U.S. Department of Energy. Weather station wind speed data were obtained from the British Atmospheric Data Centre (http://badc.nerc.ac.uk/home/index.html).

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