How do genetically modified (GM) crops contribute to background levels of GM pollen in an agricultural landscape?

Authors


*Correspondence author. E-mail: Claire.lavigne@avignon.inra.fr

Summary

  • 1It is well established that pollen-mediated gene flow among natural plant populations depends on a complex interaction between the spatial distribution of pollen sources and the short- and long-distance components of pollen dispersal. Despite this knowledge, spatial isolation strategies proposed in Europe to ensure the harvest purity of conventional crops are based on distance from the nearest genetically modified (GM) crop and on empirical data from two-plot experiments. Here, we investigate the circumstances under which the multiplicity of pollen sources over the landscape should be considered in strategies to contain GM crops.
  • 2We simulated pollen dispersal over eighty 6 × 6 km simulated landscapes differing in field characteristics and in amount of GM and conventional maize. Pollen dispersal was modelled either via a Normal Inverse Gaussian (NIG, currently used for European coexistence studies) or a bivariate Student (2Dt) kernel. These kernels differ in their amount of short- and long-distance dispersal. We used linear models to analyse the impact of local and landscape variables on impurity rates (i.e. proportion of seeds sired by pollen from a transgenic crop) in conventional fields and quantified their increase due to dispersal from other than the closest GM crops.
  • 3The average impurity rate over a landscape increased linearly with the proportion of GM maize over that landscape. The increase was twice as fast using the NIG kernel and was governed by the short-distance dispersal component.
  • 4Variation in impurity rates largely depended on the distance to the closest GM crop and the size of the receptor field. However, impurity rates were generally underestimated when only dispersal from the closest GM field was considered.
  • 5Synthesis and applications. Distance to the closest GM crop had most impact on impurity rates in conventional fields. However, impurity rates also depended on intermediate- to long-distance dispersal from distant GM crops. Therefore, isolation distances as currently defined will probably not allow long-term coexistence of GM and conventional crops, especially as the proportion of GM crops grown increases. We suggest strategies to account for this impact of long-distance dispersal.

Introduction

As the area of GM crops grown continues to increase globally, the proportion of fields cultivated with GM crops (termed ‘GM fields’ for brevity in this article) is also likely to increase within areas where both GM and conventional crops coexist. However, the impact of distant fields is not taken into account in the definition of isolation distances required to ensure conventional harvest purity.

Several pollen dispersal models attempt to predict rates of adventitious transgene presence in a conventional crop (i.e. here, the proportion of seeds sired by pollen from a transgenic crop and termed ‘impurity rate’ throughout this article) at various spatial distributions of GM and conventional fields within a landscape (Colbach et al. 2005; Ceddia, Bartlett & Perrings 2007). Other models consider pairs of fields consisting of one GM and one conventional field. The latter use data only from the closest GM field to each particular conventional field (e.g. Klein et al. 2006b; Hoyle & Cresswell 2007; Kuparinen et al. 2007) without accounting for the possible impact of more distant fields. The paired-field approach might be an inappropriate method for predicting impurity rates and deriving effective isolation distances between fields. In the presence of long-distance pollen dispersal, higher impurity rates are expected in conventional fields when there are more GM fields distributed throughout the local landscape (Ramsay, Thompson & Squire 2003; Shaw et al. 2006). Such long-distance dispersal is a widespread phenomenon: (i) estimations of pollen dispersal using a neighbourhood model (Adams & Birkes 1991) have confirmed the existence of a large ‘external’ pollen cloud (Smouse & Sork 2004 in natural populations; Devaux et al. 2007 for oilseed rape), (ii) observations have demonstrated the presence of pollen far away from any source (Shaw et al. 2006) and (iii) landscape measures of impurity rates have produced values independent of distance from the source (Rieger et al. 2002).

The relative impact of distant GM fields compared to that of the neighbouring GM field is, in fact, likely to depend on the amount of long-distance dispersal (LDD) of pollen. LDD can be quantified as the proportion of dispersal farther than a specified threshold distance (absolute distance definition) or by the distance beyond which only a specified fraction of dispersal is found, for example, 1/1000 (proportional distance definition) (Nathan 2005). LDD can also be quantified by the shape of the tail of the dispersal kernel. In this case, the main distinction is whether the functions are exponentially bounded (i.e. thin-tailed) or not (i.e. fat-tailed) (Mollison 1977). Fat-tailed kernels generally provide better fits to pollen dispersal data (e.g. Devaux et al. 2007 for oilseed rape).

The impurity rate in a conventional field depends both on the proportion of seeds sired by a pollen cloud from outside the field (and thus on its opposite, local pollination), and on the composition of that pollen cloud. For self-compatible pollen sources of large areas such as maize fields, it is likely that short (and medium) distance dispersal governs the amount of local pollination as opposed to point sources where local pollination depends on selfing. Indirect evidence was reported by Robledo-Arnuncio & Austerlitz (2006) who found that for dense clumps of point sources, more leptokurtic pollen dispersal resulted in more intra-clump pollination. The diversity of outcrossing pollen clouds (i.e. the number of effective fathers) depends on plant density, pollination vector and population size (Sork & Smouse 2006). Moreover, isolated individuals tend to receive a more diverse pollen cloud as isolation from pollen sources increases (i.e. smaller rates of correlated paternities within sibship in Robledo-Arnuncio et al. 2004 for trees; lesser contribution of closest field in Ramsay et al. 2003 for oilseed rape). This can be explained by an increasing expected number of sources located at increasing distances (Adams & Birkes 1991). Additionally, such behaviour is expected under pollen dispersal via fat-tailed but not thin-tailed dispersal kernels (Klein, Lavigne & Gouyon 2006a). As a consequence, the frequency of GM pollen in the external pollen cloud over an isolated conventional field should mostly depend on its distance to the nearest GM field under thin-tailed dispersal kernels and on the proportion of GM crops over the landscape under fat-tailed kernels.

The regulation concerning coexistence of GM and conventional crops in Europe is mainly based on keeping a minimum distance from conventional to the closest GM field but there is a margin for regional adaptation of rules. Given existing empirical results and landscape models, isolation distances should be adjusted to the regional GM frequency. Two major differences between agricultural landscapes and natural populations are that: (i) sources have shapes and areas, and (ii) two sources may have the same genetic composition. Thus, results from point sources cannot be directly extrapolated and a better understanding of the interactions between the spatial pattern of pollen sources and the components of the dispersal kernel would help to design isolation strategies. We investigate how characteristics of pollen dispersal kernels and field patterns affect impurity rates in conventional fields by first simulating such rates over realistic landscapes using two dispersal kernels that differ in their amount of LDD. We then compare these simulated impurity rates to those obtained assuming conventional plots that only receive pollen from their closest GM crop neighbour, using a paired-field approach. Finally, we fit linear models (including or excluding landscape factors) on simulated impurity rates and assess their ability to correctly classify fields as being above or below threshold values of impurity rates.

Material and methods

simulation of field patterns

To simulate realistic maps in terms of field sizes and distances among fields, and to test whether impurity rates were sensitive to systematic variation among maps (i.e. largely different field sizes) as well as small random variations, we digitized aerial photographs from five 1·5 × 1·5 km contrasted French landscapes. Random variation was introduced with respect to the spatial distribution of field centroids in these original maps. This spatial distribution was modelled using a point process model of centroids that accounts for: (i) mean number of centroids per unit area, and (ii) the distribution of the distance of each centroid to its nearest neighbour. This model was fitted for each of five original maps (Adamczyk et al. 2007). To generate a simulated map, we first simulated a point pattern using the above model with parameters estimated on the corresponding original map and we then delineated field margins by performing a Voronoi tessellation using these points as seeds (Okabe, Boots & Sugihara 1992). This procedure has been shown to simulate different maps with numbers of fields and distribution of distances among field centroids close to those of the original map (Adamczyk et al. 2007). It was implemented using r2·4·1 (r Development Core Team 2006) and the genexp software (http://www.loria.fr/~jfmari/GenExP/). We thus obtained 10 (5 original maps × 2 replicates) 1·5 × 1·5 km simulated maps that were set in a raster format with 5 × 5 m cells.

We simulated a total area of maize of either 70% or 20% of landscape area, mimicking production areas where maize is either a major crop or not. The proportion of GM maize was set either to 10% or 50%, simulating low or high farmer acceptance of the GM crop. The resulting proportions of GM maize over the landscape were thus 2%, 7%, 10% and 35%. GM and conventional fields were allocated randomly. Two replicate allocations of maize were performed per proportion of GM crop per simulated map. We thus obtained 80 (8 × 10) field patterns (Fig. 1).

Figure 1.

Simulation design.

To clearly see the impact of long-distance dispersal and to avoid border effects due to the fewer neighbours available to edge fields, we transformed each 1·5 × 1·5 km field pattern into a 6 × 6 km field pattern by pasting it 16 (4 × 4) times.

We recorded local characteristics of each conventional field, namely its size (Area_target_field), its border-to-border distance to the closest GM field (Dist_gm) and the size of that GM field (Area_GM_field).

simulations of pollen dispersal and cross-pollination

Pollen dispersal was simulated with a modified version of the mapod software used to estimate cross-pollination in maize (Angevin et al. 2008). To maintain focus on the impact of spatial characteristics of landscapes, we kept agronomic and climatic inputs constant and identical for the GM and conventional crops. Both crops were assumed to be homozygous. Impurity rates were simulated over each 6 × 6 km field pattern but they were only recorded within the 3 × 3 km central area to avoid border effects.

We used two dispersal kernels (Fig. 2) differing in their behaviour over long distances. The first kernel is a NIG (Normal Inverse Gaussian). This is the default kernel in MAPOD which was used in coexistence studies (Angevin et al. 2002; Messéan et al. 2006). It has a power-law decrease at short distances and an exponential decrease at long distance, and is given as follows:

Figure 2.

Probability functions used as dispersal kernels over three ranges of distances. Full line, 2Dt; dotted line, NIG.

image(eqn 1)

with inline image and inline image and where (x, y) is the pollination position relative to the emission position. The values of λz, λx, λy, δx, δy were estimated at λz = 0·027; λx = 0·165; λy = 0; δx = δy = 0·499 in Klein et al. (2003).

The second kernel is a 2Dt (bivariate Student) (Clark 1998):

image(eqn 2)

where (x, y) is the pollination position relative to the emission position, r = (x2 + y2)0·5 and θ is the angle of the direction from (0, 0) to (x, y). The parameters were estimated at a = 1·55; b = 1·45; κ = 1·12; θ0 = 0.

Unlike the NIG, this kernel is fat-tailed, with a power-law decrease at every distance. The NIG and 2Dt only differ at very short and very long distances (Fig. 2).

Parameters for anisotropic versions of these two kernels were estimated on an approximately 1-ha maize dispersal experiment by maximum likelihood (Klein et al. 2003 for the NIG; same method, unpublished for the 2Dt). Parameters were estimated on anisotropic versions of these kernels because the experimental data presented a clear anisotropy due to dominant winds. However, to maintain the focus on spatial issues, we modelled isotropic dispersal by setting the wind angle to 16 regularly spaced values from 0 to 2Π each with a 1/16 frequency. This procedure models isotropic dispersal accounting for actual wind speed.

For each simulation, we computed the average impurity rate of each conventional field as the mean of the proportions of GM pollen grains above each 5 × 5 m pixel of the field, which we assumed equal to the proportion of seeds sired by GM plants.

Over each pixel of conventional field, this proportion is written as:

image(eqn 3)

where γ is the dispersal kernel, GM is the area of GM maize, ext non-GM is the area of non-GM maize excluding the target field, and local non-GM is the area of the target field (Lavigne et al. 1998).

The average landscape impurity rate was computed as the mean of rates from all conventional fields weighted by field area.

statistical analyses

To compare impurity rates of individual conventional fields obtained over landscape-level field patterns to those obtained considering only the closest GM field, all fields were erased from field patterns except one pair involving a conventional field and its closest GM neighbour. Impurity rates were thus calculated by considering only two fields.

image

where closest-GM is the area of the closest GM field, and local non-GM is the area of the target field.

This was repeated for each conventional field in a 1·5 × 1·5 km central area of the 80 field patterns. The ratio µclosest-GM over µ was computed for each field. We expected the landscape influence to be smaller when GM and conventional fields were close; therefore, we distinguished average ratios for neighbouring (i.e. borders less than 2 pixels (= 10 m) distant) and non-neighbouring pairs of fields.

The relative importance of landscape versus local characteristics on impurity rates of individual conventional fields was assessed through the fit of linear models including or excluding landscape effects (PROC MIXED in sas 8·01, SAS Institute, Cary, NC, USA). Models were run independently for each dispersal kernel. Impurity rates were log- transformed to stabilize the variance. In the most complete models, landscape factors were original map (five levels), simulated map (two levels per original map, nested within original map), proportion of maize (two levels), proportion of GM maize (two levels) and their interaction, as well as field pattern (nested within all landscape factors). Local variables were target field size (quantitative), size of closest GM field (quantitative), and distance to the closest GM crop (quantitative). Both distance and log(distance) were introduced in the model which allowed both a power-law and an exponential component to be included in the model describing the decline in impurity rates with distance to the closest GM field.

For comparisons of model fit, models were adjusted independently for each kernel on the 10 692 field impurity rates from all 80 field patterns, including or excluding landscape factors. Comparisons were performed using the Akaike Information Criterion (AIC). In the complete model, mean square ratios (the usual F statistics) were used to assess the importance of landscape and local variables relative to their respective residual variability. Because of computational limits, the default denominator degrees of freedom provided in sas PROC MIXED were used.

We also checked the accuracy of field classification with respect to threshold values, by comparing simulated impurity rates used as a reference to either (i) simulations based on two plots only, or linear predictions based on (ii) local variables only, or (iii) all variables and their interactions. We used 0·01%, 0·05%, 0·1% and 0·9% as threshold values. Values of 0·05% and 0·1% approximate the detection and quantification limits of 0·045% and 0·09%. The value 0·9% corresponds to the EU labelling threshold and 0·01% was used for a better understanding of model behaviour (http://gmo-crl.jrc.it/doc/Method%20requirements.pdf).

Results

simulated landscapes and field patterns

Simulated maps differed in field number, ranging from 42 to 180 per 1·5 × 1·5 km, field size and variability of field size. Voronoï tessellations homogenized field sizes within a landscape (Fig. 3). Distances between conventional fields and their closest GM field were comparable among maps and depended largely on crop allocation, increasing with the decreasing proportion of maize over the landscape or of GM maize among maize. In most configurations, the closest GM field to a target conventional field was farther than 100 m, except for landscapes with the highest proportion of GM maize where the closest GM fields were on average at distances ~15 m (Table 1).

Figure 3.

Field area distribution of the five real and their two associated simulated 1·5 × 1·5 km maps (Xr = real mapX, Xsy = yth simulation of map X). Within each box plot: horizontal line, median; box limits, first and third quartiles; whiskers limits, 10th and the 90th percentiles.

Table 1.  Average distances between conventional fields and their closest GM neighbour (m) on simulated landscapes as a function of the original map, the proportions of maize area and of GM maize
Maize20%70%
GM in maize10%50%10%50%
Original mapA129010312615
P1284 6712320
S140310115313
S447914622515
T1346206 9910

impurity rates

The average landscape impurity rates (i.e. mean of rates from all conventional fields weighted by field areas) varied mainly with the dispersal kernels and with the proportions of conventional and GM maize but little with the original map (Fig. 4). Within the range of simulated situations, the average landscape impurity rate increased linearly with the proportion of GM maize per unit area (i.e. proportion of maize × proportion of GM maize) for both kernels. This increase was twice as fast with the NIG as with the 2Dt model (Fig. 4).

Figure 4.

Average landscape impurity rates simulated either with the 2Dt or the NIG dispersal kernel. Error bars represent standard errors calculated over the 5 × 4 simulated field patterns.

Field impurity rates were systematically lower when simulated with the 2Dt kernel (mean ± SD = 0·87 × 10−3 ± 1·13 × 10−3 for the 2Dt and 1·71 × 10−3 ± 2·03 × 10−3 for the NIG) and showed larger variation (CV = 1·30 for the 2Dt and CV = 0·84 for the NIG) (Fig. 5). Absolute differences between predictions with 2Dt and with NIG kernels decreased with increasing distance between the target conventional field and the nearest GM crop because the predicted values decreased (Fig. 6). However, the ratio NIG:2Dt, indicating whether the difference is in the order of magnitude of the predicted impurity rate, increased until distances were about 300 m and then stabilized around 3·3 (see Supplementary Material Fig. S1).

Figure 5.

Impurity rates simulated with the NIG versus those simulated with the 2Dt, one point per conventional field.

Figure 6.

Difference between impurity rates predicted with the NIG and with the 2Dt versus distance to closest GM field.

Field impurity rates were smaller when simulated only from the closest GM field rather than from the whole landscape (Fig. 7). Despite the large difference in predicted rates outlined above, underestimation was similar for the NIG and the 2Dt kernels. As expected, underestimation was stronger (smaller ratios) when there was more GM maize in the landscape and when the closest GM field was not a neighbour of the target conventional field (Fig. 7). Surprisingly, even when the closest GM field was a neighbour of the target field, ratios were smaller than 1 as soon as there was more than 2% GM maize over the landscape. As a consequence, when considering pairs of fields instead of all fields to determine whether impurity rates were above the defined thresholds, error rates were high, in particular for small thresholds (Fig. 8). Most errors were false negatives, i.e. declaring values below the threshold when they were above the threshold (see Supplementary Material Table S1).

Figure 7.

Ratio of impurity rates predicted with two-field simulations over those predicted with landscape simulations as a function of the proportion of GM maize over the landscape. (a) neighbouring GM and conventional plots; (b) non-neighbour GM and conventional plots.

Figure 8.

Error rates when classifying fields with respect to four thresholds using simulations at the landscape level as the reference. Classification was based either on linear predictions (model with local variables only (local) or model with all variables (total)) or on values simulated with two fields only (two fields).

linear modelling of impurity rates in conventional fields

Local variables were more important than landscape variables in the linear models on field log-impurity rates as indicated by a much larger reduction in AIC (Table 2). When considering the most complete model (Table 3), the distance to the closest GM field was the main local factor affecting these rates. As expected, rates decreased with increasing distance to the closest GM field. Whatever the dispersal kernel, this decrease was a geometric type but somewhat slower than expected under a purely geometric decrease, with unexpectedly, a faster decrease for the 2Dt than for the NIG (not shown). As expected, impurity rates also decreased with increasing size of the target field and increased with increasing size of the closest GM field, although this latter effect was very small. Landscape variables had some importance in interaction with distance. The (log) distance effect in particular was less important when the proportion of GM maize in the landscape was high, probably because there was less variability in distances.

Table 2.  Akaike Information Criterion (AIC) values for linear models on log-impurity rates including either no factor (Intercept), landscape factors, local factors or both (see Materials and methods for details). Models were fitted separately for the two kernels. N = 10 692 conventional fields
ModelAIC
NIG2Dt
Intercept2·72 × 1043·18 × 104
Landscape only2·70 × 1043·17 × 104
Local only1·19 × 1041·55 × 104
Local + landscape1·18 × 1041·53 × 104
Local + landscape + interactions1·08 × 1041·45 × 104
Table 3.  Analysis of variance on landscape and local variables affecting log-impurity rates
Model term d.f. Numd.f. DenMean square ratio
NIG2Dt
  1. Largest mean square ratios in bold. Orig. map, original map; Distgm, distance to closest GM field; d.f. Num, degrees of freedom of numerator, d.f. Den; degrees of freedom of denominator.

Landscape variablesOrig. map4    671·01·1
rep(orig. map)5    671·00·8
% maize1    674263
% gm1    6770115
% maize*% gm1    672020
Estimated landscape residual variance   0·0500·052
Local variablesLog(distgm)110 60213 40111 360
Distgm110 602629416
Area target field110 602804711
Area GM field110 6022041
InteractionsLog(Distgm)*orig. map410 6021515
Log(Distgm)*% gm110 602645731
Log(Distgm)*%maize110 602613499
Estimated local residual variance  0·150·22

The quality of linear predictions from models including only local variables confirmed their adequacy: they were generally correct regarding the defined thresholds and almost as good as predictions from the complete linear model (Fig. 8). Furthermore, error rates were smaller than those obtained by simulating dispersal from two fields only (Fig. 8).

Discussion

It has been known for some time that small-scale experiments would not be appropriate to assess the safety of GM crops and that effort should be put into large-scale field tests (Stone 1994). Some experiments have estimated long-distance dispersal (e.g. Halsey et al. 2005; Bannert & Stamp 2007 on maize), and some models have considered long-distance dispersal (e.g. Aylor, Schultes & Shields 2003 for mechanistic approaches on maize). Despite these, most recommendations about isolation distances between GM and conventional crops are based on two-plot experiments that fail to account for the multiplicity of GM pollen sources (see however Shaw et al. 2006 on Brassica napus). Here we show that background pollen increases the potential for cross-pollination between GM and conventional crops. Moreover, as the proportion of GM crops increases, so will the failure to keep impurity rates below thresholds when using isolation distances derived from two-plot experiments.

To ensure realism in our study, we used (i) simulated landscapes based on real contrasted landscapes and (ii) pollen dispersal kernels estimated from real data on maize. The NIG kernel in particular was used for studies on the coexistence of GM and non-GM crops in Europe (Angevin et al. 2002; Messéan et al. 2006). Nevertheless, we made a number of simplifications. First, landscapes were characterized by field patterns and maize crop allocations, ignoring physical discontinuities such as hedges, forests or slopes because modelling the impact of landscape discontinuities on pollen dispersal is still ongoing research (Aylor et al. 2003, Dupont, Brunet & Jaroz 2006). Secondly, the Voronoi tessellation used for simulating fields resulted in convex and compact fields with a small variance in area. Thirdly, we assumed isotropic pollen dispersal although this is clearly not usual for wind-dispersed species. However, our results concerning landscape-level pollination rates can probably be extrapolated to anisotropic dispersal provided that orientations of GM and conventional fields are random with respect to wind direction. Our conclusions at the field level can also probably be extrapolated by considering distances to the closest GM field that account for wind direction. Furthermore, the impact of the prevailing wind direction has been specifically studied elsewhere (Hoyle & Cresswell 2007). Finally, we assumed synchronous flowering of GM and conventional crops as it is already recognized that their asynchronous flowering is efficient in reducing cross-pollination rates in maize (e.g. Angevin et al. 2002; Halsey et al. 2005). Given these simplifications, and also because the dispersal kernels used were not validated at the longest distances considered here, precise values of simulated impurity rates should be considered with caution. However, compared to the highly variable data (e.g. in Halsey et al. 2005; Bannert & Stamp 2007), the range is correct. Further, as in coexistence studies (Messéan et al. 2006), our simulations indicate that in general, the 0·9% regulatory threshold is achievable (Fig. 5).

As expected, at the landscape level, given some proportion of GM maize, the average impurity rate increased with an increasing maize area and, for a given amount of maize, also increased with the proportion of GM fields. Interestingly, these rates increased linearly with the proportion of GM maize throughout the whole landscape. Within a conventional field, the impurity rate is the ratio of GM pollen over all pollen (i.e. both conventional local pollen and external GM and conventional pollen; equation 3). Contrary to expectations for point sources, it is likely that for most dispersal kernels, the vast majority of pollen over a field comes from the field itself and that the quantity of external pollen (GM and non-GM) is negligible compared to local pollen. These large differences in the amount of pollen would explain the linear increase of landscape impurity rates with the GM crop area. Equation 3 indicates that the slope of the linear relationship is in fact determined by the ratio of GM pollen received over local pollen. Our results are consistent with the fact that the amount of local pollen was on average about twice as large for the 2Dt kernel than for the NIG kernel (impurity rates and slope are twice as small) and that the external pollen cloud was small and of similar magnitude for both dispersal kernels (similar ratios of simulations with two versus all fields). Unexpectedly, both individual field impurity rates and the slope of the linear relation between landscape average impurity rates and GM crop area were thus mostly affected by the shape of the dispersal kernel at short distances rather than by long-distance dispersal. However, more contrasted types of dispersal kernels should be considered before drawing general conclusions.

A linear response of landscape average impurity rate with GM crop area was also reported in a simulation study with oilseed rape (Ceddia et al. 2007). The slope of the relationship was 0·033 (i.e. a faster increase than the 0·011 and 0·006 found here). Results are difficult to compare because of differences in crop biology and simulation design. It follows nevertheless that the linear increase can probably be considered a general property but that the slope of increase is highly dependent on conditions of simulations. Caution should be exercised when deriving advice about maximum acceptable GM crop surface area over a landscape (as in Ceddia et al. 2007).

It is recognized that species with a high dispersal ability interact with the landscape structure at a larger spatial scale (Keitt, Urban & Milne 1997). We thus expected that impurity rates predicted with the 2Dt dispersal kernel would depend to a larger extent on landscape variables but we found no evidence of this. The ratios of impurity rates simulated from all fields in the landscape over the rate from only the closest GM field were similar for both kernels. This results from a complex interplay between the probability of dispersing at each distance and the presence of pollen sources at each of these distances (Adams & Birkes 1991). The 2Dt kernel exhibited a higher probability of long-distance dispersal than the NIG kernel but a lower probability for intermediate-distance dispersal. Pollen sources were placed randomly over the landscape and it is likely that the lesser contribution from long-distance sources to the external pollen cloud with the NIG kernel was compensated by a larger contribution from sources placed at intermediate distances. This result questions the way long-distance dispersal is defined for pollen data. The two dispersal kernels we used indeed differ considering either the absolute distance definition of LDD (NIG: 1045 m and 2Dt: 3245 m for a quantile of 0·999) or its proportional distance definition (2Dt: 0·55% and NIG: 0·40% at 500 m or 2Dt: 0·29% and NIG: 0·11% at 1000 m). They also differ if the dispersal tail is considered (exponential-like versus fat-tailed), although the NIG is not typical of an exponential decrease because it is governed by a power-law function over short and intermediate distances. However, none of these three characteristics ensured that more effective long-distance cross-pollination occurred here with the 2Dt, because long-distance cross-pollination events depend both on short- and long-distance dispersal and on the spatial distribution of pollen sources.

Numerous studies investigating the effect of field size or isolation distances (i.e. distance from a conventional field to its closest GM neighbour) on impurity rates are based on two-plot experiments (e.g. Halsey et al. 2005; Weekes et al. 2007 on maize). Our results clearly indicate that considering only two fields may lead to an underestimation of these rates and that dispersal should be considered from multiple sources. It is worth noting that the larger the isolation distances, the larger the underestimation, and thus, the more important it is to consider the impact of GM fields farther away. We thus advocate a new strategy for defining isolation distances that includes the contribution made by background GM pollen at a landscape level. There are a few non-exclusive options for this. First, if the goal is to predict the impurity rate for a specific field, and information about the location of surrounding GM crop fields is known, simulations should be performed considering all fields; if only the area of GM crop over the landscape is known (the minimum information necessary), the impurity rate may be simulated from the closest GM field only, and it may be increased to account for the impact of other fields. Here, for example, we found that rates should be multiplied by a factor of about 1·05 for neighbouring GM and conventional fields and 2% GM maize over the landscape, and by a factor of 6·7 if fields are not neighbours and the GM proportion is 35%. However, these ratios most probably depend on dispersal kernels together with landscape and should be considered with caution. Secondly, if the goal is to test the impact of field characteristics and isolation distance on impurity rates, abacuses could be built based on numerous simulations performed at the landscape scale for various sizes and isolation distances from the target field and random allocation of other GM and conventional fields, although computational time is high. As reported here, a final option would be to predict impurity rates using linear models that only consider local variables but have been previously fitted to a moderate number of pertinent simulations at the landscape scale.

Obvious choices for variables in such linear models are the size of the source and the target fields, as well as the distance between them (e.g. Ingram 2000; Smouse, Robledo-Arnuncio & Gonzalez-Martinez 2007; Weekes et al. 2007). More sophisticated approaches also consider field shape (e.g. Klein et al. 2006b; Kuparinen et al. 2007), which we omitted here because the Voronoi tesselation does not provide elongated fields. We did not expect such a small impact of the size of the closest GM field compared to that of the receptor field. The large effect of the size of the receptor field, basically a dilution effect, is usual (Devos, Reheul & De Schrijver 2005). The minor effect of the source field is more controversial. A small effect of source field area has been reported on cross-pollination data (Devaux et al. 2008 for oilseed rape) and deduced from models (Gustafson et al. 2005). In the latter case, it was argued that this was due to a lack of LDD in the dispersal model (Willenborg & Van Acker 2006). Other studies, however, report a substantial effect of source field size (Kuparinen et al. 2007). Discrepancies might, to a certain extent, depend on the range of variability tested. Here, field sizes in simulated maps were realistic as compared to the original maps, but variability was somewhat reduced (Fig. 3). This is, however, not the only explanation because variability was sufficient for conventional field size to have a large effect. Most probably, the whole GM area within a buffer surrounding the conventional field should have been considered, rather than restricting the analysis to the closest GM field. Testing the ability of these models to correctly predict classification of fields over independent landscapes is the next necessary step before their use can be recommended for decision purposes.

Acknowledgments

Original maps were provided by the Institute for the Protection and Security of the Citizen (Joint Research Centre of the European Union) and AUP-ONIGC–ex ONIC (Office National Interprofessionnel des Céréales). We thank F. Austerlitz for very useful comments, M. Leclaire for his technical help and Suzette Tanis-Plant for editorial advice. We acknowledge support from the French national programme ‘ACI OGM et environnement’.

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