Appendix B. Effect of field geometry on cross-pollination rates C(i).
The general form of the cross-pollination rate, C(), is a function of the shapes and the relative orientation () ( here assumed continuous) of the source GM and sink non-GM fields, and is given by Equation A1. In this section, we investigate the effect of field geometry on the cross-pollination rate, C(). Consider the field geometry shown in Figure 1 in the main text. The fields are orientated at compass direction i , the fields are separated by distance R, and the length of the facing edges of the fields is H. First suppose that the depth of both fields (L) is small compared to the length of the facing edges (H). Defining;
Then in the continuous and discrete cases, approximately;
(B.1a) |
|
(B.1b) |
or
(B.1c) |
where
(B.1d) |
where is the proportion of the length of the facing borders contributing to cross-pollination when the wind blows from angle () (Fig. 1, main text). We assume that C(i ) is a linear function of fL(i ), since cross-pollination rates are generally small (Appendix I). The minimum and maximum values of are = -tan-1(H/R) and = tan-1(H/R). As an approximation, we assumed that d(r) is a negative exponential function, with decay parameter . This has been found to fit empirical data reasonably well over relatively short distances (Loos et al. 2003, McCartney and Fitt 1985). The distance between any two points situated on the adjacent borders at angle is R/cos(), hence the expression in the exponential function. The wj are the weighting factors applied to . The cross-pollination rates are then a smoothed version of fL(i ) . The greatest degree of smoothing occurs when the range [ , ] is large, and when the weighting factors wj are roughly constant. As the field depth L increases, the weightings become more variable, and hence C(i) is less smoothed.
Case 1: Greatest effect of wind direction: narrow, distant fields
Suppose we assume R=1,000m, H=200m, so that R/H is 5, which is large (Fig. 2 centre right panel, main text). The facing edges of the fields are short compared to the distance between the fields, i.e. the range [ , ] = [-11°,11°] is narrow. The pollen decay function, d(r), (Eq. B.1) varies little across the range of , regardless of the value of . As the depth of the fields, L, increases, the degree of spreading decreases, as the angles between any two points in the fields decreases. We make the approximation;
since wj is approximately zero for j i (Fig. B1).
Case 2: Least effect of wind direction: long, close fields
Suppose we assume R = 100 m, H = 800 m, so that R/H = 1/8, which is small (Fig. 2 upper right panel, main text). The facing edges of the fields are long compared to the distance between the fields, i.e., the range [ , ] = [-83°,83°] is wide. The pollen decay function (Eq. B.1) varies considerably over the range of , for any . The weighting factors wj are affected by the characteristic pollen dispersal distance (Fig. B2). Nevertheless, even when is small, e.g., 20 m, the smoothing is applied over a reasonably wide range of angles. As increases, smoothing is applied over an increasing range of angles (Fig. B2). has been measured to vary widely both among and within species. For example, for oilseed rape, the reduction in pollen concentration from 0 m to 100 m from the source has been measured as 2–11% (McCartney and Lacey 1991) and 27–69% (Timmons et al. 1995). We estimated a from several field trials, and found the greatest value of a from the data of Timmons et al. (1995), when a is approximately 60 m (Hoyle, unpublished results). In the case of maximum smoothing, we choose = 60 m. In this case, C( i ) was estimated from Eq. B.1b. As the depth of the fields (L) increases, the degree of spreading decreases (as in Case 1), as the angles between any two points in the fields decreases.
FIG. B1. wj against for (a) R = 1000 m, H = 200 m (steeper line) and (b) R = 100 m, H = 800 m (shallower line). In both cases, = 60 m. |
FIG. B2. wj against , for R = 100 m, H = 800 m. As increases (20 m, 40 m, 60 m), wj flattens and spreads out. |
LITERATURE CITED
Loos, C., R. Seppelt, S. Meier-Bethke, J. Schiemann, and O. Richter. 2003. Spatially explicit modelling of transgenic maize pollen dispersal and cross-pollination. Journal of Theoretical Biology 225:241–255.
McCartney, H. A., and B. D. L. Fitt. 1985. Mathematical Modelling of Crop Disease. Pages 107–143 in C. A. Gilligan, editor. Advances in Plant Pathology, Vol. 3. London: Academic Press.
McCartney, H. A., and M. E. Lacey. 1991. Wind dispersal of pollen from crops of oilseed rape (Brassica napus L.). Journal of Aerosol Science 22:467–477.
Timmons, A. M., E. T. O’Brien, Y. M. Charters, S. J. Dubbels, and M. J. Wilkinson. 1995. Assessing the risks of wind pollination from fields of genetically modified Brassica napus ssp. oleifera. Euphytica 85:417–423.