Appendix A2

Appendix A2 : Additional information about spatial models and spatial component of the risk Melen Leclerc, Emily Walker, Antoine Mess´ean and Samuel Soubeyrand In this document we provide a full mathematical descriptions of our spatial model for simulating the spread of pollen grains in the landscape, and some simulation results that show the importance of using spatially explicit models for assessing the risk.

1

Spatial model for pollen dispersal

The amount of toxic pollen grains in the ambient air and located at position (x, y) is defined by the following convolution product: Z Z Ra (x, y) =

E(x0 , y 0 )K(x − x0 , y − y 0 )dx0 dy 0 = E ⊗ K(x, y),

(B-1)

where K is a dispersal kernel function (Kot et al., 1996) modeling the density probability function of the deposit locations of particles released from a point source, and E is the number of pollen grains emitted by square metre. In the example of Lepidoptera larvae, as susceptible individuals ingest transgenic pollen grains that are present on the leaves of host-plants (Lang and Otto, 2015), an individual located at position (x, y) is not exposed to the amount of pollen Ra (x, y) of the ambient air. Indeed, only a fraction of the total amount of pollen adheres on leaves, and, climatic events (e.g. rain or strong wind) can cause significant loss (Pleasants et al., 2001; Holst et al., 2013; Hofmann et al., 2016). For the sake of simplicity the amount of toxic Bt pollen R(x, y) at position (x, y) is obtained from: R(x, y) = Ra (x, y) ω (1 − ψ)

(B-2)

where ω and ψ are respectively the percentage of pollen that adheres to leaves and the total loss of pollen. Albeit some dispersal kernels have already been fitted to experimental field data (Klein et al., 2003; Lavigne et al., 2008), pollen dispersal at long-distances is still a controversial and discussed topic, especially for the ecological risk assessment of the Bt maize (Hofmann et al., 2014; EFSA, 2015). Hofmann et al. (2014) fitted a power-law dispersal gradient on a large data set of pollen concentration and suggested that long dispersal events are likely. However, this is 1

not a dispersal kernel (i.e. not a probability density function) and it does not integrate the contribution of every sources. Similar patterns of dispersal gradient that describe the change in the concentration of pollen with the distance to the closest source might be obtained, for instance, with kernels that decrease faster than a power-law decrease (e.g. exponential decrease) at long-ditances when associated with a strong emission of numerous sources in the landscape (Lavigne et al., 2008). In this study we use three contrasted dispersal kernel functions and assessed their effects on the risk. The first kernel is a NIG (Normal Inverse Gaussian) fitted by Klein et al. (2003). It has a power-law decrease at short distances and an exponential decrease at long distance and is given by:

K(x, y) =

δx δy eλz q(x, y)−0.5 + p0.5 −√pq(x,y) δx λx x+δy λy y e e 2π q(x, y)

(B-3)

with p = λ2z + λ2x + λ2y and q(x, y) = 1 + δx2 x2 + δy2 y 2 , and where (x, y) is the pollen deposition in Cartesian coordinates relative to the emission position and λx , λy , λz , δx , δy are parameters adjusted by wind condition as follows:

λz =

0.027h , 0.831

λx = 0.165

h µ cos(θ) , 2 0.831

λy = 0.165

h µ sin(θ) , 2 0.831

δx = δy = 0.499

The parameter h is the height between emitting flowers and receiving host-plants and (µ, θ) are the daily average wind speed and direction at ten meters high for each day. For the simulations we fixed h to 2m and we consider two contrasted situations : the isotropic case without wind (µ = 0) and an anisotropic case with parameters µ = 2ms−1 and θ ∼ U [0, 2π]. The second kernel is a 2Dt (bivariate Student) fitted by Lavigne et al. (2008) which, unlike the NIG, is a fat-tailed with a power-law decrease at every distance (Lavigne et al., 2008): K(x, y) =

b−1 πa2

 1+

r2 a2

−b

eκcos(θ−θ0 )

(B-4)

p where r = (x2 + y 2 ) is the Euclidean distance, θ is the angle of the direction made by (x, y). For the simulations we used the estimated parameters (a = 1.55, b = 1.45, κ = 1.12, θ0 = 0) and we considered the isotropic case with identical wind condition for every direction (Lavigne et al., 2008). The third one is a geometric fat-tailed kernel with a power-law decrease (Devaux et al., 2005): K(x, y) =

(γ + 1) (γ + 2) γ (1 + r) 2π

(B-5)

where r is still the Euclidean distance and γ is the power shape parameter of the kernel. γ was estimated to −2.59 from the power-law relationship published by Hofmann et al. (2014)(see below). 2

0.831 . h

The convolution product of E and K was obtained by using the convolution theorem: E ⊗ K = F −1 (F(E)F(K))

(B-6)

where F and F −1 are respectively the Fourier and the inverse Fourier transforms. Equation (B-6) was solved numerically by computing Fast Fourier Transforms (FFT) on a 212 × 212 regular rasterized domain with periodic boundary conditions.

2

Parameter estimation of the Geometric kernel from a dispersal gradient

Hofmann et al. fitted a power-law gradient on a large data set of maize concentration. Power dispersal gradient used by Hofmann et al. (2014): g pow (x) = a||x||b , with a ≥ 0 and b < 0 (a = 1.271x106 and b = −0.585 for the estimated values). This is not a kernel and it does not take into account the multiple source issue (only the distance to the closest source is used). Consider a generic isotropic dispersal kernel f modeling the density probability function of the deposit locations of particles released from a point source located at the origin 0 = (0, 0) ∈ R2 : f : R2 → R+ , x 7→ f (x) = f (||x||). If n point sources located at (x1 , . . . , xn ) ∈ R2n release particles with dispersal kernel f and constant intensity α, then the expected concentration of particles at the origin 0 is: C1 (0; x1 , . . . , xn ) = α

n X

f (||xi ||).

i=1

Now, consider a situation with less information. Suppose that (i) sources form an homogeneous Poisson point process with intensity λ > 0 over R2 , and (ii) the distance between the closest source and the origin is known and equal to r. Then, the expected concentration of particles at the origin 0 is: Z C2 (0; r) = αf (r) + αλf (||x||)dx, ¯ B(0,r)

¯ r) is the complementary set of the ball with center 0 and radius r. where B(0, It follows that: Z ∞ C2 (0; r) = αf (r) + 2παλ uf (u)du. (B-7) r

C2 can be interpreted as an infinite mixture concentration and is thereafter called mixture concentration.

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Consider the following geometric kernel: f geo (x) =

(β + 1)(β + 2) (1 + ||x||)β , 2π

with β < −2. This (well-defined) kernel has a finite value at the origin and is asymptotically equivalent to the power dispersal gradient used by Hofmann et al.: when ||x|| → ∞, g pow (x) = O(1). f geo (x) Then, the expected concentration C2 satisfies:   (β + 1)(β + 2) C2geo (0; r) = α(1 + r)β + λ(1 + r){1 − (β + 1)r} . 2π

(B-8)

Estimations of parameters of B-8 were obtained by fitted the model to simulated gradient of Hofmann.

4

Figure B-1: Fit of the mixture concentration C2geo (x) to the Hofmann gradient (g pow (x) = a||x||b with a = 1.271 × 106 and b = −0.585). The fit was done over values of g pow (x) for ||x|| belonging to a regular grid of size 104 covering the interval of distances [0.2m; 5000m]. Estimates of (α, β, λ) are (3.77 × 106 , −2.59, 0.226).

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Figure B-2: Plot of the Hofmann gradient (g pow (x) = a||x||b with a = 1.271×106 and b = −0.585; black line), the fitted mixture concentration (C2geo (0; r); red dots) and the contribution of the closest source αf geo (x). The x- and y-axes are log-scaled.

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3

Importance of being spatial for landscape-scale risk assessment

It is now well established that the mean-field assumption might not be appropriate for describing and predicting the behaviour of ecological systems that involve particle dispersal, especially at the landscape level (Durrett and Levin, 1994; Dieckmann and Law, 2000; Filipe and Maule, 2003, 2004; Klein et al., 2017). The mean-field approximation assumes a perfect mixing and in systems where individuals are immobile it can overestimate the intensity of the interactions between individuals. As in GM crops - NTOs systems the interacting individuals (emitting GM plant, exposed larvae) can be considered to be both immobile it is likely that common temporal ODE models can, everything else being equal, sometimes lead to inaccurate estimates of the mean risk at the landscape level.

3.1

Simulations

To assess the influence of a non-spatial approximation on our results we performed a GSA by considering i) an experimental design of 150 points with 3 replicates of each point, ii) the same priori distributions used in our study and, iii) 5 dispersal kernel functions: the 4 dispersal kernels presented above (isotropic NIG, anisotropic NIG, Bivariate Student, Geometric) and a Uniform dispersal kernel (K(d) = 1/5000) which is equivalent to a non-spatial mean-field model (Filipe et al., 2004).

3.2

Results

This numerical experiment corroborates the influence of parameters observed in the main study and also exhibits a strong influence of the dispersal kernel (Figure 4). This new substantial effect is due to the uniform dispersal kernel which, as expected, over-estimates the mean risk in the landscape (Figure 3). Thus, these results confirm the importance of considering spatial models, instead of mean-field models, improving risk assessment at the landscape level.

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Figure B-3: Boxplot showing the distribution of the mean risk of mortality for the 5 dispersal kernel functions considered for the simulations.

8

Figure B-4: Sensitivity Indices obtained through this numerical experiment. First first-order SI are represented in dark-grey while second-order indices are in light-grey.

9

Below are given R outputs with deviance analysis and estimates of the parameters. > mod anova(mod,test="Chisq") Analysis of Deviance Table Model: Gamma, link: log Response: data$mrisk Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev Pr(>Chi) NULL 2216 2610.18 omega 1 28.90 2215 2581.27 < 2.2e-16 p 1 19.77 2214 2561.51 < 2.2e-16 u 1 53.68 2213 2507.83 < 2.2e-16 E 1 272.19 2212 2235.64 < 2.2e-16 psi 1 137.28 2211 2098.35 < 2.2e-16 tau 1 186.79 2210 1911.57 < 2.2e-16 I 1 0.00 2209 1911.57 0.9876417 rho 1 13.60 2208 1897.96 2.378e-14 kernel 4 520.94 2204 1377.02 < 2.2e-16 omega:p 1 4.08 2203 1372.94 2.956e-05 omega:u 1 0.30 2202 1372.64 0.2544714 omega:E 1 2.31 2201 1370.32 0.0016613 omega:psi 1 3.36 2200 1366.96 0.0001486 omega:tau 1 2.52 2199 1364.44 0.0010277 omega:I 1 0.01 2198 1364.43 0.8308707 omega:rho 1 3.10 2197 1361.33 0.0002734 omega:kernel 4 0.21 2193 1361.12 0.9244879 p:u 1 0.08 2192 1361.04 0.5539611 p:E 1 0.03 2191 1361.01 0.7386620 p:psi 1 7.51 2190 1353.50 1.437e-08 p:tau 1 0.71 2189 1352.79 0.0809682 p:I 1 12.87 2188 1339.92 1.184e-13 p:rho 1 0.00 2187 1339.92 0.8964426 p:kernel 4 3.03 2183 1336.89 0.0115015 u:E 1 1.34 2182 1335.55 0.0166417 u:psi 1 2.15 2181 1333.40 0.0024311 u:tau 1 9.58 2180 1323.82 1.543e-10 u:I 1 0.00 2179 1323.82 0.9098264 u:rho 1 0.01 2178 1323.81 0.8411760 u:kernel 4 18.21 2174 1305.60 4.910e-16 E:psi 1 4.48 2173 1301.13 1.208e-05 E:tau 1 0.62 2172 1300.50 0.1028291 E:I 1 0.87 2171 1299.64 0.0537735 E:rho 1 0.46 2170 1299.18 0.1629256 E:kernel 4 0.28 2166 1298.90 0.8809918 psi:tau 1 0.74 2165 1298.17 0.0757219 psi:I 1 0.48 2164 1297.69 0.1518703 psi:rho 1 2.90 2163 1294.79 0.0004302 psi:kernel 4 0.70 2159 1294.09 0.5594687 tau:I 1 3.00 2158 1291.09 0.0003444 tau:rho 1 155.75 2157 1135.34 < 2.2e-16 tau:kernel 4 220.30 2153 915.04 < 2.2e-16 I:rho 1 4.28 2152 910.76 1.884e-05 I:kernel 4 0.23 2148 910.53 0.9135987 rho:kernel 4 18.76 2144 891.77 < 2.2e-16 --Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

*** *** *** *** *** *** *** *** *** ** *** ** ***

*** . *** * * ** ***

*** *** .

. *** *** *** *** *** ***

> summary(mod) Call: glm(formula = data$mrisk ~ (omega + p + u + E + psi + tau + I +

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rho + kernel)^2, family = Gamma(link = "log")) Deviance Residuals: Min 1Q Median -4.1465 -0.3370 -0.0601

3Q 0.2468

Max 1.4108

Coefficients: (Intercept) omega p u E psi tau I rho kernelNIG kernelNIGw kernel2Dt kernelUniform omega:p omega:u omega:E omega:psi omega:tau omega:I omega:rho omega:kernelNIG omega:kernelNIGw omega:kernel2Dt omega:kernelUniform p:u p:E p:psi p:tau p:I p:rho p:kernelNIG p:kernelNIGw p:kernel2Dt p:kernelUniform u:E u:psi u:tau u:I u:rho u:kernelNIG u:kernelNIGw u:kernel2Dt u:kernelUniform E:psi E:tau E:I E:rho E:kernelNIG E:kernelNIGw E:kernel2Dt E:kernelUniform psi:tau psi:I psi:rho psi:kernelNIG psi:kernelNIGw psi:kernel2Dt psi:kernelUniform tau:I tau:rho

Estimate Std. Error t value Pr(>|t|) -8.168e-01 4.525e-01 -1.805 0.071226 . 1.491e+00 7.992e-01 1.865 0.062311 . -1.421e+00 5.441e-01 -2.612 0.009064 ** -9.517e-02 1.183e-02 -8.047 1.39e-15 *** 4.627e-08 2.418e-08 1.913 0.055838 . -5.096e+00 4.921e-01 -10.356 < 2e-16 *** -1.594e+00 9.155e-01 -1.741 0.081752 . 3.870e-03 6.573e-04 5.887 4.54e-09 *** -1.298e-04 8.030e-05 -1.617 0.106039 2.028e-01 2.607e-01 0.778 0.436609 1.785e-01 2.668e-01 0.669 0.503573 -1.251e-01 2.607e-01 -0.480 0.631437 1.219e-01 2.607e-01 0.468 0.640106 6.671e-01 8.763e-01 0.761 0.446558 6.710e-02 1.628e-02 4.121 3.91e-05 *** -1.089e-07 3.760e-08 -2.897 0.003800 ** 6.791e-01 7.390e-01 0.919 0.358243 -4.155e-01 1.297e+00 -0.320 0.748703 -2.832e-03 1.039e-03 -2.725 0.006481 ** 9.831e-06 1.125e-04 0.087 0.930386 -2.193e-02 3.724e-01 -0.059 0.953045 -2.373e-01 3.809e-01 -0.623 0.533406 9.555e-02 3.724e-01 0.257 0.797506 1.212e-01 3.724e-01 0.325 0.744935 2.033e-02 9.716e-03 2.092 0.036566 * 1.045e-07 2.010e-08 5.198 2.20e-07 *** 3.540e+00 4.672e-01 7.577 5.23e-14 *** 1.848e+00 7.568e-01 2.442 0.014685 * -3.621e-03 6.062e-04 -5.973 2.72e-09 *** -5.196e-05 6.567e-05 -0.791 0.428962 -1.114e-01 2.238e-01 -0.498 0.618692 3.486e-02 2.308e-01 0.151 0.879969 -4.213e-02 2.238e-01 -0.188 0.850715 4.848e-01 2.238e-01 2.166 0.030418 * 1.661e-09 3.650e-10 4.552 5.62e-06 *** 2.137e-02 7.545e-03 2.833 0.004658 ** 6.147e-02 1.340e-02 4.588 4.73e-06 *** 2.458e-05 1.162e-05 2.114 0.034602 * 3.996e-07 1.168e-06 0.342 0.732414 -7.588e-03 3.852e-03 -1.970 0.049005 * -2.591e-03 3.963e-03 -0.654 0.513382 -1.620e-02 3.852e-03 -4.205 2.72e-05 *** 1.717e-02 3.852e-03 4.456 8.77e-06 *** 6.660e-08 1.694e-08 3.932 8.69e-05 *** -3.636e-08 2.805e-08 -1.296 0.195066 -5.574e-11 2.370e-11 -2.352 0.018770 * -2.299e-13 2.477e-12 -0.093 0.926068 -1.019e-08 8.155e-09 -1.250 0.211493 -1.008e-08 8.459e-09 -1.191 0.233684 -1.118e-09 8.155e-09 -0.137 0.890941 -4.723e-09 8.155e-09 -0.579 0.562552 -1.172e+00 6.333e-01 -1.850 0.064443 . -2.451e-04 5.067e-04 -0.484 0.628700 1.418e-04 5.585e-05 2.538 0.011218 * 1.260e-01 1.862e-01 0.677 0.498628 2.055e-01 1.908e-01 1.077 0.281577 -2.485e-02 1.862e-01 -0.133 0.893844 2.295e-01 1.862e-01 1.233 0.217873 -4.665e-03 8.141e-04 -5.730 1.15e-08 *** -3.037e-03 8.807e-05 -34.487 < 2e-16 ***

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tau:kernelNIG -1.359e+00 3.104e-01 tau:kernelNIGw -7.386e-01 3.158e-01 tau:kernel2Dt -1.116e+00 3.104e-01 tau:kernelUniform 6.671e+00 3.104e-01 I:rho -3.350e-07 7.591e-08 I:kernelNIG -1.216e-04 2.482e-04 I:kernelNIGw -1.373e-04 2.531e-04 I:kernel2Dt -5.541e-05 2.482e-04 I:kernelUniform 1.799e-05 2.482e-04 rho:kernelNIG -5.075e-05 2.722e-05 rho:kernelNIGw -3.964e-05 2.770e-05 rho:kernel2Dt -3.550e-05 2.722e-05 rho:kernelUniform 1.678e-04 2.722e-05 --Signif. codes: 0 *** 0.001 ** 0.01 * 0.05

-4.378 -2.339 -3.595 21.492 -4.413 -0.490 -0.542 -0.223 0.072 -1.865 -1.431 -1.304 6.167 . 0.1

1.25e-05 0.019414 0.000332 < 2e-16 1.07e-05 0.624222 0.587608 0.823387 0.942223 0.062367 0.152598 0.192314 8.32e-10

*** * *** *** ***

.

***

1

(Dispersion parameter for Gamma family taken to be 0.2337837) Null deviance: 2610.18 Residual deviance: 891.77

on 2216 on 2144

degrees of freedom degrees of freedom

References Devaux, C., Lavigne, C., Falentin-Guyomarc’h, H., Vautrin, S., Lecomte, J., and Klein, E. (2005). High diversity of oilseed rape pollen clouds over an agroecosystem indicates long-distance dispersal. Molecular Ecology, 14(8):2269– 2280. Dieckmann, U. and Law, R. (2000). The geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press. Durrett, R. and Levin, S. (1994). The importance of being discrete (and spatial). Theoretical population biology, 46(3):363–394. EFSA (2015). Panel on genetically modified organisms. updating risk management recommendations to limit exposure of non-target lepidoptera of conservation concern in protected habitats to bt-maize pollen. EFSA Journal, 13(7):4127. Filipe, J. and Maule, M. (2003). Analytical methods for predicting the behaviour of population models with general spatial interactions. Mathematical Biosciences, 183(1):15–35. Filipe, J. and Maule, M. (2004). Effects of dispersal mechanisms on spatiotemporal development of epidemics. Journal of theoretical biology, 226(2):125– 141. Filipe, J., Otten, W., Gibson, G. J., and Gilligan, C. A. (2004). Inferring the dynamics of a spatial epidemic from time-series data. Bulletin of Mathematical Biology, 66(2):373. Hofmann, F., Kruse-Plass, M., Kuhn, U., Otto, M., Schlechtriemen, U., Schr¨ oder, B., V¨ ogel, R., and Wosniok, W. (2016). Accumulation and variability of maize pollen deposition on leaves of european lepidoptera host plants 12

and relation to release rates and deposition determined by standardised technical sampling. Environmental Sciences Europe, 28(1):1–19. Hofmann, F., Otto, M., and Wosniok, W. (2014). Maize pollen deposition in relation to distance from the nearest pollen source under common cultivationresults of 10 years of monitoring (2001 to 2010). Environmental Sciences Europe, 26(1):24. Holst, N., Lang, A., L¨ ovei, G., and Otto, M. (2013). Increased mortality is predicted of inachis io larvae caused by bt-maize pollen in european farmland. Ecological Modelling, 250:126–133. Klein, E. K., Lagache-Navarro, L., and Petit, R. J. (2017). Demographic and spatial determinants of hybridization rate. Journal of Ecology, 105(1):29–38. Klein, E. K., Lavigne, C., Foueillassar, X., Gouyon, P.-H., and Lar´edo, C. (2003). Corn pollen dispersal: quasi-mechanistic models and field experiments. Ecological monographs, 73(1):131–150. Kot, M., Lewis, M. A., and van den Driessche, P. (1996). Dispersal data and the spread of invading organisms. Ecology, 77(7):2027–2042. Lang, A. and Otto, M. (2015). Feeding behaviour on host plants may influence potential exposure to bt maize pollen of aglais urticae larvae (lepidoptera, nymphalidae). Insects, 6(3):760–771. Lavigne, C., Klein, E. K., Mari, J.-F., Ber, F. L., Adamczyk, K., Monod, H., and Angevin, F. (2008). How do genetically modified (gm) crops contribute to background levels of gm pollen in an agricultural landscape? Journal of Applied Ecology, 45(4):1104–1113. Pleasants, J. M., Hellmich, R. L., Dively, G. P., Sears, M. K., Stanley-Horn, D. E., Mattila, H. R., Foster, J. E., Clark, P., and Jones, G. D. (2001). Corn pollen deposition on milkweeds in and near cornfields. Proceedings of the National Academy of Sciences, 98(21):11919–11924.

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