Supplementary materials 1

Model parameters

Landing on crops

Aphids landing rate (named ) depends on the proportion of wheat fields and of their disposition :

where p is the proportion of wheat fields and η is spatial autocorrelation of wheat fields. a, a1, a2and k are parameters required to fit the model (Ciss et al., 2013) and are without units:

  • a1 =1.004855
  • a2 =1.652655
  • a =-0.267819
  • k =6.214583

Fromthe field counts of Vialatte et al.(2007) the mean number of winged S. avenae landing on wheat in theRennesareain 2004 was 2.65 /m²/day (range:0 – 16) during population growth period.Our model predicts comparable values of C.: mean number of 1.84 aphid/m²/day (range: 0 – 31).

For all our simulations, the mean of the landing rate at the France scale is 0.1265 and the variance is 0.03440516

Multiplication on crops

Aphid growth rate (named r) is derived from weekly aphid counts in 29 wheat fields, done between 1975 and 2011 in two important French cereal growing areas, the Rennes basin representative of the oceanic zone (cumulated minimum temperatures of January >0) where aphids commonly overwinter in cereal fields, and the Paris basin, representative of the ‘continental zone’ (France ‘mainland’ where cumulative minimum temperatures of January <0) where aphids do not overwinter or overwinter weakly.

r(aphids/aphid/day) depends on mean temperature θ (in °C) and wheat growth stage s (in numerical Zadoks scale(Zadoks et al., 1977):

Where M is the maximum temperature for S. avenae reproduction (Dean, 1974), sM the later wheat growth stage allowing aphid feeding, sm the position of the left inflexion point for the response to the wheat growth stage. a1, a2, b and k are parameters required to fit the model.

As each zone involves a particular dynamics of aphid populations during spring and summer, two different r were calculated separately with each of the two series of aphid counts, and used for modelling populations in the two different zones respectively (Ciss, unpublished)

For oceanic zone: sm=91.747, b=0.041, k=-0.113, a1=-0.132, a2=0.132, M =30 and sM=92. These parameters have no units

and for continental zone: sm =32.90865, b =-0.05322, k =1.10676, a1=-0.53372, a2=0.53372, M =30 and sM=92. These parameters also have no units

From laboratory experimentsof Dean (1974) andSalman (2006),the range in the values of S. avenae daily rate of increase is [0.11-0.31] and [0.10-0.30] respectively, depending on temperature. In the results of our model, the range of r values during the growing period of aphid populations is [0.18-0.33], which is close from the above laboratory results.

For all our simulations, the mean of aphid growth rate at the France scale is -0.2024 and the variance is 0.223. The negative value of the mean is explained by the sharp decrease of aphid populations after their culmination at the whole national scale during simulations.

Take-off

We define a take-off rate named α2. It depends on:

  • which represents the proportion of aphids taking-off in function of the temperature (Walters and Dixon, 1984). If temperature is lower than 14°C, aphids do not take off.
  • which represents the proportion of winged larvae produced in function of the total number of aphids in the fields (A). This variable is derived from field counts (see above)

With:

where:

  • θ is the temperature in °C
  • s is the wheat growth stages in numerical Zadoks scale (Zadoks et al., 1977)
  • A is the apterous aphids number

a, a0, a1, a2, a3 and a4 are parameters required to fit the model: a=0.034733, a0=1.417143, a1=0.048163, a2=-0.543663, a3=-6.93706 and a4=0.3430935 (these parameters have no unit)

The second part of the coefficientα2() is directly taken from the literature, as explained above and the first one is calculated using field data (Ciss, unpublished). These data are very similar to those of Watt and Dixon (1981) and both give a [0-100%]range of winged S. avenae produced in the field, depending on the size of the total aphid population and on wheat growth stage.

For all our simulations, the mean of the take-off rate at the France scale is 0.0618 and the variance is 0.026

Supplementary materials 2

Numerical scheme

In this section, we use an operator splitting with two half-time steps. For the first half-time step, we solve the part linked to the reaction through a Runge-Kutta 4 approximation; whereas for the second half-step, we solve the part linked to the diffusion, with an upwind scheme for the term of order 1. For the second half-step, we use the alternating direction methods of Douglas, Peaceman, and Rachford (Douglas, 1955; Peaceman and Rachford, 1955; Douglas, 1962) with .Let consider the spatial domain Ω=[a,b] x [c,d] and the temporal domain [0,T]. Let I, J and N be positive integers andk=T/(N-1) and ?=(?−?)/(?−1)=(?−?)/(?−1); h is named the time step (in day) and k the space step (in km). We divide the spatial domain in (I-1) x (J-1) subdomains and the temporal in domain N-1 subintervals with the grid points x1i=a+(i-1)k; x2j=c+(j-1)k and tn=(n-1)h for i=1,2,…,I, j=1,2,…,J and n=1,2,…,N.

The spatial derivatives of order 2 are approximated by finite difference quotients:

  • Reaction:

Where: , and represents the classical Runge-Kutta applied to the function f

  • Diffusion

.

References

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Dean GJ (1974) Effect of temperature on the cereal aphids Metopolophium dirhodum (Wlk.), Rhopalosiphum padi (L.) and Macrosiphum avenae (F.)(Hem., Aphididae). Bulletin of Entomological Research 63: 401-409

Douglas J (1955) On the numerical integration of uxx+ uyy= utt by implicit methods. Journal of the Society of Industrial and Applied Mathematics 3: 42-65

Douglas J (1962) Alternating direction methods for three space variables. Numerische Mathematik 4: 41–63

Peaceman DW, Rachford HH (1955) The numerical solution of parabolic and elliptic differential equations. Journal of the Society for Industrial and Applied Mathematics 3: 28–41

Salman AMA (2006) Influence of temperature on the developmental rate and reproduction of the English grain aphid, Sitobion avenae (Fabricius) (Homoptera : Aphididae). Arab Univ. J. Agric. Sci., Ain Shams Univ., Cairo 14: 789-801

Vialatte A, Plantegenest M, Simon JC, Dedryver CA (2007) Farm-scale assessment of movement patterns and colonization dynamics of the grain aphid in arable crops and hedgerows. Agricultural and Forest Entomology 9: 337-346

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Watt AD, Dixon AFG (1981) The role of cereal growth stages and crowding in the induction of alatae in Sitobion avenae and its consequences for population growth. Ecological Entomology 6: 441-447

Zadoks JC, Chang TT, Konzak CF (1977) Un code décimal pour les stades de croissance des céréales. Phytiatrie-Phytopharmacie 26: 129-140