## Numerical Experiments

The Complete Grape Growing System

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An example of field data is available (Calonnec et al., personal communication) of a powdery mildew epidemic over a 5-row vineyard. It shows that without fungicide treatment the disease invades the entire vineyard within 3 months. We make a simulation of this particular vineyard. Each row is 66 m long and 0.5 m wide, and the distance between two rows is 1.5 m.

We choose a rectangular computing domain ^ such that the 5 rows are located at the center of the domain and ^ is 3 times larger than the vineyard. As mentioned before, by doing so, the Dirichlet conditions at the boundary of ^ describe the fact that the spores may freely disperse out of

The parameters of the model as well as roughly realistic values are listed in Table 2.1. With these parameters, the basic reproductive rate of the disease in the homogeneous case is Ro = 10. We now explain how the values of the dispersal parameters Sa = 82 and Di were estimated.

All spores lifted up in the atmosphere fall within half an hour so the deposition rates 8 are more or less equal to 50 day-1.

To estimate the diffusion coefficients D1 and D2, we focus on the spore dispersal mechanism alone. Let D be a diffusion coefficient and 8 a deposition rate; the density S of spores dispersed in the atmosphere and produced by a single source obeys the following equation:

I dS(x, t) = V. (DVS(x, t)) - 8S(x, t), V(x, t) e R2 x R+ \ S(x, 0) = Dirac(x), Vx e R2 '

where Dirac (x) is the Dirac function. Then the total amount of fallen spores upon the vineyard at some point x e R2 is r d(x) = 8S(x, t) dt,

where d(x) is the the probability density of fallen spores. It can be explicitly computed and its variance is a = The values of D1 and D2 in Table 2.1 have been chosen so that a = 1m for the short range dispersal, and a = 20 m for the long range dispersal.

We start the infection at t = 0 with one latent colony at the center of the vineyard over one vine stock. For simplicity, we take an initial uniform site density for all the vine stocks. Hence, the initial conditions are H0 (x) = 4 m 2 colony sites and L0(x) = I°(x) = R0(x) = 0 for x in the rows except for x e [-1/4; 1/4]2 where H0(x) = 10(x) = R0(x) = 0 and L0(x) = 4 m-2 colony site density. We also set S0(x) = S0(x) = 0 for all x.

### The duration of the simulation is 90 days.

Results of the simulation for these parameters are displayed on Fig. 2.1-2.5. They show the proportion of diseased colony sites with respect to spatial location P(x) = D(x)/N(x) = (L(x) + I(x) + R(x))/N(x)) 30, 60 and 90 days after the beginning of the infection (Fig. 2.1-2.3). The epidemic first invades the central row of

2 Modeling of the Invasion of a Fungal Disease over a Vineyard 17 Table 2.1. Model parameters.

 Parameter Description Value ¿1 short range deposition rate 50 day-1 ¿2 long range deposition rate 50 day-1 D1 short range diffusion coefficient 50 m2 day-1 D2 long range diffusion coefficient 20,000 m2 day-1 rP spore production 104 spores day-1 colony site-1 F short range vs. long range dispersion 0.8 E inoculum effectiveness 0.1% P latency period duration 10 days i infectious period duration 10 days K carrying capacity of the colony sites 40 m-2 colony sites r growth rate of the colony sites 0.1 day-1

the vineyard (day 30) then it reaches the other rows until almost all the vineyard has been contaminated at day 90.

We also display the short and long range spore density with respect to the spatial location at day 90 in Fig. 2.4 and 2.5. Short range spores mostly stay over the row where they are produced whereas the distribution of long range spores is more uniform. The lower spore density in the central row is due to the fact that the corresponding colonies have attained the postsporulating phase.

proportion of diseased surface at day 30

proportion of diseased surface at day 30

Fig. 2.1. Proportion of diseased colony sites in the vineyard at day 30.

proportion of diseased surface at day 60

proportion of diseased surface at day 60

Fig. 2.2. Proportion of diseased colony sites in the vineyard at day 60.

Fig. 2.2. Proportion of diseased colony sites in the vineyard at day 60.

Finally, we investigate the influence of the parameter F over the intensity of the epidemic, keeping other parameters of the simulation at the same values as above. If F = 0 only long range dispersion takes place. The proportion of diseased colony sites is displayed in Fig. 2.6 at day 90. Compared with Fig. 2.3, the disease intensity is very low in each row. If F = i only short range dispersion takes place. As shown in Fig. 2.7, the epidemic has attained its maximum intensity but only in the main part of the central row whereas the other rows have not been contaminated.

proportion of diseased surface at day 90

proportion of diseased surface at day 90

Fig. 2.3. Proportion of diseased colony sites in the vineyard at day 90.

density of spores S1 (short range) at day 90

density of spores S1 (short range) at day 90

Fig. 2.4. Density S1 of short range dispersal spores over the vineyard at day 90.

As pointed out in [11,13], the rate of expansion of the epidemic needs both short and long range dispersal of its vectors to reach an optimal value. This is even more evident in the case of separate rows of vine: without long distance dispersal, the disease hardly reaches the rows where the initial contamination did not take place, while without short distance dispersal, local extension of the disease is not strong enough to ensure a high level of contamination.

density of spores S2 (long range) at day 90

density of spores S2 (long range) at day 90

Fig. 2.5. Density S2 of long range dispersal spores over the vineyard at day 90.

proportion of diseased surface at day 90

proportion of diseased surface at day 90

Fig. 2.6. Proportion of diseased colony sites at day 90—long range dispersal only.

Fig. 2.6. Proportion of diseased colony sites at day 90—long range dispersal only.