Details

Submodels

Cropping practices

Sowing and harvesting of aphid host crops is determined in input data. A practice delay can be set, that defines a sowing and harvesting window after the initial sowing and harvesting dates. For each parcel, a practice delay is attributed, either following a uniform distribution defined by its window, or following the practice delay given in the crop succession csv file. It defines the number of days after the crop initial sowing date at which the parcel is to be sown. The same delay from the crop initial harvest date is applied to the parcel harvesting date.

Crop practice parameters

Crop	Sugar beet	Legumes	Oilseed rape	Maize	Grass
Sowing (julian day)	82	77	237	98	1
Harvesting (julian day)	266	190	189	281	365
Possible delay (days)	15	15	10	10	0

Crop growth

Crop growth is explicitly modeled only for M. persicae main hosts: sugar beet, oilseed rape, potato and grass. Additionally, because a relay crop was needed to host the aphids after their dispersion from sugar beet, maize is also modeled as an aphid host. Crop development influences several processes: aphid growth and dispersion, and virus transmission efficiency from its vector to the plant.

Crop phenology is governed by daily mean temperature, as we consider the sum of degree-days between each crop's base and maximum temperature.

Crop development parameters. sumDD: sum of degree-days since sowing. emergence : sum of degree-days above base temperature from sowing to emerging. decline : from sowing to the start of aphid growth rate decline due to the decrease in plant quality. ^{1 2}

Crop	Sugar beet	Potato	Oilseed rape	Maize	Grass
Tbase (°C)	0	0	0	6	0
Tmax (°C)	28	28	27	24	22
\(sumDD_{emerge}\) (°C)	150	150	80	80	154
\(sumDD_{decline}\) (°C)	310	310	1200	240	314
\(\alpha\) (/°d)	0.029	0.029	0.029	0.029	0.029

The parcel is considered as not covered by vegetation until it reaches the sum of degree-days necessary for the current crop emergence ( \(sumDD_{emerge}\)). The crop is then suitable for aphid growth, which starts to decline when the sum of degree-days reaches \(sumDD_{decline}\).

Crop developmental stage is computed, for crop c and time step t, as : $$ D_{c,t} = 2 + (sumDD_{c,t} - sumDD_{c,decline}) \times \alpha $$

Aphid growth

Aphid can develop in different crop, mainly dicotyledons. Main suitable crops for aphid growth in sugar beet production areas in France are: sugar beet, oilseed rape, potatoes and grass. Aphid growth is dependent on daily mean temperatures, host crop family and development stage as follows.

Temperature

Literature shows that Myzus persicae growth is optimal on Brassica plants, and suboptimal on other plant families³. Following several different studies^{4 5 6 7}, growth rate of the aphid is modelled using Logan’s equation ⁸.

When the temperature is below the optimal temperature for M. persicae, on crop c: $$ r_c(T) = \psi_c \exp{(\rho_c \times T)} $$

With \(T\) the mean temperature up to the optimal temperature (25°C), \(\psi\) the developmental rate at the base temperature above the developmental threshold (Tb = 4.0°C) and \(\rho\) the rate increase to optimal temperature.

Above the optimal temperature: $$r_c(T) = Co_c [1 - exp{(-\tau)}] $$

With $$ \tau = \frac{(T_M - T)}{T_M - T_{opt}} $$

Were \(T_M\) is the lethal, maximum temperature in degrees above the base temperature and \(Co\) is a constant.

Parameters for aphid growth

	Brassica	Other families
\(\psi\)	0.3	0.09
\(\rho\)	0.07	0.07
\(Co\)	2.1	0.6
\(T_M\)	36.0
\(T_{opt}\)	25.0
\(T_{base}\)	4.0

Plant quality / Plant development

Crops nutritional quality for M. persicae growth decreases with the development of the plant⁹. To take this phenomenon into account, we modelled the aphid intrinsic growth rate (\(Ri\)) following Willliams 1995⁹ when the crop reaches a defined development stage as:

\[ Ri_t = r_c(T) - 0.00557 \times D_t \]

With \(r_c(T)\) the intrinsic growth rate on host crop \(c\) at temperature \(T\), and \(D_t\) the development stage of the host plant. Once maturity, or the 20-leaves stage for sugar beet, is reached, the crop is considered unsuitable for aphid growth (carrying capacity is set to zero). We do not consider any decline in crop nutritional quality due to the aphid feeding on the plant.

Population growth

The stock growth was then modelled as : $$A_{t+1} = A_t \times (1 + Ri_t) $$

With \(A_t\) the density of aphids at step t and \(Ri\) the intrinsic growth rate (depending on the temperature, the host plant family and development stage).

Winter survival}

Following Howling et al. 1994¹⁰, aphid survival during winter was modeled as depending on the number of days at low (from -1 to -6°C) and very low (<-6°C) mean temperature.

Survival (%) is computed as:

\[ S = \left\{ \begin{array}{ll} 93-2.16 \times nbD & \text{if } T \in [-6 : -1] ^\circ C \\[3pt] 88-10.77 \times nbD & \text{if } T \in [-9 : -6] ^\circ C \\[3pt] -50 & \text{if } T < -9 ^\circ C \\ \end{array} \right. \]

With \(nbD\) the number of consecutive days during which mean temperature was in the temperature interval.

Aphid movement

We consider two types of aphid movements across the landscape: migration from far away (large scale) and dispersion at the local scale.

Migration

During spring and autumn aphids are carried from far away by the wind and can land on the landscape. We use the model developed by Luquet et al.¹¹ to predict the aphid flight activity during spring. This model predicts flight onset dates, length of flight period and cumulative abundance of flying aphids using climatic and land-use predictors as well as geographical position. From these data, a gaussian was computed that represent daily migrating aphids, representing the number of flying aphids that would be caught into a succion trap. Following Fabre et al. 2010¹², and SimBAL's sensitivity analysis and calibration, for one aphid caught in a trap, we set to 2666 the number of aphids that colonize 1 ha of crop that is suitable for aphid growth. Migrating aphids starts reproducing as soon as they arrive in their landing cells, but are only able to disperse in the following time step.

Dispersion

The number of aphids leaving the stock of a cell depends on the number of aphids composing the stock, the developmental stage of the host plant and the presence of aerial natural enemies. The sink cell of aphids dispersing is determined by the distance between the source and sink cell, considering a dispersion kernel. When the nutritional quality of the crop host declines (with plant development), a fraction of the aphid stock will disperse. When natural enemies are detected, aphids produce winged morphs that will later disperse on other cells when they reach the adult stage (200 degree-days after birth). The proportion of migrating aphids that go to a specific cell depends on the distance to the source cell. Dispersion from a source cell to sink cells causes the decrease of the number of individuals of the aphid stock of the source cell and an increase of the stocks of the sink cells (if those cells are suitable for aphid growth).

Short-range movements are triggered mainly by the decline in quality of the host and the detection of natural enemies. Crowding does not seem to trigger aphid movement in M. persicae³. Winged aphids disperse on cells belonging to parcels not presenting bare soils, with preference for its host crops.

When a dispersal event occurs, all aphids from source cells take flight at the same time, and land on all the cultivated cells with a vegetation cover, depending on their attractivity relative to the source cells. The attractivity of each cell is computed for each dispersal event, considering the mean attractivity of the cell to each source cell weighted by the number of aphids leaving the source cells. We consider that an aphid can land in its source cell, the attractivity of a cell relative to itself being set to 0.99. Only host crops that have already emerged and that is not yet mature are attractive to aphids.

The attractiveness of a sink cell relative to a source cell is the inverse of the average distance an aphid can travel when flying. According to¹³, given that an aphid can fly between 0.8 and 3.3 km/h for a duration of 100 to 200 minutes¹⁴, and considering a repulsive parameter depending on the crop (host = 1; non-host = 0) the attractiveness of a cell cultivated with crop c at a distance d from the source cell is calculated as:

\[ A(d) = exp{(-\frac{3.6}{2 \times 60 \times 150} \times d)} \times r_c \]

Production of winged aphids when crop quality declines

To determine the number of aphids that disperse, following Muller et al. 2001³, a proportion of winged morphs produced is calculated depending on the crop developmental stage. We consider that winged aphids start to be produced at flowering, or at the 6-leaves stages for sugar beet, considering that crop quality starts declining then, and that the maximum production of winged morphs arrives at the start of grain filling, or the 21-leaves stage for sugar beet (approx. when the growth rate reaches zero following Williams 1995⁹). At plant maturity the aphids stop producing winged morphs.

Parameters of crop development: sum of degree-days necessary to reach the start, maximum and end of winged aphid production. Data for oilseed rape are derived from the calibration process

Crop	Sugar beet	Oilseed rape	Legumes	Maize	Grass
DD start production	350	1333	350	950	1400
DD max production	900	1517	900	1550	1900
DD end production	1100	1958	1100	1850	2170

The maximum proportion of winged aphid produced is set to 6% in oilseed rape, and 8% on other crop (following Williams et al. 2000¹⁵ and derived from the calibration process).

The number of dispersing aphids (\(W\)) due to crop quality decline in cell i is then computed as:

\[ W_i = \left\{ \begin{array}{ll} W_{max} \times \frac{DD_t - DD_{start}}{DD_{max} - DD_{start}} & \text{if } DD_t \in [DD_{start}:DD_{max}] \\[9pt] W_{max} \times (1 - \frac{DD_t - DD_{max}}{DD_{end} - DD_{max}}) & \text{if } DD_t \in [DD_{max}:DD_{end}] \\ \end{array} \right. \]

With \(DD_t\), \(DD_{start}\), \(DD_{max}\) and \(DD_{end}\) the sum of degree-days from sowing of time t, of the start, maximum and end of winged morph production respectively (see above table).

Production of winged morphs when aphids detect natural enemies

Aphids produce winged progenies when they detect natural enemies. We assume that aphids are homogeneously spread over the cell, therefore the proportion of aphids that detect the presence of flying enemies depend on the number of natural enemies and the area they cover. We consider that one natural enemy can be detected in a 0.1m\(^2\) area, which correspond to one sugar beet plant considering a crop density of 10 per m\(^2\). Therefore, the production of winged morphs is computed as:

\[NW_i = \sum_{j} (NE_{i,j} \times \frac{0.1}{A_i}) \times N_i \times MW\]

With \(i\) the cell, \(NW\) the number of winged morphs produced, \(j\) the aerial natural enemies, \(NE\) the density of the natural enemy, \(A\) the crop area of the cell, \(N\) the total number of aphids in the stock and \(MW\) the maximum proportion of winged morph produced. We consider a development time of 200 degree-days for the development of aphids from birth until adulthood, after which the aphid takes flight and joins the aphid stock of its landing cell.

Biocontrol

The formalism of natural enemy’s biocontrol on aphid population is adapted from Jonsson et al. 2014¹⁶. We consider families of natural enemy that feed on \textit{M. persicae} (ground beetles, wolf and sheet-web spiders, lady beetles, rove beetles, hoverflies and lacewings) and parasitize them (parasitoids). The abundance of each of these natural enemies is computed depending on the surrounding landscape composition and configuration following Martin et al. 2019¹⁷, and corrected using data on natural enemies abundance on sugar beet fields in France. Predators consume aphids following a Holling type III response, while parasitoids parasitize following a Holling type II response.

Functional group of each family of natural enemy¹⁷ and condition for the end of their wintering. \(T_m\) : mean temperature over two days. \(DD\) : degree-days.

Family	Activity	Overwintering habitat	Dispersal mode	Condition end wintering
Parasitoids	Parasitism	Non crop	Flight	\(d=1\) *
Ground beetles	Predation	Crop, non crop	Ground	\(T_m \geq 12\)°C after 15/03
Wolf spiders	Predation	Non crop	Ground	\(T_m \geq 10\)°C after 01/02
Sheet-web spiders	Predation	Non crop	Wind	\(T_m \geq 10\)°C after 01/02
Rove beetles	Predation	Crop, non crop	Ground	\(\sum_{T_0 = 5.6} DD = 400\)°C from 01/04
Lady beetles	Predation	Non crop	Flight	\(\sum_{T_0 = 12} DD = 60\)°C from 01/01
Hoverflies	Predation	Crop, non crop	Flight	\(T_m \geq 12\)°C after 15/03
Lacewings	Predation	Non crop	Flight	\(T_m \geq 10\)°C after 01/05

Landscape effect

Following Martin et al. 2019¹⁷, we consider the proportion of semi-natural habitats (\(SNH\)) and the mean edge density (\(ED\)) at a 2km radius around the focal cell. The landscape predictors are ln(x+1)-transformed.

Abundance \(Y\) is computed as :

\[ Y = b_0 + b_1 \times SNH + b_2 \times SNH^2 + b_3 \times ED + b_4 \times ED^2 + \\ \hspace{8mm} b_5 \times ED \times SNH + b_6 \times SNH \times ED^2 + b_7 \times ED \times SNH^2 \]

where \(b_0\)...\(b_7\) are estimates for each term, specific to each family of natural enemy depending on their functional group.

We used data on natural enemies abundance in sugar beet fields in France from the PNRI project IAE-B to correct the predictions given by the model, as the study area is quite intensive compared to the ones used in Martin et al. 2019. To do so, SimBAL computes the enemies abundances with the mean of the landscape descriptors over the three landscapes used for the calibration process, which is used in combination with mean field data to correct the predicted enemies abundances in each cell as follows:

\[Y'_{f,i} = Y_{f,i} \times \dfrac{IAE_{f}}{\overline{Y}_{f,c}} \]

with \(f\) the natural enemy family, \(i\) the cell, \(Y\) the abundance prediction, \(\overline{Y}_{c}\) the mean prediction results over the calibration landscapes and \(IAE\) the field data.

Ground beetles are an important group of natural enemies that overwinter in oilseed rape fields. According to Marrec et al. 2017¹⁸, presence of oilseed rape, winter cereals and spring crops the preceding year and current year influence the abundance of carabid beetles.

Therefore, we compute the abundance \(Y^*\) of cell \(i\) as :

\[ Y^*_i = Y'_i + \sum_{y} \sum_{c} a_{y,r,c} \times \%C_{i,r,y} \]

with \(y\) the year (previous or current), \(C\) the considered crop (oilseed rape, winter cereal or spring crop), \(a\) the estimate and \(\%C\) the standardized proportion of the crop \(C\) in the surrounding landscape at a distance \(r\).

Activity

Natural enemies exit wintering with the rise in daily mean temperatures. Conditions for overwintering end depend on the natural enemy family. When predators are active they forage for aphid preys.

Parasitoids are potentially critical for aphid biocontrol as they emerge early in the season and are therefore able to control aphid populations before they grow exponentially. Following Zamani et al. 2007¹⁹, we compute a developmental index as follows.

We compute the daily development rate as: $$ R(T) = a \times T (T - T_0) \times \sqrt{T_{max} - T}$$

with \(T\) the daily mean temperature, \(a\) a constant, \(T_0\) the lower developmental threshold, \(T_{max}\) the lethal maximum temperature.

From Zamani et al. 2007¹⁹, we computed the mean of each parameter predicted by the model (a = 0.0001, \(T_0\) = 4.66 and \(T_{max}\) = 32.86). From the 1st of November, when daily mean temperatures fall below 10°C, we consider that parasitoids stop their parasitic activity²⁰, and the offsprings start developing within the aphids. The development index \(d\) is then set to zero. Each following day, \(d\) is incremented if \(T \geq T_0\)

\[ d_{t} = d_{t-1} + R(T) \]

When \(d\) reaches 1, parasitoids emerge and start parasitizing aphids, only when daily mean temperatures are above 10°C.

Developmental stages

Natural enemies can have differentiated regulatory effects depending on their developmental stage. In this model we explicitly consider the life cycle of lady beetles, following Xia et al. 1999²¹. The following table gives the sum of degree-days necessary to complete the different stages, and the survival rate of each stage. Adult disperse, before oviposition and after emerging as an adult, if the density of aphids in the cell is insufficient (aphid/m² \(<\) 10). They disperse at maximum distance of 300m, with a dispersion kernel of 0.02. An individual will lay 100 eggs.

Life history of lady beetles. sumDD with Tbase 0°C. ^ stages that consume aphids.

adult^	oviposition	egg	larvae^	pupa
sum DD	622	313	91	318
%survival		59	81	52
sum DD for dispersal	313

Biocontrol effect

The per capita consuming or parasitism rate depends on the maximum attack rate and the handling time of each family of natural enemies. We use a Holling type II functional response for parasitoids, considering that parasitoids are specialists and therefore use cues to detect their host²⁰.

$$ \Delta N = P \times \frac{a \times N}{1 + a \times Th \times N}$$ with \(\Delta N\) the number of aphids parasited, \(P\) the number of parasitoids in the cell, \(a\) the attack rate and \(Th\) the handling time.

We use a Holling type III response for predators, since they are generalist species that will be able to consume other preys and will therefore be less efficient at low aphid densities to reduce aphid stocks. $$ \Delta N = P_p \times \frac{a \times N^2}{1 + a \times Th \times N^2}$$ with \(\Delta N\) the number of aphids parasited, \(P_p\) the number of predator p in the cell, \(a\) the attack rate and \(Th\) the handling time.

As the per capita attack rate is dependent on aphid densities, natural enemies affect aphid stock sequentialy. Parasitoids being more efficient at finding ther prey, they are the first to reduce aphid stocks. Then, predators' order is set randomly at each time step.

Parameters for biocontrol (from references and calibration process)

Enemy	Response	Attack rate (a)	Handling time (Th)	reference
Parasitoids	II	0.025	0.05	20
Ground beetles	III	0.007	0.1	22
Wolf spiders	III	0.0055	1	23
Sheet-web spiders	III	0.0023	0.25	24
Lady beetle (adult)	III	0.0023	0.015	25 26
Lady beetle (larvae)	III	0.0045	0.0125
Rove beetles	III	0.0057	0.1	27
Hoverflies	III	0.001	0.03	28
Lacewings	III	0.0025	0.03	29

Pesticide effect

We consider that insecticide spray affects aphid stock of all the cells of a parcel. A parcel is sprayed when an aphid density threshold is reached in at least one of its cells, provided that this parcel was not treated previously (return period) and that the maximum number of treatments has not been reached. When a treatment occurs, aphid stocks are reduced by the treatment efficiency (set to 80%³⁰). To consider residual efficiency of the treatment, for 14 days the arriving aphids’ numbers (from the aphid rain and dispersion, value set after calibration) are also reduced by 80\%. We consider that these treatments have negative effects on aerial natural enemies. Their densities are reduced by a different %, which were set from the calibration process. After the residual period is over, natural enemies densities are computed depending on the surrounding landscape as before.

Negative effect of pesticide treatments on natural enemies, values derived from the model calibration

Ground beetle	Lady beetle	Parasitoid	Wolf spider	Sheet-web spider	Hoverfly	Rove beetle	Lacewing
0.67	0.5	0.5	0.33	0.33	0.5	0	1

Disease spread

Following Werker et al. 1998³¹, the spread of the virus depends on a primary infection (virus coming from outside the cell: brought by migrating and dispersing aphids) and the secondary infection (virus spreading within the cell through walking aphids).

Primary infection

Virus reservoirs are not explicitly localized in the model. To introduce the virus in the landscape, we consider that a fixed proportion of flying aphids (aphids colonizing the landscape through the aphid rain and aphids dispersing from cell to cell) is viruliferous. In addition, aphids dispersing from infected cells will be able to inoculate plants in their arrival cell. In order to determine how many viruliferous aphids will land in each cell cultivated with sugar beet, sink cells are randomly sampled with replacement, weighted by the mean attractiveness of each sink cell relative to each source cell multiplied by the number of viruliferous aphids taking flight from these cells.

The viruliferous proportion of total migrating aphids is calculated depending on the infection degree of the source cells weighted by the number of migrants taking off from these cells. Cells on which these viruliferous aphids land are sampled depending on the cells attractiveness (weight). The proportion of plants infected from the primary infection on cell i at time t, entering a latency state, is computed as: $$ Pc_{i,t} = [A_{i,t} \times \%V_{migr} + M_{i,t} \times \%V_{disp}] \times H_{i,t} \times \tau_{i,t} $$

with \(A\) the number of aphids immigrating from the aphid rain, \(\%V_{migr}\) the proportion of viruliferous aphids from the aphid rain, \(M\) the number of aphids immigrating from other sugar beet cells of the landscape, \(\%V_{disp}\) the mean infection degree of dispersal sugar beet source cells, \(H\) the proportion of healthy plants in the cell and \(\tau\) the transmission efficiency.

Secondary infection

Considering that from each infectious plant, a maximum number of eight neighbouring plants can be infected at each time step, and that several viruliferous aphids can reach the same plant, the proportion of plants infected by walking viruliferous aphids, entering a latency state, in cell i at time t is: $$ Pw_{i,t} = I_{i,t} \times 8 \times [1 - exp{(-b_t \times [w_{i,t} \times (1 - \%w_{NE,i,t}) \times N_{i,t} \times \tau_{i,t}]^k)}] \times H_{i,t} $$

With \(I\) the proportion of infectious plants in the cell, \(w\) the proportion of walking aphids in cell i, \(\%w_NE\) the proportion of aphid walking due to the detection of natural enemies, \(N\) the total number of aphids in the cell, \(b\) and \(k\) parameters of a Poisson distribution, \(H\) the proportion of healthy plants and \(\tau\) the transmission efficiency.

Aphids can walk from plant to plant through touching leaves when plants are developed enough. They can also go through the ground, with a lower success rate due to the presence of natural enemies. We consider that at any time step 15% of the aphid population is walking (^{32 33} and calibrated). When neighbouring plants are not in contact (developmental stage below 8-leaves), b is equal to 0.07, otherwise b equals 0.11 (calibrated). k is set to 0.9.

When aphids detect the presence of aerial natural enemies, a greater proportion of them move. We consider that all aphids that detect a natural enemy will be walking. Therefore, the proportion of aphids that will move due to natural enemies’ detection is: $$ N_{NE} \times \frac{0.1}{surfcell} $$

Transmission efficiency

Considering that infection can occur after the 2-leaves stage, that the maximum efficiency of inoculation of a plant from one viruliferous aphid is 0.6 (Brault et al., unpublished), and that the transmission efficiency decreases with plant age, the transmission efficiency in cell \(i\) at time \(t\) is computed as: $$ r_{i,t} = 0.6 \times \frac{112.9 - 10.06 \times D_{i,t}}{100} $$

with \(D\) the developmental stage of sugar beet plants (ie. the number of leaves).

\paragraph{Latency period} Continuous phenomenon: we consider that at each time step, 6.88\% of plants in the latency state become infectious (Jacquot et al., unpublished).

\[ L_{i,t} = L_{i,t-1} \times (1-l) + Pc_{i,t} + Pw_{i,t} \]

\[ I_{i,t} = I_{i,t-1} + L_{i,t-1} \times l \]

Yield loss

Yield loss due to infection with yellows virus depend on the developmental stage of the crop at the time of infection as follows ^{34 35 36}:

\[ YL_{i,t} = \left\{ \begin{array}{ll} YL_{i,t-1} + 0.3 \times (Pc_{i,t} + Pw_{i,t}) & \text{if } D_{i,t} < 5 \\[3pt] YL_{i,t-1} + (0.3 - 0.016875 * (D_{i,t}-5)) \times (Pc_{i,t} + Pw_{i,t}) & \text{if } D_{i,t} \geq 5 \\ \end{array} \right. \]

with \(D_{i,t}\) the developmental stage of sugar beet plants (ie. the number of leaves) in cell \(i\) at time \(t\).

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