In this section, we first present some biological features of the two pathogens that we deemed necessary to understand our modelling choices. We then describe the implementation of the model before finally presenting the simulation scenarios.

## Wheat leaf rust and septoria tritici blotch

Wheat leaf rust (*Puccinia recondita* f. sp. *tritici*, hereafter WLR) and Septoria tritici blotch (*Zymoseptoria tritici*, hereafter STB) share biological features common to many leaf pathogenic fungi (van Maanen and Xu 2003; Caubel et al. 2012; Garin et al. 2014). In particular, they cause polycyclic diseases: spores that fall on leaves of susceptible plants germinate and infect the leaf where they create new lesions. The time between infection and the onset of reproduction of lesions is called the latent period. At the end of this period, lesions become mature and start releasing spores. Spores disperse and can initiate new lesions on the same leaf, on other leaves of the same plant or on the leaves of new susceptible hosts nearby (in the field or further). The number of infection cycles determines the intensity of the annual epidemic.

Based on previously published comparisons between WLR and STB (Robert et al. 2005; Garin et al. 2014), we considered four main differences between their life cycles that may lead to contrasted responses to crop diversification and weather variables: (i) duration of the latent period, (ii) dispersal ability (dispersal mode and range), (iii) start date of the epidemic depending on weather conditions, and (iv) trophic behaviour and associated capacity to survive the interculture. Trait differences are reflected by differences in parameter values in our model (Table 1). In this section we explain the four main differences considered and how we expressed these differences in our model.

First, STB has a longer latent period, and thus a longer infection cycle than WLR (Précigout et al. 2020a). The duration of an infection cycle, and the number of infection cycles during a cropping season, are key pathogen characteristics interacting with crop growth, determining the outcome of the crop-pathogen interaction and therefore the amount of damage caused by the pathogen (Robert et al. 2008, 2018; Précigout et al. 2017; Garin et al. 2018). This is why we aimed to incorporate this important aspect in our classical SEIR model by implementing an age structure of the lesions when they are latent (see Eqs. 21-23 below). In doing so, we ensure that any amount of leaf surface newly infected will remain asymptomatic during a full latent period (\(\lambda\) parameter) before becoming infectious. To our knowledge, this is the first combination of a semi-continuous SEIR model and a discrete modelling of the latent period of the pathogen. The latent period of pathogenic fungi is particularly sensitive to temperature (Précigout et al. 2020a). This is why in our model, we chose to express both plant and pathogen development (and thus latent periods) in thermal time (degree-days, dd) (Robert et al. 2008, 2018; Garin et al. 2014; Précigout et al. 2017).

Second, STB asexual spores mainly disperse through raindrop splashes from infected leaves (Eyal 1987) while WLR asexual spores are mainly dispersed by wind (Sache 2000). Consequently, STB asexual spores disperse over shorter distances (up to one meter within a field (Saint-Jean et al. 2004)) while wind allows WLR asexual spores to disperse over short (within a field) and long (outside the field) distances (Sache 2000; Mundt et al. 2011). Furthermore, because the dispersal of STB asexual spores occurs only when rainfall is of sufficient intensity (Walklate 1989), dispersal events of STB are less frequent than those of WLR, for which dispersal takes place almost every day as long as air humidity is not too low (Duvivier et al. 2016). In a much lower proportion, STB also produces sexual spores dispersed by wind that can leave their native field. Their role in epidemic propagation was considered as low (Suffert and Sache 2011), but recent studies question their importance in particular for long distance dispersal, survival in dry weather and inoculum production. Suffert and Sache (2011) and Suffert et al. (2019) indeed showed that though usually rare, these long-distance dispersal events could be of significance since they can lead to the infection of neighboring fields. This is why in our model we consider both asexual and sexual spores for STB. This is an additional originality of our model. We modelled the specific spore dispersal characteristics via different dispersal types (*d* function, Eqq. 10 and 11), different maximum dispersal distances (Δ parameter, Eq. 12) and different behaviours concerning dispersal outside the native field (α parameter, Eq. 10).

Third, in addition to the impact on dispersal, weather conditions also influence the onset of epidemics. In western Europe, winter wheat is sown in late October and STB epidemics start after seedling germination, when inoculum is splashed from the local crop residues (Suffert and Sache 2011; Morais et al. 2016). In our model, wheat seedlings can be infected from the first rainfall after plant germination if any inoculum is present in the field (as in Robert et al. (2008, 2018) and Baccar et al. (2011)). By contrast, WLR epidemics usually begin between late March (early epidemics, 800 dd after sowing) and May (late epidemics, 1300 dd after sowing (El Jarroudi et al. 2014; Duvivier et al. 2016)). The date of the onset of the epidemic depends on weather conditions (weather should be warm enough) and also it requires the presence of an inoculum, which can be external or internal to the field (Sache 2000). Moreover, the date of the onset of the epidemic is known to determine the intensity of epidemics: the earlier they start, the more intense they are (Garin et al. 2018). In our model, this difference is reflected in the onset date (*k**s*, Eq. 9) of the epidemic that depends on the first rain for STB and that is set between March and May for WLR, reflecting different climatic spring conditions.

Fourth, STB is a hemibiotrophic pathogen that infects living tissue but kills the infected tissue to reproduce, while WLR is a biotrophic pathogen that needs living host tissue to live and reproduce (Perfect and Green 2001; Précigout et al. 2020a). This trophic characteristic has an impact on spore survival. STB can indeed survive the interculture and overwinter on dead crop residues (Suffert and Sache 2011), whereas WLR can only survive the interculture period in the presence of infected residual live plants or volunteer plants (Roelfs and Bushnell 1985; Eversmeyer and Kramer 1998). This difference is reflected in our model by a higher survival capacity during the interculture of STB (\(\theta\)parameter, Eq. 14).

## Model overview

In our model, the landscape is a grid of 21 x 21 square fields. Fields are the elementary units of our model. Each field is planted with a resistant or susceptible crop, or a mixture of both. The resistant crop, which could be a resistant wheat cultivar or a different crop species, is assumed to be totally resistant to the disease. Both susceptible and resistant crops follow the same annual growth pattern. A year in the model corresponds to a crop growing season. We express both plant and pathogen growth in thermal time (degree-days, dd), facilitating the description of epidemics, whose development follows plant growth and its response to temperature. The disease dynamics, modelled at the field scale, correspond to a classical SEIR epidemiological model with inclusion of an age structure for the latent (E) compartment in order to model the pathogen’s latent period as a discrete time period rather than a transmission rate between E and I. The model keeps track of the number of spores released during each dispersal event, both within the field and outside the field. We consider different routes of infection depending on the type of inoculum. At the landscape scale, infected fields are linked together through pathogen dispersal. Table 1 gives the list of the model parameters along with their values and biological interpretation.

## Crop growth and seasonality of susceptible crops

The healthy canopy of the susceptible crop is represented by its green leaf area index S ("susceptible" leaf area). Based on published empirical data (Hinzman et al. 1986; Benbi 1994; Forsman and Poutala 1997; Baccar et al. 2011; Huang et al. 2016), the wheat growth curve is simulated using a logistic model. *S* increases from sowing (*k**init* = 0 dd) to \({k}_{growth}=1400\) dd, where *k* represents the discrete time index within a year. After \({k=k}_{growth}\), *S* becomes senescent at a rate \(\mu\) and is transformed into *R* (“removed” leaf surface). Hence, in a healthy canopy, the total leaf area index at time *k* is \({LAI}_{k}={S}_{k}+{R}_{k}\). The dynamics of \({S}_{k}\) and \({R}_{k}\) are given by:

$${S}_{k+1}={S}_{k}+g\left(k\right)-\mu {S}_{k}{1}_{k\ge {k}_{growth}}\left(k\right)$$

(Eq. 1)

with \(g\left(k\right)\) being the growth rate of the crop corresponding to a logistic equation. *K* represents the field carrying capacity (corresponding to the maximum value of the leaf area index) and \(\beta\) the crop growth parameter of the logistic function.

$$g\left(k\right)=\beta {S}_{k}\left(1-\frac{{S}_{k}}{K}\right){1}_{k<{k}_{growth}}\left(k\right)$$

(Eq. 2)

and

$${R}_{k+1}={R}_{k}+\mu {S}_{k}{1}_{k\ge {k}_{growth}}\left(k\right)$$

(Eq. 3)

Note that in the following,

$${1}_{condition}\left(k\right)=\{\begin{array}{c}1 \text{when the condition is fulfilled}\\ 0 \text{otherwise}\end{array}$$

## Disease dynamics

*SEIR dynamics.* Fields planted with susceptible crop display SEIR epidemiological dynamics. In the following, *S*, *E*, *I* and *R* denote the surface (in square meters per square meter of ground, LAI unit) of healthy, latent, sporulating and senescent (removed) plant tissue, respectively. Susceptible tissue becomes exposed at a rate \(c\left(k\right)\). Exposed tissue becomes infectious at a rate \(h\left(k\right)\). As time goes on and spore dispersal occurs, older sporulating structures get progressively empty at a rate \(\psi \left(k\right)\). Natural senescence affects all parts of the canopy at the same rate \(\mu\). The dynamics of the different leaf compartments can thus be given by:

$${S}_{k+1}={S}_{k}+g\left(k\right)-c\left(k\right){S}_{k}-\mu {S}_{k}{1}_{k\ge {k}_{growth}}\left(k\right)$$

(Eq. 4)

$${E}_{k+1}={E}_{k}+c\left(k\right){S}_{k}-h\left(k\right)-\mu {E}_{k}{1}_{k\ge {k}_{growth}}\left(k\right)$$

(Eq. 5)

$${I}_{k+1}={I}_{k}+h\left(k\right)-\psi \left(k\right){I}_{k}-\mu {I}_{k}{1}_{k\ge {k}_{growth}}\left(k\right)$$

(Eq. 6)

$${R}_{k+1}={R}_{k}+\psi \left(k\right){I}_{k}+\mu ({S}_{k}+{E}_{k}+{I}_{k}){1}_{k\ge {k}_{growth}}\left(k\right)$$

(Eq. 7)

$${LAI}_{k}={S}_{k}{+E}_{k}+{I}_{k}+{R}_{k}$$

(Eq. 8)

where \(g\left(k\right)\) is the crop growth rate introduced in Eqs. 1 and 2 and \({LAI}_{k}\) corresponds to the total leaf area index of the field canopy. The functions \(c\left(k\right)\), \(h\left(k\right)\) and \(\psi \left(k\right)\) depend on the pathogen species and the dynamics of the disease in the neighbouring fields, especially the production of inoculum (spores), and will be explained below (Eqs. 18, 23 and 24 respectively).

*Infection and spore dispersal.* Our model keeps track of the number of spores released during each dispersal event, whether they stay in their native field or not. We thus compute the number of spores present in each field and potentially able to infect the crop at every time step. In our model, we distinguish between spores of four different origins: (i) the external primary inoculum, i.e. spores entering the landscape from the outside, corresponding to long-distance dispersal events \({P}_{external}\left(k\right)\); (ii) spores produced within the field during the current year’s epidemic \({P}_{released}\left(k\right)\); (iii) incoming spores produced by infectious neighbours in the landscape \({P}_{neighbours}\left(k\right)\); and (iv) spores accumulated in the field’s spore pool over time that still participate in the current’s year epidemic \({P}_{-pool}\left(k\right)\).

(i) The arrival of external spores occurs every year in our simulations, but is limited in both space and time. Only a fraction *p* of the *N* fields gets infected by the external inoculum, and the arrival of that inoculum is limited to a temporal window of \({k}_{cl}=200\) dd long, starting at *k* = *k**start*. The value of *k**start* depends on the nature of the disease (Table 1). Fields get inoculated at a constant rate *P**ext,0*. This leads to the following number of spores in each of the *Np* inoculated fields:

$${P}_{external}\left(k\right)={P}_{ext,0}.{1}_{{k}_{start}\le k<{k}_{start+{k}_{cl}}}\left(k\right)$$

(Eq. 9)

The *Np* fields receiving external inoculum are randomly chosen every year. The first season’s epidemic in each simulation is initiated by the arrival of the external inoculum. After that, during the following cropping seasons, infection starts at *k* = *k**start* mainly through infection by spores inherited from the previous year’s epidemic (see below), although arrival of external inoculum continues to occur at that date.

(ii) Once an epidemic has started, the number of spores produced and dispersed within an infected field at every time step *k* is given by:

$${P}_{released}\left(k\right)=\left(1-\alpha \right)\sigma {I}_{k}.d\left(k\right)$$

(Eq. 10)

where \(\sigma\) is the pathogen spore production rate. The interpretation of the parameter \(\alpha\) depends on the pathogen species. In the case of WLR, \(\alpha\) corresponds to the fraction of spores (urediospores) leaving the field. Thus, \(\left(1-\alpha \right)\) corresponds to the fraction of spores remaining in their natal field. In the case of STB, \(\alpha\) corresponds to the fraction of spores dispersed by wind (ascospores), not rain (pycnidiospores). Consequently, the \(\left(1-\alpha \right)\) rain-splashed pycnidiospores are also unable to leave their native field. The dispersal function \(d\left(k\right)\) thus differs between the two pathogens. Since WLR urediospores are airborne and are released daily, we implemented a regular dispersal every 20 degree-days. Since STB pycnidiospores are rain-splashed, we used several weather time series recorded at Grignon experimental station between 1994 and 2006 to generate several annual rain patterns (the \(rain\) function in Eqs. 11 and 15) corresponding to more or less favourable weather conditions for pathogen development. An example of rain pattern is given in Supplementary Fig. 1 (lower panel). The dispersal function \(d\left(k\right)\) is given by:

$$d\left(k\right)=\left\{\begin{array}{c}wind\left(k\right)=\left\{\begin{array}{c}1 \text{if} k \equiv 0\left(20\right)\\ 0 \text{ot}\text{herwise}\end{array}\right. \text{for WLR}\\ \\ rain\left(k\right) \text{for STB}\end{array}\right.$$

(Eq. 11)

(iii) Infectious fields are a source of inoculum for their neighbours. The quantity of spores received by a given field depends on the number, distance, and spore production rate of its infectious neighbours. Let A be a receptor field and B one of its infectious neighbours. Let \({\delta }_{AB}\le \varDelta\) be the distance between A and B, where \(\varDelta\) is the maximum dispersal distance of the pathogen. The number of spores transmitted from B to A decreases with increasing \({\delta }_{AB}\). But here again, we must distinguish between STB and WLR. In the case of WLR, all (uredio)spores produced could theoretically be blown away by wind and leave their natal field. We denote by \(\alpha\) the fraction of spores produced in a given field that leaves it and contributes to the landscape-scale spread of the disease. To simulate that, we define \({\varGamma }_{WLR,A}\) as the set of fields which centre is within the Euclidian distance \(\varDelta\) of A, A not included. The number of spores received by A is the sum of the spores emitted by its neighbours towards it. In the case of STB, we distinguish between rain-splashed spores (a fraction \(\left(1-\alpha \right)\) of the spores produced) and wind-dispersed spores (the remaining \(\alpha\)). Only the latter can potentially leave their natal field and contribute to landscape-scale disease spread. But many of them will undoubtedly land within their natal field before some of them are able to disperse further (Frezal et al. 2009). To simulate that, we define \({\varGamma }_{STB,A}\) as the set of fields which centre is within the Euclidian distance \(\varDelta\) of A, A included (in that case, \({\delta }_{AA}=0\)). It follows that:

$${P}_{neighbours}^{A}\left(k\right)=\left(\sum _{B\in {\varGamma }_{*,A}}\frac{1}{{W}_{A}}.\frac{1}{{\delta }_{AB}+1}\alpha \sigma {I}_{k}^{B}\right).d\left(k\right)$$

(Eq. 12)

with

$${W}_{A}=\sum _{B\in {\varGamma }_{*,A}}\frac{1}{{\delta }_{AB}+1}$$

(Eq. 13)

where * denotes the type of pathogen (STB or WLR).

(iv) Finally, all spores received from the outside of the field or released within the field join a general pool of spores *P* corresponding to the spores that did not succeed in infecting plants and have fallen to the ground or on non-susceptible plant surfaces. The pool of spores plays an important role at the beginning of epidemics, since a fraction \(\theta\) of these spores survives the interculture period between two successive growing seasons and joins the external inoculum to create the first lesions at \({k}_{start}\). We model this survival of spores during the interculture as an instantaneous projection of the number of spores in the pool at harvest (\({k=k}_{end}\)) to the start of the next growing season (\(k={k}_{init}\)). With *T* and *T+1* being two consecutive growing seasons, we get:

$${P}_{{k}_{init}}^{T+1}=\theta {P}_{{k}_{end}}^{T}$$

(Eq. 14)

Depending again on the pathogen species, spores from that pool can also be remobilized and participate to the creation of new lesions. Spores from STB survive well on the ground and on crop residues and can be rain-splashed from the ground onto seedlings for \({k}_{start}\le k<{k}_{pool}\). Spores from WLR have a lower survival rate (because the biotrophic pathogen needs volunteer plants to survive the interculture) but can be remobilized throughout the course of the epidemic.

$${P}_{-pool}\left(k\right)=\left\{\begin{array}{c}{P}_{k}\left(1-\frac{k-{k}_{start}}{{k}_{pool}-{k}_{start}}\right)\bullet rain\left(k\right)\bullet {1}_{{k}_{start}\le k\le {k}_{pool}} \text{f}\text{o}\text{r}\text{ }\text{S}\text{T}\text{B}\\ { P}_{k}{\bullet wind\left(k\right)\bullet 1}_{k\ge {k}_{start}} \text{f}\text{o}\text{r}\text{ }\text{W}\text{L}\text{R}\end{array}\right.$$

(Eq. 15)

Therefore, at any time *k* during the year, the number of spores present in a field can be calculated as:

$${N}_{sp}\left(k\right)={P}_{external}\left(k\right)+{P}_{-pool}\left(k\right)+{P}_{released}\left(k\right)+{P}_{neighbours}\left(k\right)$$

(Eq. 16)

In a given field, the inoculum \({N}_{sp}\left(k\right)\) spreads within the canopy where it is intercepted at a rate \(\epsilon \left(k\right)\) by crop leaves. Spore interception by the crop canopy follows a Beer-Lambert equation:

$$\epsilon \left(k\right)=1-{e}^{-bLAI\dot{}\left(k\right)}$$

(Eq. 17)

where *b* is equivalent to the molar extinction coefficient of the Beer-Lambert law. Note that spores intercepted by the resistant plants, either in crop mixtures or in a resistant field in mosaics and rotations, do not contribute to the epidemics and are removed from the system. The spores that are not intercepted by any part of the canopy fall to the ground and build the pool of spores *P*. The number of spores intercepted by the canopy at time *k* is thus \(\epsilon \left(k\right){N}_{sp}\left(k\right)\). To simulate crop infection, we subdivided the susceptible part *S* of the canopy into elementary surfaces of size \({s}_{0}\). Each elementary surface can be infected only once. Intercepted spores will create lesions in the susceptible canopy with a probability \({\pi }_{inf}\). Hence, the number of lesions produced in the canopy at a given time follows a Poisson distribution. The proportion of the canopy that intercepts infecting spores and thus becomes infected is given by Eq. 18.

$$c\left(k\right)=1-exp\left(\frac{{\pi }_{inf}\epsilon \left(k\right){N}_{sp}\left(k\right)}{\frac{{S}_{k}}{{s}_{0}}}\right)$$

(Eq. 18)

Spores that were not intercepted by the canopy and spores that were intercepted by the canopy but did not cause lesions join the pool of spores *P*. Spores in *P* decay at a constant rate \(\rho\) and contribute, at each time step, to the within-field infection of susceptible tissue via the function \({P}_{-pool}\left(k\right)\) (Eq. 15). The dynamics of *P* are thus given by:

$${P}_{k+1}={P}_{k}\left(1-\rho \right)-{P}_{-pool}\left(k\right)+\left(1-\epsilon \left(k\right)\right){N}_{sp}\left(k\right)+\left(1-{\pi }_{inf}\right)\epsilon \left(k\right){N}_{sp}\left(k\right)$$

(Eq. 19)

*Age-structured latent period.* Symptoms of the disease do not appear immediately after leaf infection. They appear after an incubation period. Spore production begins after an even longer period called the latent period (\(\lambda\) in our model). To ensure that the elementary surfaces \({s}_{0}\) remain exposed during the latent period, we must keep a record of the surface of the exposed tissue over time. In our model, we describe latent infected surfaces using an age-structured vector \({\eta }_{k}\):

$$\eta \left(k\right)=\left({\eta }_{k,1} {\eta }_{k,2} ⋮ {\eta }_{k,\lambda } \right)$$

(Eq. 20)

where \({\eta }_{k,t}\) is the fraction of \({E}_{k}\) that got infected *t* time steps ago. Thus, \({\eta }_{k,1}\) corresponds to the fraction of \({E}_{k}\) that just got infected and \({\eta }_{k,\lambda }\) is the fraction of \({E}_{k}\) that will become infectious. Thus, at any time step *k*, the exposed surface can be calculated as:

$${E}_{k}={\sum }_{i=1}^{\lambda }{\eta }_{k,i}$$

(Eq. 21)

The vector \(\eta\) thus corresponds to the age structure of the exposed crop surfaces. The components of the vector \(\eta\) change according to the following rules:

$$\left\{\begin{array}{c}{\eta }_{k+\text{1,1}}={c\left(k\right)S}_{k}\\ {\eta }_{k+1,i}=\left(1-\mu \right){\eta }_{k,i-1} \forall i\in ⟦2,\lambda ⟧\end{array}\right.$$

(Eq. 22)

where \(c\left(k\right)\) is the infection rate from Eq. 18.

Thus, the transition rate between infected tissue *E* and infectious tissue *I* is given by:

$$h\left(k\right)={\eta }_{k,\lambda }\left(1-\mu \right)$$

(Eq. 23)

*Emptying of reproductive structures.* As time goes on and spore dispersal occurs, older sporulating structures get progressively empty at a rate \(\psi \left(k\right)\). This rate depends on the frequency of dispersal events and on the average efficiency of individual dispersal events, which is, in turn, linked to the number of spores per reproductive structure.

$$\psi \left(k\right)={\psi }_{0}.d\left(k\right)$$

(Eq. 24)

with *d(k)* the dispersal function described by Eq. 11. But owing to the difference in the number of spores between pycnidia of STB and uredia of WLR, we decided that the number of spores per uredium was not likely to be limiting (\({\psi }_{0}=0\) for WLR).

## Simulating crop diversification

We use the model to simulate three strategies of crop diversification: within-field crop mixtures, crop rotations and landscape-scale crop mosaics (Fig. 1).

*Crop mosaics.* Crop mosaics correspond to landscape grids where fields planted only with the susceptible crop (hereafter called “susceptible fields”) and others planted only with the resistant crop (hereafter called “resistant fields”) coexist. Resistant fields display a S-R structure identical to that of a healthy canopy since we only consider complete (qualitative) resistance. The dynamics of resistant fields are thus given by Eq. 1-3. The dynamics of susceptible fields are given by Eq. 4-24. In a mosaic landscape, resistant and susceptible fields are randomly distributed across the landscape (Fig. 1). The spatial structure of mosaics remains the same throughout the 10 years of the simulations.

*Crop rotations.* We modelled crop rotations by implementing two types of rotations (Fig. 1). In synchronous rotations, all the fields in the landscape are either susceptible or resistant. In asynchronous rotations, susceptible and resistant fields coexist in the landscape in constant proportions every year. In asynchronous rotations, susceptible and resistant fields are randomly selected at the beginning of each simulation. Since a given field is alternatively planted with susceptible wheat and resistant crops during one simulation, crop rotations therefore correspond to crop mosaics changing through time.

*Crop mixtures.* Crop mixtures correspond to landscapes where all fields are similar. Diversification takes place at the field scale. Within a field, the canopy of the crop is divided into two parts: qualitatively resistant and susceptible. We will denote by Sr and Rr the healthy and removed parts, respectively, of the canopy corresponding to resistant crops. The total leaf area index (LAI) of a field is then divided into six parts: *S*, *E*, *I*, *R*, *S**r* and *R**r*, so that:

$${LAI}_{k}={S}_{k}{+E}_{k}+{I}_{k}+{R}_{k}+{S}_{k}^{r}+{R}_{k}^{r}$$

(Eq. 25)

Mixtures have the same global carrying capacity *K* as susceptible and resistant fields but the carrying capacity is divided between the susceptible and the resistant parts of the canopy according to the percentage of susceptible and resistant hosts in the mixture \(\omega\). Thus, for a fraction \(\omega\) of susceptible crop in the mixture, one can derive the dynamics of the system by substituting *K* for \({K}^{\text{'}}=K\omega\) in Eqs. 1, 2 and 4. Note however that there is no difference between \({S}_{k}\) and \({S}_{k}^{r}\) in terms of spore interception: both parts of the canopy contribute to spore interception even though the pathogen can only infect and reproduce on susceptible plants.

*Effects of crop diversification strategies on fungal diseases.* Since the spores intercepted by the resistant plants (either in crop mixtures or in resistant fields in mosaics and rotations) do not contribute to the epidemics and are removed from the system, a major regulating mechanism involved in all three crop diversification strategies is a dilution effect. Indeed, the resistant crop reduces the density of susceptible crop and intercepts pathogen spores, thus limiting pathogen spread between susceptible plants (processes reviewed by Mundt 2002). In our model, the dilution effect might be stronger in mixtures than in mosaics and rotations since a large fraction (\(1-\alpha\)) of the spores do not leave their native field.

Another regulating mechanism involved in rotations is inoculum mortality in the absence of the host (Hossard et al. 2018). We modelled the inoculum-suppression effect of rotations through a constant decay rate of the spore pool in every field of the landscape (*ρ* parameter, Eq. 19). In susceptible fields, new epidemics increase the pool of spores and compensate for the spore pool decay rate so that inoculum accumulates. In resistant fields, no epidemic occurs and the spore pool only decays until the field becomes susceptible and infected again. Moreover, every year, only a fraction *θ* of spore pool survives the interculture (Eq. 14).

*Percentage of susceptible and resistant crop in the landscape.* For each of the three crop diversification strategies, we simulate landscapes with different proportions of susceptible wheat. For mixtures, we compare epidemics when decreasing the proportion of susceptible wheat in the field from \(\omega\)=100% (entirely susceptible) to \(\omega\)=0% (entirely resistant). Note that for mixtures, we assume that each field in the landscape has the same fraction of susceptible wheat (Fig. 1). Consequently, for mixtures, the proportion of wheat in each individual field in the landscape is the same as the proportion of wheat in the landscape. The percentage of wheat in mixtures thus also corresponds to the fraction of susceptible plants in the whole landscape. For crop mosaics, we compare epidemics when decreasing the proportion of susceptible wheat fields from \(\omega\)=100% (susceptible fields only) to \(\omega\)=0% (resistant fields only) in the landscape. For both synchronous and asynchronous rotations, we vary the proportion of wheat in the landscape by simulating two-year cyclical models (resistant-susceptible, r-s) and three-year cyclical models (s-s-r and s-r-r). This corresponds to average proportions of resistant crops in the landscape (over a full rotation cycle) of \(\omega\)=50%, \(\omega\)=33%, and \(\omega\)=67% respectively.

## Simulation equilibrium and output variable

During each simulated cropping season, we follow the pathogen development in each field of the landscape. After 10-12 seasons, the system reaches a one-season epidemiological equilibrium cycle in the case of crop mixtures and mosaics, and a two-season or three-season equilibrium cycle in the case of two-year and three-year crop rotations, respectively. To estimate the epidemic levels at equilibrium in landscapes with crop mixtures, mosaics and rotations, we compute the area under disease progress curve (AUDPC, Madden et al. 2007) over the duration of the equilibrium cycle. Commonly used in plant epidemiology, AUDPC corresponds to the total area of crop covered by the disease in a given field during one cropping season. It is a quantitative summary of disease intensity over time that allows between-year and between-field comparisons. In our simulated landscapes, it is calculated as the average value over the *N* wheat-containing fields of the landscape of the total diseased area *I* of each field:

$$AUDPC=\frac{1}{N}{\sum }_{i=1}^{N}{\sum }_{j=0}^{{k}_{e}}{I}_{i,j}$$

(Eq. 26)

For crop mixtures, AUDPC values correspond to the landscape-scale average epidemic levels of the wheat plants in the mixtures at equilibrium. For crop mosaics, AUDPC values correspond to the landscape-scale average epidemic levels of the wheat fields at equilibrium. For crop rotations, AUDPC corresponds to the landscape-scale average epidemic levels in wheat fields averaged over the duration of the rotation cycle.

## Simulation schedule

For the three diversification strategies, we simulate epidemics in the fields of the landscape under three sets of meteorological conditions that are more or less favourable to STB and WLR (Fig. 2). We used temporal patterns of rainfall recorded at the experimental site of Grignon (Fr-78850) between 1993 and 2006 to select favourable and unfavourable rain patterns for STB simulations. For this, following Robert et al. (2008) and Garin et al. (2014), we analysed rainfall patterns that take the form of "rain combs" in which each "tooth" represents a pathogen dispersal event (Supplementary Fig. 1). We chose three of these rain patterns to represent favourable, average and unfavourable annual weather for STB epidemics: 1994-95 was a favourable year for STB: very rainy winter and regular rain events during plant growth in spring. Conversely, 1996-97 was unfavourable: there was little rainfall throughout the year. We also use the more realistic sequence corresponding to the years 1994 to 2006, which includes both favourable, average and unfavourable years as an average condition (Supplementary Fig. 1). For WLR, following Duvivier et al. (2016) and Garin et al. (2018), we use the date of the onset of the epidemics to simulate favourable and unfavourable weather scenarios. Favourable weather leads to epidemics starting 800 dd after sowing while unfavourable weather leads to epidemics starting 1200 dd after sowing. Average weather conditions lead to epidemics starting 1 000 dd after sowing (Supplementary Fig. 2).

To extend the robustness of our results, we explore the impact of crop diversification by varying different parameters of the infection cycles of the pathogens: inoculum survival (\(\theta\), Fig. 3) and the intra-season spore mortality rate (\(\rho\), Supplementary Fig. 3). These pathogen traits are important for disease development. Moreover, these traits could respond to weather conditions. For instance, spore overwintering could become easier for pathogens as winters become milder. Summer droughts could make survival during the interculture more difficult if it reduced the availability of volunteer plants or increased spore decay caused by UV radiation and high temperatures. Favourable climatic conditions on a broader scale may lead to a global increase in pathogen populations, thereby increasing the inoculum pressure at a regional scale and the pressure of an external inoculum at the beginning of epidemics.

## Model parameters and implementation

The model was parameterized according to our knowledge of the two pathosystems in order to allow for qualitatively consistent results with epidemiological data from the literature or data collected at the Grignon experimental site (Robert et al. 2004, 2005; Frezal et al. 2009; Pariaud et al. 2009; Baccar et al. 2011) and already used in previous modelling studies (Robert et al. 2008, 2018; Garin et al. 2014). Hence, parameters of the model were chosen so that maximum disease severity did not exceed two-thirds of the LAI for STB and one-half of the LAI for WLR (Bancal et al. 2007). This difference explains why maximum disease severity in our simulations is higher for STB than for WLR. Differences in parameter values are inspired by findings of Robert et al. (2008, 2018), Garin et al. (2014, 2018), and Précigout et al. (2020b). Examples of disease dynamics and sensitivity to weather conditions can be found in Supplementary Figs. 1 and 2. The model was developed with MatLab (2020).

Table 1

model parameters, values and interpretation. dd: degree-days, [LAI]: leaf area index unit (m2 of leaf per m2 of ground).

Symbol | Value | Unit | Interpretation |

**WLR** | **STB** |

**Landscape parameters** |

*N* | 1089 | fields | Number of fields in the landscape |

*p* | 0.01 | - | Fraction of fields infected at the beginning of each simulation |

\({\delta }_{AB}\) | - | fields | Euclidian distance between two fields A and B |

**Seasonal dynamics** |

*k* | \({k}_{init}\)-\({k}_{end}\) | dd | Time index within a year |

*T* | 1-12 | - | Year (crop growing season) index |

\({k}_{init}\) | 0 | dd | Start of season |

\({k}_{start}\) | 800-1300 | 0 | dd | Arrival date of primary inoculum (annual start date of epidemic) |

\({k}_{growth}\) | 1400 | dd | End of period of canopy growth |

*k**cl* | 200 | dd | Time period during which the crop is exposed to external primary inoculum |

\({k}_{pool}\) | \(\varnothing\) | 700 | dd | Time period during which the crop is exposed to primary inoculum by spores from the spore pool |

\({k}_{end}\) | 2500 | dd | End of season |

**Crop dynamics** |

*K* | 6 | [LAI] | Maximum value of Leaf Area Index (LAI) |

\(\beta\) | 0.09 | [LAI]/(10 dd) | Growth rate of the green Leaf Area Index (gLAI) |

\(\mu\) | 0.03 | [LAI]/(10 dd) | Mortality rate of plant tissue |

\(\omega\) | 0.1 - 1 | - | Fraction of susceptible crop in mixtures/landscapes |

**Pathogen dynamics & dispersal** |

\({P}_{ext,0}\) | 20 000 | spores | External primary inoculum |

*λ* | 100 | 200 | dd | Latent period |

\(b\) | 1 | [LAI]−1 | Spore interception rate by the canopy |

\({s}_{0}\) | 1 | cm2 | Individual lesion size |

\({\pi }_{inf}\) | 0.0002 | - | Infection probability |

\(\sigma\) | 1.5x106 | 5.0x107 | [LAI]−1 | Spore production rate |

\(\theta\) | 0.01 | 0.15 | - | Spores survival rate during the intercropping |

\(\rho\) | 0.01 | 0.002 | - | Spore mortality rate |

\(\alpha\) | 0.5 | 0.02 | - | Fraction of spores leaving their natal field (WLR); fraction of sexual airborne spores produced (STB) |

\({\psi }_{0}\) | 0.3 | 0 | - | Emptying rate of pathogen reproductive structures |

\(\varDelta\) | 5 | 2 | fields | Maximum dispersal distance of the pathogen |