Quantitative plant resistance in cultivar mixtures: wheat yellow rust as a modeling case study


  • Natalia Sapoukhina,

    Corresponding author
    1. INRA, UMR1345 Institut de Recherche en Horticulture et Semences – IRHS, SFR 4207, PRES UNAM, Beaucouzé Cedex, France
    2. AgroCampus-Ouest, UMR1345 Institut de Recherche en Horticulture et Semences – IRHS, Angers, France
    3. Université d'Angers, UMR1345 Institut de Recherche en Horticulture et Semences – IRHS, Angers, France
    Search for more papers by this author
  • Sophie Paillard,

    1. INRA, UMR 1349, Institut de Génétique, Environnement et Protection des Plantes - IGEPP , Le Rheu Cedex, France
    Search for more papers by this author
  • Françoise Dedryver,

    1. INRA, UMR 1349, Institut de Génétique, Environnement et Protection des Plantes - IGEPP , Le Rheu Cedex, France
    Search for more papers by this author
  • Claude de Vallavieille-Pope

    1. INRA, UR1290 BIOGER-CPP, BP 01, Thiverval-Grignon, France
    Search for more papers by this author


  • Unlike qualitative plant resistance, which confers immunity to disease, quantitative resistance confers only a reduction in disease severity and this can be nonspecific. Consequently, the outcome of its deployment in cultivar mixtures is not easy to predict, as on the one hand it may reduce the heterogeneity of the mixture, but on the other it may induce competition between nonspecialized strains of the pathogen.
  • To clarify the principles for the successful use of quantitative plant resistance in disease management, we built a parsimonious model describing the dynamics of competing pathogen strains spreading through a mixture of cultivars carrying nonspecific quantitative resistance.
  • Using the parameterized model for a wheat–yellow rust system, we demonstrate that a more effective use of quantitative resistance in mixtures involves reinforcing the effect of the highly resistant cultivars rather than replacing them. We highlight the fact that the judicious deployment of the quantitative resistance in two- or three-component mixtures makes it possible to reduce disease severity using only small proportions of the highly resistant cultivar.
  • Our results provide insights into the effects on pathogen dynamics of deploying quantitative plant resistance, and can provide guidance for choosing appropriate associations of cultivars and optimizing diversification strategies.


In plant epidemiology, quantitative or partial plant resistance that reduces the disease severity rather than conferring immunity offers an alternative to qualitative or total resistance (Singh et al., 2004), which does confer immunity to disease. Quantitative resistance can be used to reduce the size of the pathogen population whilst also avoiding exerting high selective pressure (Zhan et al., 2002; Chen, 2005). For the same reason, cultivars carrying quantitative resistance are potential components of low-selective pressure cultivar mixtures intended to control not only the spread of disease, but also the adaptive dynamics of the pathogen population and its genetic diversity (Sommerhalder et al., 2011). However, so far, most of the theoretical and empirical research on the use of cultivar mixtures for disease reduction has focused on diseases caused by biotrophic pathogens that interact with their hosts on a gene-for-gene basis (Flor, 1956), which has made it possible to identify major resistance genes conferring cultivar immunity to disease. The presence of major resistance genes in a cultivar is still one of the most common criteria for selecting it as a mixture component, despite the fact that this can lead to rapid pathogen evolution and the emergence of a super-virulent pathogen strain (Mundt, 2002).

The principles of the mixture theory were originally elaborated for a two-component mixture, consisting of one susceptible cultivar and one totally resistant cultivar (Leonard, 1969). Cultivar mixtures including qualitative resistance reduce the rate of disease spread by eliminating large numbers of spores that are deposited on resistant cultivars, thus diluting the inoculum falling on the susceptible hosts. In this way, the spatial heterogeneity of a cultivar mixture creates a physical barrier to disease spread (Garett & Mundt, 1999). Studies of the epidemiology and theory of biological invasions have contributed considerably to our understanding of the impact of the spatial heterogeneity of a host population on the dynamics and rate of spread of disease. It has been shown that there is an epidemic threshold, a minimum percentage of suitable hosts, of c. 10–40%, which is required for disease to spread (Collingham & Huntley, 2000; Otten et al., 2004; Dewhirst & Lutscher, 2009; Mundt et al., 2011). The development of theoretical approaches to integrating spatial heterogeneity into models has made it possible to demonstrate that the epidemic threshold depends on various spatial factors, such as the degree of landscape fragmentation, the spatial arrangement of the mixture components, the spatial scale, and the dispersal capacity of the disease (Kinezaki et al., 2010; Papaïx et al., 2011; Suzuki & Sasaki, 2011). Understanding the process of disease spread over a genetically and spatially heterogeneous cultivar mixture has made it possible to identify the key characteristics that determine the effectiveness of a mixture: genotype unit area, dispersal gradient of pathogen spores, disease pressure, sowing density, and the number of components (Garett & Mundt, 1999). In the case of wheat yellow (stripe) rust (caused by Puccinia striiformis f.sp. tritici), it has been shown that mixing cultivars carrying different major resistance genes slows the spread of epidemics, and that this effect can be enhanced by using an appropriate proportion of a resistant cultivar. The 133 empirical studies analyzed by Huang et al. (2012) show that 83% of wheat cultivar mixtures produced disease intensities that were lower than the mean values found for pure stands. Overall, the reduction ranged from 30 to 50%, with an average of 28%. Empirical studies demonstrated that the reduction of rust severity depended on characteristics that modify the degree of heterogeneity of the mixture. For instance, Mundt et al. (1995) showed that two-component mixtures produce a smaller mixture effect than mixtures of larger numbers of components.

Most empirical studies focusing on mixtures of susceptible and partially resistant cultivars have analyzed the dynamics of splash-dispersed necrotrophs on cereals: Rynchosporium secalis on barley, Stagonospora nodorum on wheat, Bipolaris sorokiniana on wheat, and Mycosphaerella graminicola on wheat. They have shown that mixtures including cultivars with partial resistance provide relatively low amounts of disease control. Overall, the results of empirical studies of quantitative resistance deployment range from not effective, as in wheat Cephalosporium stripe (Mundt, 2002), M. graminicola (Cowger & Mundt, 2002), wheat eyespot (Mundt et al., 1995) and barley scald (Abbott et al., 2000), to significantly effective, as in wheat Septoria nodorum blotch (Jeger et al., 1981b), wheat yellow rust (Huang et al., 2011), wheat leaf (brown) rust (Mahmood et al., 1991), wheat Septoria tritici blotch (Mille et al., 2006), barley powdery mildew (Newton & Thomas, 1992), and potato late blight (Andrivon et al., 2003). To understand the key factors that determine the success of deploying quantitative plant resistance in mixtures, we need to distinguish between specific and nonspecific quantitative resistance. Specific resistance is effective against some pathogen strains, and so can create more pronounced mixture heterogeneity than nonspecific quantitative resistance, which can be infected by all pathogen strains, but with different degrees of severity. We would expect the effectiveness of these mechanisms to depend on the type and degree of quantitative resistance used in a mixture. For instance, an association of a nonspecific, moderately resistant cultivar with a susceptible one can result in little difference between the susceptibilities of the cultivars, and therefore less pronounced heterogeneity of the mixture. The physical barrier and dilution effects can be dramatically reduced if cultivars carrying strong specific resistance are excluded from the mixture. The degree of heterogeneity in susceptibility of the components can explain why mixtures are more effective in reducing specialized pathogens than nonspecialized ones (Xu & Ridout, 2000). However, Jeger et al. (1981a), modeling the dynamics of a single pathogen strain, showed that some differences in the susceptibility of the components can create functional heterogeneity in the mixture that is sufficient to control a nonspecialized pathogen. Although competition between pathogen strains may play an important role in determining the effectiveness of mixtures, including quantitative resistance (Garett & Mundt, 1999; Lannou et al., 2005; Abang et al., 2006), it has not received much attention in theoretical studies. In their two-dimensional stochastic spatial contact model of fungal pathogens in cultivar mixtures, Xu & Ridout (2000) considered the dynamics of specialized and nonspecialized pathogen strains, but not the competition between them, as a host unit was assumed to be occupied by a single pathogen strain. Moreover, this assumption generated additional spatial heterogeneity for pathogen strains that was not related to the degree of host resistance. There has been almost no attempt to adapt the mixture theory to the situation of host diversification, in which quantitative resistance leads to a continuum in host susceptibility and competitive interactions among pathogen strains.

Our goal was to study the dynamics of competing pathogen strains spreading over a cultivar mixture, the components of which carry nonspecific quantitative resistance, and thereby to clarify the role of the quantitative plant resistance in disease management involving the use of cultivar mixtures. Spatiotemporal models of the propagation of the airborne fungal diseases include the system of differential equations representing continuous epidemic dispersal by diffusion (Yang et al., 1991) or of integrodifference equations, where the dispersal kernel can represent both short- and long-distance disease dispersal (Skelsey et al., 2005). These models are spatially explicit, and so can be used to study pathogen spread over heterogeneous host populations. Moreover, being mechanistic, they can easily be parameterized and can mimic disease dynamics quite well (Yang et al., 1991). However, the abundance of parameters and processes in the available epidemiologically relevant models make them unsuitable for use for studying the dynamics of diversified host–pathogen systems, where both the host and pathogen populations can be divided into a large number of classes with different properties. Here, we constructed a parsimonious, spatially explicit, host–pathogen model describing pathogen spread over a genetically diversified host population distributed over a two-dimensional landscape. The simplicity of our reaction–diffusion model means that it can be used to describe the dynamics of a host–pathogen system in which the host and pathogen populations are divided into numerous classes, depending on their susceptibility and infection efficiency, respectively. This general model can be used to represent the dynamics of both specialized and nonspecialized pathogen strains. We used field data from wheat plots inoculated with yellow rust to study the effect of host diversity on the dynamics of four competing, nonspecialized pathotypes of P. striiformis f.sp. tritici. First, we parameterized the model from field data for the spread of wheat yellow rust over a homogeneous, susceptible wheat cultivar (de Vallavieille-Pope & Goyeau, 1996; Finckh et al., 2000; de Vallavieille-Pope, 2004). Second, we used the parameter estimates obtained from the field studies to investigate the effectiveness of two- and three-component random cultivar mixtures in which the degrees of susceptibility and proportions of mixture components were varied. Finally, we discuss what constitutes the effective management of quantitative plant resistance.


The model

In this section we present a model of the focal expansion of a fungal disease propagated by airborne spores over a host population spatially distributed in a two-dimensional domain Ω = [0, Lx] × [0, Ly] (m2), Ω ⊂ R2. The spatial scale of the domain, Ω, depends on the pattern of disease dispersal being studied. We assume that the pathogen and host populations are both genetically diverse, and include n and m distinct genotypes, respectively. We use a reaction–diffusion model to describe the spread of a diverse pathogen population over a heterogeneous host population:

display math(Eqn 1)

where, Pi(x,t) is the density of the leaf area (m2 m−2) infected by the ith pathogen strain, and Hj(x,t) is the density of the healthy leaf area (m2 m−2) of the jth host genotype, i = 1, …, n,= 1, …, m. The terms ‘pathogen’ and ‘host’ are used below to refer to infected and healthy leaf areas, respectively. We consider the dynamics of interacting populations during one cropping season, from stem elongation until the ripening stage, that is, during the period when green vegetation is available. Time t is measured in days (d). The parameter ei,j is the infection efficiency of the jth host genotype by the ith pathogen genotype. All pathogen genotypes are assumed to have the same diffusion coefficient, δ (m2 d−1). Similarly, the densities of healthy leaf area of all host genotypes are assumed to have the same growth rate, r (d−1), and carrying capacity, K. The plausible assumption of logistic host growth matches the dynamics of the host–pathogen interactions at the beginning of the cropping season. However, the continuous growth of healthy leaves also compensates for the loss of diseased tissues during the last stage of the epidemic. Consequently, while the density of healthy leaf area is maintained below the K threshold, the densities of the infected leaf area, inline image, and of the total leaf area, inline image, can exceed the carrying capacity of the healthy leaf area, K (Supporting Information, Notes S1). While theoretically this can happen, under field conditions it does not, as the growth of diseased plants is slowed (M. Trottet, pers. comm.). As we use relative measurements for the assessment of the mixture efficiency, exceeding K by inline image does not affect either the simulation results or the conclusions. Moreover, the density of the infected plant area is regulated naturally by the availability of the healthy leaf area and the short duration of the season, T = 150, and so we do not limit it by the carrying capacity (Fig. S1). However, in order to track host–pathogen dynamics over a longer season, it would be necessary to include a capacity parameter that limits disease development during the last stage of the epidemic.

The definitions and units of variables and parameters are summarized in Table 1. We assume that (Eqn 1) has a reflecting boundary condition:

display math

meaning that the habitat boundary is completely impermeable for the pathogen population, so that disease propagation is forced to turn back when it reaches the boundary. For a detailed description of the model behavior, see Notes S1. The resulting model, the methods, and numerical simulations listed in the following sections were implemented in Borland Delphi V4.5.

Table 1. Definition and values of the parameters used to model wheat yellow rust dynamics, and results of the sensitivity analysis
ParameterDefinition (units)Measured valueEstimated valueElasticity, E
  1. Estimated values are reported with standard errors between parentheses. The elasticity, E, of predicted disease severity to model parameters (Supporting Information Notes S1). Parameters varied by ± 5%.

Hj (x,0)Initial density of healthy leaf area of the jth host genotype (m2 m−2)0.1± 0.23
Pi (x,0)Initial density of the leaf area infected by the ith pathogen genotype (m2 m−2)7.8E-5 (1.4E-5)± 0.03
T Season duration (d)150
ΔxSideways distance between two field points (m)0.2
ΔyLengthways distance between two field points (m)0.4
L x Field width (m)9
L y Field length (m)18
e Nominal infection efficiency of the pathogen on a susceptible wheat cultivar (d−1)0.296 (0.023)± 0.42
δDiffusion coefficient (m2 d−1)0.029 (0.008)± 0.11
K Carrying capacity of the density of the healthy leaf area of the mth host genotype (m2 m−2)2.7 (0.2)0
r Growth rate of host population (d−1)0.055 (0.015)0

Case study: wheat yellow rust system

In this study, six single seed descent (SSD) wheat (Triticum aestivum L.) lines, issued from the cross between two winter bread wheat cultivars, cv Renan (Yr17), the susceptible cv Récital (Yr6), and two parental lines, were assessed for adult-plant resistance in a 3 yr field experiment. The SSD lines carried different combinations of the specific resistance gene (Yr17) and the two quantitative trait loci (QTLs), QYr.inra-2BS (Q2) and QYr.inra-6B (Q3), issued from cv Renan, and a QTL QYr.inra-2AS1 (Q1) derived from cv Récital (Table 2). Q1 was located in wheat region 2AS, which is homologous to the Yr17 introgression (Aegilops ventricosa segment) carried by cv Renan (Dedryver et al., 2009).

Table 2. Adult-plant resistance reaction to yellow rust of the single seed descent wheat (Triticum aestivum) lines and the resistant parent cv Renan, characterized by the presence or absence of three quantitative trait loci (QTLs), QYr.inra-2ASI (Q1), QYr.inra-2BS (Q2) and QYr.inra-6B (Q3), evaluated under field conditions, with four Puccinia striiformis f.sp. tritici pathotypes (S, highly susceptible; MS, moderately susceptible; MR, moderately resistant; R, highly resistant), according to Dedryver et al. (2009)
Degree of resistanceSMSMSMRMRRR

The presence of QTLs in these different SSD lines was determined using both phenotype (seedling tests and field assays) and molecular marker data (Dedryver et al., 2009). QTLs were detected by mapping a population of 194 (F6 and F7) SSD lines issued from the cross between cv Renan and cv Récital. The identified adult plant resistance (APR) QTLs Q1, Q2, and Q3 were studied here in the SSD lines L32, L138, L64, L110, and L22, and the resistant cv Renan (Table 2). Line 15 and cv Récital were highly susceptible. The lines were evaluated during the period 2006–2008. The experiments were conducted at the INRA-Versailles (Ile de France) experimental station. Four pathotypes of P. striiformis f.sp. tritici were chosen on the basis of their virulence phenotype to overcome all the specific resistance genes carried by cv Renan (Yr17), and two pathotypes had a virulence of 6, which allowed them to overcome the specific resistance of cv Récital (Yr6). The pathotypes inoculated in the field plots during three consecutive years were V1,2,3,9,17,Sd; V1,2,3,4,6,9,17,Sd,Su; V1,2,3,4,6,9,17,Sd,Su; and V1,2,3,4,9,17,32,Sd,Su (de Vallavieille-Pope et al., 2012). Two lines of each SSD line were planted with 30 seeds per line in a 1.2 m row at the end of October, and arranged in a completely randomized block design with four replications. Each field plot was located at least 700 m from any other plot to prevent cross-contamination with the pathotypes. A susceptible disease spreader line, consisting of three susceptible cultivars, Victo, Slejpner (Yr9) and Audace (Yr17), was planted perpendicularly to the test lines at both ends of the test lines. Only one pathotype was inoculated in each plot. The field inoculation was done with cv Victo seedlings, cultivated in a growth chamber to the two-leaf stage, and then artificially inoculated with P. striiformis f.sp. tritici isolates. In March, the inoculated seedlings were planted out just before sporulation (one seedling every 2 m) in the spreader rows. Visual scoring of disease severity was carried out three times during the growing season, using a 0–12 scale, with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 corresponding to 0.3, 0.7, 2, 8, 12, 16, 24, 33, 50, 66, 82, and 100% sporulating leaf area, respectively. Disease severity was assessed on five tillers of each of the four replicates for each leaf of the main stem of the plants at the time of rust appearance, corresponding to the growth stages DC32, DC39, and DC79, and a diseased surface area was calculated per plant at the last scoring date.

Fig. 1 shows differential severity of the pathogen pathotypes averaged over 3 yr (2006–2008). No specificity was found between the QTL lines and the four P. striiformis f.sp. tritici isolates tested in field tests (data not shown). After assigning (Eqn 1) notations, P. striiformis f. sp. tritici pathotypes V1,2,3,9,17,Sd, V1,2,3,4,6,9,17,Sd,Su, V1,2,3,4,6,9,17,Sd,Su and V1,2,3,4,9,17,32,Sd,Su became P1, P2, P3, and P4, respectively. Among the host genotypes, we distinguished resistant {L22, Renan}, moderately resistant {L110, L64}, moderately susceptible {L32, L138}, and susceptible {L15} lines. Hereafter, we use ‘S’ for susceptible, ‘M’ for moderately susceptible, and ‘R’ for resistant. Table 2 summarizes the notations used for the genetic compositions of the lines and their susceptibility.

Figure 1.

Disease severity on the seven wheat (Triticum aestivum) lines carrying different combinations of three quantitative trait loci (QTLs), QYr.inra-2ASI, QYr.inra-2BS, and QYr.inra-6B (Table 2), and the parent cultivar, Renan, averaged over three cropping seasons (2006–2008) after inoculating with four pathotypes of Puccinia striiformis f.sp. tritici: V1,2,3,9,17,Sd (P1); V1,2,3,4,6,9,17,Sd,Su (P2); V1,2,3,4,6,9,17,Sd,Su (P3); and V1,2,3,4,9,17,32,Sd,Su (P4). Field data were obtained at the end of an epidemic season.

Parameter estimation

We parameterized the model using epidemiological data for wheat yellow rust. Most of the parameter values were derived from the results of field experiments performed with wheat yellow rust (de Vallavieille-Pope & Goyeau, 1996; Clevers et al., 2002; de Vallavieille-Pope, 2004). A local sensitivity analysis was performed to identify which parameters need to be estimated more accurately (Notes S1).

Measured parameters

Based on experimental data on the dynamics of yellow rust severity in pure susceptible wheat stands (de Vallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope, 2004), and on the dynamics of the leaf area index for a wheat field (Clevers et al., 2002), we fixed the values of some of the model parameters. We assumed the wheat field size to be Lx = 9 (m), Ly = 18 (m), which corresponds to 162 m2 of real cropping surface of a pure susceptible stand (de Vallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope, 2004). According to the numerical method used to solve the system (1), the field contained 2116 cells of Δ× Δ= 0.2 m × 0.4 m size. We assumed the duration of the growing season, T, to be 150 d, and the initial density of healthy leaf area, H0, to be equal to 0.1 (Clevers et al., 2002). In other words, we assumed that the disease arrives at the beginning of the growing season, when the healthy leaf area available has reached 4% of its carrying capacity.

Model fitting

We first estimated the values of the r and K parameters using data for the temporal variation of the leaf area index for a wheat field (Clevers et al., 2002). By setting infection rates equal to zero, ei,j = 0, i = 1, …, 4, = 1, …, 7, we obtained a pure wheat growth model. We fixed the value of H0 (Table 1), and assessed the values of r and K by nonlinear fitting. We fitted the wheat growth model with M = 2 adjustable parameters ρ = (r, K) to N1 = 5 field measurements of the leaf area index, inline image inline image. At a given time, ti, the model leaf area index was calculated as the space-averaged density of the host population inline image. The maximum likelihood estimate of the model parameters was obtained by minimizing the merit function as follows: inline image, N = N1, defined as the sum of squared residuals between the experimental data and the corresponding model output. The parameters of the model were then adjusted to minimize the merit function, thus yielding the best-fitting parameters. To minimize the merit function over M parameters, we used the simplex-simulated annealing approach for the global optimization, which combines the downhill simplex and simulated annealing algorithms (Press et al., 1992).

We then fixed the values of three parameters, r, K, H0 (Table 1), and assessed the values of e, δ, and P0, the initial density of the infected leaf area. We fitted the complete (Eqn 1), e ≠ 0, with M = 3 adjustable parameters ρ = (e, δ, P0) to N2 = 8 observations of the mean disease severity of the pure, homogeneous, susceptible plot, inline image inline image (Fig. 2). At a given time, ti, the model disease severity was calculated as the percentage of the leaf area infected over the total leaf area, inline image. We obtained the best-fitting parameters by minimizing the merit function, F(ρ), with N = N2.

Figure 2.

The dynamics of yellow rust severity on a susceptible wheat (Triticum aestivum) cultivar in a pure stand – observed (circles) (de Vallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope, 2004) and predicted (solid line) by (Eqn 1). Simulation results were obtained with the parameter values presented in Table 1.

To assess the accuracy of the evaluated parameter values, we used residual bootstrapping (Efron & Tibshirani, 1993). To assess the fit of the model, we calculated the coefficient of determination as inline image, where inline image denotes the mean of the observations and inline image R2 estimates the fraction of the observed variability that is explained by the model. The higher the value of R2, the better the fit of the model, with R2 = 1 denoting a perfect fit. Since it is risky to use R2 alone as a model-fitting criterion for nonlinear models (Kvålseth, 1985), we supplemented it by the estimation of the concordance coefficient, ρc, measuring the degree of agreement between the observed and predicted variables (Lin, 1989). The concordance coefficient assesses the degree to which data pairs fall on the 45° line through the origin. It is estimated by

display math

where yi are experimental data, y(tiρ) are the model outputs, and inline image and inline image are their respective means. ρc = ± 1 if, and only if, the readings are in perfect agreement or perfect reversal.

Differential pathogen infection efficiency, (ei,j)1 ≤ i ≤ n, 1 ≤ j ≤ m. Fig. 1 shows that four P. striiformis f.sp. tritici pathotypes produced different average disease severity on different wheat lines. We split seven wheat lines into four groups according to their degrees of severity: ‘R’, resistant including L22 and Renan lines; ‘MR’, moderately resistant, including L110 and L64 lines; ‘MS’, moderately susceptible, including L32 and L138 lines; and ‘S’, susceptible, including L15 line. We assumed that the differences in severity were caused by differing pathogen infection rates, ei,j, i = 1, …, 4, = 1, …, 4. The estimated value of the pathogen infection efficiency, e = 0.296, on a susceptible wheat cultivar is a nominal value that we used to assess the infection rates, ei,j, of all the pathogen pathotypes in four line groups. The value e = 0.296 resulted in 100% disease severity, while pathotype P2 on susceptible cultivar H1 led to 72% disease severity at the end of the season. Thus, starting at e2,1 = 0.296, and decreasing the e2,1 value, while the other model parameters were fixed, we found e2,1 = 0.082, corresponding to 72% disease severity. Repeating this procedure for all the pathotypes and four line-groups, we obtained the matrix (ei,j)1 ≤ i ≤ 4, 1 ≤ j ≤ 4, summarized in Table 3. Differentiated values of infection efficiencies determine the degree of susceptibility or resistance of the interactions between pathogen and host genotypes.

Table 3. Matrix of estimated differentiated infection efficiencies (ei,j)1 ≤ i ≤ 4, 1 ≤ j ≤ 4 of Puccinia striiformis f.sp. tritici pathogen genotypes P1, P2, P3, P4 on susceptible (S), moderately susceptible (MS), moderately resistant (MR), and resistant (R) lines
  P 1 P 2 P 3 P 4
  1. The matrix corresponds to the disease severities reported in Fig. 1.


Note that the differing infection rates of the strains naturally resulted in competition between them, since the healthy leaf area available was limited (Notes S1).

Numerical simulations

Using biologically relevant parameters for the wheat–P. striiformis f.sp. tritici pathosystem (Table 1), we performed numerical experiments in order to derive effective mixture compositions. The spatial arrangement of the mixture components was random and invariable in the numerical simulations. The initial inoculum of the pathogen population was focal, consisted of four pathotypes with equal density, and was located in the top-left corner of the field, P1(10Δx,10Δy,0) = P2(10Δx,10Δy,0) = P3(10Δx,10Δy,0) =P4(10Δx,10Δy,0) = 7.8E-5. We varied the proportions of the components from 10 to 90% in six two-component mixtures (R/MR, R/MS, R/S, MR/MS, MS/S, and MR/S) and four three-component mixtures (MS/MR/R, S/MS/R, S/MS/R, and S/MS/MR). For all 378 combinations, we calculated the disease severity of a cultivar mixture at the end of the growing season, inline image inline image, and the absolute disease reduction (%) = inline image, where γi(T) is the severity in the ith mixture component grown in a pure stand, i = 1, …, k (where k is the number of mixture components), αi is frequency of ith component in the mixture, and y(T) is the severity of the mixture. Absolute disease reduction reflects the absolute gain in severity decrease resulting from the use of a cultivar mixture in comparison to the weighted mean of severity of components grown in pure stands. Negative values of absolute disease reduction indicate a negative mixture effect, whereas positive values indicate that the mixture deployment was beneficial. Further, we define an effective mixture as one displaying positive absolute disease reduction, and a disease severity, y (T ), of < 20% (the acceptable disease severity).

In contrast to the relative disease reduction (%) (inline image) used in mixture theory as a measure of the mixture performance (Finckh & Mundt, 1992; Akanda & Mundt, 1996), we used the absolute disease reduction, as this seemed to fit better with the estimation of the efficiency of a mixture, which has some components that can be partially resistant. Indeed, in this case, relative disease reduction can take high values with only a small gain in severity decrease, when the weighted mean severity of components grown in pure stands is small. For example, the relative disease reduction for the mixture of 80% R and 20% MR is c. 23%. This high mixture effect value was obtained with a low disease severity gain, resulting from 6.5% weighted mean of severity in pure stands, and 5% mixture severity.


Model fitting

The smooth line in Fig. 2 depicts the fit between (Eqn 1) and the experimental data of wheat yellow rust spread over the homogeneous susceptible plot. The smooth curve generated by the model fits the field data quite closely, and the values obtained for the merit function were sufficiently small: F(eδP0) = F(0.296, 0.029, 7.8E–5) = 107.7, F(r, K) = F(0.055, 2.7) = 0.1 (Fig. 2). This is a qualitative indicator of the goodness of fit of the model. When we quantitatively tested the goodness of fit for (Eqn 1) (df = 5, df = 3), we did not reject the null hypotheses that the data were derived from the model, as R2 = 0.98 and inline image. The bootstrap estimates of the errors in an estimated parameter set are given in Table 1.

Table 1 shows that, overall, the model dynamics are moderately sensitive to changes in infection efficiency, e, and initial density of the healthy leaf area, H0. The impact of the variation of the diffusion coefficient, δ, and initial infected leaf area, P0, on the system dynamics was very low and negligible. (Eqn 1) is insensitive to both the host growth rate, r, and the host carrying capacity, K.

Effective deployment of quantitative resistance

Two-component mixtures

Fig. 3 shows that the absolute disease reduction increased from negative values to positive ones for mixture R/S. It remained negative for mixtures MR/S, MS/S, and positive for R/MR and R/MS. It fluctuates around zero for MR/MS. Like the disease severity, it declined, as the proportion of the resistant components, MS, MR, or R, increased in the mixtures. The rate of increase or decline in absolute disease reduction or severity curve depends on mixture composition.

Figure 3.

Absolute disease reduction (a) and disease severity (b) are plotted for various percentages of the first mixture component (20–90%) in six two-component mixtures, R/MR, R/MS, R/S, MR/MS, MS/S, MR/S, where ‘S’ denotes susceptible, ‘M’, moderately susceptible, and ‘R’, resistant. Solid curves correspond to percentages for which the absolute disease reduction is positive. Absolute disease reduction was calculated as the weighted mean severity (pure stands) – severity (cultivar mixture). For instance, to be effective, the mixture of resistant and susceptible lines should contain < 37% of the susceptible component. In this case, the mixture severity becomes < 30%, which corresponds to the weighted mean severity of the mixture components grown in pure stands. Simulation results were obtained with the parameter values presented in Tables 1 and 3.

A susceptible line needs to be combined with > 63% of a resistant line (Fig. 3a) to make up an effective mixture. Using this minimum allowable percentage of a resistant line reduced the disease severity by 31% (Fig. 3b). Progressively increasing the proportion of a moderately susceptible line in a mixture with a susceptible one slowly reduced disease severity. Moving from 60 to 80% of MS reduced the severity by only 10%, while the same change in the MR line resulted in a 15% reduction. However, the progressive severity decline was not enough for either MR/S or MS/S to obtain a positive absolute disease reduction. An increase in the proportion of the resistant line in a mixture with a susceptible line produced the fastest decrease in severity: reducing R from 80 to 60% reduced the severity from 34 to 14%.

The mixture of moderately resistant and resistant lines was the most effective, as disease severity was reduced to 20% or even less by adding small proportions of the resistant line (Fig. 3b). A mixture of resistant and moderately susceptible lines attained the acceptable degree of disease severity with 45% of a resistant line. In both mixtures, the severity decreased slowly as the proportion of R increased. The severity values of R/MS drew closer to those of R/MR as the proportion of R increased, and from 60% of the resistant line, the severities of the two mixtures become almost identical and extremely low.

When the proportion of MR was increased, the absolute disease reduction for the MR/MS mixture oscillated closely around zero. Disease severity declined slowly from 35% as the proportion of the moderately resistant line increased, and was 21% with 90% of the MR line.

Only three mixtures satisfied our criterion for mixture effectiveness – a severity at the end of the growing season of < 20% and positive absolute disease reduction: R/S, R/MS, and R/MR, where the R variety had to exceed values of 71, 42, and 5%, respectively. The MS/S and MR/S mixtures could not reduce disease severity to the acceptable value, while the MR/MS mixture approached 21% severity when the proportion of MR was increased.

Three-component mixtures

Fig. 4 shows that the mixture of moderately susceptible, moderately resistant, and resistant lines was the most effective, since a reduction of disease severity to 20–10% could be obtained over a broader range of proportions of the components. The higher the proportion of the MS line, the higher the proportion of the R line had to be in the mixture in order to preserve the effectiveness of the mixture (Fig. 4a). The maximum allowable proportion of the MS cultivar was 50%, and this could be combined with 30–40% of the R line and 20–10% of the MR line. To maintain disease control at 20% severity, low proportions of the R line should be offset by low proportions of the MS line and high proportions of the MR line. To attain a disease severity of 10%, we would have to increase the proportion of the R cultivar to at least 40%. The advantage of the MS/MR/R combination is that it consists of a well-balanced mixture: 30% of MS, 40% of MR and 30% of R lines can reduce the severity of disease by 17%.

Figure 4.

Percentage of the three components of a cultivar mixture for which the absolute disease reduction is positive, and the disease severity is < 30%: light gray, component proportions for which the disease severity belongs to a set of [0%; 10%]; mid-gray, proportions for which the disease severity belongs to a set of [10%; 20%]; dark gray, proportions for which the disease severity belongs to a set of [20%; 30%]. Outside the depicted areas, the target level of control cannot be attained, as either the absolute disease reduction becomes negative or the disease severity exceeds 30%. (a) MS/MR/R mixture, %MR = 100% – (%R + %MS); (b) S/MR/R mixture, %MR = 100% – (%R + %S); (c) S/MS/R mixture, %MS = 100% – (%R + %S) (S, susceptible; M, moderately susceptible; R, resistant). Simulation results were obtained with the parameter values presented in Tables 1 and 3.

The use of the susceptible line in three-component mixtures greatly reduces the range of component proportions at which the disease control can be effective (Fig. 4b,c). To reduce disease severity to 20%, the proportion of the susceptible line should not exceed 20% in mixtures with moderately resistant, or moderately susceptible and resistant lines (Fig. 4b,c). The higher the proportion of the susceptible line in a mixture, the higher the proportion of the resistant or moderately resistant line that has to be deployed to maintain effective disease control. To obtain 20% disease severity, an amount of only 10% of the S cultivar should be offset by including at least 30% of R and 60% of MR lines, while including 20% of the S cultivar requires the inclusion of at least 60% of the resistant line. Moderately resistant and resistant lines complement each other in reducing the negative impact of an S cultivar. In comparison with the MS/MR/R mixture, with which 20% severity can be obtained with well-balanced cultivar proportions, the S/MR/R mixture can provide the same result with 10% of S, 60% of MR, and 30% of R.

Fig. 4(a,c) shows that a moderately susceptible line should be deployed in combination with moderately resistant and resistant lines. A high proportion of 50% of a moderately susceptible line can be successfully offset by 30% of resistant and 20% of moderately resistant lines, and results in 20% disease severity. The combination of a moderately susceptible line with a susceptible one requires the deployment of > 50% of a resistant line to reduce disease severity to 20% (Fig. 4c).


We built a parsimonious host–pathogen model describing the dynamics of competing pathogen strains spreading over a genetically diversified host population distributed in a two-dimensional environment. Applying our model to the wheat–yellow rust pathosystem, we identified the conditions under which cultivar mixtures that divide the pathogen population into nonspecialized pathotypes can result in both an acceptable degree of disease severity and a positive absolute disease reduction. In particular, we conclude that the best way to use moderately resistant and moderately susceptible cultivars is to associate them with a highly resistant cultivar, rather than with a susceptible one. Moreover, our results reveal that the effective deployment of quantitative resistance makes it possible to reduce disease density using small proportions of the highly resistant cultivar. The significance of our findings is that they demonstrate that the role of quantitative plant resistance in cultivar mixtures is to reinforce the effect of the highly resistant cultivars rather than being a substitute for them.

The model is parameterized to approximate to yellow rust spatial dynamics, but our work is generic in nature, and the model is relevant to any plant–pest system where the pest population has a diffusive pattern of dispersal. According to the test of goodness of fit, our model accurately captures the dynamics of the development of yellow rust epidemic. The results of the local sensitivity analysis suggest that the infection efficiencies of pathogen strains and the initial density of healthy leaf area must be estimated accurately to provide a reliable model output. The result yielded by the model agrees with previous theoretical findings demonstrating that disease spread can be slowed markedly when at least 60% of the landscape area becomes unsuitable for the disease (Collingham & Huntley, 2000; Otten et al., 2004; Dewhirst & Lutscher, 2009). The fact that our theoretical findings are consistent with empirical results showing that rust dynamics can be controlled by a mixture consisting of one-third of a susceptible cultivar and two-thirds of a resistant cultivar (de Vallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope, 2004) is further evidence of the credibility of the model. Moreover, the model confirms empirical findings reporting that the severity-reducing effect of a resistant cultivar mixed with a moderately susceptible cultivar on wheat yellow rust was greater than that obtained when it was mixed with the same proportion of a susceptible cultivar (Aslam & Fischbeck, 1993; Huang et al., 2011). Thus, we can conclude that the model is biologically relevant, and that it can be used for educational and research purposes to illustrate the principles of mixture theory and generate hypotheses about the deployment of resistant cultivars. In the present formulation, the model is valid for locations with short growing seasons, such as cereal crops grown in a temperate climate. It has been shown that the length of the growing season can influence the effects of cultivar mixtures on disease dynamics (Garrett et al., 2009). Incorporating the carrying capacity of the pathogen strains could provide a model applicable to locations with longer seasons. We intend to complete the validation of the model by carrying out experiments to test the performance of the suggested deployment strategies of wheat cultivars carrying nonspecific quantitative resistance. This will increase confidence in the model for further practical application in designing cultivar mixtures including quantitatively resistant cultivars to provide sustainable wheat rust management.

Despite the fact that the model does not account for the processes by which the pathogen adapts to quantitative resistance, it does allow us to derive evolutionarily stable strategies, if we use the following theoretical criterion: in order to slow the process of pathogen adaptation, the proportion of a resistant cultivar in a mixture with a susceptible cultivar should be either < 30% or > 70%, as intermediate proportions lead to the rapid emergence of super-virulent rust strains (van den Bosch & Gilligan, 2003; Bourget, 2013; Bourget et al., 2013). According to this criterion, our model shows that two-component mixtures consisting of a resistant cultivar and a susceptible, moderately susceptible, or moderately resistant cultivar can be evolutionarily stable, if the proportion of the resistant cultivar exceeds 70%. Linking the characteristics of the plant resistance to epidemic dynamics in a theoretical model, Fabre et al. (2012) have shown that low cropping ratios of a resistant plant can prolong the resistance durability. We therefore also highlight the advantages of three-component mixtures, such as MS/MR/R and S/MS/R, as they can provide evolutionarily stable disease control with low proportions of the resistant cultivar.

Modeling the dynamics of a nonspecialized pathogen in a cultivar mixture, Jeger et al. (1981a) showed that its dynamics can be retarded if the components of the mixture create sufficiently heterogeneous susceptibility. Our model allowed us to extend this finding by determining what amounts of component resistance need to be combined to create a mixture possessing this degree of functional heterogeneity. According to our results, the heterogeneity created by a mixture of highly resistant and moderately resistant/susceptible cultivars is functional, as it can reduce pathogen density to the target low value even with an intermediate proportion of the highly resistant cultivar. Moreover, moderately resistant/susceptible cultivars can be used to reduce the proportion of the highly resistant cultivar required in mixtures, and thus to reduce the selective pressure on the pathogen population. They can be substituted for a susceptible line, but not for a highly resistant cultivar. Our results reinforce the view that the proportions of the different components of the mixture can compensate for weak heterogeneity in the susceptibility of the components, or reinforce the effect of strong heterogeneity (van den Bosch, 1993). The model showed that the higher the proportion of the highly resistant cultivar present in a mixture with a moderately susceptible cultivar, the less severe the yellow rust, which is consistent with the recent empirical results reported by Huang et al. (2011).

Our model shows that three-component mixtures including differing amounts of resistance make it possible to design a broader range of effective control strategies. Finckh et al. (2000) argued that mixture performance depends on the mean amount of resistance of all the components of the mixture, rather than on the number of components. However, empirical studies demonstrate that four-way mixtures are more effective than two-way mixtures in reducing severity of septoria tritici blotch (Mille et al., 2006). Furthermore, in the case of Rhynchosporium secalis, Newton et al. (1997) showed that increasing the number of barley cultivars increased the mixture effect. Cultivar mixtures in practical use tend to include up to five cultivars carrying qualitative or quantitative resistance. Our work shows that successful disease control can be achieved with a three-component mixture, as they can create well-balanced diversity and functional heterogeneity in component susceptibility. Furthermore, in mixtures including combinations of a highly resistant line with a moderately resistant one, the proportion of the resistant line can be reduced to 10–30% without any loss of mixture effectiveness. Our results suggest that it is worth increasing the number of components in the mixture when a susceptible cultivar is unavoidable, or when we need to reduce the proportion of the resistant cultivar in order to obtain more lasting control.

Recently, cultivar mixtures have been promoted by Chinese government agencies and researchers to control yellow rust in mountainous areas where P. striiformis can oversummer, but they are not yet being used on extensive areas, such as those that would be required for controlling rice blast (Huang et al., 2011). More than 30% of the wheat acreage in Washington State was planted with mixtures of two or three cultivars, and this has reduced the yield losses caused by stripe rust (Kolmer et al., 2009). Despite the high potential value of cultivar mixtures in disease management, the current standards of uniformity, marketing restrictions, and processing quality impede their wide use. In addition to technical barriers, there is a dearth of the mathematical models that could help in selecting component cultivars and in tailoring cultivar mixtures to local pathogen populations. Here, we demonstrate that a parsimonious, spatially explicit model of a reaction–diffusion type could fill this gap. Our study contributes to the development of a general framework of mixture theory that could allow us to design diversified agroecosystems that exert low selective pressure on the pathogen populations. The model can be used in multidisease control by intercropping as well. It provides theoretical support for future experimental research intended to develop disease control strategies based on diversified host populations.


The authors are grateful to Laurent Gérard and Marc Leconte for their technical assistance, and to Christophe Montagnier and his team (INRA Experimental Unit, Thiverval-Grignon) for field plot management. The research was funded by the ENDURE project (2008–2010) under the 6th European Framework Programme.