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.