The proportion of resistant hosts in mixtures should be biased towards the resistance with the lowest breaking cost.

Current agricultural practices facilitate emergence and spread of plant diseases through the wide use of monocultures. Host mixtures are a promising alternative for sustainable plant disease control. Their effectiveness can be partly explained by priming-induced cross-protection among plants. Priming occurs when plants are challenged with non-infective pathogen genotypes, resulting in increased resistance to subsequent infections by infective pathogen genotypes. We developed an epidemiological model to explore how mixing two distinct resistant varieties can reduce disease prevalence. We considered a pathogen population composed of three genotypes infecting either one or both varieties. We found that host mixtures should not contain an equal proportion of resistant plants, but a biased ratio (e.g. 80 : 20) to minimize disease prevalence. Counter-intuitively, the optimal ratio of resistant varieties should contain a lower proportion of the costliest resistance for the pathogen to break. This benefit is amplified by priming. This strategy also prevents the invasion of pathogens breaking all resistances.

Impact of ontogenic changes on the dynamics of a fungal crop disease model motivated by coffee leaf rust.

Ontogenic resistance has been described for many plant-pathogen systems. Conversely, coffee leaf rust, a major fungal disease that drastically reduces coffee production, exhibits a form of ontogenic susceptibility, with a higher infection risk for mature leaves. To take into account stage-dependent crop response to phytopathogenic fungi, we developed an SEIR-U epidemiological model, where U stands for spores, which differentiates between young and mature leaves. Based on this model, we also explored the impact of ontogenic resistance on the sporulation rate. We computed the basic reproduction number [Formula: see text], which classically determines the stability of the disease-free equilibrium. We identified forward and backward bifurcation cases. The backward bifurcation is generated by the high sporulation of young leaves compared to mature ones. In this case, when the basic reproduction number is less than one, the disease can persist. These results provide useful insights on the disease dynamics and its control. In particular, ontogenic resistance may require higher control efforts to eradicate the disease.

Improving sustainable crop protection using population genetics concepts.

Growing genetically resistant plants allows pathogen populations to be controlled and reduces the use of pesticides. However, pathogens can quickly overcome such resistance. In this context, how can we achieve sustainable crop protection? This crucial question has remained largely unanswered despite decades of intense debate and research effort. In this study, we used a bibliographic analysis to show that the research field of resistance durability has evolved into three subfields: (1) "plant breeding" (generating new genetic material), (2) "molecular interactions" (exploring the molecular dialogue governing plant-pathogen interactions) and (3) "epidemiology and evolution" (explaining and forecasting of pathogen population dynamics resulting from selection pressure[s] exerted by resistant plants). We argue that this triple split of the field impedes integrated research progress and ultimately compromises the sustainable management of genetic resistance. After identifying a gap among the three subfields, we argue that the theoretical framework of population genetics could bridge this gap. Indeed, population genetics formally explains the evolution of all heritable traits, and allows genetic changes to be tracked along with variation in population dynamics. This provides an integrated view of pathogen adaptation, in particular via evolutionary-epidemiological feedbacks. In this Opinion Note, we detail examples illustrating how such a framework can better inform best practices for developing and managing genetically resistant cultivars.

Taking Advantage of Pathogen Diversity and Immune Priming to Minimize Disease Prevalence in Host Mixtures: A Model.

Host mixtures are a promising method for agroecological plant disease control. Plant immunity is key to the success of host mixtures against polymorphic pathogen populations. This immunity results from priming-induced cross-protection, whereby plants able to resist infection by specific pathogen genotypes become more resistant to other pathogen genotypes. Strikingly, this phenomenon was absent from mathematical models aiming at designing host mixtures. We developed a model to specifically explore how priming affects the coexistence of two pathogen genotypes in host mixtures composed of two host genotypes and how it affects disease prevalence. The main effect of priming is to reduce the coexistence region in the parameter space (due to the cross-protection) and to generate a singular mixture of resistant and susceptible hosts corresponding to the maximal reduction disease prevalence (in absence of priming, a resistant pure stand is optimal). The epidemiological advantage of host mixtures over a resistant pure stand thus appears as a direct consequence of immune priming. We also showed that there is indirect cross-protection between host genotypes in a mixture. Moreover, the optimal mix prevents the emergence of a resistance-breaking pathogen genotype. Our results highlight the importance of considering immune priming to design optimal and sustainable host mixtures.

Estimating virus effective population size and selection without neutral markers.

By combining high-throughput sequencing (HTS) with experimental evolution, we can observe the within-host dynamics of pathogen variants of biomedical or ecological interest. We studied the evolutionary dynamics of five variants of Potato virus Y (PVY) in 15 doubled-haploid lines of pepper. All plants were inoculated with the same mixture of virus variants and variant frequencies were determined by HTS in eight plants of each pepper line at each of six sampling dates. We developed a method for estimating the intensities of selection and genetic drift in a multi-allelic Wright-Fisher model, applicable whether these forces are strong or weak, and in the absence of neutral markers. This method requires variant frequency determination at several time points, in independent hosts. The parameters are the selection coefficients for each PVY variant and four effective population sizes Ne at different time-points of the experiment. Numerical simulations of asexual haploid Wright-Fisher populations subjected to contrasting genetic drift (Ne ∈ [10, 2000]) and selection (|s| ∈ [0, 0.15]) regimes were used to validate the method proposed. The experiment in closely related pepper host genotypes revealed that viruses experienced a considerable diversity of selection and genetic drift regimes. The resulting variant dynamics were accurately described by Wright-Fisher models. The fitness ranks of the variants were almost identical between host genotypes. By contrast, the dynamics of Ne were highly variable, although a bottleneck was often identified during the systemic movement of the virus. We demonstrated that, for a fixed initial PVY population, virus effective population size is a heritable trait in plants. These findings pave the way for the breeding of plant varieties exposing viruses to stronger genetic drift, thereby slowing virus adaptation.

Influence of the components of propagule pressure, Allee effects, and stochasticity on the time to establish introduced populations.

Forecasting whether individuals of an introduced population will succeed to establish is a challenge in invasion and conservation biology. The present work aims to decouple the impact of the components of propagule pressure on the time for population establishment in the presence of Allee effects and stochasticity in propagule sizes. The mean first passage time (MFPT) for a population to reach a viable size is used as a measure of the establishment success for the introduction processes involving periodic introductions. By fixing the introduction rate (mean number of introduced individuals per unit time) and varying the period of introduction from small ranges (small and frequent introductions) to large ones (infrequent and large releases), we study the influence of introduction distribution over time. These patterns of introduction are compared in a semi-stochastic model by observing which factors minimize the MFPT from an initially absent population, and hence, ensure the fastest population establishment. We investigate the influence on these minima of the introduction rate, variability in the introduction sizes, and occurrence of catastrophes that temporarily wipe out the population. Whereas most investigated cases show that infrequent and large introductions favor population establishment as expected, small and frequent introductions are preferred when the introduction rate is large and/or the variability in the introduction size is strong. Moreover, we observed counterintuitively that catastrophes strongly increase MFPT at small periods of introduction. In addition, we showed that stochasticity in introduction tends to increase the MFPT except when the introduction rate is small and introductions are evenly spread out in time.

Quantitative trait loci in pepper control the effective population size of two RNA viruses at inoculation.

Infection of plants by viruses is a complex process involving several steps: inoculation into plant cells, replication in inoculated cells and plant colonization. The success of the different steps depends, in part, on the viral effective population size (Ne), defined as the number of individuals passing their genes to the next generation. During infection, the virus population will undergo bottlenecks, leading to drastic reductions in Ne and, potentially, to the loss of the fittest variants. Therefore, it is crucial to better understand how plants affect Ne. We aimed to (i) identify the plant genetic factors controlling Ne during inoculation, (ii) understand the mechanisms used by the plant to control Ne and (iii) compare these genetic factors with the genes controlling plant resistance to viruses. Ne was measured in a doubled-haploid population of Capsicum annuum. Plants were inoculated with either a Potato virus Y (PVY) construct expressing the green fluorescent protein or a necrotic variant of Cucumber mosaic virus (CMV). Newas assessed by counting the number of primary infection foci on cotyledons for PVY or the number of necrotic local lesions on leaves for CMV. The number of foci and lesions was correlated (r=0.57) and showed a high heritability (h2=0.93 for PVY and h2=0.98 for CMV). The Ne of the two viruses was controlled by both common quantitative trait loci (QTLs) and virus-specific QTLs, indicating the contribution of general and specific mechanisms. The PVY-specific QTL colocalizes with a QTL that reduces PVY accumulation and the capacity to break down a major-effect resistance gene.

Augmentative biocontrol when natural enemies are subject to Allee effects.

Intraspecific interactions such as Allee effects are key properties that can guide population management. This contribution considers component Allee effects that are elementary mechanisms leading to declines of fitness at the population scale, i.e. demographic Allee effects. It especially focuses on the consequences of such properties in predator populations, and investigates their repercussions in a biological control context. A modelling framework able to account for reproductive and/or foraging component Allee effects is proposed. From this, four models of augmentative biological control, corresponding to the periodic introduction of natural enemies, have been investigated. This is done using semi-discrete models: ordinary differential equations are used to depict predator-prey dynamics and a discrete equation describes the abrupt augmentation of predators at periodic intervals. In that context, stability of a prey-free solution corresponding to pest eradication has been analyzed. It has been found that rare but large introductions should be preferred over frequent and small ones, when Allee effects influence predator populations. In particular, the occurrence of foraging, rather than reproducing, Allee effects significantly hinders pest eradication. Cases where the pest-free solution is locally, but not globally, stable were also observed and were shown to be favored by the occurrence of reproductive Allee effects among predators.

Competition between phytoplankton and bacteria: exclusion and coexistence.

Resource-based competition between microorganisms species in continuous culture has been studied extensively both experimentally and theoretically, mostly for bacteria through Monod and Contois "constant yield" models, or for phytoplankton through the Droop "variable yield" models. For homogeneous populations of N bacterial species (Monod) or N phytoplanktonic species (Droop), with one limiting substrate and under constant controls, the theoretical studies indicated that competitive exclusion occurs: only one species wins the competition and displaces all the others (Armstrong and McGehee in Am Nat 115:151, 1980; Hsu and Hsu in SIAM J Appl Math 68:1600-1617, 2008). The winning species expected from theory is the one with the lowest "substrate subsistence concentration" s([star]), such that its corresponding equilibrium growth rate is equal to the dilution rate D. This theoretical result was validated experimentally with phytoplankton (Tilman and Sterner in Oecologia 61(2):197-200, 1984) and bacteria (Hansen and Hubell in Science 207(4438):1491-1493, 1980), and observed in a lake with microalgae (Tilman in Ecology 58(22):338-348, 1977). On the contrary for aggregating bacterial species described by a Contois model, theory predicts coexistence between several species (Grognard et al. in Discrete Contin Dyn Syst Ser B 8(1):73-93, 2007). In this paper we present a generalization of these results by studying a competition between three different types of microorganisms: planktonic (or free) bacteria (represented by a generalized Monod model), aggregating bacteria (represented by a Contois model) and free phytoplankton (represented by a Droop model). We prove that the outcome of the competition is a coexistence between several aggregating bacterial species with a free species of bacteria or phytoplankton, all the other free species being washed out. This demonstration is based mainly on the study of the substrate concentration's evolution caused by competition; it converges towards the lowest subsistence concentration s([star]), leading to three different types of competition outcomes: (1) the best free bacteria or phytoplankton competitor excludes all other species; (2) only some aggregating bacterial species coexist in the chemostat; (3) A coexistence between the single best free species, with one or several aggregating species.

The functional response predicts the effect of resource distribution on the optimal movement rate of consumers.

Understanding how often individuals should move when foraging over patchy habitats is a central question in ecology. By combining optimality and functional response theories, we show analytically how the optimal movement rate varies with the average resource level (enrichment) and resource distribution (patch heterogeneity). We find that the type of functional response predicts the effect of enrichment in homogeneous habitats: enrichment should decrease movement for decelerating functional responses, but increase movement for accelerating responses. An intermediate resource level thus maximises movement for type-III responses. Counterintuitively, greater movement costs favour an increase in movement. In heterogeneous habitats predictions further depend on how enrichment alters the variance of resource distribution. Greater patch variance always increases the optimal rate of movement, except for type-IV functional responses. While the functional response is well established as a fundamental determinant of consumer-resource dynamics, our results indicate its importance extends to the understanding of individual movement strategies.

How optimal foragers should respond to habitat changes: a reanalysis of the Marginal Value Theorem.

The Marginal Value Theorem (MVT) is a cornerstone of biological theory. It connects the quality and distribution of patches in a fragmented habitat to the optimal time an individual should spend exploiting them, and thus its optimal rate of movement. However, predictions regarding how habitat alterations should impact optimal strategies have remained elusive, with heavy reliance on graphical arguments. Here we derive the sensitivity of realized fitness and optimal residence times to general habitat attributes, for homogeneous and heterogeneous habitats, retaining the level of generality of the MVT. We provide new predictions on how altering travel times, patch qualities and/or relative abundances should affect optimal strategies, and study the consequences of habitat heterogeneity. We show that knowledge of average characteristics is in general not sufficient to predict the change in the average rate of movement. We apply our results to examine the conditions under which the optimal strategies are invariant to scaling. We prove a previously conjectured form of invariance in homogeneous habitats, but show that invariances to scaling are not generic in heterogeneous habitats. We also consider the relative exploitation of patches that differ in quality, clarifying the conditions under which it is adaptive to stay longer on poorer patches.

Self-financing model for cabbage crops with pest management.

Smallholder farmers rely on their farm earnings to cover operating costs and generate income. That is not an easy task because of the pests, which reduce yields and generate plant protection costs. The farm yield and plant protection depend on the budget capacity of the farmer. In this work, we want to explore conditions for a sustainable and self-financing cabbage farm. We propose then a non-linear mathematical model for cabbage crops by considering the current account of the plantation as a dynamic variable. We assume that this variable increases due to the sale of cabbages, and provides for the seedling purchase, the plant protection costs, and the grower's income. In the first part, we analyze the model without pest management. We determine how the budget must be spent and we show the existence of a double transcritical bifurcation. We quantify the seasonal yield and income, and estimate the damage due to pest herbivory. In the second part, we analyze a slightly simplified version of our model and obtain the existence of a backward bifurcation. Furthermore, we show that botanical pesticides can be used to prevent pest spread with relatively low plant protection costs.

Impulsive modelling of rust dynamics and predator releases for biocontrol.

Fungal diseases cause serious damages in crop worldwide. In particular, coffee leaf rust (CLR), caused by fungus Hemileia vastatrix attacks coffee leaves and reduces coffee yield. This paper presents a multi-seasonal model of the CLR development in the coffee plantation with continuous dynamics during the rainy season and a discrete event to represent the simpler dynamics during the dry season. Biological control using predators through one or more discrete introduction events over the year is then added. Analytical and semi-numerical studies are performed to identify how much and how frequently predators need to be introduced through the definition of a threshold value, as a function of various parameters. We show that the best strategy to efficiently control the disease depends on the predator mortality: low mortality parasites need be released only once a year, while high mortality parasites should be released more frequently to ensure their persistence in the plantation. This work hence provides qualitative and quantitative bases for the deployment of predator-based biocontrol, a promising alternative to fungicides for rust control.

Host mixtures for plant disease control: Benefits from pathogen selection and immune priming.

Multiline and cultivar mixtures are highly effective methods for agroecological plant disease control. Priming-induced cross protection, occurring when plants are challenged by avirulent pathogen genotypes and resulting in increased resistance to subsequent infection by virulent ones, is one critical key to their lasting performance against polymorphic pathogen populations. Strikingly, this mechanism was until recently absent from mathematical models aiming at designing optimal host mixtures. We developed an epidemiological model to explore the effect of host mixtures composed of variable numbers of single-resistance cultivars on the equilibrium prevalence of the disease caused by pathogen populations polymorphic for virulence complexity. This model shows that a relatively large amount of resistance genes must be deployed to achieve low disease prevalence, as pathogen competition in mixtures tends to select for intermediate virulence complexity. By contrast, priming significantly reduces the number of plant genotypes needed to drop disease prevalence below an acceptable threshold. Given the limited availability of resistance genes in cultivars, this mechanism of plant immunity should be assessed when designing host mixtures.

Influence of intrapredatory interferences on impulsive biological control efficiency.

In this paper, a model is proposed for the biological control of a pest by its natural predator. The model incorporates a qualitative description of intrapredatory interference whereby predator density decreases the per capita predation efficiency and generalises the classical Beddington-DeAngelis formulation. A pair of coupled ordinary differential equations are used, augmented by a discrete component to depict the periodic release of a fixed number of predators into the system. This number is defined in terms of the rate of predator release and the frequency at which the releases are to be carried out. This formulation allows us to compare different biological control strategies in terms of release size and frequency that involve the same overall number of predators. The stability properties of the zero-pest solution are analysed. We obtain an upper bound on the interference strength (the biological condition) and a minimal bound on the predator release rate (the managerial condition) required to eradicate a pest population. We demonstrate that increasing the frequency of releases reduces this minimal rate and also increases the rate of convergence of the system to the zero-pest solution for a given release rate. Additionally, we show that other conclusions are to be expected if the interferences between predators have weaker or stronger effects than the generalised Beddington-DeAngelis formulation proposed in this paper.

Modelling and control of a banana soilborne pest in a multi-seasonal framework.

We study the infestation dynamics of banana or plantain plants by Radopholus similis, a plant-parasitic nematode that causes severe damages. Two control strategies are implemented in our model: pesticides, which are widely used, and fallows, which are more environmentally friendly. To represent the host-parasite dynamics, two semi-discrete models are proposed. During each cropping season, free nematodes enter the plant roots, on which they feed and reproduce. At the end of the cropping season, fruits are harvested. In the first model, the parent plant is cut down to be replaced by one of its suckers and pesticides are applied. In the second model, the parent plant is uprooted and a fallow period is introduced, inducing the decay of the free pest populations; at the beginning of the next cropping season, a pest-free vitroplant is planted. For both models, the effective reproduction number of pests is computed, assuming that the infestation dynamics are fast compared to the other processes, which leads to the model order reduction. Conditions on the pesticide load or the fallow duration are then derived to ensure the stability of the periodic pest free solution. Finally, numerical simulations illustrate these theoretical results.

Virus epidemics, plant-controlled population bottlenecks and the durability of plant resistance.

Plant qualitative resistances to viruses are natural exhaustible resources that can be impaired by the emergence of resistance-breaking (RB) virus variants. Mathematical modelling can help determine optimal strategies for resistance durability by a rational deployment of resistance in agroecosystems. Here, we propose an innovative approach, built up from our previous empirical studies, based on plant cultivars combining qualitative resistance with quantitative resistance narrowing population bottlenecks exerted on viruses during host-to-host transmission and/or within-host infection. Narrow bottlenecks are expected to slow down virus adaptation to plant qualitative resistance. To study the effect of bottleneck size on yield, we developed a stochastic epidemic model with mixtures of susceptible and resistant plants, relying on continuous-time Markov chain processes. Overall, narrow bottlenecks are beneficial when the fitness cost of RB virus variants in susceptible plants is intermediate. In such cases, they could provide up to 95 additional percentage points of yield compared with deploying a qualitative resistance alone. As we have shown in previous works that virus population bottlenecks are at least partly heritable plant traits, our results suggest that breeding and deploying plant varieties exposing virus populations to narrowed bottlenecks will increase yield and delay the emergence of RB variants. This article is part of the theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes'. This issue is linked with the subsequent theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control'.

Impact of genetic drift, selection and accumulation level on virus adaptation to its host plants.

The efficiency of plant major resistance genes is limited by the emergence and spread of resistance-breaking mutants. Modulation of the evolutionary forces acting on pathogen populations constitutes a promising way to increase the durability of these genes. We studied the effect of four plant traits affecting these evolutionary forces on the rate of resistance breakdown (RB) by a virus. Two of these traits correspond to virus effective population sizes (Ne ) at either plant inoculation or during infection. The third trait corresponds to differential selection exerted by the plant on the virus population. Finally, the fourth trait corresponds to within-plant virus accumulation (VA). These traits were measured experimentally on Potato virus Y (PVY) inoculated to a set of 84 pepper doubled-haploid lines, all carrying the same pvr23 resistance gene, but having contrasting genetic backgrounds. The lines showed extensive variation for the rate of pvr23 RB by PVY and for the four other traits of interest. A generalized linear model showed that three of these four traits, with the exception of Ne at inoculation, and several pairwise interactions between them had significant effects on RB. RB increased with increasing values of Ne during plant infection or VA. The effect of differential selection was more complex because of a strong interaction with VA. When VA was high, RB increased as the differential selection increased. An opposite relationship between RB and differential selection was observed when VA was low. This study provides a framework to select plants with appropriate virus evolution-related traits to avoid or delay RB.

Optimal life-history strategies in seasonal consumer-resource dynamics.

The interplay between individual adaptive life histories and populations dynamics is an important issue in ecology. In this context, we considered a seasonal consumer-resource model with nonoverlapping generations. We focused on the consumers decision-making process through which they maximize their reproductive output via a differential investment into foraging for resources or reproducing. Our model takes a semi-discrete form, and is composed of a continuous time within-season part, similar to a dynamic model of energy allocation, and of a discrete time part, depicting the between seasons reproduction and mortality processes. We showed that the optimal foraging-reproduction strategies of the consumers may be "determinate" or "indeterminate" depending on the season length. More surprisingly, it depended on the consumers population density as well, with large densities promoting indeterminacy. A bifurcation analysis showed that the long-term dynamics produced by this model were quite rich, ranging from both populations' extinction, coexistence at some season-to-season equilibrium or on (quasi)-periodic motions, to initial condition-dependent dynamics. Interestingly, we observed that any long-term sustainable situation corresponds to indeterminate consumers' strategies. Finally, a comparison with a model involving typical nonoptimal consumers highlighted the stabilizing effects of the optimal life histories of the consumers.

Global stability and optimisation of a general impulsive biological control model.

An impulsive model of augmentative biological control consisting of a general continuous predator-prey model in ordinary differential equations, i.e. a meta-model, augmented by a discrete part describing periodic introductions of predators is considered. The existence of an invariant periodic solution that corresponds to prey eradication is shown and a condition ensuring its global asymptotic stability is given. An optimisation problem related to the preemptive use of augmentative biological control is then considered. It is assumed that the per time unit budget of biological control (i.e. the number of predators to be released) is fixed and the best deployment of this budget is sought in terms of release frequency. The cost function to be minimised is the time needed to reduce an unforeseen prey (pest) invasion occurring at a worst time instant under some harmless level. The analysis shows that the optimisation problem admits a countable infinite number of solutions. An argumentation considering the required robustness of the optimisation result with respect to the invasive prey population level and to the model parameters is then conducted. It is shown that the cost function is decreasing in the predator release frequency so that the best deployment of the biocontrol agents is to carry out as frequent introductions as possible.