Olivia Pointurier

Stéphanie Gibot-Leclerc

Delphine Moreau

Carole Reibel

Eric Vieren

Nathalie Colbach
email: nathalie.colbach@inrae.fr

Designing a model to investigate cropping systems aiming to control both parasitic plants and weeds

Keywords: branched broomrape, weed, agroecology, modelling, cropping systems, Phelipanche ramosa, PHERASYS, biological regulation 2

Branched broomrape (Phelipanche ramosa (L.) Pomel) is a parasitic plant, which causes severe yield losses in major crops worldwide. Because of its broad host range, including numerous non-parasitic weed species, the persistence of its seeds in the soil, and the poor efficiency of available management techniques, broomrape management is complex. The objective of the present paper was to develop a broomrape-dynamics model to support the design of management strategies combining multiple techniques aiming at long-term control of broomrape. Towards this goal, we developed a simulation model with formalisms and parameters based on data from our own experiments and the literature. This model called PHERASYS combines 1) a demographic submodel to predict broomrape seed bank dynamics, 2) a trophic-relationships submodel to predict the effect of parasitism on crops and weeds, and 3) a submodel of weed dynamics in agroecosystems to predict the growth of crops and weeds from cropping techniques and pedoclimate. Thanks to an individual representation of each host plant, PHERASYS is able to simulate complex heterogeneous canopies. This model can be used as a tool to test management strategies including crop mixtures and relying on biological regulations by weeds.

Introduction

Broomrapes are parasitic plants that threaten major crops worldwide (Parker, 2013). As holoparasites, they are unable to photosynthesize and entirely rely on host resources to survive [START_REF] Heide-Jørgensen | Introduction: The Parasitic Syndrome in Higher Plants[END_REF]. They must germinate close to a host root, after being stimulated by its exudates [START_REF] Yoneyama | Induction of germination[END_REF], to connect to its vascular system and derive the resources they need (Heide-Jørgensen, 2013). Among broomrapes, branched broomrape, Phelipanche ramosa (L.) Pomel, is particularly devastating since it is found on every continent and is able to infect crops in more than 10 botanical families including Solanaceae, Brassicaceae and Asteraceae (Parker and Riches, 1993;[START_REF] Molenat | Introduction des cultures pièges ou non-hôtes dans les systèmes culturaux afin de lutter contre l'orobanche rameuse : du laboratoire au terrain[END_REF]. Poaceae species are usually not concerned (Qasem and Foy, 2007;[START_REF] Molenat | Introduction des cultures pièges ou non-hôtes dans les systèmes culturaux afin de lutter contre l'orobanche rameuse : du laboratoire au terrain[END_REF]. In France, it is a major pest of winter oilseed rape, where it can cause up to 90% yield losses [START_REF] Gibot-Leclerc | What are the traits of Phelipanche ramosa (L.) Pomel that contribute to the success of its biological cycle on its host Brassica napus L.?[END_REF], and it also infects hemp, sunflower and tobacco (Terres Inovia, 2018).

To date, the only curative method available in arable crops is the application of herbicides on herbicideresistant crops (e.g, varieties tolerant to imidazolinones) (Fernández-Aparicio et al., 2016a;Données Ephy -Anses, 2018). This approach is used in some countries (e.g., USA) but not available in Europe. Moreover, this technique clashes with current policies aiming to reduce the use of pesticides because of their impacts on human health and environment (Potier, 2014). Most herbicides efficient on broomrape are systemic herbicides such as inhibitors of aromatic (glyphosate) or branched-chain amino acid synthesis (imidazolinones and sulfonylureas) applied to the leaves of the crop (e.g., [START_REF] Eizenberg | Imazamox application timing for small broomrape (Orobanche minor) control in red clover[END_REF]. To date, none of these are both tolerable to the host plant (with the exception of herbicide-resistant varieties, see above) and lethal to the parasite. Consequently, several preventive techniques with partial effects must be combined to control broomrapes, such as reduced host-crop frequency in the rotation, delayed host-crop sowing or deep tillage (Grenz et al., 2005a;[START_REF] Rubiales | Innovations in parasitic weeds management in legume crops. A review[END_REF]Fernández-Aparicio et al., 2016b). They must provide long-term control because broomrape seeds are assumed to persist up to 20 years in the soil, although this assumption is based on a few studies on species other than branched broomrape (Murdoch and Kebreab, 2013). Moreover, branched-broomrape management must be thought along with non-parasitic weed management (hereafter, the word "weeds" refers to non-parasitic weeds) because several dozens of weed species are hosts [START_REF] Boulet | Weed species present various responses to Orobanche ramosa L. attack[END_REF][START_REF] Gibot-Leclerc | Potential role of winter rape weeds in the extension of broomrape in Poitou-Charentes[END_REF]Simier et al., 2013;[START_REF] Gibot-Leclerc | Unexpected fast development of branched broomrape on slow-growing Brassicaceae[END_REF]. Weeds might increase broomrape infestation, as they serve as alternative hosts in the absence of host crops. However, they can also deplete the broomrape seed bank, as some species stimulate broomrape germination without supporting further parasite development [START_REF] Gibot-Leclerc | Potential role of winter rape weeds in the extension of broomrape in Poitou-Charentes[END_REF]. Such weed species can potentially provide biological regulation of branched broomrape.

As a consequence, branched-broomrape management must be thought at the cropping-system scale, defined by the crop sequence and the techniques implemented in a field [START_REF] Sebillote | Système de culture, un concept operatoire pour les agronomes[END_REF], using multiple techniques and exploiting biological regulations by weeds. Such strategies are complex and require specific tools for decision support. The field experimental approach, for example, is insufficient, since it allows to test only a limited number of techniques, in a few pedoclimatic and floristic contexts for a few years [START_REF] Jeuffroy | Agronomic model uses to predict cultivar performance in various environments and cropping systems. A review[END_REF].

Simulation models are complementary tools to design cropping systems because they allow evaluating cropping-system performances in the long term in various pedoclimatic and floristic contexts [START_REF] Bergez | Designing crop management systems by simulation[END_REF]Colbach et al., 2014a;[START_REF] Colbach | A methodology for multi-objective cropping system design based on simulations. Application to weed management[END_REF]. Several models of broomrape dynamics have been developed to date which include the effects of cropping techniques on multiannual parasite dynamics [START_REF] Schnell | Simulation of the seed bank dynamics of Orobanche crenata forsk. in some crop rotations common in Northern Syria[END_REF][START_REF] López-Granados | Modelling the demography of crenate broomrape (Orobanche crenata) as affected by broadbean (Vicia faba) cropping frequency and planting date[END_REF][START_REF] Regan | Optimally managing under imperfect detection: a method for plant invasions[END_REF]. However, these models usually consider only one crop species, disregard interactions with weather and weeds and do not represent the host plant's root system (i.e., the host's organ on which the parasite attaches). The one notable exception is the model developed by Grenz et al. (2005a). Their model was, though, parameterized for another broomrape species, Orobanche crenata, and does not take into account interactions with weeds. A first attempt to adapt it to P. ramosa was made with the PHERASYS model (for Phelipanche ramosa in cropping systems) [START_REF] Colbach | A preliminary approach for modelling the effects of cropping systems on the dynamics of broomrape (Phelipanche ramosa) in interaction with the non-parasitic weed flora[END_REF]. It integrated the interactions with weeds thanks to a connection with FLORSYS, a model simulating the dynamics of multispecies weed floras in agroecosystems (Colbach et al., 2014a;[START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF]. However PHERASYS remains largely based on the model developed by Grenz et al. (2005a), with most parameter values borrowed from O. crenata, and processes determining interactions with crops and weeds only partially modelled. In particular, PHERASYS does not simulate the effect of parasitism on host growth, and so it cannot predict yield losses due to parasitism, even though the latter is a major criterion when designing cropping systems.

Consequently, the objective of the present study was to develop a model to simulate branched broomrape dynamics in interaction with multiple crop and weed species in agroecosystems. This model was inspired by the existing PHERASYS but parameters were specifically measured for branched broomrape (hereafter simply called "broomrape"), and the formalisms that describe the broomrape life cycle and the interactions with other plants in the field were redesigned. We (1) proposed a structure for the redesigned PHERASYS version, (2) translated the results of experiments set up purposely for this aim into equations to represent biological processes, and (3) completed missing data from the literature. In a companion paper (Pointurier et al., 2021a), we ran simulations with the model to check the consistency of the model with literature, and to evaluate the potential of cropping systems to manage broomrape and weeds in interaction.

Model structure

The structure for the redesigned PHERASYS was based on the structure of FLORSYS which simulates multispecies crop and weed growth and reproduction at a daily time-step and in 3D from cropping system in interaction with pedoclimate [START_REF] Gardarin | Modeling the dynamics and emergence of a multispecies weed seed bank with species traits[END_REF]Munier-Jolain et al., 2013;Colbach et al., 2014b;Munier-Jolain et al., 2014;[START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF]Pointurier et al., 2021b). The present paper focuses on the specific formalisms that were added for parasite life-stages and its impact on host plants. Details on the FLORSYS model can be found in section S1 in supplementary material online.

The user provides inputs to characterize the field (daily weather, latitude and soil characteristics), the crops and cropping techniques (with their dates, tools used and options) and the initial weed seed bank (including broomrape and non-parasitic weed species). From these, PHERASYS predicts the number of broomrape individuals at different stages and the biomass of broomrape shoots every day depending on biophysical processes (Figure 1). Each day, a part of the seeds in the soil dies; non-dormant seeds can germinate only if stimulated by neighbouring plant root exudates; the germinated seeds must attach to nearby susceptible plant roots before emerging, flowering and producing new seeds. Attached broomrapes die either when they have finished their life cycle, or when their host dies of old age or is destroyed by cultural operations. PHERASYS also simulates the effect of parasitism on host plant growth.

As FLORSYS, PHERASYS represents the canopy in 3D and discretized with voxels ("3D pixels") whose size (e.g. 444 cm³) is chosen by the user. Each non-parasitic plant is represented in 3D, albeit simplified as a cylinder (above-ground) on top a spilled cone (below-ground). The plant's above-ground height, width and leaf area are explicitly located in 3D to simulate competition with neighbouring plants for light. Similarly, the plant's root system is represented in 3D, distributing root length density and biomass across voxels. Broomrape plants are not explicitly represented in 3D, only the location of their host as well as the number and biomass of shoots infecting this host are computed. Seed position in the soil is simulated in a 1D vertical dimension, with 30 successive 1-cm-thick soil layers across which seeds are distributed, with heat and water transfer functions across soil layers. Seeds are assumed to be distributed homogeneously within a soil layer and their future to depend on conditions (temperature, water potential, proximity to host roots) within this soil layer.

Connections with FLORSYS to model interactions with crops and weeds

PHERASYS was connected to FLORSYS to predict the growth of crops, non-parasitic weeds and soil conditions from cropping system components and pedoclimate (Figure 1). PHERASYS and FLORSYS connect two ways, with variables from FLORSYS influencing broomrape dynamics in PHERASYS, and variables from PHERASYS driving the effect of parasitism on host growth in FLORSYS. Soil temperature and moisture (which are predicted from weather, soil characteristics and management techniques, [START_REF] Brisson | An overview of the crop model STICS[END_REF] determine dormancy relief of broomrape seeds and germination progress. The root volume of host plants determines the probability that parasitic seeds encounter roots that stimulate them and support their subsequent attachment. Host biomass determines the amount of resources available for broomrapes and thus the number of broomrapes the host can potentially support. Host phenology determines when root exudates that trigger broomrape germination are released and when attached broomrapes start to grow from host resources. Host growth, which is driven by wphotosynthesis and respiration processes in FLORSYS, is reduced by parasitism. Biomass allocation between roots and above-ground organs is also modified due to parasitism.

Effect of tillage on broomrape-seed movements in the soil is simulated by the seed-movement submodel of FLORSYS (see principle in [START_REF] Gardarin | Modeling the dynamics and emergence of a multispecies weed seed bank with species traits[END_REF]. Any effect of cropping technique on host plants in FLORSYS indirectly affects broomrape dynamics in PHERASYS since it results in host mortality or host biomass reduction (e.g., effect of herbicides on weed) or host biomass increase (e.g., weed management reduces competition with crops).

Data origin

Published data

The first simplistic PHERASYS version pointed out gaps in knowledge of broomrape biology [START_REF] Colbach | A preliminary approach for modelling the effects of cropping systems on the dynamics of broomrape (Phelipanche ramosa) in interaction with the non-parasitic weed flora[END_REF]. Experiments were set up to study and quantify insufficiently known processes, i.e. seed mortality, dormancy and production of the parasite as well as host-parasite trophic relationships. Most results were published (see 1 in Figure 1) and are summarized in Table 1. The experiment quantifying broomrape seed production was described in the present article (see 2 in Figure 1 and section2.3.2). Further data was taken from experiments carried out by other teams (see 3 in Figure 1).

Branched broomrape consists of several pathovars i.e. genetically distinct populations with different host preferences [START_REF] Brault | Variability of Orobanche ramosa populations in France as revealed by cross infestations and molecular markers[END_REF]. Our study focussed on the pathovar predominant in oilseed rape which causes the main damages in France (Terres Inovia, 2018). When no data on this pathovar were available, data on other pathovars were used instead (Appendix A 4).

Measuring broomrape seed production

Broomrape at fructification stage were collected in July 2017 in two oilseed rape fields infested with broomrape in Fontenay-le-Comte (46°26'44" N, 00°46'09" W; Vendée, France) and Saint-Ouenne (46°25'44" N, 00°28'04" W; Deux-Sèvres, France) at crop harvest. Each broomrape consisted of a stem possibly bearing several branches. In total, 36 broomrapes were collected. They were split into two samples to estimate (1) the mean number of seeds produced per seed capsule (structure containing seeds in broomrapes) on six broomrapes (three from each field), and (2) the number of capsules produced per broomrape on the 33 remaining broomrapes.

In the first sample, closed capsules (still comprising their seeds) of each broomrape were counted and dried for 48 hours at 80°C. Then, they were opened to collect the seeds. Seeds and empty capsules were weighted separately. The number of seeds was counted under a stereoscopic microscope (1.95x -250x). Some capsules could not be opened because they were atrophied, contained immature seeds (black seeds that stayed attached in the capsule), or seeds had been eaten by insects (with insect or larvae still inside the capsule). They were counted and weighed separately.

In the second sample, the number of capsules was counted on each broomrape, discriminating closed, empty (having already released their seeds) and missing capsules (leaving an abscission mark on the stem). Closed capsules and broomrape stems from which capsules had been removed were dried for 48 hours at 80°C and weighted separately.

In samples collected in Fontenay-le-Comte, broomrapes were counted on each host to analyse the tradeoff between number of broomrapes attached on a host and the biomass of each broomrape. Biomass per broomrape was calculated as the total biomass of broomrapes (including stems, open and closed capsules) collected on a given host, divided by the number of broomrapes attached on this host. Since the open capsules had not been weighed in the second sample, the mean open capsule weight measured in the first sample was used instead and multiplied by the number of opened capsules counted.

Statistical analysis

Data from published data (section 2.3.1) and from the field samples (section 2.3.2) were analysed using R (R Core Team, 2019). Linear models were fitted with the R package "lm" and non-linear models with "nls" and "nls2" of R. The algorithm "brute-force" was used to solve cases of failed convergence with function "nls2". A pseudo-R² was calculated to estimate the predictive quality of non-linear models (UCLA: Statistical Consulting Group, 2015).

Results

Modelling broomrape dynamics

The following subsections describe how the redesigned PHERASYS models the different stages of the broomrape life-cycle and which data and hypotheses were used. The detailed equations can be found in Appendix A 1, Appendix A 2 and Appendix A 3.

Seed mortality in the soil

Broomrape life-cycle starts in the soil as a seed. Some seeds die due to aging, diseases or predation resulting in annual seed bank decline of 7% (eq. [1] in Appendix A 1) (Pointurier et al., 2019). This annual mortality rate was converted into a daily rate using the same equation as in FLORSYS for nonparasitic weeds [START_REF] Gardarin | Modeling the dynamics and emergence of a multispecies weed seed bank with species traits[END_REF].

Seed dormancy

Fresh broomrape seeds are dormant. They require (1) a period of dry storage, followed by (2) a period of moist storage ("conditioning"), and finally (3) a stimulation by root exudates of potential host plants to relieve dormancy. In field conditions, seed dormancy varies moreover with the season (Murdoch and Kebreab, 2013). Dormancy was modelled as two successive phases in PHERASYS: (1) dormancy relief of new seeds, as a function of soil temperature and moisture; and (2) seasonal dormancy, as a function of time since seed shed.

According to data from [START_REF] Gibot-Leclerc | Responsiveness of Orobanche ramosa L. seeds to GR 24 as related to temperature, oxygen availability and water potential during preconditioning and subsequent germination[END_REF], once moisture requirements are met (≥ -2MPa), dormancy relief of fresh seeds depends solely on temperature during the conditioning period (section S1 in supplementary material online). No seed germinates if temperature during conditioning is too cold or too hot (< 0°C or > 37°C, even if temperature during exposure to root exudates are adequate). Inbetween these temperature thresholds, the closer the temperature is to the optimum, the faster the seeds lose dormancy (section S1 online). Thus, the proportion of non-dormant seeds among viable seeds was modelled as a function of thermal time accumulated during conditioning using a Weibull equation (eq. [3] in Appendix A 1, Figure 2). Thermal time is calculated in each soil layer as a function of soil temperature and cardinal temperatures of conditioning [2] (equation adapted from [START_REF] Bradford | Applications of hydrothermal time to quantifying and modeling seed germination and dormancy[END_REF]. The latter were calculated with the method for calculating cardinal temperatures for germination [START_REF] Bradford | Applications of hydrothermal time to quantifying and modeling seed germination and dormancy[END_REF].

Older seeds (but still younger than one year) follow a seasonal pattern of dormancy, with a high dormancy (> 80% dormant seeds) in winter-spring and a low dormancy (< 10% dormant seeds) in summer-autumn (Pointurier et al., 2019). Seeds older than one year enter dormancy two months later. This seasonal dormancy was modelled as a function of seed age [4], using the same equation as for nonparasitic weeds [START_REF] Gardarin | How much of seed dormancy in weeds can be explained by seed traits?[END_REF].

Stimulation of germination

Dormancy relief ends with germination triggering by root exudates of plants. The stimulatory activity of root exudates depends on the species emitting them [START_REF] Fernández-Aparicio | Recognition of root exudates by seeds of broomrape (Orobanche and Phelipanche) species[END_REF][START_REF] Gibot-Leclerc | Assessment of phylogenetic signal in the germination ability of Phelipanche ramosa on Brassicaceae hosts[END_REF]Perronne et al., 2017). Data from germination tests from the literature were compiled to calculate the ability of crop and weed species to stimulate broomrape relatively to GR24, a synthetic stimulant commonly used as a synthetic control (Mangnus et al., 1992) (Appendix A 4). This method allowed us to compare data from different experiments. When experiments did not include GR24 but oilseed rape as a positive control, the ability of the species to stimulate broomrape was first calculated relatively to oilseed rape, and then multiplied by the ability of oilseed rape to induce germination relatively to GR24. When no data were available for a species, its ability to stimulate broomrape germination was estimated from a close species (e.g., the data of Medicago lupulina were used for Medicago sativa). When no data on broomrape germination were available, the number of broomrapes attached per plant relative to a sensitive species (e.g. oilseed rape, tomato or Geranium dissectum) was used instead. If absolutely no data could be found in the literature, the species was considered to be unable to stimulate broomrape germination.

Broomrape seeds must be close to a stimulating root to be able to perceive germination stimulants (≤ 36 mm, [START_REF] Goldwasser | Differential induction of Orobanche seed germination by Arabidopsis thaliana[END_REF]. So, every day in PHERASYS, a part of the non-dormant seeds are stimulated (eq. [6] in Appendix A 1), corresponding to the proportion of soil volume comprised in the "stimulating zone" [5], i.e. close enough to stimulating roots (section S2 online). As root exudates are mainly released at root tips [START_REF] Brown | The germination of the seed of Striga lutea: I. Host influence and the progress of germination[END_REF][START_REF] Dennis | Are root exudates more important than other sources of rhizodeposits in structuring rhizosphere bacterial communities[END_REF], broomrape seeds are only stimulated by the "new" stimulating zone predicted each day, i.e. the stimulating zone on the present day minus the one from the day before [6]. Plants are considered to be able to induce germination from emergence to the end of flowering [START_REF] Auger | Germination stimulants of Phelipanche ramosa in the rhizosphere of brassica napus are derived from the glucosinolate pathway[END_REF], which is also approximately the time when plants stop emitting new roots [START_REF] Gregory | Root systems and root:mass ratio-carbon allocation under current and projected atmospheric conditions in arable crops[END_REF].

Germination progress

Once stimulated, seeds start to germinate. The wetter and the warmer the soil layer is (i.e. the higher above base temperature and base water potential, [START_REF] Bradford | Applications of hydrothermal time to quantifying and modeling seed germination and dormancy[END_REF], the faster the seeds germinate (Pointurier et al., 2019). Germination dynamics of broomrape seeds are simulated using the same principle as [START_REF] Gardarin | Prediction of germination rates of weed species: relationships between germination parameters and species traits[END_REF], i.e. as a function of hydrothermal time accumulated since germination triggering by root exudates (eq. [START_REF] Arslan | Potency of some synthetic stimulants and root exudates on the germination of Phelipanche spp[END_REF] in Appendix A 1). Too cold temperatures (below base temperature) halt germination temporarily. When the soil is too dry (i.e. below base water potential), or after a tillage operation which dilutes root exudates in the soil, germination stops altogether and new root exudates are necessary to trigger germination again [8]. Base temperature and base water potential of germination were taken from [START_REF] Gibot-Leclerc | Responsiveness of Orobanche ramosa L. seeds to GR 24 as related to temperature, oxygen availability and water potential during preconditioning and subsequent germination[END_REF] (section S1 online), and germination dynamics parameters from Pointurier et al. (2019).

Every day, germination flushes are triggered by new stimulating roots in the model. Consecutive flushes are merged to avoid storing data from several simultaneous germination flushes in the computer memory. To avoid large under or overestimation (see details in section S3 online), only close enough flushes (i.e. the new flush occurs before the ongoing flush has run 2.5 times its time to mid-germination) are merged. Otherwise, only the most recent flush is kept [START_REF] Arslan | Potency of some synthetic stimulants and root exudates on the germination of Phelipanche spp[END_REF].

Germinated seeds are removed from the seed bank daily [9].

Attachment

Germinated seeds produce a radicle which grows towards the host root to attach and establish vascular connections to take up water and nutrients from the host (Joel, 2013b). Yet not all species that stimulate broomrape germination also support broomrape attachment. The ability of a species to allow broomrape attachment after stimulation and consecutive development was compiled from the literature (Appendix A 4). Radicle elongation is limited, so only germinated seeds within a 4 mm "attachment zone" around host roots and younger than three days can attach, otherwise they die [START_REF] Gibot-Leclerc | What are the traits of Phelipanche ramosa (L.) Pomel that contribute to the success of its biological cycle on its host Brassica napus L.?[END_REF].

First, the volume of the attachment zone is calculated for each host plant (eq. [10] in Appendix A 1) and cumulated over all host plants to determine the proportion of soil volume where broomrape seeds are close enough to host roots to attach, and to deduce the total number of broomrape attachments [11]. Then, attachments are distributed over the hosts proportionally to their contribution to the total attachment zone [12].

Survival on the host

Only a few attached broomrapes survive on the host until flowering [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF], depending on the biomass the host allocates to them (Figure 3) (Grenz et al., 2005b;[START_REF] Lins | Effect of small broomrape (Orobanche minor) on red clover growth and dry matter partitioning[END_REF][START_REF] Grenz | Interactions of sunflower (Helianthus annuus) and sunflower broomrape (Orobanche cumana) as affected by sowing date, resource supply and infestation level[END_REF]. Host biomass at rosette stage represents a potential biomass to be used by broomrapes since, at this stage, parasitism has not yet an effect on host biomass [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF]. Above a minimum host biomass threshold, the total number of emerged broomrapes per host plant increases linearly with increasing host biomass at rosette stage. Hosts whose biomass is below the threshold do not have enough biomass to support the growth of emerged broomrapes.

The relationship of Figure 3 was used to calculate host carrying capacity in PHERASYS (eq. [13] in Appendix A 1). Moreover, this capacity is limited to a maximum of 20 broomrapes per host plant in our model, which corresponds to the maximum number of broomrapes observed per oilseed rape plant in heavily infested fields [START_REF] Gibot-Leclerc | What are the traits of Phelipanche ramosa (L.) Pomel that contribute to the success of its biological cycle on its host Brassica napus L.?[END_REF]. The host-carrying capacity allows introducing density-dependence into the model, i.e. the more broomrapes compete on a host, the less likely each of them is to survive. To represent this, we adapted a relationship used to simulate intra-specific competition between oilseed rape seedlings for emergence [START_REF] Colbach | GENESYS: a model of the influence of cropping system on gene escape from herbicide tolerant rapeseed crops to rape volunteers. I. Temporal evolution of a population of rapeseed volunteers in a field[END_REF]. This relationship [15] calculates the number of emerged broomrapes from the number of (unemerged) broomrapes attached at host rosette stage [14], when host-carrying capacity is determined [13]. Since broomrape can still attach after host rosette stage, the competition relationship [15] was derivated to calculate the survival probability for broomrapes attached daily after rosette stage [16]. The surviving emerged broomrapes are summed until broomrape reproduction [17]. They may though die before reaching seed production if the host dies earlier due to old age, frost or management operations (harvest, herbicide, tillage etc) [18].

Broomrape biomass

Broomrape phenology is predicted solely from temperature, since it does not depend on host phenology (Gibot-Leclerc et al., 2013a), year or experimental conditions (fields or greenhouse) (Gibot- [START_REF] Gibot-Leclerc | Etude épidémiologique, écophysiologique et agronomique du couple Orobanche ramosa L./Brassica napus[END_REF]. So, once attached, broomrape biomass accumulation is simulated from thermal time since host emergence (eq. [20] in Appendix A 1). Broomrape biomass becomes noticeable at about 700°C•days (base 5°C) after host emergence (Figure 4). The maximum biomass that the broomrapes can derive from the host is reached approximately at double this time, and 50% of this maximum biomass is reached after about 1100 °C•days [21].

This maximum biomass depends on the number of emerged broomrapes competing with each other for resources on the same host at the end of their life cycle (see section 3.1.6). The more emerged broomrapes survive on a host, the less biomass is available for each of them [19] [START_REF] Hibberd | Can source-sink relations explain responses of tobacco to infection by the root holoparasitic angiosperm Orobanche cernua? Plant[END_REF]Grenz et al., 2005b;[START_REF] Mauromicale | Effect of Branched Broomrape (Orobanche ramosa) Infection on the Growth and Photosynthesis of Tomato[END_REF]Fernández-Aparicio et al., 2016a;[START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF] (Figure 5). The later intraspecific-competition relationship was parameterized from data from the greenhouse experiment of [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF] and not from our field measurements (section 2.3.2) because data from [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF] presented a larger range of values (on the x-axis, Figure 5). However, our field measurements are consistent with data from [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF] (Figure 5).

Seed production

Assuming that broomrape phenology is mainly driven by temperature (see section 3.1.7), broomrape seed production occurs at 1709°C•days (base 5°C, section S4 online) after host emergence in the model, whatever the host, the location or the cropping system (eq. [20] in Appendix A 1).

Our measurements (section 2.3.2) on broomrape reproduction demonstrated that the number of capsules containing seeds increased linearly with broomrape biomass (Figure 6), while seed biomass per capsule (0.55 ± 0.082 mg) and individual seed biomass (2.1 ± 0.26 µg) were relatively constant. A small proportion (8%) of the analysed capsules did not produce mature seeds as they were either eaten by insects, atrophied or did not have the time to reach maturity. As a result, the model calculates broomrape seed production by ( 1) calculating the number of capsules from broomrape biomass and the proportion of unproductive capsules, (2) multiplies the capsules by their average seed weight to get seed biomass, (3) divides it by average seed weight to obtain the number of broomrape seeds per host plant, and (4) multiplies it by seed viability in fresh seeds to produce the number of viable broomrape seeds [22]. The seed viability (0.93 seeds•seeds -1 ) was measured in a previous experiment (Pointurier et al., 2019).

Finally, seeds are released and added to the surface layer of the broomrape seed bank [23].

Modelling the effect of parasitism on host growth

The following subsections describe the effect of parasitism on host growth and reproduction as well as the data used to justify and fit the equations. As host phenology is not affected by parasitism (Gibot-Leclerc et al., 2013b;[START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF], the relevant equations used by FLORSYS (Colbach et al., 2014b) remain unchanged.

Pathosystem biomass loss

The model considers broomrape as a supplementary sink for the host [START_REF] Manschadi | Quantitative aspects of Orobanche crenata infestation in faba beans as affected by abiotic factors and parasite soil seedbank[END_REF][START_REF] Grenz | Interactions of sunflower (Helianthus annuus) and sunflower broomrape (Orobanche cumana) as affected by sowing date, resource supply and infestation level[END_REF]Fernández-Aparicio et al., 2016a) and simulates the total pathosystem biomass consisting of both the host and the attached broomrapes. The amount of biomass lost by infected hosts is not fully invested in broomrapes, resulting in a reduction of the pathosystem biomass compared to the biomass of healthy plants [START_REF] Barker | Interactions between the parasitic angiosperm Orobanche aegyptiaca and its tomato host: Growth and biomass allocation[END_REF][START_REF] Eizenberg | A predictive degree-days model for small broomrape (Orobanche minor) parasitism in red clover in Oregon[END_REF][START_REF] Lins | Effect of small broomrape (Orobanche minor) on red clover growth and dry matter partitioning[END_REF][START_REF] Grenz | Interactions of sunflower (Helianthus annuus) and sunflower broomrape (Orobanche cumana) as affected by sowing date, resource supply and infestation level[END_REF][START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF]. Up to host flowering, parasitism has little or no effect on the pathosystem biomass [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF]. Then, broomrape starts to emerge and reduces the pathosystem biomass by 27 % (Figure 7). This loss rate is applied in our model to the daily net biomass produced after photosynthesis and respiration predicted in FLORSYS (eq. [24] in Appendix A 1).

Reallocation among vegetative organs

Before rosette stage, parasitism has no effect on the proportion of the pathosystem biomass allocated to roots (p=0.67). After that key stage, it reduces the root to pathosystem biomass ratio by 21% (Figure 8). Biomass allocation to roots is reduced accordingly in PHERASYS (eq. [25] in Appendix A 1). Biomass allocation to leaves in the pathosystem is not affected by parasitism [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF], so the FLORSYS equations determining the leaf biomass ratio remain unchanged (Colbach et al., 2014b) [26].

Host seed production

Since broomrape competes mostly with the reproductive compartment of the host [START_REF] Manschadi | Quantitative aspects of Orobanche crenata infestation in faba beans as affected by abiotic factors and parasite soil seedbank[END_REF][START_REF] Grenz | Interactions of sunflower (Helianthus annuus) and sunflower broomrape (Orobanche cumana) as affected by sowing date, resource supply and infestation level[END_REF]Fernández-Aparicio et al., 2016a;[START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF], the combined biomass of broomrapes and host reproductive organs (flowers and fruits) was compared to the biomass allocated to reproduction in healthy host plants (Figure 9). In healthy host plants, a part of the above-ground biomass is allocated to seeds (Colbach et al., 2014b) (eq. [27] in Appendix A 1). In infected hosts, this part, together with broomrape biomass, is 1.7 times higher [28]. In the model, the biomass allocated to host seeds in infected hosts is computed by subtracting the broomrape biomass (calculated in section 3.1.7) from the biomass allocated to host reproduction and broomrapes [29]. Although host plants did not reach full maturity in the data we used from [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF], we considered that our formalisms were still valid at later stages because the proportion of above-ground biomass allocated to the reproductive compartment varies only little then [START_REF] Weiner | Allocation, plasticity and allometry in plants[END_REF].

This formalism takes into account that hosts that reproduce early suffer less from broomrape competition [START_REF] Manschadi | Quantitative aspects of Orobanche crenata infestation in faba beans as affected by abiotic factors and parasite soil seedbank[END_REF]Grenz et al., 2005b;[START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF] (section S5 online).

Effect of host-plant phenology on host-parasite interactions

Host-plant phenology (which discriminates five stage: cotyledon, plantlet, vegetative, flowering, maturation, see details in section S1.4 online) influences parasite growth and reproduction as well as the impact of parasitism on host growth and reproduction, as seen in the previous section. In summary:

-Host-plant emergence determines the onset of root emission and growth, which sets off hostroot exudation triggering parasite germination as well as parasite attachment to host roots; -When a host plant reaches the "rosette" stage (approximately 25% of the duration of its vegetative stage), the parasite starts to affect host growth and biomass allocation to different plant compartments; -When a host plant starts to flower, biomass allocation to roots slows down and stops, which shuts off host-root exudation (and the resulting parasite germination) as well as parasite attachment to host roots; -The duration of the host's life cycle determines whether the parasite has the time to reproduce but the time from parasite emergence to parasite maturity does not depend on the host plant.

Modelling the effects of cropping systems on broomrape dynamics

The different processes modelled in PHERASYS allowed integrating the effects of several cropping techniques (Table 2). For example, soil tillage moves seeds in the soil, which makes them more or less close to stimulating root exudates and to host roots and thus determines the number of stimulated broomrape seeds and attachments.

Discussion

The first cropping-system model of broomrape dynamics specific to branched broomrape…

The redesigned PHERASYS is the first model of broomrape dynamics in agroecosystems specifically designed and parameterized for branched broomrape. The life-stages chosen as key steps to model broomrape dynamics are basically the same as those included in other models [START_REF] Schnell | Simulation of the seed bank dynamics of Orobanche crenata forsk. in some crop rotations common in Northern Syria[END_REF][START_REF] López-Granados | Modelling the demography of crenate broomrape (Orobanche crenata) as affected by broadbean (Vicia faba) cropping frequency and planting date[END_REF][START_REF] Eizenberg | A predictive degree-days model for small broomrape (Orobanche minor) parasitism in red clover in Oregon[END_REF]Grenz et al., 2005a;[START_REF] Ephrath | Quantification of the dynamics of Orobanche cumana and Phelipanche aegyptiaca parasitism in confectionery sunflower[END_REF]Pérez-de-Luque et al., 2016). However, seed dormancy was rarely included in previous models and when it was, this was done very simplistically (i.e., a constant proportion of the seed population was considered to be dormant, López-Granados and García-Torres, 1997; Grenz et al., 2005a). We modelled dormancy more precisely as two successive phases. First, dormancy relief in fresh seeds depended on temperature, moisture and duration of conditioning. [START_REF] Kebreab | A quantitative model for loss of primary dormancy and induction of secondary dormancy in imbibed seeds of Orobanche spp[END_REF] used a similar approach to model both dormancy release and induction of seeds in other broomrape species under laboratory conditions. We preferred using a seasonal, more realistics model for older seeds, built from a field experiment. This approach was shown to produce satisfactory predictions in non-parasitic weeds [START_REF] Colbach | Uncertainty analysis and evaluation of a complex, multi-specific weed dynamics model with diverse and incomplete data sets[END_REF].

Some PHERASYS formalisms were inspired from the model developed for O. crenata (Grenz et al., 2005a), e.g. the calculation of the "stimulating zone" around roots and the estimation of broomrape seed production. Here, we not only estimated parameter values for branched broomrape, we also improved the initial concepts. For instance, we added a dynamic and more realistic approach to the stimulating zone by restricting it to root tips. As Grenz et al. (2005a), we found that the number of seed-containing capsules was related to broomrape biomass. From this relationship, we estimated that each broomrape produces 10000 to 55000 seeds, which is at the lower end of the range of 10000 to 500000 seeds reported for other broomrape species in the literature across different continents (Grenz et al., 2005a, Turkey;Joel, 2013a;Prider, 2015, Australia). However, our mean number of seeds per capsule (200-300) was lower than the 500-700 seeds generally found in broomrapes, including branched broomrape [START_REF] Gibot-Leclerc | What are the traits of Phelipanche ramosa (L.) Pomel that contribute to the success of its biological cycle on its host Brassica napus L.?[END_REF]Joel, 2013a;Prider, 2015). Indeed, because of experimental constraints, our measurements of seeds per capsule only included the closed capsules from the top of the inflorescence, which contain fewer seeds [START_REF] Miegel | Banched broomrape seed production[END_REF]. If future sensitivity analyses demonstrate that this parameter is essential for predicting branched-broomrape dynamics and the resulting yield loss, we will carry out additional measurements with an improved protocol.

We used the well-known concept of thermal time [START_REF] Bonhomme | Bases and limits to using "degree.day" units[END_REF] to predict broomrape phenology, which had already been successfully applied to several other broomrape species [START_REF] Eizenberg | A predictive degree-days model for small broomrape (Orobanche minor) parasitism in red clover in Oregon[END_REF][START_REF] Eizenberg | PICKIT-a decision support system for rational control of Phelipanche aegyptiaca in tomato[END_REF][START_REF] Ephrath | Quantification of the dynamics of Orobanche cumana and Phelipanche aegyptiaca parasitism in confectionery sunflower[END_REF]Eizenberg et al., 2012a;Eizenberg et al., 2012b;Pérez-de-Luque et al., 2016;[START_REF] Hosseini | A modelling approach for predicting the initial phase of Egyptian broomrape (Phelipanche aegyptiaca) parasitism in potato[END_REF]. Using this concept here involved assuming that broomrape phenology depends mostly on temperature and not on host phenology. This assumption was supported by observations on O. crenata [START_REF] Manschadi | Quantitative aspects of Orobanche crenata infestation in faba beans as affected by abiotic factors and parasite soil seedbank[END_REF]Grenz et al., 2005b) as well as field observations on branched broomrape (Gibot-Leclerc et al., 2012).

… in interaction with several species of crops and weeds …

The major originality of PHERASYS consists in the interactions between broomrape and several species of crops and weeds. At present, more than 60 species are included in the model and its generic structure makes it easier to add new species in the future [START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF]. Broomrape dynamics models in the literature include generally only one host crop, three to four at the most [START_REF] Schnell | Simulation of the seed bank dynamics of Orobanche crenata forsk. in some crop rotations common in Northern Syria[END_REF]Pérezde-Luque et al., 2016), and no weed species.

We characterized parasite interactions with crops and weeds at the individual host plant level by predicting the number of germinated broomrape seeds, attachments and mature broomrapes as a function of daily host biomass and root volume. This allowed modelling, for the first time, the effects of heterogeneous multispecies stands on broomrape dynamics. For example, our model takes into account the fact that hosts located close to germination-stimulating non-attaching plants are more infected, or that shaded hosts are less infected. This is crucial for designing management strategies using biological regulation by either weeds or crop mixtures associating complementary species.

Our species parameters need, though, to be improved. First, the stimulatory activity of species was estimated from in vitro germination tests. It is likely to be different in the field since seasons [START_REF] López-Granados | Effects of environmental factors on dormancy and germination of crenate broomrape (Orobanche crenata)[END_REF][START_REF] Fernández-Aparicio | Low strigolactone root exudation: a novel mechanism of broomrape (Orobanche and Phelipanche spp.) resistance available for faba bean breeding[END_REF], nutrient availability [START_REF] Gaudin | Place de l'azote dans l'interaction plante -plante parasite : Brassica napus L. -Phelipanche ramosa (L.) Pomel[END_REF] or interactions with microorganisms [START_REF] Fernández-Aparicio | Colonisation of field pea roots by arbuscular mycorrhizal fungi reduces Orobanche and Phelipanche species seed germination[END_REF] may influence the stimulatory plant activity. Moreover, data from a related species or from other broomrape pathovars were used to parameterize some species. This is acceptable for Fabaceae (Perronne et al., 2017), but not for Brassicaceae where similar species induce different responses in branched-broomrape germination which also depend on the broomrape pathovar [START_REF] Gibot-Leclerc | Assessment of phylogenetic signal in the germination ability of Phelipanche ramosa on Brassicaceae hosts[END_REF]. Even with such proxys, some species known to stimulate branched broomrape could not be parameterized for stimulatory activity (e.g., Fallopia convolvulus, [START_REF] Boulet | Adventices du colza et orobanche[END_REF] and were considered as non-host. Misestimating the stimulatory activity of a species can result in misestimating broomrape seed bank depletion due to germination losses and/or broomrape reproduction due to successful germination and attachments. Future sensitivity analyses will identify which model parameters should be measured as a priority.

… and including a submodel for host-parasite trophic relationships

Another novelty of PHERASYS is that interactions between broomrape and host plants are two-ways, i.e. host plants have an effect on broomrape dynamics while parasitism influences host growth. This allows estimating yield losses due to parasitism and improves the prediction of plant-plant competition for light and other resources in FLORSYS. Broomrape dynamics are also better predicted, via a feedback loop, as host growth and biomass determine broomrape infestation in PHERASYS. Grenz et al. (2005a) also included a submodel for host-broomrape trophic relationships as we did here, but on a different broomrape species and with a single host species in homogeneous stands. Grenz et al's model focused more on within-plant variability on an average host plant; for instance, parasitism caused abortion of young fruits whereas older, more competitive fruits survived. We included little within-plant variability of broomrape effects, only considering that late broomrape attachments rarely survived, in order to focus on between-plant variability. This 3D individual-based representation of each host (and non-host) plant was better adapted to the multispecies nature of FLORSYS and still allows modelling the competitive advantage of hosts reproducing early.

Although most formalisms that we proposed here to model host-broomrape trophic relationships are new, the concepts involved are supported by the literature (see references in the result section). Some less well-known properties emerged from our work and also found support in the literature. For example, broomrape affects the allocation of resources within host vegetative organs before deriving biomass [START_REF] Barker | Interactions between the parasitic angiosperm Orobanche aegyptiaca and its tomato host: Growth and biomass allocation[END_REF], reducing the allocation to roots in the pathosystem [START_REF] Lins | Effect of small broomrape (Orobanche minor) on red clover growth and dry matter partitioning[END_REF]. However, our formalisms were based on data from only three host species and need to be checked on more species since they may vary between hosts (Fernández-Aparicio et al., 2016a).

Agronomic implications

The structure of the FLORSYS-PHERASYS model complex makes it possible to simulate a large range of contrasting cropping systems, including both direct effects on the parasite (e.g., tillage influences broomrape-seed movements) as well as indirect effects via the crop and non-parasitic weeds (e.g., plant species, host plant location and biomass, triggering of suicide broomrape germinations), and this in interaction with pedoclimate (e.g., parasite attachment success also depends on soil temperature and water potential). Thus, the effects of cropping techniques potentially efficient for controlling broomrape can be evaluated, sometimes resulting from a combination of underlying processes (Table 2). Promising techniques such as trap and catch crops aiming to deplete the soil seed bank, delayed sowing or reduced host crops frequency in the rotation aiming to limit broomrape reproduction can be evaluated indidivudally in terms of broomrape control in various weather scenarios. More importantly, combinations of individual techniques can be evaluated and fine-tuned to fit to different pedoclimates, weed-flora contexts or cropping-system types.

The model will also allow investigating how to exploit interactions between broomrape and nonparasitic weeds for biological regulation of broomrape. Though weeds can allow broomrape to reproduce as shown in our previous experiments [START_REF] Moreau | Screening studies on the host range of branched broomrape (Orobanche ramosa)[END_REF], these same experiments illustrate that non-parasitic weeds can potentially play the role of catch plants. Indeed, some weed hosts have a shorter life cycle than broomrape and thus preclude seed-bank replenishment after triggering broomrape germinations. This effect will though be highly dependent on the quality of the weed phenology predictions in FLORSYS, which was shown to be deficient at some latitudes [START_REF] Colbach | Uncertainty analysis and evaluation of a complex, multi-specific weed dynamics model with diverse and incomplete data sets[END_REF].

Perspectives

The previous discussion advertises the many potential uses of PHERASYS. In addition to investigating the effects of cropping techniques, a sensitivity analysis to model parameters would allow identifying the key processes in broomrape dynamics and, incidentally which model parameters need to be measured more precisely and which formalisms must be improved.

To extend the range of broomrape-controlling strategies, more management techniques could be included in the model, such as parasite-tolerant crop varieties, fertilization or biological control with fungi and insects. The process-based modular structure of PHERASYS facilities the addition of techniques and processes, providing that data for parameterization are available. For example, techniques having a direct toxic effect on broomrape, such as herbicides, could be modelled by eliminating a part of the broomrape population when applied. Indirect effects influencing root exudation, such as fertilization (Fernández-Aparicio et al., 2016b), would modulate the stimulatory activity of plants. PHERASYS also benefits from FLORSYS improvements. Now that competition for nitrogen has been included in FLORSYS [START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF], we will be able to investigate how fertilization strategies could be adapted to make hosts more competitive towards broomrape.

Finally, as suggested above, processes modelled in PHERASYS are quite generic. The same model structure can be used for other pathovars of branched broomrape or other broomrape species, albeit with different parameter values. We started to collect parameter values for the hemp pathovar (Appendix A 4), which is the second most frequent in France (Terres Inovia, 2018).

Conclusion

We developed PHERASYS, a mechanistic model of broomrape dynamics in agroecosystems, specifically designed and parameterized for branched broomrape. It allows, for the first time, to simulate interactions between broomrape and heterogeneous multispecies stands of crops and weeds. It integrates the effects of multiple cropping techniques, including complex ones playing on the competition between broomrapes and hosts for resources. The next step will be to determine how to combine partially efficient cropping techniques into a consistent cropping system to control broomrape harmfulness and, in particular, how to manage non-parasitic weeds in order to contribute to biological regulation of broomrape. 2016), Table 1), 2 experiment described in the present article (section 2. x SeBHhphd Seed biomass in healthy plants p (Colbach et al., 2014b) g•host

Acknowledgements

8 IllustrationsFigure 1 :

 81 Figure 1: Processes of branched-broomrape life cycle modelled in the redesigned PHERASYS model (in bold, section2.1) and connection with FLORSYS. Inputs given by the user are in grey. Variables used to connect both models are in italics (section 2.22.2; green and purple: biotic variables from FLORSYS and PHERASYS respectively, brown: abiotic variables). Numbers indicate the origin of the data used with 1 own data published ( 1a Pointurier et al. (2019), 1b Gibot-Leclerc et al. (2004), and 1c Moreau et al. (2016), Table1), 2 experiment described in the present article (section 2.3.2) and 3 other literature (Olivia Pointurier 2019 )

  Figure 1: Processes of branched-broomrape life cycle modelled in the redesigned PHERASYS model (in bold, section2.1) and connection with FLORSYS. Inputs given by the user are in grey. Variables used to connect both models are in italics (section 2.22.2; green and purple: biotic variables from FLORSYS and PHERASYS respectively, brown: abiotic variables). Numbers indicate the origin of the data used with 1 own data published ( 1a Pointurier et al. (2019), 1b Gibot-Leclerc et al. (2004), and 1c Moreau et al. (2016), Table1), 2 experiment described in the present article (section 2.3.2) and 3 other literature (Olivia Pointurier 2019 )

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :

 23456789 Figure 2: Dormancy relief of branched-broomrape seeds as a function of thermal time accumulated during conditioning (TTcond ld, base 0°C). Each dot represents the proportion of seeds (pNDlda) germinating after being stimulated with a synthetic stimulant (GR24 at 1 mg.L -1 at 20°C, i.e. optimal GR24 concentration and temperature for broomrape germination) after being stored in moist conditions ("conditioning") for different durations and temperatures of conditioning respectively (see legend on the figure). The line represents the model fitted to the data. Based on data from Gibot-Leclerc et al. (2004) (Olivia Pointurier 2019 )

- 2 NA

 2 Parameters are estimated by linear or non-linear regressions (parameter value ± standard error), or by calculating mean values ± standard deviation (*) or from the literature (** (Gibot-Leclerc et al., 2004) ***(Gibot-Leclerc, 2004)). Appendix A 3: Name, meaning and unit of variables in PHERASYS. Meaning of indices: a = seed age class (young vs. old), d = day, l = soil layer, ph = plant p of species h. biomass of the pathosystem p (including attached broomrapes in infected hosts) (Colbach et al., attachments among germinated broomrape seeds in soil layer l before competition between attachments for host resources Attachments •host -1 x Fldph Number of attachments on roots of host plant p in soil layer l before competition between attachments for host resources Attachments •host -above-ground biomass ratio in host plant p (Colbach et al., non-dormant broomrape seeds stimulated by root exudates of all stimulating plants in soil layer l seeds•seeds -1 x RBDldph Root biomass density of stimulating plant p in soil layer l (Pagès et al., allocated to roots in healthy plant p (Pointurier et al.

  volume in layer l where broomrape seeds are close enough to the roots of host plant p to attach it m volume in layer l occupied by the root system of stimulating plant p (Pagès et al., by host plant p on day d through photosynthesis after deduction of respiration loss(Colbach et al.,

Table 1 :

 1 Summary of own published data working with Oilseed rape pathovar[START_REF] Brault | Variability of Orobanche ramosa populations in France as revealed by cross infestations and molecular markers[END_REF][START_REF] Le Corre | Development of microsatellite markers in the branched broomrape Phelipanche ramosa L. (Pomel) and evidence for host-associated genetic divergence[END_REF] used to build and parameterize PHERASYS

	Data	Experimental conditions	Treatment	Measures
				Proportion of germinated
	Gibot-Leclerc et al. (2004)	Germination on glass microfibre paper in vitro	Different durations of conditioning at different temperatures and water potentials	seed after stimulation with GR24 (synthetic growth regulator used to stimulate broomrape
				germination)
			3 hosts species (Brassica napus,
			Capsella bursa-pastoris,
		Co-cultivation of	Geranium dissectum) grown	Biomass of broomrape
	Moreau et	branched broomrape	under 3 light conditions in	and host organs and
	al. (2016)	with hosts in pots in	substrate heavily infested with	number of attached
		greenhouse	broomrape seeds or uninfested	broomrapes.
			and harvested at 4 phenological
			stages
		Seeds buried in the		Proportion of viable and
	Pointurier	field and put to	Germination after different	germinated seeds after
	et al.	germination on glass	durations of burial in the soil up	different durations of
	(2019)	microfibre paper in	to 2 years	burial and stimulation
		vitro		with GR24

Table 2 :

 2 Effects of cropping techniques on branched-broomrape dynamics in PHERASYS

	Cropping technique	Effect	Consequences on broomrape dynamics
			Determines the proximity of broomrape seeds to
	Tillage	Buries and excavates seeds	stimulating/attaching roots, and thus the number
			of germinated seeds/attachments
		Crop root distribution	Same as tillage
		Broomrape seed stimulation	Determines the number of stimulated broomrape
		ability	seeds
		Broomrape seed attachment	Determines the number of attached broomrape
	Crop species	ability; host plant biomass	seeds that will survive up to maturity
	and variety		Broomrapes die before reproducing if the host
		Crop plant life duration	dies. Hosts reproducing suffer less from
			broomrape competition.
		Sowing season	Determines broomrape seed dormancy level when crops stimulate broomrape germination.
	Sowing date	Determines crop emergence and life duration	The later broomrape emergence is, the less time there is for broomrapes to damage the crop and to reproduce
	Crop sowing density	Number of crop seeds in the soil	Same as tillage
	Weed management	Indirect effects via the non-parasitic weed flora (see crop species and variety).

ramosa (L.) and different hosts depending on host phenological stage and host growth rate. Frontiers in Plant Science 7, https://doi.org/10.3389/fpls.2016.01033. Moreau, D., Abiven, F., Busset, H., Matejicek, A., Pagès, L., 2017. Effects of species and soil-nitrogen availability on root system architecture traitsstudy on a set of weed and crop species. Annals Of Applied Biology 171, 103-116, https://doi.org/10.1111/aab.12355. Moreau, D., Pointurier, O., Perthame, L., Beaudoin, N., Villerd, J., Colbach, N., 2021. Integrating plantplant competition for nitrogen into a 3D individual-based model simulating the effects of cropping systems on weed dynamics. Field Crops Research 268, 108166, https://doi.org/10.1016/j.fcr.2021.108166. Munier-Jolain, N. M., Guyot, S. H. M., Colbach, N., 2013. A 3D model for light interception in heterogeneous crop:weed canopies. Model structure and evaluation. Ecological Modelling 250, 101-110, http://dx.doi.org/10.1016/j.ecolmodel.2012.10.023. Munier-Jolain, N. M., Collard, A., Busset, H., Guyot, S. H. M., Colbach, N., 2014. Investigating and modelling the morphological plasticity of weeds in multi-specific canopies. Field Crops Research 155, 90-98, http://dx.doi.org/10.1016/j.fcr.2013.09.018. Murdoch, A. J., Kebreab, E., 2013. Germination ecophysiology. In: Joel, D. M., Gressel, J. and Musselman, L. J. (Eds.) Parasitic Orobanchaceae : Parasitic mechanisms and control strategies, Springer-Verlag Berlin Heidelberg, Pagès, L., Bécel, C., Boukcim, H., Moreau, D., Nguyen, C., Sterckeman, T., Voisin, A.-S., 2014. Calibration and evaluation of ArchiSimple, a parsimonious model of the root system architecture. Ecological Modeling 290, 76-84, https://doi.org/10.1016/j.ecolmodel.2013.11.014. Pagès, L., Pointurier, O., Moreau, D., Voisin, A.-S., Colbach, N., 2020. Metamodelling a 3D architectural root system model to provide a simple model based on key processes and species functional groups. Plant and Soil 448, 231-25, https://doi.org/10.1007/s11104-019-04416-z. Parker, C., Riches, C., 1993. Parasitic Weeds of the World: Biology and Control. CAB International, Wallingford, UK, Parker, C., 2013. The parasitic weeds of the Orobanchaceae. In: Joel, D. M., Gressel, J. and Musselman, L. J. (Eds.) Parasitic Orobanchaceae: Parasitic mechanisms and control strategies, Springer-Verlag, Berlin Heidelberg Pérez-de-Luque, A., Flores, F., Rubiales, D., 2016. Differences in crenate broomrape parasitism dynamics on three legume crops using a thermal time model. Frontiers in Plant Science 7, 1910, https://doi.org/10.3389/fpls.2016.01910. Perronne, R., Gibot-Leclerc, S., Dessaint, F., Reibel, C., Le Corre, V., 2017. Is induction ability of seed germination of Phelipanche ramosa phylogenetically structured among hosts? A case study on Fabaceae species. Genetica 145, 481-489, https://doi.org/10.1007/s10709-017-9990-x. Pointurier, O., Gibot-Leclerc, S., Le Corre, V., Reibel, C., Strbik, F., Colbach, N., 2019. Intraspecific seasonal variation of dormancy and mortality of branched broomrape seeds. Weed Research, https://doi.org/10.1111/wre.12378. Pointurier, O., Gibot-Leclerc, S., Moreau, D., Colbach, N., 2021a. How to pit weeds against parasitic plants. A simulation study with Phelipanche ramosa in arable crops. European Journal of Agronomy accepted with moderate revisions Pointurier, O., Moreau, D., Pagès, L., Caneill, J., Colbach, N., 2021b. Individual-based 3D modelling of root systems in heterogeneous plant canopies at the multiannual scale. Case study with a weed dynamics model. Ecological Modelling 440, 109376, https://doi.org/10.1016/j.ecolmodel.2020.109376. Pointurier, O., Moreau, D., Pagès, L., Caneill, J., Colbach, N., submitted. Individual-based 3D modelling of root systems in heterogeneous plant canopies at the multiannual scale. Case study with a weed dynamics model. Ecological Modelling Potier, D., 2014. Pesticides et agro-écologie. Les Champs du possible. 252 p. p. Prider, J., 2015. The reproductive biology of the introduced root holoparasite Orobanche ramosa subsp. mutelii (Orobanchaceae) in South Australia. Australian Journal of Botany 63, 426-434, https://doi.org/10.1071/BT14134.

Appendixes

Appendix A 1. Equations of the PHERASYS model representing branched broomrape dynamics and the effect of parasitism on host growth. Meaning of indices: a = seed age class (young vs. old), d = day, l = soil layer, ph = plant p of species h. Parameters are in bold. For explanations on parameters and state variables, see Appendix A 2, Appendix A 3 and Appendix A 4.

Eq.

when Process Equation Broomrape dynamics [1] ∀d Seed mortality da = 1-(365-am•agea)/(365-am•(agea-1)) SBlda= (1-da)•SBl(d-1)a [2] ∀d ] pNDlda = pND'lda [4] If [deph,dfleph] If

Parameter Meaning and unit

Estimated value Oilseed rape pathovar Hemp pathovar

Appendix A 4. Crop and weed parameters characterizing interaction with branched broomrape. In case of missing data, data from a close species (*), from another pathovar (**) or from number of attachments instead of germination rate for αh/GR24 (***, section 3.1.3) were used. Root tip diameters originate from parameters of the root architecture model ArchiSimple (Pagès et al., 2014, Pagès, pers. comm.;Moreau et al., 2017;[START_REF] Seneze | Etude de la diversité intra-et inter-spécifique de l'architecture racinaire des légumineuses, en lien avec la croissance et la nutrition azotée[END_REF][START_REF] Guinet | Quantification des flux d'azote induits par les cultures de légumineuses et étude de leurs déterminants : comparaison de dix espèces de légumineuses à graines[END_REF]. When it was unknown, the mean diameter calculated over all other species was used.