Olivia Pointurier

Stéphanie Gibot-Leclerc

Delphine Moreau

Nathalie Colbach
email: nathalie.colbach@inrae.fr

How to pit weeds against parasitic plants. A simulation study with Phelipanche ramosa in arable cropping systems

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

Branched broomrape (Phelipanche ramosa (L.) Pomel) is a parasitic plant, which causes severe yield losses in major crops worldwide. Due to 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. In a previous paper, we developped a broomrapedynamics model called PHERASYS to support the design of management strategies combining multiple techniques aiming at long-term control of broomrape. Here, the objective is to use this simulation model to (1) check the consistency of simulations vs. literature data, (2) evaluate the potential of cropping systems to manage the combination of branched broomrape and weeds, (3) investigate whether weeds can biologically regulate parasitic plants in agroecosystems. Five contrasting cropping systems including different levers known to influence broomrape dynamics were simulated with different weather series. Four simulation series were run, with or without broomrape as well as with or without weeds, to discriminate the individual effects of weeds and broomrape on crop production as well as the effect of weeds on broomrape dynamics. Simulations with PHERASYS showed that delayed sowing in combination with the use of trap and catch crops are promising for reducing broomrape infestation and yield losses in the long term. Tolerating a temporary and/or low-density weed flora in such cropping systems could improve broomrape management because spring/summer weeds could reduce broomrape seed bank by triggering broomrape germinations that would not reproduce. During cash-crop growth, weed contribution to broomrape infection would be negligible. However, these conclusions are only valid if broomrape-attaching weeds reproduce before broomrape has time to do so, which needs to be checked with field experiments for most weed species.

Introduction

Broomrapes are parasitic plants that threaten major crops worldwide [START_REF] Parker | The parasitic weeds of the Orobanchaceae[END_REF]. As holoparasites, they are unable to photosynthesize and entirely rely on host resources to survive and reproduce [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 [START_REF] Heide-Jørgensen | Introduction: The Parasitic Syndrome in Higher Plants[END_REF]. 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, though it is rare in Poaceae (Parker and Riches, 1993a;[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 (Fernández-Aparicio et al., 2016a;Données Ephy -Anses, 2018), but this technique clashes with current policies aiming at reducing the use of pesticides because of their impacts on human health and environment [START_REF] Potier | Pesticides et agro-écologie[END_REF]. 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 and lethal to the parasite. Biocontrol is a promising option to complement cultural techniques, but applicable and efficient products are still under development (Fernández-Aparicio et al., 2016c;[START_REF] Cartry | Main drivers of broomrape regulation. A review[END_REF]. Consequently, several preventive techniques with partial effects must be combined to control broomrapes (Grenz et al., 2005a;[START_REF] Rubiales | Innovations in parasitic weeds management in legume crops. A review[END_REF]. They must provide long-term control as broomrape seeds die off very slowly (other broomrape species: [START_REF] Murdoch | Germination ecophysiology[END_REF]branched broomrape: Pointurier et al., 2019).

Reducing the frequency of host-crops in the rotation is the first broomrape-preventive option that comes to mind. Another option is to choose varieties that stimulate fewer germination, reduce attachments and/or growth of attached shoots [START_REF] Gaudin | Place de l'azote dans l'interaction plante -plante parasite : Brassica napus L. -Phelipanche ramosa (L.) Pomel[END_REF]. Trap crops and catch crops deplete the broomrape seed bank by inducing broomrape germination without allowing further development of the parasitic plant because they are resistant (trap crop) or because they are destroyed before broomrape reproduction (catch crop) [START_REF] Goldwasser | Integrated Agronomic Management of Parasitic Weed Seed Banks[END_REF]. Flax (Linum usitatissimum L.) and birdsfoot trefoil (Lotus corniculatus L.) are interesting trap-cop candidates (Parker and Riches, 1993b;[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], white mustard (Sinapis alba L.) and oilseed rape volunteers can be used as catch crops during summer fallow [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]. Delayed sowing is consistently reported to reduce infection of winter crops such as oilseed rape (Gibot-Leclerc et al., 2006b;Fernández-Aparicio et al., 2016c) but not of summer crops [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]. Evidence on tillage effects (Rubiales et al., 2009a;Fernández-Aparicio et al., 2016c) and mineral fertilization [START_REF] Goldwasser | Integrated Agronomic Management of Parasitic Weed Seed Banks[END_REF] is contradictory.

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][START_REF] Simier | Variabilité génétique de l'orobanche rameuse et son spectre d'hôtes[END_REF][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, using multiple techniques and exploiting biological regulation by weeds. Therefore, simulation models are essential 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][START_REF] Jeuffroy | Agronomic model uses to predict cultivar performance in various environments and cropping systems. A review[END_REF][START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF]. To our knowledge, to date only one model answers to the requirements for evaluating cropping systems for branched-broomrape management. The mechanistic (process-based) PHERASYS (Pointurier et al., 2021a), connected to the multi-species weed dynamics model FLORSYS [START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF], simulates branched-broomrape dynamics at a daily time-step and in 3D over the years depending on cropping techniques and pedoclimate, in interaction with growth and development of crops and weeds.

Consequently, the objective of the present study was to use PHERASYS to evaluate the potential of cropping systems to manage branched broomrape (hereafter simply called "broomrape") and weeds in interaction. In particular, we investigated whether weeds can contribute to biologically regulate parasitic plants in agroecosystems. The simulations were also used to check the consistency of the model with literature.

Materials and methods

This section first summarily presents the model to be used, followed by the choice of the cropping systems to be tested, the simulation plan and the statistical methods needed to evaluate cropping-system performance and understand the reasons for this performance.

2.1

The PHERASYS and FLORSYS model association PHERASYS (Pointurier et al., 2021a) describes broomrape dynamics from cropping system in interaction with crop and weed growth as well as pedoclimate. It is a submodel 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][START_REF] Munier-Jolain | A 3D model for light interception in heterogeneous crop:weed canopies. Model structure and evaluation[END_REF][START_REF] Colbach | Assessing innovative sowing patterns for integrated weed management with a 3D crop:weed competition model[END_REF][START_REF] Munier-Jolain | Investigating and modelling the morphological plasticity of weeds in multi-specific canopies[END_REF][START_REF] Colbach | The FLORSYS crop-weed canopy model, a tool to investigate and promote agroecological weed management[END_REF]Pointurier et al., 2021b). This model associated can be considered to be a virtual field on which cropping systems can be experimented with a large range of virtual measurements of crop, weed and environmental state variables. Only the main features were summarized here and additional information can be found in sections S1 and S2 of supplementary material online.

Input variables and soil state variables

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). These input variables drive various soil state variables (i.e.

structure, temperature, water potential) in 30 1-cm-thick soil layers.

Crop and weed life-cycle

The input variables and the soil-state variables influence the annual life cycle of annual weeds and crops, with a daily time-step (Figure 1).

Pre-emergence stages (surviving, dormant and germinating seeds, emerging seedlings) are driven by seed depth, soil structure, temperature and water potential. The crop-weed canopy is represented in 3D, with each crop and weed plant schematized as a cylinder (above ground) and top of a spilled cone (below ground). Post-emergence processes (e.g. photosynthesis, respiration, growth, shade response) are driven by light availability and air temperature. At plant maturity, weed seeds are added to the soil seed bank; crop seeds are harvested to determine crop yield. Nitrogen stress is disregarded in the present model version, and water stress only considered for pre-emergent processes.

Domain of validity

The model is currently parameterized for 26 frequent and contrasting annual weed species and 30 crop species, including cash crops as well as cover and forage crop species (section S1.5 online). This limits its use in conditions with dominant annual weeds. In other situations, the comparison of simulations to independent field data from different French regions showed that crop yields, daily weed species densities and, particularly, densities averaged over the years were generally well predicted and ranked as long as a corrective function was added to keep weeds from flowering during winter at more southern latitudes [START_REF] Colbach | Uncertainty analysis and evaluation of a complex, multi-specific weed dynamics model with diverse and incomplete data sets[END_REF]Pointurier et al., 2021b).

Parasite life-cycle

In parallel, the PHERASYS submodel predicts the number of broomrape individuals per host plant at different stages and the biomass of broomrape shoots every day depending on biophysical processes.

Each day, a part of the seeds in the soil dies; non-dormant seeds can germinate only if stimulated by neighbouring host-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.

When a broomrape produces seeds, the new broomrape seeds are dormant and are added to the superficial layer of the soil seed bank.

Effect of parasitism on host plant growth and reproduction

Between attaching to host roots and emerging from the soil, broomrape biomass and its effect on hostplant growth are negligible. Once broomrape starts to emerge, the daily net biomass produced by an infected host after photosynthesis and respiration is reduced. Once the host plant reaches the rosette stage, parasitism reduces biomass allocation to roots. Biomass allocation to leaves in the pathosystem is not affected by parasitism. Host seed production is the most affected by infection: the larger broomrape biomass is, the less biomass can be allocated to host seed production. But, as the broomrape life-cycle duration is independent of host species, hosts that reproduce early suffer less from broomrape competition for host resources and allow less broomrape reproduction (or even none at all).

Modelling the effects of cropping systems on broomrape dynamics

The different processes modelled in PHERASYS allowed integrating the effects of several cropping techniques. 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.

Simulation plan

To understand broomrape-weed interactions as a function of cropping system, we chose to investigate in detail a small number of contrasting cropping systems differing in terms of cropping techniques that potentially influence broomrape. Five cropping systems practiced by farmers in western France (Colbach et al., 2017a), where broomrape is the most problematic on oilseed rape (Terres Inovia, 2018), were simulated (Table 1). They included a typical local system, i.e. crop rotation including wheat, sunflower and oilseed rape, with mouldboard ploughing and pesticide treatments (Agreste, 2012). In addition, four alternative systems implementing three techniques known to control broomrape were simulated, namely diversified crop rotations, delayed sowing and reduced tillage (Fernández-Aparicio et al., 2016b). The crops used to diversify the rotations, flax and mustard, are known to be "trap" and "catch" crops" respectively, which reduce parasite seed bank by stimulating parasite germination but preclude parasite reproduction either because they are resistant to parasite attachment (the former) or destroyed before parasite maturity (the latter).

Each system was simulated over 30 years with the typical soil, weather and a typical weed flora from Poitou-Charentes (south-western France) given as inputs in FLORSYS. Simulations were repeated ten times with ten weather repetitions, each repetition consisting of a series of 30 years randomly chosen in the regional weather database. A moderate broomrape seed density was included in the initial weed seed bank, corresponding to 2000 seeds/m 2 per centimetre of soil layer [START_REF] Jestin | Évaluation de l'efficacité de différentes pratiques culturales dans la lutte contre l'orobanche rameuse par une méthode de quantification du stock grainier et cartographie du parasite sur le territoire français[END_REF] in the top 10 cm of soil [START_REF] Prider | Molecular diagnosis of parasite seed banks[END_REF].

Four sets of simulations were run: (a) with weeds but no broomrape, (b) without weeds but with broomrape, (c) with weeds and broomrape and (d) without weeds nor broomrape. They allowed to deduce the yield losses due to weeds only, due to broomrape only and due to both weeds and broomrape by comparing (a), (b) and (c) to (d) respectively. In each simulation set, the starting point in terms of weed and broomrape seed-bank densities was the same for all five cropping systems and 10 weather repetitions, i.e., an average of 171 seeds/m² per weed species (varying from 0.02 to 598 seeds/m² depending on the species) and zero broomrape seeds in (a), 20000 broomrape seeds/m 2 and zero weed seeds in (b), 171 weed seeds/m² and 20000 broomrape seeds/m² in (c) and zero weed and broomrape seeds in (d). Further details can be found in section S.3 online.

Statistical analysis

The effects of the type of infestation (i.e. with weeds and/or broomrape) and of cropping systems on annual crop yield losses due to pests (i.e. weeds and/or broomrape) were analysed with a linear model with the function "lm" of R (R Core Team, 2019):

Liscyr = cte + Ii + Ss + Cc + Yy + Rr + (I×S)is + (I×C)ic + eiscyr
Liscyr is the yield loss due to pests in crop c of year y in cropping system s and weather repetition r, starting with infestation i. Ii is the effect of type of infestation i on yield loss, with i{only weeds, only broomrape, both, none}. Similarly, Ss, Cc,Yy and Rr are the effects of, respectively cropping system s{reference, …}, crop c{oilseed rape, …}, year y{1,…} and weather repetition r{1,…} on yield loss. (I×S)is is the interactive effect of infestation i and system s on yield, and (I×C)ic the effect of system i in interaction with crop c. Finally, cte is a constant and eiscyr the residual.

Yield loss (%), i.e. the difference in yield (MJ/ha) in simulations with and without pests, was calculated relatively to yield obtained in simulations without pests. When analysing broomrape-caused yield loss, only host crop species were included in the analysis. Data were squared-root-transformed before analysis to achieve normality of residuals. Least significant difference tests were performed to compare yield losses due to broomrape between cropping systems for each crop species with the function LSD.test from the package agricolae of R.

Finally, we analysed which weed species and parameters drive broomrape infection. RLQ analyses were used to identify significant relationships between broomrape-infection indicators and weed species parameters, using the library ade4 [START_REF] Chessel | The ade4 package. I. One-table method[END_REF] of R (R Core Team, 2016). The RLQ analysis was initially developed to analyse correlations between cultural techniques (R matrix) and species traits (Q matrix) via weed species densities (L matrix). Here, we used annual indicator values of broomrape shoot density, biomass, seed production and soil seed bank from the 30 simulated years and 10 weather repetitions for the R matrix. The Q matrix consisted of the 142 parameters driving the annual life-cycle of the 26 weed species in FLORSYS, two parameters determining the species ability to trigger broomrape germination and allow broomrape attachment. The L matrix comprised the plant density of each weed species for each of the 30 years and the 10 repetitions, using the maximums of the daily weed species densities between crop sowing and harvest. In all three matrixes, we distinguished summer fallow from in-crop variables (e.g., broomrape shoots during summer fallow vs broomrape shoots during cash crop).

Only parameter-indicator relationships significant at p<0.05 after a 4 th corner analysis were considered, using the fourthcorner() function of R. This analysis tests whether species are distributed independently of their effect on indicators and of their traits, retaining for each indicator × trait combination the highest p values of models permuting either indicators or traits. To check whether weed species could be aggregated into functional groups in terms of impact on crop production related to plant morphology and shading response, species were grouped based on a Ward ascendant hierarchy classification using the hclust() function of R according to the Euclidian distances separating coordinates of species in the RLQ multidimensional space.

Results

Broomrape dynamics

In simulations with broomrape only, all cropping systems led to a continuous increase of the broomrape seed bank over 30 years, with regular sharp increases of the seed bank after host crops and slow decreases in-between (Figure 2). This dynamic was slower in cropping systems with delayed sowing and diversified crop rotation (scenarios 3 and 2 in Figure 2), where the seed bank was multiplied by only 4.3 and 7.3 respectively over 30 years, compared to 18-20 in the other three systems (Table 2).

Seed bank increased mostly after oilseed rape (Figure 2). Broomrape is highly reproductive on this crop (Figure 3.A and C), except in the cropping system with delayed sowing where broomrape did not have time to reproduce on oilseed rape (Figure 3.B). Indeed, the more oilseed rape sowing was delayed (i.e.

from system 1 to 2 to 3), the later broomrapes matured (from mid-April, mid-June and the end of June respectively, Figure 3.A-B). Sunflower (which stimulates broomrape germination less than oilseed rape, Figure 4.A) increased the broomrape seed bank in cropping systems with delayed sowing and diversified rotation (systems 3 and 2 in Figure 2), but depleted it in other systems. There, the number of germinated seeds exceeded broomrape seed production, resulting in a net decrease of the broomrape seed bank (e.g., system 1 in Figure 2). Flax and mustard achieved their goal as trap and catch crops: they induced broomrape germination (Figure 4.B-C) but did not allow its reproduction (Figure 3.A-C). However, the mustard benefits were largely cancelled out by broomrape seed production on the following sunflower (Figure 2). Overall, most broomrape germination occurred in spring, after a smaller germination peak in autumn in winter crops (Figure 4.A-C). Broomrapes started to reach maturity in June in oilseed rape crops sown at the end of August (Figure 3.A-C).

How weeds change broomrape dynamics

Generally, broomrape dynamics followed the same interannual pattern (i.e. succession of increases and decreases) in weed-infested (broken lines in Figure 2) vs weed-free simulations (continuous lines). The final broomrape seed bank was always lower than in the weed-free simulations (Table 2). The reduction was strongest in the delayed sowing and no-till systems; it was the weakest in the reference and the noplough systems whereas there was no difference in the diversified rotation.

Indeed, weeds stimulated broomrape germination over all seaons, regardless of the crops in place (Figure 4.D-F) but did not allow more broomrape reproduction (Figure 3.D-F vs A-C). As a consequence, the broomrape seed bank decreased even in crops such as wheat, which is neither a catch nor a trap crop.

The main drivers of crop yield loss

All factors tested in the analysis of variance (type of infestation, cropping system, crop species, weather repetition, simulation year, section 2.3) had a significant effect on annual yield losses due to broomrape and/or weeds predicted by PHERASYS (p < 0.05, section S4.4 online). Crop yield losses were 38% averaged over all crop species and situations (type of infestation, cropping system, weather repetitions, simulation year). They depended mostly on the crop species (partial R² = 0.28+0.03 for primary effect and interactions), on the cropping system (partial R² = 0.15+0.05), and to a lesser extent on the type infestation (i.e. with weeds, broomrape, or both, partial R² = 0.00+0.09). In other words, cropping techniques, including crop choice, were more important than the pest community composition (i.e., including weeds and/or broomrape). Although weeds contributed to deplete the broomrape seedbank (section 3.1), this did affect yield loss only in some crops and cropping systems (as shown by the negligible primary-effect R²) and this effect was small on average compared to effects of cropping techniques.

Which management techniques drive broomrape-caused yield loss?

Broomrape caused on average 16 % yield losses in oilseed rape and 45% yield losses in sunflower in the reference cropping system (system 1 in Figure 5.C and D), with a high variability in oilseed rape depending on the year and weather repetition (size of boxes in Figure 5). Only the cropping system with delayed oilseed rape sowing and mustard catch crop significantly reduced yield losses due to broomrape to almost zero in oilseed rape (system 3 in Figure 5.C). Diversified rotation and no plough (systems 2 and 4) also tended to reduce yield loss compared to the reference system, particularly its variability, but this was not significant.

In sunflower, whose management differed little between cropping systems, broomrape-caused yield loss was similar in all systems, though it was lower in system 3 where sunflower was sown after a mustard catch crop (Figure 5.D). The variations in yield loss (which is the relative difference of simulations running with and without broomrape) observed in wheat (Figure 5.B) were entirely due to stochasticity in the simulations as this crop cannot be infected by broomrape.

In which conditions do weeds increase broomrape impacts?

Overall, the combination of weeds and broomrape caused more yield losses than weeds alone (i.e. symbols are significantly above the y=x line in Figure 6.A). Adding broomrape did not change the ranking of the cropping systems in terms of yield loss (i.e. the symbols are placed along a straight line in Figure 6.A). The increase in yield loss caused by adding broomrape was the most important (i.e. symbols were increasingly above y=x when moving to the right of Figure 6.A) in those cropping systems where weeds already caused large yield losses, i.e. the systems with little or no tillage.

When doing the opposite, i.e. when adding weeds to broomrape, the cropping system ranking did not change either (i.e. the symbols were again on an approximate straight line in Figure 6.B). However, adding weeds in the delayed-sowing system (system 3) decreased yield loss compared to a field infested with broomrape only (i.e. blue diamond below the y=x line in Figure 6.B). This was due to weeds inducing broomrape germination year-round (Figure 4.F) which reduced the final broomrape seed bank by 93% compared to the final seed bank in weed-free simulations (Figure 2). Conversely, in simplified (system 4) or no till (system 5), adding weeds increased yield loss tremendously (i.e. triangles above the y=x line), because yield losses due to weeds cancelled out the benefits of broomrape seedbank depletion by weeds (-30% and -89% for systems 4 and 5, respectively).

Which weed traits drive broomrape infection

The same simulated situations favoured broomrape seed bank densities at the end of summer fallow and at crop harvest as the arrows for SeedBank0 and SeedBank1 were very close in the lower right quadrant on Figure 7.A and the Pearson correlation coefficient was high (0.86 in section S4.6 online). There was little correlation between broomrape seedbank densities with broomrape shoots, biomass or seed production during summer fallow ("0" suffix) as the corresponding arrows were approximately perpendicular to the seedbank arrows (non-significant Pearson correlations). Correlations were higher during the cash-crop period correlations between 0.13 and 0.43) but difficult to see on Figure 7.A (suffix "1") as these variables were less linked to the first two axes of the correlation circle. Seed bank was only linked to the seed rain during cash crop (correlation = 0.50) which was linked more to shoots on crop plants (correlation = 0.40) than to total shoots (correlation 0.26), indicating that seed bank densities resulted mostly from seed production from infected crop plants, and this cumulated over several years rather than only from the most recent seed rain (i.e., the correlation was not that high).

The shoot/biomass/seed rain arrows during cash crops were shorter than the corresponding summerfallow arrows, indicating that the latter were more linked to weed species that the former. During cash crops, most broomrape shoots were attached to crop plants (ShootsonCrop1 close to Shoots1 on Figure 7.A, correlation = 0.75) whereas any configuration was possible during summer fallow (the corresponding arrows were far away from each other), depending on whether a cover crop was grown during this period as in cropping system 3.

The simulated weed community was not structured according to its interactions with broomrape (Figure 7.B). The 26 simulated species were clustered into five groups which, except for group B, comprised both germination-stimulating and/or broomrape-attaching species (red, blue and grey labels) as well as species that neither attached nor triggered broomrape (white labels). Broomrape seed bank densities were actually the highest in the situations with group-B weeds (both were located in the right lower quadrant in Figure 7.A and B), which consisted exclusively of winter grass weeds (which neither trigger nor attach broomrape). For each weed species, labels with "0" and "1" suffixes were located closely, indicating that the same simulated situations favoured densities during summer fallow and during cash crops.

Generally, the correlation between the weed parameters and broomrape indicators were very low (< 0.10 in absolute values, Table 3), again indicating that broomrape infection was not linked to individual weed species or parameters, but rather to an interacting weed community. The two infection-specific weed parameters, the ability to let broomrape attach ("attachingAbility") and the ability to trigger broomrape germination ("alpha_h_GR24") were not among the 10 weed parameters that were the most correlated to broomrape indicators (Figure 7.C, Table 3). Weed species able to trigger broomrape germinations increased broomrape shoots, biomass and seed production, but mostly during summer fallow (negative correlations in Table 3, arrows in the same quadrant in Figure 7.A and C). But these species were associated to a decrease in broomrape seed bank indicating that they triggered more suicide broomrape germinations and successful host infections.

Unexpectedly, broomrape shoots, biomass and seed rain during cash crop as well as broomrape seed bank were negatively correlated to the weed ability to let broomrape attach (Table 3). But general weed parameters were more correlated to broomrape infection than the two infection-specific parameters.

Broomrape infection during summer fallow (which was mostly on weed species) was increased in the presence of weeds whose emergence period includes May and June, with a high base temperature and a high frost sensitivity, which are all characteristics of spring and summer weed annuals. In addition, these weed species presented a rather uniform leaf area distribution (low "RLH" values), usually not needing to avoid shadowing by crops. However, these are the opposite characteristics of the weed species linked with a high broomrape seed bank density, indicating that broomrape seedbank replenishment was not linked to weeds but resulted mostly from infected crops co-occurring with autumn/winter weeds, particularly those with the ability to let broomrape attach such as the winter grass weeds of group B in

Discussion

Novelty

PHERASYS is the first model of broomrape dynamics in agroecosystems specifically designed and parameterized for branched broomrape (Pointurier et al., 2021a). Compared to other broomrape 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][START_REF] Pérez-De-Luque | Differences in crenate broomrape parasitism dynamics on three legume crops using a thermal time model[END_REF], it includes several major novelties:

(1) interactions between broomrape and several dozens of annual crop and weed species, (2) broomrape impacts on crop growth and yield production, (3) a mechanistic description of the effects of all cropping techniques on broomrape, crops and weeds, (4) a 3D individual-based representation of the crop-weedbroomrape canopies allowing to investigate complex plant-plant interactions such as a biological regulation of broomrape by non-parasitic weeds or crop diversification resulting from the introduction of catch/trap crops into rotations and crop associations.

Consistency of PHERASYS with literature

During model design, we already checked that the various model formalisms were consistent with literature reports (Pointurier et al., 2021a) (Pointurier et al., 2021a). Here, we could go one step further and compare model output to field observations. The effects of the cropping techniques that we tested by simulation were generally consistent with the literature (Table 4), demonstrating that despite the many simplifications, the model produces realistic predictions.

The model produces many state variables that allow checking finer aspects of model consistency. For instance, in the simulations, broomrapes started to reach maturity in June in oilseed rape crops sown at the end of August (Figure 3.A-C), which is consistent with field observation [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].

Decomposing the effect of cropping techniques into a series of processes also allows to explain contradictory effects reports in literature. For instance, deep vs superficial is reported to both decrease broomrape infection because buried seeds germinate badly (Rubiales et al., 2009b) and increase infection because it places seeds closer to host roots (Fernández-Aparicio et al., 2016b). Potentially, both effects can cancel each other out, which is consistent with our simulations where tillage effects on broomrape dynamics were not significant.

Another critical point is the prediction of the weeds' life-cycle duration. Weed flowering and maturity dates simulated by FLORSYS were shown to be too early 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]. So, even though a corrective patch was used in the present simulations (section 2.1.2), the catch-crop effect simulated here for weeds (section 4.3) could be overestimated. This could explain why, in contrast to our simulations, recommendations usually advise to control host weed species as they might relay broomrape infestation in the absence of crop [START_REF] Gibot-Leclerc | Potential role of winter rape weeds in the extension of broomrape in Poitou-Charentes[END_REF][START_REF] Jestin | Évaluation de l'efficacité de différentes pratiques culturales dans la lutte contre l'orobanche rameuse par une méthode de quantification du stock grainier et cartographie du parasite sur le territoire français[END_REF]. Moreover, PHERASYS is still missing some common host species, such as Aphanes arvensis [START_REF] Gibot-Leclerc | Potential role of winter rape weeds in the extension of broomrape in Poitou-Charentes[END_REF].

Agronomic implications

Our simulations showed that cropping techniques must be combined (e.g., trap and catch crops aiming to deplete the soil seed bank, with delayed sowing or reduced host crops frequency in the rotation aiming to limit broomrape reproduction) for an efficient control of broomrape in various weather scenarios.

These conclusions on which techniques or systems may not be valid in other pedoclimates, weed floras contexts or cropping system types. This may also contribute to explaining why, contrary to the literature, we did not observe any effect of soil tillage on broomrape dynamics (Table 4).

Our simulations also show that interactions between broomrape and weeds can be exploited to regulate broomrape. Weeds helped to deplete the broomrape seed bank by inducing germination, particularly during summer fallow. Although they sometimes allowed broomrape to reproduce, they mostly played the role of catch plants, as they generally had a shorter life cycle than broomrape so they died before it could reproduce. This effect is highly dependent on the type of weed species present and might also be overestimated because of some deficiencies in the weed phenology predictions in FLORSYS (section 4.2).

Finally, the catch-crop effect of weeds was only beneficial in terms of crop yield loss when weeds did not compete for light with the crop. Using biological regulation by weeds to control broomrape involves finding the right balance between tolerating a residual weed flora to stimulate broomrape germination, and controlling it to avoid competition with the crop. One way would be to let weeds grow unchecked during summer fallow (where they do not hinder crop production) and destroy them before they and/or broomrape reproduce (to avoid seed bank replenishment). Dealing with such complex questions shows the need of simulation models such as PHERASYS.

Perspectives

The present simulation results are encouraging as they are mostly consistent with literature reports. However, they must still be interpreted carefully since the model has not yet been evaluated with independent field data. A sensitivity analysis will also be needed to identify which model parameters need to be measured more precisely and which formalisms must be improved. The present simulation study can be considered as a first step of a very basic sensitivity analysis insofar as it ranked the main levers influencing broomrape dynamics, identified the main species traits interacting with broomrape dynamics via the RLQ/4 th corner analyses as well as key plant stages whose prediction quality must be checked (e.g., host flowering dates, broomrape life-cycle duration). But the present study needs to be completed by a large-scale simulation study investigating the huge range of combinations of models inputs as well as model parameters.

Once the model has been evaluated, a larger set of cropping systems should be simulated in order to track innovative broomrape-management strategies that benefit from biological regulation by weeds.

The model could even be used as a decision support system to help designing cropping systems by testing prospective systems and improving them based on simulation results (Colbach et al., 2017b;Colbach et al., 2021).

Conclusion

The PHERASYS model was developed in a companion paper to simulate branched-broomrape dynamics in agroecosystems. It allowed here, for the first time, to simulate [START_REF] Ephrath | Quantification of the dynamics of Orobanche cumana and Phelipanche aegyptiaca parasitism in confectionery sunflower[END_REF] interactions between broomrape and heterogeneous multispecies stands of crops and weeds, (2) the effects of multiple cropping techniques, including complex ones playing on the competition between broomrapes and hosts for host resources. Although the model needs to be evaluated, simulation results are consistent with the literature.

Combining delayed sowing, which prevents broomrape from reproducing, to the use of trap and catch crops, to deplete the broomrape seed bank, allows a long-term control of the parasitic plant. Tolerating weed flora during summer fallow, could also help to accelerate the seed bank depletion by stimulating broomrape germination. Fewer attachments in obligate parasitic plants (Grenz et al., 2005b;[START_REF] Díaz | Characterization of the holoparasitism of Orobanche ramosa on tomatoes under field conditions[END_REF]Gibot-Leclerc et al., 2006a;[START_REF] Tippe | Delayed or early sowing: Timing as parasitic weed control strategy in rice is species and ecosystem dependent[END_REF], but benefits in terms of yield loss can be cancelled out by reduced yield potential [START_REF] Díaz | Characterization of the holoparasitism of Orobanche ramosa on tomatoes under field conditions[END_REF]Gibot-Leclerc et al., 2006a).

Acknowledgements

Crops sown later grow faster so they are more competitive than broomrapes (Fernández-Aparicio et al., 2016b). Fewer broomrape seeds germinate at crop sowing as they are more dormant later in the season [START_REF] Tippe | Delayed or early sowing: Timing as parasitic weed control strategy in rice is species and ecosystem dependent[END_REF].

Trap and catch crops

Seed bank depletion by flax, sunflower (if high infestation) and, to a lesser degree, mustard

Flax and mustard are trap and catch crops [START_REF] Goldwasser | Integrated Agronomic Management of Parasitic Weed Seed Banks[END_REF]; sunflower depletes broomrape seed bank [START_REF] Jestin | Évaluation de l'efficacité de différentes pratiques culturales dans la lutte contre l'orobanche rameuse par une méthode de quantification du stock grainier et cartographie du parasite sur le territoire français[END_REF]; summer crops and flax deplete more than autumn cover crops [START_REF] Jestin | Évaluation de l'efficacité de différentes pratiques culturales dans la lutte contre l'orobanche rameuse par une méthode de quantification du stock grainier et cartographie du parasite sur le territoire français[END_REF].

The ability of a plant to stimulate broomrape germination 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][START_REF] Perronne | Is induction ability of seed germination of Phelipanche ramosa phylogenetically structured among hosts? A case study on Fabaceae species[END_REF]. The longer the crops grow, the larger their root systems are, the more germination they induce (Grenz et al., 2005a). The spring germination flush simulated here in oilseed rape is not systematically observed in the field [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][START_REF] Gaudin | Place de l'azote dans l'interaction plante -plante parasite : Brassica napus L. -Phelipanche ramosa (L.) Pomel[END_REF]. The quantity and quality of germination stimulants exudated by plants varies over season [START_REF] López-Granados | Effects of environmental factors on dormancy and germination of crenate broomrape (Orobanche crenata)[END_REF][START_REF] Auger | Germination stimulants of Phelipanche ramosa in the rhizosphere of brassica napus are derived from the glucosinolate pathway[END_REF], whereas it only depends on stage in PHERASYS. Sunflower slightly infected by broomrape Sunflower only rarely allows broomrape to reproduce in the field [START_REF] Jestin | Évaluation de l'efficacité de différentes pratiques culturales dans la lutte contre l'orobanche rameuse par une méthode de quantification du stock grainier et cartographie du parasite sur le territoire français[END_REF].

Unknown mechanisms

Soil tillage

No effect

Deep tillage vs superficial or no tillage reduces the number of broomrape germinated seeds (Rubiales et al., 2009b) Lack of oxygen in deep soil layers prevents broomrape germination (Rubiales et al., 2009b), not modelled.

Reduced tillage vs deep tillage can reduce the number of broomrape germinated seeds and attachments (Fernández-Aparicio et al., 2016b) Prevents seeds from being buried close to host roots where they could be stimulated and attach (Fernández-Aparicio et al., 2016b).

Volume of host roots in superficial soil layers overestimated by our simplistic representation of root systems, which grow directly from the soil surface instead of from the seed.

Figure

  Figure 7.B.

  This project was funded by INRAE, the research program APR 2011-Ecophyto 2018 "Assessing and reducing environmental risks from plant protection products" funded by the French Ministries in charge of Ecology and Agriculture, the French project CoSAC (ANR-15-CE18-0007), the European Union's Horizon 2020 Research and innovation programme under grant agreements N 727321

Figure 1 :

 1 Figure 1: Processes of branched-broomrape life cycle modelled in the PHERASYS submodel of FLORSYS. Inputs given by the user are in grey. Variables used to connect both models are in italics (green and purple: biotic variables from FLORSYS and PHERASYS respectively, brown: abiotic variables) (Olivia Pointurier 2019)

Figure 2 :Figure 3 .Figure 4 .Figure 5 :

 2345 Figure2: Dynamics of the branched-broomrape seed bank in three cropping systems predicted by PHERASYS over 30 years. Each line shows the mean number of broomrape seeds in the soil after crop harvest averaged over 10 weather repetitions in a given cropping system. Thick lines show data from simulations of infestation with broomrape only, and dashed lines from simulations with both nonparasitic weeds and broomrape. Bars represent standard deviations. Each colour shows a cropping system (black: 1-reference; red: 2-diversified rotation; 3-green: delayed sowing, details in Table1). Letters indicate the harvested crop (O = winter oilseed rape, W = winter wheat, S = sunflower, mS = sunflower was sown after a mustard cover crop, F = flax) (Olivia Pointurier2019)

Table 1 :

 1 Characteristics of the simulated cropping systems

	Cropping system	Rotation*	Oilseed rape sowing date	Tillage

Table 3 .

 3 (Nathalie Colbach 2021 )

Table 4 :

 4 Effects of cropping techniques on branched-broomrape dynamics and crop yield losses in simulations with PHERASYS and consistency with the literature (green cells indicate consistent results, yellow limited consistency)

	Cropping	Effects on
	technique	broomrape and
		crop yield losses
		in simulations

Consistency of the effects with the literature Consistency of the mechanisms involved with the literature

	Delayed	Fewer broomrape
	sowing	attachments on
		host crops and
		less yield loss


Laure Guy

Reiko Oda

N Avarvari

Jeanne Crassous

Nina Berova

Gennaro Pescitelli

Oliver Trapp

Simona Collina

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Special issue: Chirality in France

Laure Guy, Reiko Oda, Narcis Avarvari, Jeanne Crassous, Nina Berova, Gennaro Pescitelli, Oliver Trapp, Simona Collina Dedicated to Jean-Pierre Dutasta and Christian Roussel on the occasion of their retirement and to all scientists who contributed to the development of Chirality in France. Historically, the tremendous role played by French scientists in the importance of chirality in chemistry, physics, biology, medicine, or materials science, starting with the pioneering works and discoveries of Augustin Fresnel, Jean-Baptiste Biot, Louis Pasteur, or Aimé Cotton, has been largely acknowledged by the international scientific community. The French chemists and physicists interested by chirality in the broadest sense continued this tradition over the years by being very actively involved in many chirality related research fields such as enantioselective synthesis and catalysis, analytical and chiral separation techniques, chiroptical spectroscopies, theoretical approaches, supramolecular chemistry, polymers, biomimetic systems, chiral materials, and fundamental physics. Witnessing the ever growing interest in France of this interdisciplinary topic, a French network in chirality supported by the CNRS (GDR Chirafun),gathering about 30 chemistry and physics teams from different Institutes, has emerged in 2015, following the organization of the "André Collet Chirality Days" in 2009 and2012, which were then continued in 2015 and 2018 under the auspices of the Chirafun network, with a new edition planned for October 2022.In this year 2022 marking the bicentenary of Louis Pasteur's birth, we are deeply pleased and honored to introduce this special virtual issue of Chirality gathering 19 very high quality articles spanning different chirality related fields covering


Serge Aubry

Gilles Abramovici

Jean-Luc Raimbault

CHAOTIC POLARONIC and BIPOLARONIC STATES IN COUPLED ELECTRON-PHONON SYSTEMS

For large electron-phonon coupling, many models describing the electron-phonon coupling have a limit called anti-integrable with trivially chaotic states. For example, let us consider the Holstein Hamiltonian which is the sum of three terms which are This new formulation of the initial Hamiltonian is specially interesting for making appearent two important physical limits which are the (standard) adiabatic limit and the anti-integrable limit. Up to now, the importance of this second limit for understanding the physical behavior of the model, was not recognized in the litterature although it has been already considered implicitly but unperfectly in few articles.

Adiabatic Limit: The standard adiabatic limit is obtained when 13 is zero. This approximation is valid when

(5-a)

g >> rico 0

that is for large enough electron coupling g. In that case, the effect of the quantum lattice fluctuations is negligible. At the adiabatic limit, the problem becomes variational since operator ui commutes with the Hamiltonian and can be considered as a scalar ui = <ui>. Note that the adiabatic Hamiltonian obtained for 13=0, is equivalent to the standard meanfield Hamiltonian which is the selfconsistent Hamiltonian obtained by replacing operator nini by ui<ni> + <ui> ni -<ui><ni> = ui ni + ui ni -ui ni in (3-b). This approximation neglects the fluctuation (ui -ui)(ni-ni) which becomes zero only when 13--0. when the electrons are in the ground-state determined by the lattice potential. We also consider situations where the electrons are in excited states as in polaronic or mixed polaronic-bipolaronic states.

Anti-Integrable Limit: When considering large electron-phonon coupling g, the relevant limit is obtained for t=O (and 13=0). This condition is fulfilled when (8-a)

g >> a] T l~co 0 the model becomes trivially soluble since the Hamiltonian H (or 1~ ) commutes with the electronic density operators c+,] , ci-], and c+$ci$ for spins T and $. For any arbitrary pseudo-spin configurations {cyi,]-} and {cyi$} with (8-b) oiT = 0 or 1 and ~i$ = 0 or 1 there exists an eigenstate of the "anti-integrable" Hamiltonian such that (8-c) <ci~ ci$> = c~i~-and <ci~ci$> = cyi$ These eigenstates are periodic, quasi-periodic, chaotic or else as the pseudo-spin configurations {oi.1-} and {cyi$} which label it. By analogy with dynamical systems where the anti-integrable limit is a limit where the chaotic behavior is perfect, we called this limit anti-integrable [1 '2].

For t=0, when a single electron is present at site i, there exists a lattice distortion at this site. The electron associated with the lattice distortion is called a polaron. This object is magnetic because of the free spin of the electron. When two electrons with opposite spins are present at site i, this pair of electrons associated with the corresponding lattice distortion is called a bipolaron. This object is non magnetic.

Thus, the eigenstates determined by pseudospin configurations fulfilling { ~i']" } = { °i$ } are bipolaronic configurations. Those where ~i$ x Gi.l, = 0 for all i, are polaronic configurations. In the general case, the eigenstates determined by {oil'} and { Gi$ } are mixed polaronic-bipolaronic configurations.

We have proven through tedious mathematics [START_REF] Aubry | Chaotic Polaronic and Bipolaronic States in the Adiabatic Holstein Model prepint Submitted to[END_REF], that these chaotic polaronic and bipolaronic states obtained trivially for t--0, survives to perturbations in t, that is for large enough electron-phonon coupling. This proof holds very generally for most arbitrary lattices which could be random or not (except for exponential lattices such as Bethe lattices), also in the presence of an arbirary uniform magnetic field (not represented in the model described here).

More precisely, for each eigenstate q~({ ~i1" }' { (Yi,l, } ) at the anti-integrable limit, there exists a positive non zero number tc (independant of the system size) such that for Itl < tc, there exists an eigenstates q~({~i$},{cyi.],}, t) of the adiabatic Holstein model (with t~0 and 13=0) which depends uniformly continuously on the parameter t and such that for t=0, we have W({~i,l-},{6i.l,},0) = q~({cYi$ },[~i$ }).This exact result means that for large enough electron-phonon coupling, there exists infinitely many local minima of the adiabatic energy.

Since they originate by continuity from the anti-integrable limit, these states are also called bipolaronic, polaronic or mixed polaronic-bipolaronic structures. However, note that these polarons and bipolarons become interacting "particles" for t;z-0 For larger t, (that is for smaller electron-phonon coupling), these structures disappear through complex cascade of bifurcations qualitatively similar to those observed for dynamical systems.

The real bounds which can be observed numerically for tc are much larger than the bounds obtained rigorously. Although these last bounds were found independent of the pseudo-spin configuration, the real bound in t of these configurations depends on the choice of the pseudo-spin configurations {~i],} and {cyi,l, }. In fact the dimensionless parameter t c ranges not very far from unity which proves for many systems the physical relevance of the domain of existence for these polaronic and bipolaronic states.

Many other exact results concerning the properties of these states are obtained. All these structures are proven to be insulating with both a non vanishing electronic gap and a non-vanishing phonon gap. These structures are "charge defectible" which means that non uniform distribution of the electronic charges are metastable as in true insulators unlike metals and semi-conductors. The local perturbations due for example to impurities or to extra-electrons decay exponentially at long distance.

For large enough t, the ground-state of the adiabatic Holstein model, is proven to correspond to an ordered bipolaronic structure which may be commensurate, incommensurate (or else?) and thus is a bipolaronic charge density wave. It is characterized by a periodic, quasi-periodic (or else) ordering of the pseudo-spin configuration {cyi,1-} --{~i,l, }' Therefore, the other bipolaronic and mixed polaronicbipolaronic structures correspond to low energy configurational excitations of this ordered ground-state. Again, this exact result holds at any dimension and for most lattices.

For small magnetic field, the ground-state remains a bipolaronic structure but for a large enough magnetic field, the ground-state is proven to become a mixed polaronicbipolaronic state which is magnetic, through complex and unexplored cascades of transitions.

The role of the dimensionless parameter [~ describing the quantum lattice fluctuations which is not strictly zero but small, has been analysed [5]. It has been shown that their effect are negligible for the polaronic and bipolaronic structures providing that the phonon gap remains large enough. By contrast, it has been proven [5-d] that the existence of a gapless phonon mode (e.g. phason), breaks down the validity of the Born-Oppenheimer approximation on which the standard theory of Charge Density Waves (CDW) is based. In that case, superconductivity is conjectured. As a result, all CDW's should be viewed as Bipolaronic CDW's, that is an incommensurate ordering of bipolarons. The physical consequences of this description sharply differ on many points from those of the widely admitted standard Peierls-Frohlich theory of CDW [START_REF] Lee | [END_REF] and could provide a new fruitful basis for understanding many experiments in CDW systems which up to now received inconsistent theoretical interpretations [START_REF]MONCEAU Recent Developments in Charge Density Wave Systems in Application of Statistical and Field Theory Methods to Condensed Matter[END_REF].

In summary, this work confirms and extends early numerical studies and conjectures on the Peierls instability in this model in one dimension model where a transition by "breaking of analyticity" toward a bipolaronic CDW was already observed [START_REF] Aubry | Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides[END_REF]. This recently obtained exact result should break down some wrong ideas concerning the effects of large electron-phonon coupling which are widely spread in the litterature. It opens a new direction for understanding globally the properties of Charge Density Waves Systems and perhaps in further stages, other structural problems concerning for example magnetic structures.

  ~ Cj,o <i,j>, which corresponds to a band of electrons (described with standard Fermion operators c'+1,~ and ci,o) propagating on an arbitrary lattice L periodic or not (<i,j> denotes neighboring sites and (r the spin of the electron which can be 1" or $) 2-Hp is the phonon Hamiltonian Z 1 (l-c) Hp = 14r.o 0 (a + a i+ ~) i which corresponds to Einstein oscillators with frequency (o 0 at the sites i of the lattice L (a + and a i are standard Boson operators) and 3-Hep is the electron-phonon coupling Hamiltonian (ci$ + ci~ci$ It couples the electronic density operator n i at site i with the oscillator positionChoosing for this operator (in appropriate units) of energy, this Hamiltonian H becomes the sum of three terms

For 1 =

 1 the adiabatic Hamiltonian, the eigenstates are characterized by the atomic configurations {ui} which are obtained as local minima of the variational form {Un})) are the eigenenergies of the tight-binding Schroedinger equation V V (6-a) -t (~R'v) n + u n q?n = Ev({Ui}) ~t'n ( (~X~)n= ~ Wm is the sum over the neighboring sites m to n on the lattice mq/h denoted mq/h).This eigenequation also determines the electronic eigenstates {Ur' nV } for the atomic configuration {Un}. Ov is the population number of state v. We have (6-b) < c'+1"v,, Cv,l" > + < Cv, +l,~ Cv,$ > = O, 1 or t5 V It can be chosen according to the standard Fermi rule (7-a) cy v = 1 for Ev({Un}) < EF and (7-b) 6v = 0 for Ev({Un}) > E F