Guillaume Fried
email: guillaume.fried@anses.fr

Emmanuelle Alexandre Villers
email: villers.alexandre@gmail.com

Emmanuelle Porcher
email: emmanuelle.porcher@mnhn.fr

Anses

Assessing non-intended effects of farming practices on field margin vegetation with a functional approach

Keywords: Agrotolerant species, Field margin width, Agricultural land use intensity, Nitrogen fertilizer, Pesticide drift, Plant diversity

come

Assessing non-intended effects of farming practices on field margin vegetation with a functional approach

Introduction

During the second half of the twentieth century, the conjunction of mechanization, landscape simplification and widespread use of chemical inputs has resulted in dramatic changes in the environmental conditions of arable landscapes [START_REF] Stoate | Ecological impacts of arable intensification in Europe[END_REF]. This is associated with both a loss of non-crop habitats [START_REF] Robinson | Post-war changes in arable farming and biodiversity in Great Britain[END_REF] and a massive decline in abundance and diversity of biotas associated with arable fields [START_REF] Benton | Linking agricultural practice to insect and bird populations: a historical study over three decades[END_REF][START_REF] Donald | Agricultural intensification and the collapse of Europe's farmland bird populations[END_REF][START_REF] Fried | Arable weed decline in Northern France: Crop edges as refugia for weed conservation?[END_REF]. In many agricultural landscapes, field boundaries have become the last remnants of semi-natural habitats and their functional role as refugia for a high level of botanical diversity and for supporting a diverse fauna has been increasingly recognized [START_REF] Marshall | Field margins in northern Europe: their functions and interactions with agriculture[END_REF]. However, the quality of this habitat may also be lowered by intensive agriculture, particularly due to leaked or misplaced fertilisers and herbicides in the adjacent arable land [START_REF] Boutin | Intensity of agricultural practices and effects on adjacent habitats[END_REF][START_REF] De Snoo | Effect of herbicide drift on adjacent boundary vegetation[END_REF]Kleinj and Verbeek, 2000). Assessing the effects of farming practices on non-target organisms is crucial to guide farmers towards better practices, and also as a part of official post registration monitoring of pesticides. However, detecting such effects requires disentangling the role of agricultural practices, landscape and local abiotic environmental conditions (e.g. soil type).

Previous studies have shown that the vegetation of field boundaries is influenced by the combination of both environmental conditions and management practices. Several studies found that the plant community composition of field boundaries varies primarily with landscape structure (proportion and diversity of non-arable habitat patches at the landscape level) and spatial variables [START_REF] Aavik | What is the role of local landscape structure in the vegetation composition of field boundaries?[END_REF][START_REF] Aavik | Quantifying the effect of organic farming, field boundary type and landscape structure on the vegetation of field boundaries[END_REF][START_REF] Tarmi | Importance of edaphic, spatial and management factors for plant communities of field boundaries[END_REF].

Local habitat structure (boundary width, presence of ditches and hedgerows) and boundary management were also shown to be important factors, while the influence of farming in the adjacent field was less clear [START_REF] Tarmi | Importance of edaphic, spatial and management factors for plant communities of field boundaries[END_REF]. Conversely, [START_REF] Bassa | Effects of farming management and landscape heterogeneity on plant species composition of Mediterranean field boundaries[END_REF] found significant differences between broad management type (organic versus conventional), with fewer effects of landscape structure. Similarly, [START_REF] Petersen | Effects of organic farming on field boundary vegetation in Denmark[END_REF] highlighted a rapid effect of conversion to organic farming in field boundaries of dairy farms, with fewer nitrogendemanding or ruderal species and more stress-tolerant species. [START_REF] Kleijn | Factors affecting the species composition of arable field boundary vegetation[END_REF] also found a strong effect of in-field fertilisation on species richness of the boundary vegetation, while boundary management and use of herbicides in the boundary had no effects.

Comparing unsprayed and sprayed ditches, de Snoo and van der Poll (1999) found a significant effect of herbicide drift only on dicotyledonous species of adjacent winter wheat field boundaries. Some of these previous results, and a vast literature in plant ecology (reviewed e.g. in [START_REF] Garnier | A trait-based approach to comparative functional plant ecology: concepts, methods and applications for agroecology. A review[END_REF], suggest that the impacts of both agricultural practices and the landscape on the community composition of field margins depend on some key functional traits, particularly response traits sensu [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: Revisiting the Holy Grail[END_REF]. Therefore, the use of trait-based approaches may offer opportunities for a deeper mechanistic understanding of the impacts of environmental factors, including management practices, on the vegetation of field margins.

Despite the growing number of studies, there have been few attempts so far to characterize the composition of plant functional traits in field margin communities, and its environmental correlates (but see [START_REF] Alignier | Changes in management practices over time explain most variation in vegetation of field margins in Brittany, France[END_REF][START_REF] Alignier | Two decades of change in a field margin vegetation metacommunity as a result of field margin structure and management practice changes[END_REF][START_REF] Chaudron | Functional response of plant assemblages to management practices in road-field boundaries[END_REF]. Here we use trait-based approaches to examine which environmental variables best explain the assembly of field margin plant communities, using a nationwide standardized monitoring scheme designed to detect the non-intended effects of in-field practices, particularly the use of synthetic inputs, on biodiversity. The following questions were addressed: (1) Are the links between environmental conditions, in-field management practices and plant traits in field margins consistent with expectations derived from ecological knowledge in more natural settings (see below for the details of such expectations)? (2) Are there any trait syndromes associated with gradients of management practices and/or land-use intensity? (3) What factors influence plant species richness and plant functional diversity in field margins?

Methods

Vegetation survey

The "500 ENI" network is a long-term survey initiated by the French Ministry of Agriculture.

It is designed to detect and document nationally any significant changes in wild flora and fauna (birds, beetles, and earthworms) of agricultural landscapes that could be associated with non-intended effects of agricultural practices, especially fertilization and pesticides (Andrade et al., in prep.). For this purpose, 500 fields distributed over all agricultural regions of France (including Corsica) were chosen to be representative of the main crop productions, using a stratified sampling within each administrative region. The survey focuses on three types of production systems: annual crops (wheat or maize as the main crop production), vineyards and market gardening centred on lettuce production. Here, we worked with a subset of 430 field margins (out of 500), chosen as those for which comprehensive floristic data were available in both 2013 and 2014 (see Figure 1 for a distribution map).

Three main areas are recognized in field margins [START_REF] Greaves | Field margins: definitions and statistics[END_REF][START_REF] Marshall | Field margins in northern Europe: their functions and interactions with agriculture[END_REF]: crop edge, field margin strip and field boundary. The area surveyed in the "500-ENI" program excludes the crop edge (sometimes called conservation headland) which is located in the first 1-6 meters within the crop. It also excludes the cultivated strip, outside the last row of crop, which is an area with mostly bare soil usually colonized by weed species from the field. The area of interest is the field margin strip (sensu [START_REF] Marshall | Field margins in northern Europe: their functions and interactions with agriculture[END_REF] hereafter field margin), which is the uncultivated herbaceous vegetation area between the cultivated strip and another patch in the landscape. The latter can be a field boundary embodied by a ditch and/or a hedge, or other land covers such as a road, a track, other fields or grassland-type vegetation (see below).

Wild plant species were identified in ten 1m² quadrats located in the field margin. The ten quadrats were divided into two sets of five contiguous quadrats of 0.25*2 m² separated by 30 m. The quadrats were placed in the centre of the field margin (i.e. equidistant from the field and the adjacent land cover). Their position was maintained in the same field margin across the two years but their exact location may slightly differ from year to year. Only presenceabsence of plant species was recorded in the ten quadrats, so that the abundance of each species was characterized by a frequency of occurrence (0-10) in each field margin. Surveys were performed at the peak of flowering, which is from late April to early August depending on locations (50% of the observations were made between June 6 th and July 11 th ). In order to avoid the issue of differences in rare species detection among observers, and because trait data are not available for all species, we focused on species observed in more than 1% of the surveyed margins, i.e. 186 of 331 species observed in total, representing 94.9% of the total abundance (number of quadrats). All subsequent analyses were performed on this subset of species.

Environmental data

Ten environmental variables were used to describe the field margins and were grouped into four categories: 1) two abiotic factors, 2) three landscape factors, 3) two field margin management practices and 4) three management practices within the adjacent field. (1) From the coordinates of the field margin, we retrieved eight factors from the Soilgrids dataset at 250m resolution [START_REF] Hengl | SoilGrids250m: Global gridded soil information based on machine learning[END_REF]. We performed a PCA and extracted the first two axes, which represented 65% of total inertia. Axis 1 was associated with a combination of soil pH and texture, opposing sandy acidic soils (positive values) to basic clay soils (negative values).

Axis 2 was positively associated with the level of organic matter (see Appendix A for the detailed outputs of the PCA). (2) Landscape composition was described on the basis of the proportion of land cover (crop, grassland, woodland, etc.) within 250 m of the field margin.

By summing the proportion of all non-cropped area (i.e. permanent grasslands, heaths, hedges and woodland as well as water surfaces), we characterized the amount of non-arable land cover, which is a well-established landscape metric for landscape complexity [START_REF] Gabriel | Local diversity of arable weeds increases with landscape complexity[END_REF]. The size of the field adjacent to the studied margin was also considered as a landscape variable. The local habitat structure immediately adjacent to the field margin (i.e. the field boundary sensu [START_REF] Marshall | Field margins in northern Europe: their functions and interactions with agriculture[END_REF] was characterized by the presence/absence of ditches or hedges resulting in four field boundary classes: i) ditch and hedge present, ii) ditch present, iii) hedge present, iv) no ditch or hedge. This latter category corresponds to field boundaries with no particular features, where the field margin is contiguous with another land cover type, such as roads or tracks (68%), other fields (12%), or grassland-type vegetation (16%). (3) Field margin management was characterized by the number of management events; mowing was the dominant management type (n=582 occurrences for the two years) with only few margins with use of chemicals (n=8) or grazing (n=12). Field margin width was also recorded. When the surveyed arable field was adjacent to a grassland, field margin width was measured up to the visible boundaries of cadastral parcels. (4) Management within the field was summarized by the amount of nitrogen fertilizer input, and the Treatment Frequency Index (TFI) for herbicides and insecticides. TFI is calculated as the cumulative ratio of the dose applied to the recommended dose, for all treatments applied during the growing season [START_REF] Halberg | Indicators of resource use and environmental impact for use in a decision aid for Danish livestock farmers[END_REF]. Finally, observation date (number of days since January 1 st ) and year (2013 or 2014) were also added as explanatory variables. The farming system (conventional versus organic) in the field adjacent to the surveyed margin was also incorporated in the analyses either as a supplementary variable (when in-field practices were already included) or as a primary variable accounting for management practices.

Plant trait data

Seven functional traits and three indicators of ecological performance were chosen on the basis of hypotheses related to potential non-intentional effects of farming practices and field margin management on two main ecological axes, namely fertility (resources) and disturbances (Table 1). The level of disturbances (frequency and intensity) is assumed to favour short-lived species reaching small height at maturity but having rapid growth rate and high seed output (ruderal strategy, e.g. [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: Revisiting the Holy Grail[END_REF]Garnier 2002, Grime 2006). Thus, disturbances such as the number of management events in the margins and herbicide drift related to the intensity of herbicide use are expected to promote a higher proportion of annual species, species with high Specific Leaf Area (SLA), low seed mass and short plant height.

The level of fertility is assumed to favour nitrophilous species, i.e. with high Ellenberg-N, tall competitive plants or rapid growing species with high SLA values (e.g. [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: Revisiting the Holy Grail[END_REF]Garnier 2002, Grime 2006), although trait responses to these two major environmental drivers can sometimes be more complex [START_REF] Douma | Disturbance and resource availability act differently on the same suite of plant traits: revisiting assembly hypotheses[END_REF]. Additional hypotheses were also considered. The presence of hedges can favour shade-tolerant species (low Ellenberg-L values), while the presence of ditches can favour water-demanding species (high Ellenberg-H values). Disturbance caused by insecticides are expected to favour selfing species and species with abiotic pollination over entomogamous species, via a decrease in the abundance and diversity of pollinators [START_REF] Brittain | The potential impacts of insecticides on the life-history traits of bees and the consequences for pollination[END_REF]. Landscape diversity and proximity of seminatural habitats are expected to benefit plants with animal pollination and dispersal (e.g. [START_REF] Steffan-Dewenter | Effects of habitat isolation on pollinator communities and seed set[END_REF]. A functional diversity index can be computed on the basis of these traits (see below) in addition to classical taxonomic diversity indices such as species richness. Since we have included both response and effect traits [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: Revisiting the Holy Grail[END_REF], a field margin with high functional diversity can be interpreted as supporting: i) a community including species with different ecological strategies ensuring more resistance to environmental changes and, ii) a community including species with different functions providing a larger range of resources for species of higher trophic levels.

In parallel to classic plant traits, we also used an emergent classification of species on the basis of their responses to agricultural disturbances. Following [START_REF] Aavik | Agrotolerant and high nature-value species-Plant biodiversity indicator groups in agroecosystems[END_REF], species were classified into agrotolerant (i.e. species adapted to current disturbance regimes in arable fields, under modern conventional agriculture) and hemerophobic species (i.e., species that are sensitive to soil tillage and/or herbicides). Each species was classified on the basis of its frequency of occurrence in arable fields using data of the Biovigilance Flore network (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012), which covered the same area as in the present survey, i.e. the whole of France [START_REF] Fried | Environmental and management factors determining weed species composition and diversity in France[END_REF]. The 47 species present in more than 10% of the 1440 arable fields surveyed in Biovigilance Flore were considered as agrotolerant, while the hemerophobic group was made of both grassland species and rare arable weeds that are not adapted to current farming practices (see Appendix B).

Data analysis

To analyse the relationships between the variation in plant traits of field margins and the variation in environmental factors, we followed the framework introduced by Dray et al.

(2014) combining the RLQ multivariate technique (an ordination analysis) and the fourth corner analysis (a hypothesis testing analysis). RLQ assigns scores to species, samples, traits, and environmental variables along orthogonal axes and yields a graphical summary of the main structures [START_REF] Dolédec | Matching species traits to environmental variables: a new three-table ordination method[END_REF], while the fourth corner analysis tests the multiple associations between one trait and one environmental variable at a time [START_REF] Dray | Testing the species traits-environment relationships: the fourth-corner problem revisited[END_REF].

Both methods use three tables: the R-table, which consists of the 860 samples (430 field margins x two years) described by the ten environmental and two temporal variables, the Q-table containing the 186 species described by their 10 traits and the L-table describing the floristic composition of the 860 samples via the abundance of 186 species. The L-table therefore links the R-and the Q-tables. For the RLQ analysis, a correspondence analysis (CA) was first performed on the L-table using the raw frequency of occurrence score for each species (0-10). Next, a Hill and Smith analysis (a mixed ordination method similar to PCA that allows combining quantitative variables and factors; [START_REF] Hill | Principal component analysis of taxonomic data with multi-state discrete characters[END_REF] was performed on the R-table (using the row scores of the CA on the L-table as canonical factor), and on the Q-table (using the column scores of the CA on the L-table as canonical factor).

Finally, RLQ calculates two separate co-inertia analyses on the R-L and L-Q tables. RLQ selects the axes that maximise the co-variance between the site scores constrained by the environmental variables (the R-table) and the species scores constrained by the species traits (the Q-table ). A Monte-Carlo permutation (n=999) test was used to test the null hypothesis (H0) of absence of link between the environmental table (R) and the trait table (Q).

Thereafter, the fourth-corner statistic [START_REF] Dray | Testing the species traits-environment relationships: the fourth-corner problem revisited[END_REF] was used to test the significance of the direct trait-environment relationships on these 860 samples (430 field margins x 2 years). This method measures the link between species traits and environmental variables using either (1) a Pearson correlation coefficient r for two quantitative variables, (2) a Pearson Chi-square (χ²) and G-statistic for two qualitative variables or (3) pseudo-F and a Pearson correlation coefficient r for one quantitative and one qualitative variable. A permutation model was applied to test the null hypothesis (H0) that species are distributed independently of their preferences for environmental conditions and of their traits (using the permutation model 6 of [START_REF] Dray | Combining the fourth-corner and the RLQ methods for assessing trait responses to environmental variation[END_REF]. We performed 49999 permutations and used the false discovery rate method [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF] to adjust P values for multiple testing.

One limit of the RLQ analysis is that it only tests for the existence of a general relationship between environmental gradients and combinations of species traits, which does not allow identifying precisely which environmental variable acts on which trait. Conversely, the fourth-corner analysis does not account for the covariation among traits or among environmental variables. Therefore we combined the two analyses by applying the fourthcorner tests directly on the outputs of RLQ analysis. This latter approach consists in testing the associations between individual traits and environmental gradients obtained from RLQ scores, and between individual environmental variables and trait syndromes obtained from RLQ scores. The detailed procedure can be found in [START_REF] Dray | Combining the fourth-corner and the RLQ methods for assessing trait responses to environmental variation[END_REF].

The centroids and ellipses of agrotolerant and hemerophobic species were also projected as supplementary individuals on the RLQ axes to assess the response of these two broad ecological groups on the highlighted trait-environment gradients. Similarly the conventional versus organic farming systems of the fields were projected. Significant differences between the median distribution of agrotolerant and hemerophobic species on the RLQ axes on the one hand, and between conventional versus organic fields on the other hand, were tested with a Wilcoxon test.

In a second part, we modelled species richness (S), relative proportion of agrotolerant (S A ) vs. hemerophobic (S H ) species, and functional diversity (FD, see below for the definition), three variables related to plant community diversity. This was done using generalized additive mixed models (GAMMs) and the gamm4 package [START_REF] Wood | gamm4: Generalized Additive Mixed Models using[END_REF] with appropriate likelihood and link function, i.e. respectively a Poisson error structure with a logarithm link (species richness), a binomial error structure with a logit link (proportion of agrotolerant vs. hemerophobic species) and Gaussian error with identity link (functional diversity). The identity of the field margin was included as a random effect to account for pseudo-replication, a common issue in ecological modelling [START_REF] Hurlbert | Pseudoreplication and the design of ecological field experiments[END_REF]. Three different models were built for the three different response variables, each including the twelve variables used in the RLQ/fourth corner analysis (i.e. ten environmental variables + day and year of observation) as potential explanatory variables. In addition, the spatial coordinates of the field margins (latitude and longitude in meters) were used to account for spatial heterogeneity that could not be properly modelled through other explanatory variables, thanks to a smooth term modelling the interaction between latitude and longitude. The degree of freedom of this smoother modelling spatial heterogeneity was left unconstrained (contrary to other covariates, see below). Local habitat structure, number of management events and year were considered as parametric coefficients, while soil pH, soil organic matter, percentage of non-arable patches, field size, field margin width, N-Fertilization, TFI herbicides, TFI insecticides were considered as smooth terms with a limited degree of freedom (k=5) to avoid overfitting. Correlations for all pairs of variables included in the analyses were well below the threshold of 0.7 [START_REF] Burnham | Model selection and multimodel inference: a practical information-theoretic approach[END_REF][START_REF] Freckleton | Dealing with collinearity in behavioural and ecological data: model averaging and the problems of measurement error[END_REF], see Appendix C). All explanatory variables (except spatial coordinates) were scaled to facilitate the estimation of parameters and their interpretation [START_REF] Schielzeth | Simple means to improve the interpretability of regression coefficients[END_REF]. Residuals were visually inspected to detect trends that could bias estimates but all assumptions of GAMMs were met. Besides examining the relationship with detailed in-field practices, we also compared S, S A , S H and FD between conventional and organic field margins with Wilcoxon tests.

Functional diversity was computed using the package BAT [START_REF] Cardoso | BAT -Biodiversity Assessment Tools, an R package for the measurement and estimation of alpha and beta taxon, phylogenetic and functional diversity[END_REF] and the functional diversity index introduced by [START_REF] Cardoso | A new frontier in biodiversity inventory: A proposal for estimators of phylogenetic and functional diversity[END_REF], which is the total branch length of a functional tree linking all species present. We used the Hill and Smith analysis previously performed for the RLQ analysis on the Q table (traits). The first two axes were conserved and a functional tree was built on the basis of the Euclidean distance between species in the trait multivariate space and Ward's clustering algorithm. Low functional diversity characterizes communities composed of closely related species in the trait functional space, while high functional diversity is indicative of communities with species occupying distinct and distant positions in the trait multivariate space.

Results

Impact of environmental conditions and management practices on functional composition

The first two axes of the RLQ accounted for 75.1% of the total inertia (61.2 and 13.9% respectively, Figure 2). The Permutation tests indicated that the environment influences the distribution of species with fixed traits (Model 2, P<0.001 based on 999 permutations) and that the traits influence the composition of species assemblages found in samples with given environmental conditions (Model 4, P<0.001 based on 999 permutations). The first two RLQ axes accounted for most of the variance of the corresponding axes in the separate analyses of environmental descriptors (85.1% for the Hill and Smith analysis of the R-table) and species traits (76.8% for the Hill and Smith analysis of the Q-table), which demonstrates the strength of the link between environmental filters (including management practices) and plant species traits in field margins.

The first RLQ axis discriminated sites according to a double gradient, related to landscape diversity and soil resources (Figure 2, Table 2). The proportion of non-arable land, the presence of a ditch, or both a ditch and a hedge within 5 m from the field margins as well as margin width were negatively correlated with axis 1. Soil pH and clay textures were positively correlated with axis 1. The second axis discriminated sites according to an agricultural intensification gradient with positive loadings associated with high herbicides use, high nitrogen fertilization level and large field size together with a high number of management events in the field margin (Figure 2, Table 2). Soil organic matter was also positively correlated with this second axis. Regarding species traits, the first axis was best correlated with traits related to the competition/disturbance trade-off: perennial, shadetolerant species with high requirement for soil moisture (Ellenberg-H) were preferentially associated with negative loadings and opposed to annuals, drought tolerant and lightdemanding (Ellenberg-L) species. On the second axis, species with rapid resource acquisition syndrome were on positive loadings associated with short height at maturity, early flowering, high SLA and high Ellenberg indicator values for nitrogen, while species with opposed features, pollinated by insects and dispersed by animals were on negative loadings.

When applying the fourth corner analysis, among the 208 possible combinations of bivariate associations between traits and environmental variables, 42 were found significant (significance level α= 0.05, Figure 3a). When P values were adjusted for multiple testing, 13 associations remained significant (Figure 3b). Annuals were negatively associated with margin width (r=0.053, P adj = 0.030), and observation date (r=-0.158, P adj = 0.010), while perennials showed the opposite association, and were also associated with field margins with ditches (F=4807, P adj = 0.030). Plant height at maturity was negatively correlated with the number of field margin management events (r= -0.068, P adj = 0.010), positively associated with the presence of ditches (r=0.091, P adj = 0.010) and negatively associated with margins with no hedge or ditch beside (r=-0.073, P adj = 0.010). Ellenberg indicator values for light was positively associated with margin with no hedge or ditch (r=0.072, P adj = 0.030), and negatively related to soil organic matter content (r=-0.083, P adj = 0.010). Ellenberg indicator values for soil moisture was negatively associated with soil pH (r=-0.123, P adj = 0.010) and positively correlated with date of observation (r=0.100, P adj = 0.030). Finally, Ellenberg indicator values for nitrogen was positively correlated with the level of N fertilization (r= 0.062, P adj = 0.029).

Testing directly the link between RLQ axes and traits or environment (Figure 4) showed that RLQ axis 1 was negatively correlated with soil organic matter, high proportion of non-arable habitats in the landscape, field margin width, presence of ditches, or ditches and hedges and date of observation. RLQ axis 1 was positively associated with soil pH, number of management events on the margin, and absence of ditches or hedges. Species associated with these regularly managed, narrow field margins on high soil pH in arable-dominated landscapes were heliophilous and drought resistant annuals, while shade-tolerant, hygrophilous perennials were associated with large field margins on soil with elevated organic matter and presence of ditches and/or hedges in more diversified landscapes. The second RLQ axis significantly opposed sites according to soil organic matter, field size, number of field margin management events and level of N-fertilization. Communities found in field margins with little management, adjacent to small-sized fields with low N-fertilization input and low organic matter content contained oligotrophic species as well as a higher proportion of insect-pollinated and animal-dispersed plants.

The median of agrotolerant species was significantly different from the median of hemerophobic species both on RLQ axis 1 and axis 2, with higher values on both axes for agrotolerant species (Figure 5), i.e. agrotolerant species were associated with disturbed field margins (RLQ axis 1) adjacent to fields with intensive farming practices (RLQ axis 2). The position of field margins adjacent to conventional fields was not different from those adjacent to organic fields on RLQ axis 1 but they were significantly higher on RLQ axis 2 (Figure 6), which was expected because farming system (conventional vs. organic) recapitulates farming practices encapsulated in RLQ axis 2 (N fertilization and pesticides).

Impact of environmental conditions and management practices on taxonomic and functional diversity

In all three generalized additive mixed models (whose adjusted R² values were 0.34, 0.31 and 0.30 for species richness, relative proportion of agrotolerant species and functional diversity respectively), the spatial term absorbed a lot of the deviance by delimiting areas with contrasting spatial structure for the response variable (see Appendix D). In addition to these spatial effects, species richness in field margins decreased linearly with increasing field size (Chi sq.=12.83, P<0.001) and decreasing soil organic matter (Chi sq.=5.61, P=0.018) with similar magnitude (β Field size = -0.07 ± 0.02 and β Soil organic matter = 0.06 ± 0.028) and responded negatively to the frequency index of herbicides use, although with a lower slope (β TFI herbicides = -0.028 ± 0.014, Figure 7a, Appendix E). Functional diversity was related similarly to these two variables: linearly with field size (F=12.08, P<0.001) and in a slightly non-linear way with soil organic matter (F=5.91, P=0.030), with slopes of similar magnitude (see Figure 7b).

The proportion of agrotolerant relative to hemerophobic species depended on field boundary structure, with more agrotolerant species in the absence of a ditch or hedge (z=2.32, P=0.020).

The proportion of agrotolerant species also increased with decreasing field margin width (Chi sq.=5.33, P=0.021) and increasing soil pH (Chi.sq=4.769, P=0.029), with a weaker effect for field margin width than for soil pH (β Margin width = -0.065 ± 0.03 and β Soil pH = 0.1 ± 0.045, Figure 7c).

With an average of 16.53±6.38 species, organic field margins were richer than conventional field margins (14.07±6.61 species; Student t-test, t = 3.690, P < 0.001). This difference was caused mainly by hemerophobic species whose number was significantly higher in organic (9.34±5.29) than in conventional (7.49±4.89) field margins (t = 3.509, P = 0.001) with smaller but nonetheless significant differences regarding agrotolerant species (7.19±3.24 and 6.58±3.26 respectively, Student t-test, t = 1.841, P = 0.022). Functional diversity was also higher in organic (FD=125.83±27.69) compared with conventional field margins (FD=114.95±29.63, Student t-test, t = 4.774, P < 0.001).

Discussion

The aim of the study was to test the relationships between plant traits and environmental and management conditions in field margins, and to identify the main factors structuring field margin vegetation. The first gradient discriminating field margin vegetation at the nationwide level depended on field boundary "naturalness" (landscape diversity, field boundary structure, margin width and number of management events). The second gradient was based on the intensity of in-field management (amount of nitrogen fertilizers, field size). Field margin vegetation was thus structured by two independent gradients of disturbances and fertility. We showed that these gradients were associated with particular plant trait syndromes including life cycle duration, plant height, mode of pollination and dispersal as well as responses to light, soil moisture and nitrogen. Broad classification of species into agro-tolerant versus hemerophobic species, and management practices into conventional versus organic, were also distinctly distributed on the RLQ axes showing the consistency of these groupings. Species richness and functional diversity were primarily correlated with landscape heterogeneity (field size) and abiotic factors (soil organic matter), with species richness also decreasing with the frequency index of herbicides use. The relative proportion of agrotolerant versus hemerophobic species changed with soil pH and was also related to more local factors such as margin width and field boundary structure.

Disturbances gradient and naturalness of field margins

The level of disturbances incurred by field margins via direct management appeared as the main factor structuring the functional composition of the vegetation. As expected, it opposed small annual agro-tolerant species in narrow, frequently managed margins to taller perennial hemerophobic species in wider, less frequently managed margins. The accumulation of recurring disturbances such as mowing can create bare soil for colonization and transient establishment of annual species able to use resources rapidly [START_REF] Kleijn | Species richness and weed abundance in the vegetation of arable field boundaries[END_REF].

In our dataset, this gradient of disturbance was also correlated with the proportion of nonarable crop habitats in the landscape and with the diversity of the field boundary structure, especially the presence of ditches and/or hedges. The combination of these local and landscape elements creates a diversity of environmental conditions that is suitable for various hemerophobic species. Hence, the above-mentioned factors generate a gradient from moist and shady conditions (i.e. ditch verges with tree layer) with hygrophilous and shade-tolerant species typical of wetlands such as Lysimachia vulgaris or Mentha suaveolens, to open dry conditions (i.e. open road verges) with heliophilous and drought tolerant species (Erodium cicutarium, Echium plantagineum). Our study confirms the importance of this gradient in a different context (Western Europe) and at a wider spatial scale, similarly to the gradient observed in a region of Estonia [START_REF] Aavik | What is the role of local landscape structure in the vegetation composition of field boundaries?[END_REF][START_REF] Aavik | Quantifying the effect of organic farming, field boundary type and landscape structure on the vegetation of field boundaries[END_REF]. This double gradient of boundary management intensity and naturalness was not related to a gradient of species richness but it did influence the richness of agrotolerant and hemerophobic species. General theory concerning diversity-area relations predicts that larger margins will support richer communities. Our standardized protocol measuring species richness on 10m² equidistant from the field and the adjacent habitat excludes this type of effect (that may however still exist on the larger landscape level). Instead, the width of the field margin determines higher buffering capacities relative to within field practices [START_REF] De Cauwer | Effect of margin strips on soil mineral nitrogen and plant biodiversity[END_REF] and higher environmental quality in the interior of the margin [START_REF] Aavik | Quantifying the effect of organic farming, field boundary type and landscape structure on the vegetation of field boundaries[END_REF].

Thus it is not surprising that species richness on the whole was unaffected by field margin width, while as expected, more agrotolerant species were observed on narrow margins which reflects the higher influence of in-field management practices in this type of margins.

Conventional and organic field margins were not different on this gradient, which suggests that some field margins of organic farms can be managed intensively (e.g., field margins next to roads, managed by regional or local authorities), while conversely some conventional farmers maintain margins with diversified local habitat structure.

Resource gradient associated with agricultural intensification and non-intended effects of in-field practices

The second structuring factor was related to in-field practices (fertilization, herbicides), field size and again the frequency of field margin management. It opposes smallsized nitrophilous species, with early flowering and rapid acquisition resource capacity (SLA) in margins of large, intensively managed crop fields, to taller, later flowering and less nitrogen-demanding species with slower resource acquisition in smaller, less intensively managed crop fields. Specific bivariate association between practices and species traits showed that this gradient was mainly driven by the dose of in-field nitrogen fertilizers and the proportion of nitrophilous species. Contrary to our expectations, we did not find any particular traits associated with more intensive uses of herbicides within the field. However, there was a significant negative effect of the frequency index of herbicide use within the field on species richness in the field margin. This result is consistent with several fine scale experiments that showed a delayed flowering onset and a reduced flower number for plants exposed to herbicide drift at ~1% of the field application rate [START_REF] Bohnenblust | Effects of the herbicide dicamba on nontarget plants and pollinator visitation[END_REF] or a reduction of species richness in plant communities exposed to increasing doses of glyphosate applied at drift levels, from 0 to 25% of the counselled rate (Pelissier et al., 2014). Decreased species richness combined with an absence of effects on functional diversity or composition suggests that the loss of species due to the intensity of herbicide use is not determined by the ten traits used in our classification. Other traits such as plant morphology or leaf surface traits, including cuticle characteristics, hairiness (e.g. density of trichomes), density of stomata or cell size, may be more relevant for sensitivity to herbicides, as they are related to the wetting and the penetration of foliar applied herbicides and therefore their bioavailability within the plant [START_REF] Gaba | Response and effect traits of arable weeds in agro-ecosystems: a review of current knowledge[END_REF]. Alternatively, opposite responses of plant traits, such as height or SLA, to herbicides vs. fertilizers [START_REF] Pellissier | Herbicide and fertilizers promote analogous phylogenetic responses but opposite functional responses in plant communities[END_REF]) may have hampered our ability to detect trait-based changes in community composition.

The effect of nitrogen fertilizer input to the crop on the richness or composition of the vegetation of field margin has been reported previously (e.g., [START_REF] Kleijn | Factors affecting the species composition of arable field boundary vegetation[END_REF]Pélissier et al., 2014) and this drift was also confirmed in our study. In addition to earlier studies, the fourth corner test in our study demonstrates the direct link between the abundance of high-nitrogen demanding species (species with high Ellenberg-N values) in the field margin and the amount of nitrogen fertilizer applied within the field. This suggests a strong shift in species composition following higher resource supply. However, contrary to other studies, the high levels of fertilization were not associated with a decrease in species richness.

The expected negative relationship between fertility and species richness [START_REF] Tilman | Species richness of experimental productivity gradients: how important is colonization limitation?[END_REF] is attributable both to increased living biomass and litter accumulation that reduce light to very low levels and thus inhibit germination and/or survival of seedlings, and decrease rates of establishment of new species [START_REF] Foster | Species richness in a successional grassland: effects of nitrogen enrichment and plant litter[END_REF]. The same processes are at play in arable field margins where higher nutrient application levels on the field generally increases field margin biomass [START_REF] Kleijn | The use of nutrient resources form arable fields by plants in field boundaries[END_REF]. However, disturbances created by field margin direct management can maintain gaps with less competitive annual species. Overall, the larger area covered by our study (the whole of metropolitan France) compared to previous field boundary studies, showed that species richness depends primarily on the natural soil trophic gradient, which opposes species rich communities on organic-rich acidic soils, versus poorer communities on organic-poor basic clay soils [START_REF] Manhoudt | Management regimes and farming practices enhancing plant species richness on ditch banks[END_REF].

Species depending on animals for their reproduction (obligate entomogamous species) or for their dispersal (zoochorous species) were associated with smaller less intensively managed fields in landscapes with a high proportion of non-arable habitats. Several processes may be at play, which are difficult to disentangle. Small fields mean higher configurational heterogeneity of landscapes [START_REF] Fahrig | Functional landscape heterogeneity and animal biodiversity in agricultural landscapes[END_REF] with more numerous linear elements that can act as corridors needed for the stability of some animal populations [START_REF] Marshall | Field margins in northern Europe: their functions and interactions with agriculture[END_REF][START_REF] Molina | Epigeal arthropod communities in intensively farmed landscapes: effects of land use mosaics, neighbourhood heterogeneity, and field position[END_REF]. The higher proportion of non-arable habitats has also been shown to favor animal diversity in agricultural landscapes [START_REF] Weibull | Diversity of butterflies in the agricultural landscape: the role of farming system and landscape heterogeneity[END_REF] through dispersal from these habitats to field margins. More generally, the negative link of plants depending on animals with RLQ axis 2 suggests that the absence of these species could be a consequence of agricultural intensification. According to the fourth corner analysis none of the within field practices of the contemporary years (dose of nitrogen fertilizer, intensity of herbicides or insecticides use) were related to the presence of this group of species. However, the strong link of dispersal and pollination modes with RLQ axis 2 (related to agricultural intensification) might suggest that the presence/absence of entomogamous and animaldispersed species is rather a result of the interactions of cumulative management practices over several years and their interactions with large scale factors (field size, landscape, [START_REF] Roschewitz | The effects of landscape complexity on arable weed species diversity in organic and conventional farming[END_REF].

Relative proportion of agrotolerant and hemerophobic species could not be directly related to one of these agricultural intensification factors. However, the distribution of agrotolerant species was clearly more associated with positive values of RLQ axis 2 which means a response to a combination of high input of agrochemicals (fertilizer, herbicides) and large field size. Our general result on this agricultural intensification axis is consistent with previous studies showing that among agrochemicals, the effect of fertilizers on plant community composition in field margins is always stronger than the effect of herbicides [START_REF] Kleijn | Field boundary vegetation and the effects of agrochemical drift: botanical change caused by low levels of herbicide and fertilizer[END_REF].

Implications for management of biodiversity in field margins

It is recognized that species richness is not necessarily the most suitable indicator of healthy field margins because some species are known to respond positively to disturbances and/or excess of nitrogen fertilizers. Following that idea, [START_REF] Aavik | Agrotolerant and high nature-value species-Plant biodiversity indicator groups in agroecosystems[END_REF] introduced a classification in two groups, agrotolerant species and hemerophobic species, on the basis of their capacity to persist in arable fields with modern cultivation practices (i.e., present in more than 10% of the agricultural fields of the studied region). With our trait-based approach that links directly environmental conditions and plant traits, we can confirm the utility of this broad classification and characterize the functional profile of species associated with undisturbed field margins adjacent to low input fields. Our RLQ axes reflect two independent dimensions of field margin disturbances (axis 1) and agricultural intensification (axis 2). The position of agrotolerant and hemerophobic species on these two axes is consistent with what was expected. Therefore one can identify factors promoting hemerophobic species. On the whole, their presence increases with field margin width, presence of semi-natural elements such as ditches and hedges, high proportion of non-arable land, small field size, low number of management events and low amounts of nitrogen fertilizer input. Associated traits include species with a perennial life cycle, a high stature at maturity, low nitrogen requirements, pollination by insects and dispersion by animals. As found in previous studies (e.g. [START_REF] Bengtsson | The effects of organic agriculture on biodiversity and abundance: a meta-analysis[END_REF] for a review), organic field margins have higher species richness than conventional field margins. Interestingly, the higher species richness in organic field margins is due, at 75%, to a higher number of hemerophobic species. Differences on the RLQ trait-environment multivariate space showed that organic field margins differed from conventional field margins on the agricultural intensification axis with fewer nitrogen-demanding species, more insectpollinated species and more animal-dispersed species.

Conclusions

With the aim of detecting non-intended effects of agrochemicals on non-targeted plants, our study highlighted that the composition and the diversity of vegetation in arable field margins were primarily driven by the direct field margin management and by landscape factors.

However, among farming practices, distinct non-intended effects of fertilization and herbicides were highlighted. The level of nitrogen fertilizers had the strongest effects on the functional composition of field margin vegetation with a change toward more nutrientdemanding species, while the intensity of herbicides use was related to a slight decrease in species richness with no effects on functional composition or diversity. A better understanding of the effect of herbicide drift on non-target plant communities will require a finer characterization of herbicides modes of action, as well as data on species traits related to herbicides sensitivity, which will be the aim of further analyses.

Mode of pollination

Biotic interactions, Landscape Plants dependent on animal for reproduction and dispersion may be less frequent in simplified landscapes (openfield), pollinated plants may also decrease with insecticide use.

Mode of dispersal

Tables Click here to download Tables: Table_F.docx 1 and2; codes for species in Appendix B. The same associations when P values were adjusted for multiple comparisons using the false discovery rate procedure. Codes for traits and variables are given in Tables 1 and2.

Figure captions Figure 1 .

 captions1 Figure captionsFigure1. Distribution map of the 430 field margins surveyed in France. Colors correspond to the different production systems: wheat (black), maize (red), market gardening centred on lettuce production (green) and vineyards (blue). The black lines represent the limit of departments, a French administrative unit dividing metropolitan France into 95 units.

Figure 2 .

 2 Figure 2. Results of the first two axes of the RLQ analysis: (a) scores of species, (b) coefficients for environmental variables, and (c) traits. The values of d give the grid size. Codes for variables and traits are given in Tables1 and 2; codes for species in Appendix B.

Figure 3 .

 3 Figure 3. Results of the fourth-corner tests. a) Significant (P<0.05) positive associations are represented by red cells, and significant negative associations by blue cells. Nonsignificant associations are in grey. Black lines separate different variables; white lines separate different modalities for categorical variables. b) The same associations when P values were adjusted for multiple comparisons using the false discovery rate procedure. Codes for traits and variables are given in Tables1 and 2.

Figure 4 .

 4 Figure 4. a) Fourth-corner tests between the first two RLQ axes for environmental gradients (AxcR1/AxcR2) and traits. (b) Fourth-corner tests between the first two RLQ axes for trait syndromes (AxQ1 and AxQ2) and environmental variables. Positive significant associations are represented by red cells, and negative significant associations by blue cells.

Figure 5 .

 5 Figure 5. a) Projection of agrotolerant versus hemerophobic species as supplementary variables in the RLQ analysis. The value of d gives the grid size. Comparison of the distribution of agrotolerant versus hemerophobic species on b) RLQ axis 1 and c) RLQ axis 2. The values of Wilcoxon test statistic W and the associated P-values are provided on the graphs.

Figure 6 .

 6 Figure 6. a) Projection of organic versus conventional field margin strips as supplementary variables in the RLQ analysis. The value of d gives the grid size. Comparison of the distribution of organic (org.) versus conventional (conv.) field margin strips on b) RLQ axis 1 and c) RLQ axis 2 The values of Wilcoxon test statistic W and the associated P-values are provided on the graphs.

Figure 7 .

 7 Figure 7. Scaled response of a) species richness, b) functional diversity and c) proportion of agrotolerant species to significant continuous variables identified in Generalized Additive Mixed Models (see Appendix E). TFI herbicides = Treatment frequency index of herbicides.

Figure

  Figure Click here to download Figure: Figures_F.docx

  Figure 3.

  Figure 4.

  Figure 5.

  Figure 6.

Figure 7 .

 7 Figure 7.

Table 1 .

 1 List of selected traits with their abbreviations, units, and the management practices likely to affect these traits based on ecological mechanisms: Dist. = Disturbance, Res. = Resources.

	Traits	Ecological mechanisms	Expected responses in field margin strips
	Life form	Dist.	Field margin strip management frequency favours annuals; Field
	(annual/perennial)		margin strip width favour perennials; herbicide drift favours
			agrotolerant (annual) species.
	Plant height at	Dist. (+Res.)	Field margin strip management frequency favours short species
	maturity (cm)		Plant height increases with resources (fertile soils, N Fertilisation)
	Seed mass (mg)	Dist., Res.	Seed size/number trade-off : disturbed field margins favour species
			producing numerous small seeds (ruderal strategy) while stable field
			margins favour species producing fewer seeds each with higher seed
			mass (competitive strategy)
	Specific Leaf Area,	Res.(+Dist.)	N fertilization, Soil fertility (Organic matter), and Field margin
	SLA (mm 2 .mg -1 )		strip management frequency favour species with high resource
			acquisition capacity (high SLA)
	Flowering onset	Dist. (+Res.)	Field margin strip management frequency favours early flowering
	(month)		species
	Ellenberg-N (EIV-N)	Res. (soil N)	High N fertilization, fertile soils favour nitrogen-demanding species
			(high EIV-N) or tall competitive species.
	Ellenberg-H (EIV-H)	Res. (water availability)	Presence of ditches favours hygrophilous species (high EIV-H)
	Ellenberg-L (EIV-L)	Res. (Light availability)	Presence of hedges favours shade-tolerant species (low EIV-L)

Table 2 .

 2 List of management practices and selected traits with their abbreviations, units, basic statistics and their coordinates on the first two RLQ axes.

	Environmental Variables	Mean (Min-Max) or counts	RLQ axis 1 RLQ axis 2
	Soil
	Soil pH and texture gradient (soil pH)	6.89 (5.39-7.83)	0.501	0.248
	Soil organic matter (soil OM) (ppm)	19.08 (5.71-53.29) -0.322	0.486
	Landscape
	Non-arable land [%]	16.68 (0-90.16)	-0.253	-0.218
	Field size [ha]	7.47 (0.005-40)	-0.112	0.391
	Boundary type
	Ditch	n = 46	-0.812	-1.169
	Ditch and hedge	n = 12	-1.012	0.738
	Hedge	n = 49	-0.036	0.250
	Other boundary type	n = 323	0.170	0.118
	Field Margin
	Margin width [m]	3.13 (1-10)	-0.141	0.029
	Number of management events (N. Mgt events)	1.18 (1-3)	0.171	0.311
	In-field farming practices
	Nitrogen fertilizer input in field (N Fertilizer) [kg.ha -1 .year -1 ]	119.17 (0-500)	-0.083	0.397
	TFI Herbicides (TFI Herbi)	0.94 (0-3)	-0.037	0.156
	TFI Insecticides (TFI Insec)	0.21 (0-3.90)	0.078	0.059
	Temporal variables
	Date (number of days since January 1st)	175 (116-276)	-0.607	0.165
	Year 2013	n = 430	0.101	0.024
	Year 2014	n = 430	-0.102	-0.024
	Traits
	Lifeform
	Annual	n=84	0.915	-0.281
	Perennial	n=102	-0.388	0.119
	Plant height [cm]	122.18 (20-3000) -0.170	-0.493
	Seed mass [mg]	5.09 (0.02-193.6) 0.225	-0.313
	Specific Leaf Area (SLA) [mm 2 .mg -1 ]	26.19 (6.50-53.68) 0.096	0.194
	Flowering onset (Flow. On.)	5.11 (1-9)	-0.138	-0.375
	Pollination mode (Polli.)
	Entomogamous	n=60	-0.247	-0.646
	Other (wind, water, autogamous)	n=126	0.064	0.170
	Mode of dispersal (Disp.)
	By animals	n=77	-0.242	-0.689
	By other means	n=109	0.143	0.407
	Ellenberg indicator values for Light (EIV-L)	7.20 (4-9)	0.422	-0.057
	Ellenberg indicator values for soil moisture (EIV-H)	4.78 (1-8)	-0.554	0.154
	Ellenberg indicator values for nitrogen (EIV-N)	6.24 (1-9)	-0.026	0.172

Acknowledgements:

The 500 ENI network is developed by the French Ministry of Agriculture under the Ecophyto framework with funding from the Agence Française de la Biodiversité. We thank all the people that have collected the data in the field, the farmers who provided information on their practices as well as all the people involved in the coordination of the network: Camila Andrade (MNHN), Jérôme Jullien (DGAL-Ministry of Agriculture), Nicolas Lenne (DGAL-Ministry of Agriculture), Pascal Monestiez (INRA) and the GT Stats 500 ENI. We also thank Marie Carles and Gérard Balent (INRA) for providing the landscape composition metrics and Jon Marshall and one anonymous reviewer for their constructive comments that helped us improving the manuscript.


The power to create chaos

Konrad Hinsen

Occasionally I receive feedback from readers on the articles I write for the Scientific Programming department. My recent article on technical debt [START_REF] Hinsen | Technical Debt in Computational Science[END_REF] provoked more feedback than anything else I have written here before, so I suspect the topic resonated with many readers. Most the e-mails I received were about personal experiences people had with technical debt that was recognized too late to be handled gracefully. But a few concentrated on a very specific point: my claim that computers exhibit chaotic behavior, and that this is a problem when using them in research. I will take those reactions as a pretext to elaborate a bit on that point, which is in my opinion not yet sufficiently appreciated.

I call this a pretext because I won't actually address the objections that were raised in much detail. They focused on the problems of defining chaos mathematically for discrete rather than continuous systems. The standard definitions of chaos refer to infinitesimal changes in the initial conditions, which for computers and other discrete state systems make no sense, because the smallest possible change is one bit. Readers interested in this topic can find detailed discussions in the literature on cellular automata, for which Stephen Wolfram's "A New Kind of Science" [START_REF] Wolfram | A new kind of science[END_REF] is a good entry point, in particular chapters 4 and 7. However, in the context of scientific programming, which is the topic of this department, the precise mathematical definition of chaotic behavior is much less relevant than its consequences on software development and testing.

Chaos is a mathematical concept from the theory of dynamical systems. A dynamical system is defined by a state space and a time evolution rule for the state. The system starts in some initial state, and then the time evolution rule is applied to yield subsequent states. Both the state space and the time variable can be continuous or discrete. A computer is a dynamical system with a discrete state, which consists of the computer's memory and its processor's internal state. Time is discrete as well, the elementary time step being the execution of one processor instruction. The processor's instruction set provides the details of the time evolution rule. The initial conditions of a computation are the memory contents plus processor state when the computation is started. Execution proceeds until the program reaches its end -if it ever does. The final memory contents contain the result of the computation. Note that "memory" should be understood in a wide enough sense to include all data storage available, including hard disks, network storage, etc. Note also that what I consider here is computation in the narrow sense of mechanically processing information. When you add multiple processes, communication between them, or external events, everything becomes more complicated.

The defining aspect of chaos is a strong sensitivity of a dynamical system's behavior on initial conditions: small changes in these conditions can cause large changes in the system's future behavior for which no useful bounds can be established. Chaotic behavior in nature makes the long-term evolution of many phenomena unpredictable even though they are deterministic. An often cited example is the weather, which can be predicted for only a very short time -about a week -not because of any inherently random processes, nor because a lack of computational power, but because the initial conditions that enter into the prediction can be measured only to some finite precision.

Computers are engineered dynamical systems for which such problems do not exist. We know the initial state of a computation precisely, and we can even store it for future re-use. But changes in the initial state are of interest as a way to explore the consequences of errors. Erratic behavior of the computing hardware itself is rare enough that it can safely be ignored, except for the extremely large parallel computers. But human errors in the preparation of the initial state -the program and the input data -are an important cause of wrong results. That's why it makes sense to ask the question how the behavior of a computation changes if the correct initial state is modified in some way.

Saying that a computer behaves chaotically means that the result of a computation depends strongly on the initial state, to the point that a small change in this initial state can change the result beyond any useful predictable bound. As I already mentioned, some of the criteria of traditional chaos theory do not apply: changes cannot be made infinitely small, as the smallest possible change is a one-bit flip, and deviations cannot become infinite, because the computation's state consists of a finite number of bits. The latter restriction applies to every physical system, of course, but the finiteness of the our planet's atmosphere has not prevented scientists from applying chaos theory to weather forecasting.

The mechanism that causes chaotic behavior in computation is the amplification of small changes by subsequent steps. The impact of a one-bit flip can be small, for example if the bit happens to represent the least significant digit of an input number. At the other extreme, a one-bit flip in a processor instruction can crash the program, leading to no useful result at all. In between these two extremes, a one-bit flip can lead to results that differ from the correct result in arbitrary ways. The worst case is not a program crash, nor a huge difference in the final result, but a wrong result that looks credible. Such a mistake has a good chance of going unnoticed.

I found a nice simple illustration for error amplification in computation in a lecture by Gérard Berry [START_REF] Berry | Pourquoi et comment le monde devient numérique[END_REF]. Suppose you live in a city with a grid-like street layout (see Figure 1). You want to explain to a friend at the other end of the city how to reach you by car. You provide a list of driving directions that tell your friend at every corner what to do: turn left (L), turn right (R), or continue straight on (S). Your driving instructions are thus a string made of the letters L, R, and S, which is not very different from a program written as a list of processor instructions. By following these instructions, your friend will move along the blue path from the bottom of the grid to the top.

But now suppose that some mistake happens in transmitting the driving directions, or that your friend takes a wrong turn while driving. The smallest possible mistake would be the replacement of a single letter. Two such minimally modified instruction sequences are shown in Fig 1, in red and violet, together with the resulting paths. It is clear that the places one reaches by following these modified instructions are not at all close to the real destination, nor close to each other. In fact, minimal mistakes can take you anywhere on the grid.

In this simple example, the mechanism of error amplification is easy to understand. Each minimal one-letter change rotates the remaining part of the path by 90 degrees compared to the correct one. The more steps remain to be done after the mutated instruction, the farther the arrival point will be from the intended one. That is also the general mechanism that makes the outcome of a computation in the presence of mistakes so hard to predict.

An error that grows linearly with the number of computational steps isn't really that bad. It doesn't yet deserve the label "chaotic behavior". After all, our "driving directions" programming language provides several useful guarantees.

No matter what mistake you make, a "program" consisting of N instructions is guaranteed to terminate after exactly N steps and yield a valid position on the grid (assuming the grid is big enough), which moreover is at most 2N grid steps away from the intended destination. There is no possibility of non-termination (driving around the grid endlessly) or a crash (hitting an invalid position). We don't get such guarantees for the programs we use in computational science. The reason why we do get them for our driving directions is that the language is very limited. It can be processed by what is known as a "finite state machine" in automata theory, whereas standard programming languages require a more powerful automaton -in fact, the most powerful type of automaton that is known today -called a "Turing machine". As in other aspects of life, more power also means more responsibility, as mistakes can have more serious consequences. Turing machines give you so much power that you can easily create chaos (see e.g. [START_REF] Mondal | Universal Computation is 'Almost Surely' Chaotic[END_REF]). If we added features such as loops or tests based on the current position to our driving directions language, we would need a Turing machine for processing it. That would give us the power to condemn our friends, intentionally or by mistake, to spend the rest of their lives driving around the city.

In writing scientific software using Turing-complete languages, we have exactly that power and we should be constantly watching out for unexpected consequences of mistakes. However, that is not how computational scientists behave in practice. The dominant attitude is to trust computational results if they "look right", i.e. if they are not in disagreement with expectations and prior knowl-edge of the problem under study. I have even heard people say that they "trust their intuition" to spot potential mistakes in the results. That is not in itself an absurd idea. Experienced experimentalists do recognize suspect results coming from instruments they know well. It is certainly possible to develop an intuition for the credibility of data.

The crucial difference is that experimental equipment is carefully designed not to exhibit chaotic behavior, such that minor damage or production defects do not lead to unexpected results. If you put a mite under a microscope, a defect in the instrument may lead to a blurred image, but not to an image showing an extra pair of legs. If the image is sharp enough to let you count the legs of the mite, then you know you can trust that observation. But this approach doesn't carry over to software. There is no typical symptom of a programming mistake -anything is possible. Even the most experienced software developers can't judge if a program is correct by inspecting the source code, or by running it on a few sample inputs, except if the program is trivially small. Moreover, in many applications of computational science, correctness of a program isn't even something one could aim for. We can only decide if a program is correct if we have some other means of specifying what the correct result is. This is often not the case in a research setting, where much code is written for computing quantities that nobody has ever computed before.

A common attitude that we should probably start to question in this context is the one of considering computation as mechanized mathematics. The focus of mathematics is on precise statements that can be proven right or wrong. In research, we often use computation for exploring imprecise statements -scientific hypotheses -whose domains of validity are not known yet. A useful complementary notion to mathematical correctness is the robustness of the computational tools we use. This is well explained in an essay by Gerald Sussman that draws on his background in electrical engineering [START_REF] Jay | Building Robust Systems[END_REF]. A robust computer program produces reasonable output for reasonable input, even if the latter is different from what the program was initially designed to handle. We have similar concepts in computational science, for example the notion of numerical stability of algorithms. But robustness criteria do not play a major role in the design and implementation of scientific software today.

In order to link these abstract considerations of potential chaotic behavior to the practice of computational science, I tried to apply them to my own work in molecular simulations. Most of my work is about extracting information from simulation trajectories, which are datasets about 1 GB in size -too big for inspection by eye, but small enough to be processed on my laptop. I spend much of my day writing, modifying, and running Python scripts that perform various geometrical and statistical analyses on these trajectories. These scripts are rather short, but rely on a collection of libraries, ranging from general and widely used ones such as NumPy (http://www.numpy.org/) or h5py (http://www.h5py.org/)) to domain-specific ones such as MOSAIC (http://github.com/mosaic-data-model/mosaicpython).

The computation that is performed when I run one of my scripts is defined by the script, the file containing the trajectory, the Python language, and all the libraries used by the script. But that's just the immediately visible part. The libraries I have cited depend on other libraries, and the Python interpreter is written in C. A different way to present this complex assembly of software is as a set of consecutive layers that transform a general-purpose computer into a tool for performing a very specific analysis of a simulation trajectory. Each of these layers can be described in terms of (1) the notation in which the additional information being added is expressed, and (2) the tool that processes this information:

1. The processor instruction set, executed by the computer.

2. The C language, translated into processor instructions by a C compiler.

3. The Python language, executed by an interpreter written in C. [START_REF] Mondal | Universal Computation is 'Almost Surely' Chaotic[END_REF]. The Python language augmented by the NumPy library, written in C and plain Python.

5. ... more libraries ...

6.

The file format for the data files, interpreted by an analysis script written in Python augmented by various libraries.

All the layers listed above define both a part of the computation and the notation in which some other part of the computation is expressed. For example, the Python language defines the data representation and memory management aspects of the computation, in addition to defining what is and is not a legal Python program. My analysis script defines all the algorithms, but also the input file format for the trajectories.

For historical and practical reasons, we use different labels to refer to these notations: the first three are called "programming languages", whereas the last one is a "file format". The intermediate ones that simply add libraries are rarely recognized as distinct notations at all. To see that they really are, consider the small Python script import numpy print(numpy.arange(5)) This is not a valid program in the Python language, but it is a valid program in the Python-plus-NumPy language, which shows that these two languages are distinct. Libraries should thus be treated as language extensions.

All the layers but the last one are general-purpose Turing-complete programming languages. There are of course significant differences between them, which is why these different layers exist. The processor instruction level is not very convenient for human programmers, being difficult to read and understand. Moreover, processor instructions are a high-risk notation: any sequence of bytes can be executed as instructions, but most byte sequences will not produce anything useful and may even cause damage to data stored in the computer. The C language is much more convenient, and also provides a better level of verification, because many possible mistakes are caught by the compiler. The Python language is even more convenient, and helps to avoid mistakes by expressing a computation in much fewer lines of code. Similarly, each library layer adds more convenience for computations made up of the kind of operations that these libraries implement, and at the same time prevents mistakes by allowing the programmer to write less code for a specific computation.

However, none of these layers adds useful guarantees about the behavior of the computation. The potential of generating chaos remains present until the very last layer, which is defined by my trajectory analysis script. From the point of view of robustness, it would be preferable to have as much of the computation as possible expressed using notations that limit the impact of mistakes. Instead of libraries that add optional shorthand notation for some operations to a generalpurpose language, we should have successive layers of languages that enforce the use of a more specialized and less dangerous notation.

This idea is sometimes advocated as the "principle of least power" [START_REF] Berners-Lee | Principles of Design[END_REF]: every aspect of a computation should be described in a language with just as much expressive power as strictly required for the task. The additional guarantees that a limited language can make also offer more opportunities for analyzing and optimizing a program. Many Domain-Specific Languages (DSLs) are based on this idea. I have written about DSLs before in this department [START_REF] Hinsen | A glimpse of the future of scientific programming[END_REF]. In contrast to a library, a DSL is both more and less than a general-purpose language. The "less" part often includes giving up Turing-completeness. A data file format is then nothing but an extreme case of a DSL, in which only constant data is allowed and no algorithm can be expressed at all.

The main obstacle to such an approach is probably psychological. We associate the term "language" in the context of programming with something complex that takes many years to master. Most computational scientists would be happy to get away with learning only one programming language in their life. But as I explained above, a language can be a small variation on another one. Just as today's libraries are extensions to general-purpose languages, we could have library-like pieces of code that remove or restrict features of languages. As an example, removing while-loops and recursive function calls would turn Python into a language in which every program is guaranteed to terminate. A quick inspection of the trajectory analysis scripts that I wrote over the last few months showed that they could all be written in such a restricted Python dialect. In fact, they already are. I rarely need all the power that Python gives me. But I cannot ask Python to verify that I didn't use that power by mistake.

These observations apply with minor variations in languages, libraries, and file formats to most computations done today in science and engineering. In the short history of computing, we can observe a general tendency towards providing more expressive power wherever some aspect of a computation is defined. In the early days, we had Fortran programs reading simply structured input files. But as soon as computers could handle larger programs, embedded scripting languagesusually Turing-complete -became a desirable feature for customizing application software. From there it was a small step to writing application software in a scripting language augmented by domain-specific libraries, as illustrated by my above example. There are even cases of languages that became accidentally Turing-complete as features were added, the best-known example being C++ templates. It is in fact possible to do arbitrary computations, including chaotic ones, as part of the compilation of a C++ program. The downsides of too much freedom in program structure have been known for a while, and have lead to the widespread adoption of structured programming in the 1980s and to the growing popularity of functional programming in recent years [START_REF] Hinsen | The promises of functional programming[END_REF]. But advocates of these approaches to safer programming were always keen to point out that no expressive power is lost in adopting them. Perhaps it is time to re-evaluate the importance of computational omnipotence. Konrad Hinsen is a researcher at the Centre de Biophysique Moléculaire in Orléans (France) and at the Synchrotron Soleil in Saint Aubin (France). His research interests include protein structure and dynamics and scientific computing. He has a PhD in theoretical physics from RWTH Aachen University (Germany). Contact him at konrad.hinsen@cnrs.fr.

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Figure 1 :

 1 Figure 1: The consequences of minimal mistakes in driving directions.