Guillaume Péron

Reproductive skews of territorial species in heterogeneous landscapes

Keywords: ideal despotic distribution, heavy-tailed distribution, Pareto, peer of a peer, socio-spatial structure

modified from published versions…): I investigated the reproductive skew resulting from the way the variation in social status and resource availability interact to shape the spatial distribution of territories and the resulting inequality in reproductive output in territorial species. Lifetime reproductive success data from 46 species indicated a common pattern in both social and solitary species, namely the emergence of a few high performers amidst a majority of mediocre performers. This first result indicated that reproductive skews are not a unique properties of social species, even if most models of reproductive skews are currently based on within-group social processes. I thus devised a I also used an alternative theoretical model that worked for solitary species, in which the socio-spatial distribution emerged from interactions between immediate neighbors only. Using the model, I predicted a correlation between three statistics of the reproductive skew (the skewness, the Gini coefficient of inequalities, and Hill's index of heavy-tailed distribution) and the degree of spatial autocorrelation in territory quality. I retrieved these three correlations when comparing the reproductive skews of the 46 species. The distribution of lifetime reproductive success was more skewed but also more egalitarian in homogeneous than heterogeneous landscapes, after controlling for the allometric effect of the body mass. In the proposed model, the increase in egalitarian properties of the LRS distribution in heterogeneous landscapes comes from the fact that some individuals "hide" behind their neighbors and secure a better territory than if the distribution was despotic. This work emphasizes that the shape of the distribution of fitness components conveys information besides the mean and the variance, and the potential for nontrivial effects to emerge from simple movement heuristics.

Introduction

Populations of territorial species [START_REF] Maher | Definitions of territoriality used in the study of variation in vertebrate spacing systems[END_REF] often exhibit a match between measures of individual quality, e.g., body mass, and measures of territory quality, e.g., resource density [START_REF] Ens | Interference Among Oystercatchers, Haematopus ostralegus, Feeding on Mussels, Mytilus edulis, on the Exe Estuary[END_REF][START_REF] Komdeur | Importance of habitat saturation and territory quality for evolution of cooperative breeding in the Seychelles warbler[END_REF][START_REF] Nystrom | Food density, song rate, and body condition in territory-establishing willow warblers (Phylloscopus trochilus). -Can[END_REF][START_REF] Pryke | Agonistic carotenoid signalling in male red-collared widowbirds: Aggression related to the colour signal of both the territory owner and model intruder[END_REF][START_REF] Candolin | Correlation between male size and territory quality: consequence of male competition or predation susceptibility?[END_REF][START_REF] Balbontín | Density-dependence by habitat heterogeneity: individual quality versus territory quality[END_REF][START_REF] Sergio | Short-and long-term consequences of individual and territory quality in a long-lived bird[END_REF]. Such patterns if they are perfect are termed ideal despotic distributions [START_REF] Fretwell | Populations in a Seasonal Environment[END_REF]. Under an ideal despotic distribution, the best individuals cumulate their intrinsic advantage and the benefits of a good territory, potentially over multiple breeding attempts. This can yield major heterogeneities in lifetime reproductive success, i.e., a few high performers and many mediocre performers (Hochachka 2006, Balbontín and[START_REF] Balbontín | Density-dependence by habitat heterogeneity: individual quality versus territory quality[END_REF], which is often referred to as reproductive skews in the behavioral ecology literature [START_REF] Johnstone | Models of reproductive skew: A review and synthesis[END_REF]. However, over iteroparous lifetimes and in the context of life-history tradeoffs, the influence of stochasticity, i.e., luck, oftentimes trumps the effect of fixed individual and environmental variation on reproductive skews [START_REF] Tuljapurkar | Skewed distributions of lifetime reproductive success: beyond mean and variance[END_REF][START_REF] Broekman | Relative contributions of fixed and dynamic heterogeneity to variation in lifetime reproductive success in kestrels (Falco tinnunculus)[END_REF]). In addition, the ideal despotic model is well-known for lacking realism.

Indeed, the model ignores that information and access to territories are constrained by distance, which hinders the ability of individuals to perfectly match their intrinsic quality and the quality of their territory [START_REF] Ens | Interference Among Oystercatchers, Haematopus ostralegus, Feeding on Mussels, Mytilus edulis, on the Exe Estuary[END_REF]Goss-Custard 1984, Hemelrijk 1999).

Territoriality and the emerging socio-spatial structure are not the only drivers of reproductive skews. Indeed, group-living species provide many examples where the main skew-generating mechanism corresponds to a within-group process rather than to cross-territory variation [START_REF] Johnstone | Models of reproductive skew: A review and synthesis[END_REF]. Within groups, dominant individuals may monopolize resources and mating opportunities [START_REF] Murray | Dominance rank influences female space use in wild chimpanzees, Pan troglodytes: towards an ideal despotic distribution[END_REF][START_REF] Spong | Factors affecting the reproductive success of dominant male meerkats[END_REF][START_REF] Saltzman | Reproductive skew in female common marmosets: What can proximate mechanisms tell us about ultimate causes?[END_REF][START_REF] Josi | Age-and sex-dependent variation in relatedness corresponds to reproductive skew, territory inheritance, and workload in cooperatively breeding cichlids[END_REF]. This observation further challenges the ideal despotic model because in most group-living species, dominants cannot be despotic, they have to make concessions to promote the continued attachment of group mates [START_REF] Johnstone | Models of reproductive skew: A review and synthesis[END_REF][START_REF] Montana | Large eastern grey kangaroo males are dominant but do not monopolize matings[END_REF]. Without concessions, subordinates leave the group (Datta 1988, Buston and[START_REF] Buston | Reproductive skew and the evolution of conflict resolution: A synthesis of transactional and tug-of-war models[END_REF]. Despite these concessions, the reproductive skews of group-living species are very similar to the reproductive skews of solitary, territorial species (Fig. 1). This suggests that a modulating mechanism also occurs in territorial species, i.e., the aforementioned spatial constraints.

Fig. 1:

Comparison of the reproductive skew in solitary, territorial species (grey) and in group-living species (white). The three skew metrics (skewness, Gini coefficient, and Hill's tail index) are explained in the method section. The data sources are presented in Appendix S1 and the method section.

To further explore how spatial constraints influence the reproductive skew of territorial species, I performed a comparative analysis across terrestrial vertebrates of the relationship between landscape configuration and three measures of the shape of the distribution of lifetime reproductive success: the skewness, the Gini coefficient of inequality, and Hill's tail index (see methods). Next, I devised a theoretical scenario designed to explain the patterns that I found in the comparative analysis.

Individuals had a fixed individual quality, representing the effect of the genotype and the conditions experienced during early life [START_REF] Albon | Early development and population-dynamics in red deer .2. Density-independent effects and cohort variation[END_REF][START_REF] Lindstrom | Early development and fitness in birds and mammals[END_REF][START_REF] Spagopoulou | Silver-spoon upbringing improves early-life fitness but promotes reproductive ageing in a wild bird[END_REF]. They scrounged for the best territory by way of a series of dyadic interactions between immediate neighbors.

At each time step, neighbors switched positions whenever their territory and individual qualities did not match (Fig. 2a). This simple model corresponds to observations in many species, such as owls [START_REF] Mcginn | Implications of non-ideal occupancy for the measurement of territory quality[END_REF], gamebirds [START_REF] Eason | New birds on the block: new neighbors increase defensive costs for territorial male willow ptarmigan[END_REF] and songbirds [START_REF] Naguib | Effects of territorial intrusions on eavesdropping neighbors: communication networks in nightingales[END_REF], and also agrees with the general common sense that individuals first interact with their immediate neighbors.

The main reason why I expected some nontrivial effects to emerge from the above-described simple movement heuristic is that some territories become effectively invisible to potential challengers, or at least less accessible than if there was no constraint on who can interact with whom (Figs. 2b-2c). More precisely, a low-quality individual may find itself protected by a shield of high-quality neighbors (Fig. 2b). An average individual may secure an above-average territory provided it is hidden behind lowquality neighbors (Fig. 2c). The frequency of these situations and the number of individuals that benefit from them expectedly depends on the grain of the landscape. A homogeneous landscape with very clinal changes in territory quality expectedly facilitates the sorting of individual and territory qualities.

Individuals would progress in a consistent direction until they hit a wall. This wall could however potentially shield many individuals behind it, leading to the emergence of modules of individuals that interact among themselves but are shielded from interaction with individuals from other modules. By contrast, in a grainy landscape, the sorting of individual and territory qualities is already hindered at small spatial scales. Therefore, homogeneous landscapes should generate more skewed and heavier- tailed distributions of lifetime reproductive success than heterogeneous landscapes, but also more egalitarian properties, as more individuals benefit from the shielding effect, and as the socio-spatial structure is increasingly modular.

Material and methods

Econometrics to quantify the reproductive skew

The Gini coefficient sums all the pairwise differences between individual LRS values. Denoting n the total number of records, 𝑥 𝑖 and 𝑥 𝑗 the LRS (or other aspect of performance) of individuals i and j, and 𝑥̅ the mean LRS, the Gini coefficient is

𝐺 = 1 2𝑛 2 𝑥̅ ∑ ∑ |𝑥 𝑗 -𝑥 𝑖 | 𝑗 𝑖
. It takes value 0 if all individuals perform the same, and value 1 if only one individual has a positive reproductive output. I used [START_REF] Davidson | Reliable inference for the Gini index[END_REF] formula for the standard error of G. The main issue with the Gini coefficient is the presence of 𝑥̅ at the denominator. In practice, the computation is more stable if the data are manipulated so that the minimum performance value is set to 1 instead of 0.

The skewness quantifies the asymmetry of the distribution and in particular the frequency and magnitude of records that are above the modal LRS value. It is positive when there is an excess of high performing individuals compared to a symmetric distribution that would have the same variance.

Because vertebrate LRS datasets are typically zero-inflated, i.e., the modal value is 0, they almost always exhibit a positive skewness [START_REF] Tuljapurkar | Skewed distributions of lifetime reproductive success: beyond mean and variance[END_REF]. Importantly, the skewness should not be computed from samples <20 records. Even if its point estimate depends on the first three moments of the distribution only, its standard error depends on the first six moments [START_REF] Fisher | The moments of the distribution for normal samples of measures of departure from normality[END_REF].

To focus on the very best performers only, and quantify how far from the rest of the population they sat and how frequent they were, I selected Hill's tail index [START_REF] Hill | A Simple General Approach to Inference About the Tail of a Distribution[END_REF]. Based on the theoretical distribution 𝐿(𝑥)𝑥 -𝛼 where 𝐿(𝑥) is any slowly varying function, and α is Hill's index, the estimator is 𝛼 ̂(𝑘) = 𝑘 ∑ log( the tail behavior occurs, and log as usual denotes the natural logarithm (base e). The standard error is 𝛼 ̂(𝑘) /√𝑘. A large α indicates a light-tailed distribution with rare high-performers and/or relatively low extremes. A small α indicates a heavy-tailed distribution with several high-performers and/or very extreme performances. The threshold k can be set arbitrarily but should be optimized instead. I chose the "AMSE" method to compute the optimal threshold [START_REF] Caeiro | Threshold selection in extreme value analysis[END_REF]. This method minimizes a criterion based on the third order Taylor decomposition of the mean squared error in 1/𝛼.

Empirically, this method was stable in small datasets and yielded results consistent with the visual aspect of the distributions. I used the package tea for R [START_REF] Ossberger | tea: Threshold Estimation Approaches, R package version 1[END_REF], which also lists many alternative tail indexes and threshold optimization methods.

Literature search for lifetime reproductive success (LRS) data

I only considered vertebrates. In addition, I excluded aquatic species, fish-eating species, and capital breeders with long-distance migration, because I was going to use the terrestrial vegetation of the breeding area to quantify the landscape configuration. Datasets had to document 20 individuals or more.

Searching for published LRS data poses a challenge because the search term "lifetime reproductive success" is frequently used in the ecology and evolution literature. For the present study, no attempt was made to put together an exhaustive list of all the LRS data ever collected. To be included, the data had to (i) be publicly available as a downloadable electronic document or as an histogram that I could digitalize without requesting additional information from the authors; and (ii) appear in one of the three sources below. This ad hoc procedure yielded a total of 71 sex-specific datasets from 25 birds, 20 mammals, and 1 reptile. The relatively large number of species (46), their large taxonomic diversity, the large array of sampled biomes and natural histories, large array of publication dates, and the fact that I retrieved all the datasets that I a priori listed before launching the search, all suggest a representative sample (Appendix S1).

The search was divided in three parts. First, I searched the Dryad data repository (datadryad.org) on July 1, 2020 with the search term "lifetime reproductive success". The search returned 98 results of which 12 met the above criteria. These studies were published between 2012 and 2020. Second I mined the two reference books of the field [START_REF] Clutton-Brock | Reproductive Success Studies of Individual Variation in Contrasting Breeding Systems[END_REF], Newton 1989), obtaining 14 studies that met the above criteria. In order to also sample datasets published between 1989 and 2012, I used webofknowledge.com and Google Scholar, last checking on July 1, 2020. To divide the task into manageable segments, and to filter out the vast majority of studies that mention lifetime reproductive success but do not provide field records of it, I combined the search term "lifetime reproductive success" with the name of specific taxa. These taxa either had lifestyles that make the collection of LRS data possible, or I deemed them to represent reference taxa likely to be mentioned by studies into other taxa: "kestrel", "tit or titmice", "flycatcher", "owl", "bat", "deer", "macaque", "chimpanzee", "mouse", and "meerkat". For each of the taxa names, I considered the first 20 results as sorted by the search engines with default sorting criterion, as well as the cited and citing literature (first 20 citing articles). I then used the title, abstract, and supplementary material titles of the articles to determine whether the articles contained publicly available LRS data. This yielded 33 studies that also met the above criteria. I acknowledge the possibility of observer and popularity biases in this procedure, but the final result suggests a representative sample nonetheless.

Although the definition of LRS is the number of offspring that fledge (for birds) or reach weaning age (for mammals), I included four datasets of lifetime production of recruits and one dataset of lifetime mating success. Another potential source of spurious variation, for socially monogamous species, was that some authors were able to perform a genetic test of paternity to sort out the extra-pair paternities, whereas most authors compiled the number of socially affiliated offspring. In this study, I did not detect any difference between the two approaches.

Lastly, I defined two broad categories of social organization: "solitary" or "group-living". I considered the social organization during the breeding period. I used the information provided in the source studies and references therein. Typical solitary species were those that defend an exclusive territory, mostly forage alone, and only associate with their current mate and recent dependent offspring. Typical group-living species were those in which group mates are not always closely related to each other and male or female offspring (or both) stay in the natal group after weaning or fledging. A few species proved hard to categorize. I categorized polar bear (Ursus maritimus) as group-living because of routine dyadic contacts between individuals and of physical competition between males for access to oestrous females; however this species is traditionally categorized as solitary. Four bird species may nest in loose colonies or alone, and may forage in groups or alone when breeding. I classified two of these as solitary on the basis that they defend small territories around the nests, and the other two as group-living on the basis that male access to reproduction depended on lek attendance, and the data documented male LRS (Appendix S1). With these decisions, all the populations that I categorized as solitary were also territorial during the breeding season. Most of the populations that I categorized as group-living are territorial too. However, group-living species will correspond to the control group. This is because, under my working hypothesis, their reproductive skews mostly come from within-territory social dynamics rather than between-territory differences.

Quantifying landscape configuration in the study areas where the LRS datasets were collected

The enhanced vegetation index (EVI) is a remote-sensed spatially-explicit measure of the absorbance of infrared light by green vegetation, available at a 250m resolution, and rescaled to take integer values between 1 (no vegetation) and 100 (greenest vegetation). I used the homogeneity of the EVI across adjacent pixels, denoted EVIHOM hereafter, to quantify landscape configuration [START_REF] Tuanmu | A global, remote sensing-based characterization of terrestrial habitat heterogeneity for biodiversity and ecosystem modelling[END_REF]. Compared to metrics that use only the greenest time of the year, or the least green, or the dynamics of the seasonal change in greenness, this choice was expected to capture information of the same level of relevance for all the species and all of the worldwide study locations [START_REF] Tuanmu | A global, remote sensing-based characterization of terrestrial habitat heterogeneity for biodiversity and ecosystem modelling[END_REF]Jetz 2015, Tucker et al. 2019). In addition, variation in EVIHOM is largely driven by heterogeneity in landcover, which does not change seasonally as much as the greenness itself does. The EVIHOM was recently demonstrated to have the best power out of a series of 14 metrics of the same type to predict macroecological patterns such as the number of American passerines in a grid cell [START_REF] Tuanmu | A global, remote sensing-based characterization of terrestrial habitat heterogeneity for biodiversity and ecosystem modelling[END_REF] and the travel speed of large birds [START_REF] Tucker | Large birds travel farther in homogeneous environments (V Boucher-Lalonde[END_REF].

Because the study animals were sampled over study regions that varied from a few hectares to a full country, I retrieved the EVIHOM value within a radius of 1 to 100km (depending on the quoted study area). I used the median EVIHOM in that circle, removing cities and open water as in the original dataset. The results did not qualitatively change if setting all circle radii to 5km. Next, because large animals range more widely than small animals, I rescaled the EVIHOM metric by dividing it by the logtransformed body mass, yielding the metric h = EVIHOM/log(mass). Most of the regressions became statistically non-significant if I removed the allometric scaling (i.e., h = EVIHOM) but they hold if I removed the log-transformation of body mass.

Comparative analysis

Each of the populations contributed a control (group-living) or a treatment (solitary) sample. Across populations, I regressed h against the skewness, the log-transformed Hill's tail index, and the logittransformed Gini coefficient, separately for control and treatment samples. I weighted the regressions by the inverse of the sampling variance of the dependent variables. I incorporated the fixed effects of the sex, the mating system ('mostly monogamous' vs. 'polygynous or promiscuous'), and their interaction. I also tentatively added the fixed effect of the mean LRS of the populations to represent any dependency on the mean in the skew metrics. Lastly, I added the fixed effect of the taxonomic class (birds vs. mammals + one reptile). These fixed effects captured most of the between-species variance. This means that I failed to fit a mixed model with non-null phylogenetic inertia or with a species random effect, as long as the aforementioned fixed effects were already in the model. This suggests that the included species were sufficiently diverse to overcome the effect of phylogenetic inertia in the dependent variables, or that phylogenetic inertia was weak. There were yet only four mammalian solitary species. To quantify the maximum extent to which this taxonomic bias could be confounding, I performed a likelihood ratio test comparing the above model and the model in which I added the interaction between h and the taxonomic class. If the test was negative (P>0.05), I concluded that it was possible that the differential effect of h in the treatment sample was caused by an excess of birds in that sample.

General model for the simulations

I considered a grid of 2500 potential territories of which the quality varied randomly with a known spatial autocorrelation range (Appendix S2, part A). At the first time step, 2500 individuals were distributed in the grid cells. Their individual quality was drawn at random from a normal distribution.

At each time step, I went through all the pairs of neighbors in the grid and switched their positions whenever one of the neighbors was of lower individual quality and but hold a better territory (Fig. 2a; Appendix S2, part A). Next, the individuals reproduced following a Poisson model and died following a binomial model, with expected fecundity and survival rates that depended on individual quality and the quality of the current territory, according to sigmoid functions. I recorded the individual lifetime reproductive success after 30 time steps (only counting individuals that had completed their lifetime).

Offspring always emigrated out of the study area. Vacant territories were settled by random individuals. These simplifying assumptions allowed the model to ignore the effect of selection on individual quality, as a first approach.

The main unrealistic aspect of the model is that in reality, territorial individuals are reluctant to change location once they have established precedence on a territory [START_REF] Komdeur | Importance of habitat saturation and territory quality for evolution of cooperative breeding in the Seychelles warbler[END_REF][START_REF] Eason | New birds on the block: new neighbors increase defensive costs for territorial male willow ptarmigan[END_REF][START_REF] Mcginn | Implications of non-ideal occupancy for the measurement of territory quality[END_REF]. However, the model applicability is not seriously challenged for two reasons. First, spatial switches similar to those considered in the model occur during the immature period [START_REF] Delgado | A statistical framework for inferring the influence of conspecifics on movement behaviour[END_REF][START_REF] Ducros | Beyond dispersal versus philopatry? Alternative behavioural tactics of juvenile roe deer in a heterogeneous landscape[END_REF]. Second, the spatial switches of the model may adequately represent any displacement of the boundary between adjacent territories following a conflict. Boundary displacements are empirically more common than changes in territory location [START_REF] Calsbeek | An experimental test of the ideal despotic distribution[END_REF][START_REF] Vanpé | Access to mates in a territorial ungulate is determined by the size of a male's territory, but not by its habitat quality[END_REF][START_REF] Mosser | Group territoriality and the benefits of sociality in the African lion, Panthera leo[END_REF][START_REF] Hartmann | Decline in territory size and fecundity as a response to carrying capacity in an endangered songbird[END_REF].

Lastly, I devised two null models. The first null model corresponded to the ideal despotic distribution.

Each time an individual died and was replaced, I shuffled the individuals so that the match between individual quality and territory quality was always perfect. The second null model did not feature any territory switch. The individuals remained in their initial territory for their whole life.

Model parameterization and sensitivity analysis

The survival and fecundity functions were chosen to represent three paces of life: fast-paced, medium, and slow-paced life histories (Appendix S2, part B). In the fast-paced species, the breeding effort was maximized and its dependency on territory and individual quality was lower than in the other two species. By contrast, in the slow-paced species, the survival probability was maximized to represent known patterns in life history variation across species [START_REF] Péron | Evidence of reduced individual heterogeneity in adult survival of long-lived species[END_REF].

Next I made the range of the spatial autocorrelation in territory quality to vary between 0 (each territory was an independent and identically distributed random variable) and 1000 (very gradual and clinal change in territory quality over the grid). I ran 10 simulations per value of the range and per species. I recorded the distribution of LRS across individuals for each simulation.

Results

Comparative analysis

In solitary species, there was evidence that the skewness increased with the h metric of landscape homogeneity (McFadden's r² = 0.32, Wald's Z statistic = 2.2; ANOVA P-value = 0.03), the Gini coefficient decreased (r² = 0.22; Z = -2.4; P = 0.02) and weak evidence that the Hill's tail index decreased (r² = 0.12; Z = -1.2; P = 0.2) (Fig. 3).

In group-living species, these correlations were in the opposite direction, the support was weak, or both (Fig. 3; skewness r² = 0.15, Z = -0.5, P = 0.7; Gini r² = 0.56, Z = 1.94, P = 0.001; Hill r² = 0.17, Z = 0.75, P = 0.1).

For the skewness and Hill's index, but not for the Gini coefficient, it was possible for these differences to be explained by the taxonomic bias, i.e., the lack of mammals among solitary species (LRT P-values: skewness 0.26, Gini 0.009, Hill 0.49).

Simulation study

In the three model species (fast-paced, medium, and slow-paced), the reproductive skew changed when the range of the spatial autocorrelation in territory quality changed (Fig. 3). As the range of the spatial autocorrelation in territory quality increased, both the skewness and the Gini coefficient increased (Fig. 3). In other words, coarse-grained landscapes with a large autocorrelation range generated more skewed distributions of lifetime reproductive success than fine-grained landscapes, but the distributions were yet more egalitarian, as also observed in the comparative analysis (Appendix S2, part C). By contrast, under the ideal despotic distribution and under the null model with no territory switch, neither the Gini coefficient nor the skewness responded to a change in the autocorrelation range (Fig. 3). Regarding Hill's tail index, when the autocorrelation range increased, the index exhibited a dip then an increase (Fig. 3). This mostly happened for the medium species, but the pattern was still detectible in the other two species. In other words, the extreme performers were most frequent for intermediate values of the autocorrelation range. This makes this prediction hard to test because what constitutes an intermediate value expectedly changes across species and is not typically known.

Discussion

Studies into reproductive skews have been dominated by group-living species, especially those with a steep effect of the social rank on the breeding output [START_REF] Johnstone | Models of reproductive skew: A review and synthesis[END_REF]. Here, I adopted a relatively basic, demographic definition of "reproductive skew", and I used three different metrics to quantify it, making it possible to focus on the reproductive skews of solitary, territorial species. In solitary, territorial species, I predicted that the socio-spatial structure would depend on the landscape configuration, yielding a correlation between the landscape configuration and the reproductive skew. In a comparative analysis of 46 species, the distribution of lifetime reproductive success was indeed more skewed, but also more egalitarian, in homogeneous than heterogeneous landscapes. My explanation for the increase in skewness is that homogeneous landscapes facilitate the sorting of individual and territory quality.

My explanation for the increase in egalitarian properties, i.e., decrease in Gini coefficient, is that homogeneous landscapes feature blocks of good territories separated by blocks of bad territories, and thereby the shielding effect depicted in Figs. 2b and 2c affects more individuals than in heterogeneous landscapes. This is supported by the fact that the theoretical model depicted in Fig. 2 reproduced the observed correlations. By contrast, the ideal despotic model cannot explain the observed correlations (Fig. 3). The proposed model is clearly too simple for me to claim that the congruence of patterns represents strong inference. Nevertheless, the exercise further brought to the fore the interest of analyzing the shape of the distribution of fitness components beyond the mean and the variance [START_REF] Tuljapurkar | Skewed distributions of lifetime reproductive success: beyond mean and variance[END_REF]. I also highlighted an intriguing aspect of the socio-spatial organization that may be general to many species of territorial animals or any situation where individuals can secure exclusive access to some resource item (Figs. 2b and2c).

On the use of econometrics in biodemography

While the skewness is familiar to ecologists, I also used in this paper two statistics that come from the field of economy: the Gini coefficient and Hill's tail index. In economy, they are used to describe and analyze the accumulation of wealth and influence, e.g., pyramid schemes, Pareto laws, and income inequalities [START_REF] Wegrzycki | Why do cascade sizes follow a power-law? -26th Int[END_REF][START_REF] Pluchino | Talent versus luck: The role of randomness in success and failure[END_REF].

There are many examples of accumulation processes in vertebrate biodemography: alpha individuals that monopolize the group reproductive output [START_REF] Johnstone | Models of reproductive skew: A review and synthesis[END_REF], dominant territory owners that combine their intrinsic advantage to the advantage associated with a good territory over multiple breeding attempts [START_REF] Hochachka | Unequal lifetime reproductive success and its implication for small isolated populations[END_REF]. Tail indexes appear especially suited to document the fitness consequences of these processes. Overall, in this study, 18% of the datasets exhibited a very heavy tail (α < 2). α < 2 means that the density function decreased slower than 1/LRS² for very large LRS values.

In other words, the situation in which a few individuals monopolize reproduction was not rare, but it is not the general case either.

Hill's tail index appeared somewhat less informative than the skewness and Gini coefficient in this study (Fig. 3). An important caveat regarding tail indices is that they are, by construction, computed from a small sample, typically <10% of the total sample. They can be instable for this reason, which is maybe why in the simulation, it was difficult to draw a clear prediction based on Hill's tail index (Fig. 3). Hill's tail index nevertheless did capture information not conveyed by the either the Gini coefficient or the skewness (both r² < 20%). For example, male Verreaux's sifakas Propithecus verreauxi exhibited one of the heaviest tail indexes (α = 1.8) but one of the most egalitarian Gini coefficients (G = 0.33) (data:

Lawler 2007). Hill's tail index seemed to capture the occurrence of a few dominant males that sired most of the offspring, whereas the Gini coefficient seemed to capture the fact that the rest of the group was egalitarian in terms of access to reproduction. These are two major aspects of the "lemur syndrome" [START_REF] Kappeler | The lemur syndrome unresolved: Extreme male reproductive skew in sifakas (Propithecus verreauxi), a sexually monomorphic primate with female dominance[END_REF]. In the typical group-living vertebrate, the social hierarchy is more linear (Appleby 1983, McDonald and[START_REF] Mcdonald | Comparative transitive and temporal orderliness in dominance networks[END_REF], which would have generated a larger Gini coefficient.

Therefore, the Gini coefficient and Hill's tail index appeared complementary in this instance.

On the other hand, when the three metrics "light up" at the same time, they may indicate a strong cumulative effect of fixed individual or environmental variation. For example, female Northern elephant-seals Mirounga angustirostris (not incuded in the present study) tested consistently skewed for all three statistics (S = 2.9; α = 2.2; G = 0.45; Le [START_REF] Boeuf | Lifetime reproductive success of northern elephant seals (Mirounga angustirostris). -Can[END_REF]. This group-living species is very iteroparous, with a strong cumulative effect of individual quality [START_REF] Oosthuizen | Positive early-late life-history trait correlations in elephant seals[END_REF]. Elephantseals do not experience a major late-life cost if their start to breed early [START_REF] Oosthuizen | Positive early-late life-history trait correlations in elephant seals[END_REF]. In a faster-living species than elephant-seals, trade-offs between the first and subsequent breeding attempts (functional or allocation) could theoretically influence the shape of the distribution of LRS more than any fixed individual trait. The outcome of the first breeding attempt is indeed expected to be stochastic, making it possible for reproductive skews to emerge by chance only [START_REF] Tuljapurkar | Skewed distributions of lifetime reproductive success: beyond mean and variance[END_REF]. For example, common kestrels (Falco tinnunculus) exhibit a moderate skew (S = 1.6; α = 4.0; G = 0.36) but it is reportedly driven by stochasticity, rather than by fixed variation in individual or territory quality [START_REF] Broekman | Relative contributions of fixed and dynamic heterogeneity to variation in lifetime reproductive success in kestrels (Falco tinnunculus)[END_REF]. This maybe suggests that combining the three skew metrics, instead of using just the skewness, can help explore and characterize the relative importance of late-life breeding attempts and in particular the influence of the stochastic outcome of the first breeding attempt.

Interpretation in graph theory terms

In Figs. 2b and2c, the neighbors of an individual protect it from challenges by third parties. In graph theory terms, one would term that an indirect connection between the focal individual and its potential competitors. The number of indirect connections, termed the social reach, and the eigenvector centrality which is computed from the adjacency matrix, both measure the influence that a focal individual garners through its direct and indirect connections [START_REF] Newman | The mathematics of networks[END_REF]. Their computation could be weighted according to the individual and territory qualities. The social reach and eigenvector centrality would then measure how many competitors are blocked from challenging a given individual, how strong they are, and how incentivized they are, i.e., the advantage that the individual gains compared to if there was no spatial constraint on who can interact with whom.

Elsewhere, in more classical types of animal social networks, the social reach and the eigenvector centrality score have been correlated to aspects of individual fitness. For example, among male manakin birds that collaborate to display for the females, the centrality of an individual in the collaboration network predicted its eventual accession to alpha status [START_REF] Mcdonald | Predicting fate from early connectivity in a social network[END_REF][START_REF] Ryder | Social networks in the lek-mating wire-tailed manakin ( Pipra filicauda )[END_REF]. In hierarchized groups, the fitness benefits associated with the social rank, combined with the transitivity of the social rank (Appleby 1983, McDonald and[START_REF] Mcdonald | Comparative transitive and temporal orderliness in dominance networks[END_REF], suggest that top individuals rely on intermediate rankers to control the individuals down the line [START_REF] Holst | Social rank, fecundity and lifetime reproductive success in wild European rabbits (Oryctolagus cuniculus)[END_REF][START_REF] Addessi | Preference transitivity and symbolic representation in capuchin monkeys (Cebus apella)[END_REF].

Lastly, the position of an individual in contact networks predicted the odds or the speed at which it picked up new information [START_REF] Aplin | Social networks predict patch discovery in a wild population of songbirds[END_REF][START_REF] Allen | Network-based diffusion analysis reveals cultural transmission of lobtail feeding in humpback whales[END_REF][START_REF] Claidière | Diffusion dynamics of socially learned foraging techniques in squirrel monkeys[END_REF][START_REF] Kulahci | Social networks predict selective observation and information spread in ravens[END_REF].

In conclusion, nontrivial patterns may emerge from simple decision rules, because of indirect effects that propagate in the socio-spatial structure of the population. These effects would be challenging to document directly in the field because of the exponential number of indirect connections each individual has in a grid network and because any missing observation could disproportionately influence the inference. On the other hand, the LRS expectedly integrates the influence of all these social interactions, which was confirmed here by the fact that the reproductive skew responded to the landscape configuration. For this reason, and following the lemur example outlined above, computing the different aspects of the reproductive skew appears promising in order to help quantify the betweenspecies variance in the social organization. Following the kestrel and elephant seal examples, the reproductive skew may also correlate with the relative contribution of late-life breeding attempts to LRS, and in particular the magnitude of the effect of the stochastic outcome of the first breeding attempt.

The fact that the three skew metrics (skewness, Gini coefficient, and Hill's tail index) appeared complementary to each other in this study is especially promising regarding their ability to help discriminate between the numerous drivers of variation in LRS across populations. I recommend that researchers interested in individual heterogeneity use not only the variance and skewness, but also the Gini coefficient and Hill's tail index.

Fig. 2 :

 2 Fig. 2: Theoretical model. (a) Neighbors switch position whenever the difference in individual quality (shades of blue) does not match the difference in territory quality (shades of orange). Here, individual C takes over the territory of individual A. (b) and (c) Two situations in which individual A secures a better territory than C even though C is of better individual quality than A.

  records x1, …, xn are sorted in increasing order, k is a threshold after which

  where A and B represent possible EVI values between 1 and 100, and 𝑃 𝑖 (𝐴, 𝐵) is the proportion, among the pairs of pixels that are in the neighborhood of the ith pixel, of pairs that take EVI values A and B. If all the pixels in the neighborhood of the focal pixel have the same EVI value, then EVIHOM=1. Large differences in EVI values within the neighborhood of a pixel cause the EVIHOM to decrease towards 0, because of the (𝐴 -𝐵) 2 term in the denominator.

Fig. 3 :

 3 Fig. 3: Left panel: Covariation between the landscape configuration (x-axis) and three metrics of the reproductive skew (y-axis) in the simulations. The shaded areas represent the 80% confidence interval of the simulation outcome (excluding the two extreme results). Right panel: Same covariation observed in the comparative analysis.

Data Availability Statement: This manuscript does not use new data. All data sources are listed in Appendix S1.

Acknowledgments: I thank all the people who contributed to the collection and curation of the datasets of lifetime reproductive success over the years.

Supplementary Materials: Appendix S1: Dataset for the comparative analysis. Appendix S2: (A) R script (B) Parameterization (C) Agreement test.

Funding: This research received no external funding.

Conflicts of Interest:

The author declares no conflict of interest


ESSCA Aymeric Hemon

Brian Fitzgerald

Frantz Rowe

Innovative Practices for Knowledge Sharing in Large-Scale DevOps

Keywords: Agile method, DevOps, Large-Scale, Innovative Practices, Knowledge Sharing

Agile development methods and DevOps require adaptation during implementation to meet the needs of a constantly changing software development environment. The emergence of knowledge-sharing practices for large-scale DevOps has not been the subject of much research. Our in-depth case study, comprising 106 interviews at a large multinational company operating in a DevOps at scale environment, identified a number of innovative practices which had emerged, principally to resolve knowledge-sharing challenges. These practices seem to be more likely to emerge in large-scale DevOps environments. While similar results might have been achieved due to the large-scale nature of the projects, it is difficult to determine definitively whether the main causal factor is project size or DevOps. We believe that self-organization and continuous improvement over a long period of time are also critical influencing factors.

Introduction

I find myself more and more exasperated with the great inflexible sets of rules that many companies pour into concrete and sanctify as methods...Use the prevailing method only as a starting point for tailoring. --Tom DeMarco (1982) Agile methods emerged from the inability of conventional (e.g. waterfall) methods to deliver software to meet the needs of a rapidly changing environment. Initially agile methods were considered to have a "home-ground" where they were best suited, namely small projects, with colocated developers, in non-critical domains. This home-ground has been challenged considerably in the past 15 years as researchers and practitioners have tailored the application of agile methods with new roles, ceremonies and artefacts to meet the needs of the development context, for example in critical and regulated domains.

In addition, a bottleneck has emerged due to a lack of alignment between the Operations function (Ops), coordinating the software release, and the Development function (Dev). Consequently, releases to customers took more time. To solve this problem, Debois 1 advocated for more collaboration between the Dev and Ops functions through a tighter integration, a rapprochement called DevOps. This term DevOps comes from the fusion of two words related to specific component activities, Development and Operations.

While agile methods can achieve a more frequent cadence of development of software and a better alignment with customer expectations, DevOps strives for a continuous delivery of value through continuous integration, delivery and deployment. DevOps is hence an extension of agile 2 to the entire software delivery pipeline, aiming to optimize lead time between code writing and its use by endusers in a real production environment.

We consider large-scale DevOps as we focused on the four DevOps pillars (Culture, Automation, Measure, Share) that involve a large number of actors, systems and interdependencies with more than two teams working in the same project [START_REF] Dingsøyr | What is large in large-scale? A taxonomy of scale for agile software development[END_REF] and applying continuous integration, delivery and deployment. Roles and responsibilities do still matter in the agile world, particularly when using prescriptive methods as Scrum. DevOps breaks silos and hence possibly blurs lines (i.e. organizational, hierarchical) regarding jobs, roles, collaborations, responsibilities, skills and practices. DevOps potentially amplifies this blurring between boundaries when applied at large-scale.

In this paper, we focus on innovative practices for knowledge sharing (KS) which is one of the four DevOps pillars and will argue that these practices are more likely to emerge in large-scale DevOps. A KS practice occurs when an individual transfer what he knows to another individual. When moving to large-scale agile, challenges of KS and related success criteria have been identified 4, 5 which raises the question about ensuring and improving learning and KS practices. Ghobadi et al. identified barriers [START_REF] Ghobadi | Perceived barriers to effective knowledge sharing in agile software teams[END_REF] and risks to effective KS in agile teams. Risk perception is even higher in DevOps than in agile [START_REF] Hemon | Job Satisfaction Factors and Risks Perception: An embedded case study of DevOps and Agile Teams[END_REF] . Consequently, KS challenges are larger and more complex, since more points of view come into consideration when using largescale DevOps. Managing knowledge dependencies become critical in a largescale DevOps context 8 . Indeed, some individuals could form a bottleneck to knowledge transfer when moving to largescale DevOps. Despite the need for innovation and tailoring, as expressed by DeMarco in the opening quote, new KS practices emerging for large-scale DevOps have not been extensively studied. In that respect Nielsen et al. proposed a DevOps knowledge sharing framework (DOKS) [START_REF] Nielsen | Closing the IT Development-Operations Gap: The DevOps Knowledge Sharing Framework[END_REF] and used the CESI dimensions (Combination, Externalisation, Socialisation, Internalisation) to increase awareness of KS modes within the organization. They concluded that the size of the team and the size of the company influence the move towards continuous delivery and DevOps. They showed that larger companies would need a more structured plan when moving to DevOps to ensure KS among their IT teams while smaller companies do not need it. Consequently, moving to large-scale DevOps might impact KS practices. In sum, the following KS challenges in DevOps have been identified in the literature:

1) More intense cross-functional collaboration between Dev and Ops 10, [START_REF] Wiedemann | A New Form of Collaboration in IT Teams-Exploring the DevOps Phenomenon[END_REF] 2) Multiple environment incompatibilities leading to specialized teams 12, [START_REF] Smeds | DevOps: a definition and perceived adoption impediments[END_REF] 3) Capability to self-organize 14 4) Loss of global vision of the project and knowledge of application due to automation [START_REF] Hemon | Job Satisfaction Factors and Risks Perception: An embedded case study of DevOps and Agile Teams[END_REF] 5) Confinement of knowledge sharing to hierarchical organisational structure and process 'red tape' [START_REF] Hemon | Job Satisfaction Factors and Risks Perception: An embedded case study of DevOps and Agile Teams[END_REF] 6) Limited sharing when parts of development or operations are outsourced [START_REF] Hemon | Conceptualizing the Transition From Agile to DevOps: A Maturity Model for a Smarter IS Function[END_REF] Moving from DevOps to large-scale DevOps amplifies the challenges linked to DevOps, i.e. coordination and collaboration improvement, dependency management, knowledge development and sharing. Three of the six KS challenges above are particularly salient. First, the capability to self-organize could be affected by the scaling-up of DevOps due to a larger number of team members covering a wider organizational perimeter.

Second, notwithstanding a higher level of automation at scale, more frequent commits (release) require manual coding adjustment [START_REF] Zhao | The impact of continuous integration on other software development practices: a large-scale empirical study[END_REF] which needs a common coordination mechanism, hence more intense cross-functional collaboration. Third, scaling DevOps could impact KS due to more frictions with organisational structure and process 'red tape'.

Context

Our study was conducted in a large multinational company (more than 100,000 employees) that has been practicing DevOps for eight years, being one of the very early adopters of the DevOps approach. We followed the case study method combining interviews, observations and documentation. To identify KS practices implemented by project teams, we carried out direct observations during field visits, attended 20 meetings plus an eighthour DevOps coaching day, and semistructured interviews with 106 employees in total. These teams included multiple functions associated with software development, such as developer, project manager, release engineers, UX designers, PO and architects. For each project, three of the four authors worked together on a deep analysis. Firstly, we collected information regarding the agile methods and actual practices supporting KS both inside and outside of teams and related to the use of space (i.e. common work area), to agile methodologies (i.e. scrum of scrum meetings, daily standup), and to collaborative tools enabling KS (i.e. Slack or Atlassian JIRA). Secondly, we compared our findings with a baseline reference (12 th State of Agile Report) to identify potential innovative practices. We studied specific adaptations of agile practices and the development of KS practices which had emerged to suit the contingencies of the development environment. Thirdly, we investigated the literature where we found practices in use in the company but not found in the baseline reference report.

Identifying innovative KS practices thus meant identifying those most advanced within the 18 projects investigated in this company. Those innovative KS practices are likely to be "new to the firm", but we do not claim that they are "new to the world". However, they help address challenges when moving towards large-scale DevOps.

Six out of the 18 projects studied are practicing DevOps, and these innovative KS practices were only found in two of the projects which were using large-scale DevOps. Project X started eight years ago, operating in DevOps mode from the beginning, and Project Y matured to eventually operate in DevOps mode since 2016. Both projects are advanced in DevOps and they practice continuous delivery or continuous deployment.

Project X involves 85 people, organized in seven teams, almost all co-located except for an Ops team located a few miles away. Teams are not necessarily the same size and do not have to be composed the same way. Project members work in DevOps and use mainly a hybrid agile method derived from Scrum, Kanban and XP. They scaled-up their agile method, first with large-scale Scrum (LeSS), then one year later they moved to the Scaled Agile Framework (SAFe). Project Y involves 65 people divided into 5 teams distributed over four geographical sites and two distant countries, distant at many levels according to Ghemawat's CAGE model [START_REF] Ghemawat | Distance still matters[END_REF] . Development activities are essentially based in the same country while Operations are mostly in another country. Project members use mainly Scrum and Scrumban and scaled-up with LeSS adopting practices such as Scrum of Scrum. They also implemented an international cross-functional daily DevOps meeting.

Innovative Practices for Knowledge Sharing

We propose three different levels of KS practices based on three levels of DevOps maturity (agile, continuous integration, continuous delivery) in the transition from agile to large-scale DevOps (see Table 1). At the first DevOps maturity level, agile teams do not cross functional silos and DevOps is inexistent. At the second level, teams practice continuous integration, hence across silos. At the third level, teams practice continuous delivery and/or deployment. We identify four practices which have emerged through tailoring of the method and which address the needs of DevOps in a large-scale development context: Cross-Functional Dynamic Role Rotation, Technical Thursdays, Heads-Up Grooming & Planning, and the Circle. We discuss each turn below. • Some large-scale frameworks are used and meet their first limits (i.e. rigidity of SAFe) along the entire pipeline

Cross-Functional Dynamic Role Rotation (DRR)

A more significant involvement of certain functions (Project Manager, PO and Scrum Master) was highlighted during Daily Standup Meetings. In some projects, this involvement was even more pronounced as several of these functions were performed by the same individuals. To overcome this difficulty, one of the large project teams at an advanced level of DevOps maturity proposed a new practice which involved a cross-functional DRR and therefore associated responsibilities. Job rotation in the software development context has been identified in a previous study [START_REF] Faegri | Introducing knowledge redundancy practice in software development: Experiences with job rotation in support work[END_REF] as a mean of addressing organizational concerns and stimulating innovation. The instantiation of this role rotation in the cross-functional DRR practice in our case enabled largescale learning and KS since all team members were able to perform several roles and became more knowledgeable. DRR is usually limited to a specific area i.e. the development team or the operations team. A developer can become a scrum master for a week or a sprint, then he goes back to his position. In our cases, DRR is crossfunctional and for instance, Production Engineers (Ops) performed code reviews. All members from both sides, Dev and Ops, were able to meet and work together alternatively. Cross-functional DRR also ensured that the 'heavy-lifting' workload did not fall on the same individuals repeatedly. It also fulfilled the role of 'succession planning' in that team composition became more flexible as individuals could be 'swapped out' without causing major perturbation to the extent that the teams could not function.

Technical Thursdays (TT)

Another practice which emerged in the context of technical knowledge-sharing was labelled "Technical Thursdays". These 'tech-talk' events took place over a half-day every two weeks. The goal was to disseminate and share technical knowledge, particularly in relation to technologies, tools, skills, or other topics. These would typically be championed and led by an individual team member. They could take the form of workshops on new technology topics (e.g., containerization, automated configuration management, infrastructure as code), and related tooling (e.g., Puppet, Docker), discussions of new initiatives (e.g., A/B experimentation) challenges (e.g. hackathons), games (e.g. Code Wars). A playful spirit and gamification were often a strong component of this practice, which had the extra benefit of boosting morale and team spirit.

The difficulty with these tech-talks lies in the fact that KS could be limited to a group a person or to communities of practice (CoP). Therefore, inter-team KS at large scale could be an issue for agile software development organizations [START_REF] Santos | Fostering effective inter-team knowledge sharing in agile software development[END_REF] . The originality of these TT events is that they go beyond CoP and the risk of developing knowledge on a tribal basis. TT are fully cross-functional, both vertically and horizontally across silos where they existed. One could have supposed the Code War game was only intended for Developers. However, POs, Architects, Designers, Developers, Testers, Ops Engineers have been invited to this game. Participants participated in teams other than their usual one. Teams had to be mixed and represent as many trades as possible at the organization level. While hackathons are mostly frequented by developers, this game fostered large-scale KS across the Dev and Ops areas and significantly helped the realization of projects, i.e. the implementation of Docker containers for deployment system proposed by Ops and accepted by Dev beyond project teams.

Heads-Up Grooming and Planning (HUGP)

Backlog Grooming, or backlog refinement, can be understood in different ways. In the 18 projects we studied, backlog grooming consisted of short-term User Story costing. However, in a large project where the team uses DevOps, one PO had instantiated a new practice to support backlog grooming and Sprint Planning, which is labelled as Heads-Up Grooming and Planning (HUGP). This practice is similar to Rolling Lookahead Planning (RLP) in the sense that there is a high-level release plan and sprints are then gradually refined as they start. In this HUGP ceremony, the PO no longer performs any immediate user story estimation with the team. Instead, the PO just provides the team with broad information on the subject. The idea is to give the team a visualization of the work to come. This meeting is short -somewhere between 10 minutes and a maximum of 30 minutes. Subsequently, the Sprint Planning meeting takes place and lasts two hours where user stories are presented, estimated and broken down into tasks. However, the effect of the previously held HUGP practice is very noticeable. The PO describes it well:

The team is more confident and self-assured when it comes to story point estimation. There is less error or difference. The team members have had time to go through the code and figure out if they had any doubts. They discuss it among themselves, around a foosball game.

We consider HUGP as an extension of RLP. Both adopt a just-in-time approach to up-front planning; both foster discussion within teams for identification and estimation of product backlog components. However, we suggest that HUGP differs because it is more than a discussion about user story estimation. HUGP strongly promotes KS, pushing team members to go into the code, to learn, to test something new, to enrich each other and even to consult Ops engineers on specific topic (i.e. production environment, deployment package). Before the discussion, there is a transfer of knowledge between members and this helps them to surface their arguments for better estimates. The benefits of such KS practices are multiple: timesaving when estimating user stories, clearer and more accurate estimations, engendering a greater sense of Dev team empowerment, at the DevOps team level, then across the project.

The Circle

One issue with Sprint Retrospectives is that, if they are done at all, they are often combined with the Sprint Review ceremony. Also, they are confined to a single team, hence limiting KS [START_REF] Santos | Fostering effective inter-team knowledge sharing in agile software development[END_REF] , thereby minimizing the opportunity for KS in achieving continuous improvement and consequently not fostering Agile and DevOps at large-scale.

The team met difficulties with retrospectives, primarily because they were quite complex to organize with a team of 85 people. They decided to set up joint reflection ceremonies, which were known as the Circle. This practice is based on discussion among a group of people who represent all the jobs and roles across multiple project teams, i.e. managers, PO, architects, developers, testers, Ops engineers. Participation is voluntary. The circle is a forum for real exchange of knowledge across all team functions. This Circle was composed of about 20 people who met twice a month. The Circle practice fulfilled the role of the Sprint Retrospective at large-scale by stimulating and animating reflection on the daily functioning of teams and the organization but had the added advantage of comprising multiple teams. This reflection was particularly useful for implementing the other innovative practices mentioned here, as the use of new practices does not always get the support of all, and therefore their implementation must be the subject of open discussion.

When the shared knowledge appeared valuable, e.g. Code Wars, it is adopted and disseminated across teams. In this way, they limit the problem of missing valuable knowledge or not capitalizing on this knowledge. The Circle allowed such discussion to take place and ensured that all these innovative practices could evolve and develop, much in the spirit of the quote by DeMarco above. This practice typically responded to KS challenge when largescale DevOps was applied (see Table 1) and went beyond DevOps boundaries integrating the Biz domain also.

KS Major Challenges Resolution

Different innovative practices have met the challenges of KS in the context of largescale DevOps. We summarize the main KS challenges and impacts generated by each of them (see Table 2).

Table 2. Major Challenges and Impacts of Innovative Practices

Innovative practices

Major challenges addressed Impact

Cross-Functional DRR

-Intense crossfunctional collaborations -Team selforganization challenges.

• Enables the breaking of knowledge dependencies [15] because knowledge is not anymore hold by an individual. However, this obliges team members to communicate amongst themselves to carry out their role effectively.

• When team members rotate, they can punctually take on responsibilities, develop skills and acquire knowledge. This fosters autonomy of the team.

Technical Thursdays

-More intense cross-functional collaboration -Complex environments and specialized teams.

• All team participants (representing almost every area, i.e. business, development, quality...) can discover and learn new approaches without any consideration regarding the belonging to a specific functional silo or the position held. This is how Ops teams by passed organizational culture related to training and Ops made Dev teams discover Rancher, an open source tool to deploy and manage containers in production environments.

• Helps to prevent from an over specialization of teams by fostering KS and by increasing the knowledge base or individuals understanding of these multiple environmental complexities and incompatibilities.

Heads-Up Grooming and Planning

-Confinement of KS due to hierarchical and organizational structure and process red tape.

• Facilitates KS at scale avoiding the repetition of red tapes and reduces the burden of agile processes and organizational rules and procedures.

•

By practicing HUGP, the manager gives the team members the possibility to obtain central information that they would not have had while respecting the processes of the agile method and the organization. Team members can therefore use information in advance which allows them to enrich and transform the information into knowledge. This new knowledge is shared and allows them to make more realistic and accurate decisions in their mission.

The Circle

-Capability to selforganize -Multiple environment incompatibilities leading teams' specialization.

• Facilitates the circulation of knowledge because the Circle is fully cross-functional with representants from all jobs linked to the project and, in the same time, because the circle is only 20 individuals representing larger teams (85 people).

•

This fosters KS at large-scale, decision-making and consequently teams' autonomy.

Large-Scale DevOps Enabler of Innovative KS Practices

While the Scrum method is the backdrop for the projects we surveyed, reality is much more complex in terms of overlapping and complementary practices [START_REF] Fitzgerald | Customising agile methods to software practices at Intel Shannon[END_REF] . Thus, we were able to identify the development of innovative KS practices only in the most advanced project (see Table 1). This raises the question of how large-scale DevOps enables innovation. Our hypothesis is that these innovation practices are the result of further continuous improvement, facilitated by stronger levels of self-organization in DevOps teams acting on a broader mandate.

As already mentioned, these KS practices were in fact observed in large-scale projects which commenced eight years ago, one of them operating in DevOps mode from the beginning, and the other maturing to eventually operate in DevOps mode since 2016. The other four DevOps projects were clearly sharing knowledge (e.g. common project management, common processes, common tools e.g. shared continuous integration and automated deployment tools, co-construction of deliverables) and used basic practices. However, KS was less developed than the other two. This may be because they had had less time to develop those, but also because three of these four were smaller projects. Interestingly, in this large company, these innovative KS practices were not shared across all projects, even at the same level of software process maturity. These innovative KS practices addressed specific challenges faced by projects and helped the organization resolving initial problems partially or fully. Large-scale DevOps may require more KS practices since additional constraints have to be integrated between the business, development and operations functions. Such KS practices respond to projects operating in both large-scale and DevOps because such joint conditions multiply coordination issues and risks related to lack of competencies [START_REF] Hemon | Conceptualizing the Transition From Agile to DevOps: A Maturity Model for a Smarter IS Function[END_REF] . Moreover, we believe that project autonomy or self-organization and continuous improvement over a long period of time are also critical influencing factors. Thus, in the teams who developed KS practices:

• Projects were more advanced in agile and tended towards large-scale DevOps

• The larger they were, the more they were self-organized and used their relative autonomy to continuously improve over a long period of time,

• Also more they were more likely to develop innovative KS practices within their teams and across functional boundaries.

Clearly the emergence and implementation of these innovative KS practices is an outcome of multiple idiosyncratic conditions and we would need more data beyond the 18 projects studied here to reach any firm conclusion. However, observations in our ongoing research on this topic in other firms tend to support a large-scale DevOps effect and a continuous improvement effect related to self-organization in the generation of innovative KS practices. Using DevOps at large-scale amplifies the KS challenge as well as related solutions through the development of innovative practices. Such effects are not fixed, as the implementation of these practices is not fixed, and is subject to continuous adjustment to avoid deterioration and rigidity, or indeed being poured into concrete!

Table 1 . Knowledge Sharing Practices in Transition to Large-Scale DevOps (After 15 ) Maturity Level Characteristic of KS Practice Path to Large-Scale DevOps

 1

	Level 1: Agile	• More frequent KS due to iterative	• Large-Scale agile methods are used (e.g.
		process and more frequent	SAFe, LeSS).
		releases.	• DevOps is not achieved because silos
		• Better communication and sharing	still exist.
		between customer and developers.	• KS is limited to specific silos (mostly Dev,
		• Limited sharing among the team	Biz to a lower level).
		and little common culture with PO.
	Level 2:	• KS through the performance of
	Continuous	various tests (unit and non-
	Integration	regression tests) synchronized with
		code development.
		• Automation as far as possible, task
		automation knowledge transfer.
		• Partial KS. Developers need quick
		feedbacks.
		• Partial common vocabulary and
		culture boosting KS.
		• Some common tools fostering KS.
		• Different metrics and measurement
		systems limiting KS.

  Aymeric Hemon is an Assistant Professor in MIS at Essca School of Management and a researcher at the LEMNA Laboratory, University of Nantes. He is also Senior Lecturer in Software Engineering at the Mines Telecom Institute. His research focus on IS-enabled organizational transformation, particularly on agile methods and DevOps. Brian Fitzgerald is Director of Lero -the Irish Software Research Centre. He also holds an endowed professorship, the Krehbiel Chair in Innovation in Business & Technology, at the University of Limerick, Ireland. His publications include 15 books and more than 150 peerreviewed articles in the leading international journals and conferences in both the Information Systems and Software Engineering fields. Prior to taking up an academic position, he worked in the software industry for about 12 years. Contact him at University of Limerick, LERO, Castletroy, Barbara Lyonnet is an Associate Professor in Management Sciences at University of Nantes and a researcher at LEMNA Laboratory, France. She has published in the field of information system, quality management and supply chain management. She has also written several books including "Lean management". Contact her at University of Nantes, LEMNA, Chemin de la Censive du Tertre, Bâtiment Erdre, 44322 Nantes, France, barbara.lyonnet@univ-nantes.fr Frantz Rowe is a Professor at University of Nantes, and a researcher at LEMNA and at KTO, SKEMA Business School, France, former CIO of his university and Professor at Telecom Paris; an AIS Fellow an Honorary member of AIM and a member of IFIP 8.2 and 8.6; He is Editor Emeritus of EJIS. He directed the research program on DevOps at LEMNA on which this paper is based. Contact him at University of Nantes, LEMNA, Chemin de la Censive du Tertre, Bâtiment Erdre, 44322 Nantes, France, frantz.rowe@univnantes.fr

	Limerick,	V94	T9PX,	Ireland,
	brian.fitzgerald@ul.ie
				Contact him at Essca School of Management, 1
				rue Joseph Lakanal -BP 40348, 49003 Angers
				Cedex	01,	France,	aymeric.hemon-
				hildgen@essca.fr


Marion Ribout

Carole Henaux

Jean-François Llibre

Frédéric Messine

Batoul Attar

Three-Dimensional Analytical Model of Servovalve Torque Motor Using Reluctance Network

Keywords: Torque Motor, 3-D Analytical Equivalent Magnetic Circuit, Magnetic Flux Leakage

Electrohydraulic servovalves used in various aircraft systems (flight control system, auxiliary power unit, etc.) are a key element in fluid flow and pressure control. This paper presents a 3-D analytical modelling method of the electromagnetic performance of servovalves torque motor. This analytical model, based on a reluctance network, allows to quickly evaluate the actuator performance taking into account the complex 3-D geometry and the high flux leakage rate. This model enables to reduce the performance analysis time by about 89% compared to the conventional 3-D finite element calculation time. This is valuable for the design and optimization of the actuator geometry.

I. INTRODUCTION

The purpose of the study is to present a more accurate analytical model of an actuator called torque motor which is commonly used in electrohydraulic servovalves. The idea is to use, in a future work, this model for torque motor design optimization. The proposed analysis method for determining motor performance saves computation time compared to a finite element software. The difficulty in implementing such a model lies in the need to represent a 3-D actuator which has many flux leakages as shown in Fig. 1. A reluctance model which takes into account the 3-D geometry of the actuator is proposed.

Fig. 1. Studied torque motor and flux leakage visualization [START_REF] Ozzello | Servovalve for Regulating the Flow or Pressure of a Fluid[END_REF] Several studies have already been carried out. Merritt [START_REF] Merritt | Hydraulic Control Systems[END_REF] has developed a simple analytical model of the actuator electromagnetic behavior which is still widely used. However, this model is not accurate as neither the magnet reluctance nor the leakage reluctances are taken into account. Urata [START_REF] Urata | On the torque generated in a servo valve torque motor using permanent magnets[END_REF], Liu [START_REF] Liu | Influence of magnetic reluctances of magnetic elements on servo valve torque motors[END_REF] and Zhang [START_REF] Zhang | High Torque Density Torque Motor With Hybrid Magnetization Pole Arrays for Jet Pipe Servo Valve[END_REF] have improved it by integrating the magnetic characteristics of the magnet, flux leakages and the permeability of the yokes. However, these latter models cannot be used in an optimization process as an adjustment with experimental data is required each time the geometry is modified, hence the interest in developing a new analytical model without any need for recalibration. The study focuses on modeling the flux leakage without considering the magnetization curves of the soft ferromagnetic parts nor their saturation limits, i.e. considering their relative permeability infinite.

Fig. 2 shows a simple schematic 2-D view of the actuator under study. The actuator consists of two fixed iron parts, two magnets and a moving iron part supporting two coils fed along the same magnetization axis and separated by the pivot axis. The developed torque in the actuator depends on the current intensity 𝐼 consumed by the coils (𝑁 turns each), as well as the reluctance effect inherent to the movement of the armature relative to its centered position (𝜃). It is calculated either by determining the forces applied to the armature by Maxwell's stress tensor or by using the principle of virtual work, whose general expressions are given in [START_REF] Pyrhonen | Design of rotating electrical machines[END_REF]. These methods are both based on the knowledge of the magnetic fluxes that passing through the actuator. In previous studies, the expression of the force applied to the armature, according to the Maxwell stress tensor, assumes that the tangential component of the magnetic field in the air gap is zero and that only the fluxes flowing in the air gaps generate a force.

II. METHODOLOGY OF ANALYTICAL MODELS DEVELOPMENT

The equivalent reluctant network developed in this study is used to model the main flux path through the air gaps and magnets and the leakage flux paths between the various ferromagnetic parts (Fig. 3). The developed method provides a progressive improvement of the analytical model by iteratively integrating different types of leakage flux (Fig. 3). Four models are presented to highlight the impact of different leakages on the actuator electromagnetic torque estimation accuracy.

The magnetic potential and reluctance of the magnet are denoted 𝑉 𝑎 and ℛ 𝑎 respectively, a coefficient 𝑘 𝑎 gives the relation 𝑉 𝑎𝑒𝑞 = 𝑘 𝑎 𝑉 𝑎 and ℛ 𝑎𝑒𝑞 = 𝑘 𝑎 ℛ 𝑎 . Φ 1 , Φ 2 , Φ 𝑎 are the magnetic fluxes passing through air gap 1, 2 and magnet respectively.

Model 1 (M1) is based on the Merritt model [START_REF] Merritt | Hydraulic Control Systems[END_REF] that considers only the main flux path. Only the air gap reluctances ℛ 1,2 (𝜃), as well as the coil and basic magnet model (𝑘 𝑎 = 1) are taken into account. The M1 is still an improvement of what is done in [START_REF] Merritt | Hydraulic Control Systems[END_REF] as the magnet reluctance has added.

Model 2 (M2) implements a non-uniform magnetic induction in the magnet as introduced in [START_REF] Urata | On the torque generated in a servo valve torque motor using permanent magnets[END_REF], so an equivalent model of the magnet will be suggested (𝑘 𝑎 ∈]0,1[).

Model 3 (M3) adds the leakage fluxes that occur between the two pole shoes represented by the reluctance ℛ 𝑝𝑠𝑝𝑠 (shown in purple in Fig. 3). Model 4 (M4) is enhanced by considering the leakage between the pole shoes and the armature. These leakages have the particularity of being introduced by variable reluctances that depend on the position of the armature. They can be divided into 3 types: ℛ 𝑝𝑠𝑎 𝑖𝑛𝑡 (𝜃), the leakage on the inner sides of the armature, ℛ 𝑝𝑠𝑎 𝑒𝑥𝑡 (𝜃), the leakage on the outer end of the armature and ℛ 𝑝𝑠𝑎 𝑙𝑎𝑡 (𝜃) , the leakage on the top and bottom of the armature (shown in brown in Fig. 3).

A. Modeling the flux leakage at magnet level (M2)

The magnet is discretized into a number of layers 𝑛 𝑎 stacked along its magnetization axis. Each layer is assumed to be crossed by an uniform induction with a symmetrical evolution of the induction with respect to the central layer representing the middle of the magnet (Fig. 4). The leakage path can be assimilated to a half-circle, whose geometric parameters depend on the dimensions of the magnet.

Using the Thévenin and Norton theorems, an equivalent model of the magnet is determined, taking into account the modelled leakage path. The coefficient 𝑘 𝑎 is expressed as a mathematical series and converges to a finite value when 𝑛 𝑎 tends to infinity.

B. Modeling of magnetic leakage between two iron parts (M3-M4)

In M3-M4, the analytical expression of the leakage reluctance ℛ 𝑐 that exist between two iron parts is required. It is therefore necessary to estimate the length 𝑙 𝑐 and crosssectional area 𝑆 𝑐 of the flux tube as: The modeling is based on a simplification of the possible shapes of a magnetic flux tube in air (Fig. 5). The shape of the tube is restricted to a 3-D ellipse denoted ℰ, which can be set mathematically according to the following equation:

ℛ 𝑐 = ∫ 𝑑𝑙 𝜇 0 𝑆 𝑐 (𝑙) 𝑙 𝑐 0 (1)
∀(𝑥, 𝑦, 𝑧) ∈ ℰ, ((𝑥 -𝑢 𝑥 ) cos 𝜑 + (𝑦 -𝑣 𝑦 ) sin 𝜑) 2 𝑎 2 + (𝑧 -𝑤 𝑧 ) 2 𝑏 2 = 1 (2)
with (𝑢 𝑥 , 𝑣 𝑦 , 𝑤 𝑧 ) the origin of the ellipse ℰ, 𝜑 the ℰ rotation along the z-axis, 𝑎 and 𝑏 the semi-major and minor axis of ellipse ℰ . To obtain a non-tubular ellipse that can be represented in a 3-D plane, the following equation is used, ∀(𝑥, 𝑦, 𝑧) ∈ ℰ :

cos 𝜑 (𝑦 -𝑣 𝑦 ) = sin 𝜑 (𝑥 -𝑢 𝑥 )

(3) It is assumed that these equations, whose unknowns are (𝑎, 𝑏, 𝜑, 𝑢 𝑥 , 𝑣 𝑦 , 𝑤 𝑧 ), can model all flux in accordance with the observed magnetic symmetries of the actuator. However, it is found that there are an infinite number of ellipses passing through random points in 3-D space.

To restrict the set of solutions of the problem, the discontinuity of the boundary condition of the tangential component of the magnetic field 𝐻 ⃗ ⃗ between any two media A and B is added.

(𝐻 𝐴 ⃗⃗⃗⃗ -𝐻 𝐵 ⃗⃗⃗⃗⃗ ) × 𝑛 𝐴𝐵 ⃗⃗⃗⃗⃗⃗ = 𝐽 𝑠 ⃗⃗⃗ (4) Considering that no free current density 𝐽 𝑠 exists at the iron-air interface and that the permeability of iron is high compared to that of air, we can assume that the magnetic field 𝐻 ⃗ ⃗ is zero in iron (because the magnetic induction ‖𝐵 ⃗ ‖ is finite in iron). Thus, the tangential component of 𝐻 ⃗ ⃗ is set to be zero at the air-side interface. This requires that the slope of the ellipse ℰ , which represents the magnetic leakage line, is normal to the leakage surface. This assumption is restricted by the fact that the iron should not be saturated at the interfaces.

The equations relating to the above assumptions are applied between two distinct surfaces of the actuator 𝑆 𝑖 and 𝑆 𝑗 , where the points considered 𝑋 𝑖 ∶ (𝑥 𝑖 , 𝑦 𝑖 , 𝑧 𝑖 ) and 𝑋 𝑗 ∶ (𝑥 𝑗 , 𝑦 𝑗 , 𝑧 𝑗 ) are the contour points of the two surfaces. Solving the problem enables to determine the length (ellipse arc perimeter) and cross-sectional area of the leakage flux tube between two distinct surfaces and thus the values of the leakage reluctances (Fig. 5).

The cases of leakage between, first, the two pole shoes and, secondly, between the pole shoe and the armature are treated individually. Indeed, this two cases lead to two distinct situations in the resolution of the system of equations.

C. Case of leakages between the two pole shoes (M3)

The two pole shoes in the actuator are both symmetrical in the (0xy) plane. Only the points 𝑋 𝑖 and 𝑋 𝑗 which are symmetrical along this plane, admit a solution to the problem set. Nevertheless, solving the above system of equations leads to an infinite number of solutions (Fig. 6). The aim of our model is to obtain a unique and repeatable model of the flux tube. The methodology applied is to consider that there are two ellipses with a minimum and maximum perimeter for each pair of surfaces studied. This defines the minimum and maximum length of the flux tube between these two surfaces. An average length is then associated with the studied flux tube.

In order to find the minimum perimeter, the ellipse is not allowed to cross any iron part, which means that the leakage flux tube must remain in the air. The minimum coordinates of the ellipse is set to (𝑥 𝑚𝑖𝑛 , 𝑦 𝑚𝑖𝑛 ,0). To find the maximum perimeter, the flux line is bound to remain in an air box of finite size, larger than the dimension of the actuator (in this case 2.5 times larger). Thus, the maximum coordinates of the ellipse is set to (𝑥 𝑚𝑎𝑥 , 𝑦 𝑚𝑎𝑥 ,0). In Fig. 7, the blue zone represents the maximum cross section area for the leakage flux outside the actuator in (0xy) plane and the yellow zone is the maximum cross section area for the leakage flux inside the actuator in (0xy) plane. The leakage flux tube is included in these areas as it passes through the (0xy) plane.

The cross section of the flux tube is assumed to vary linearly between the leakage area in the iron and the area of the tube in the (0xy) plane. This latter area depends on the previously defined limits, and is located between the actuator and the specified air box. Note that it is important to avoid considering the same leakage area multiple times, as this would lead to an underestimation of the leakage reluctance. Indeed, for different pairs of surfaces, it is possible that the flux tubes passing the (0xy) plane share the same passage zone. In this case, it is possible to simultaneously treat several pairs of surfaces whose leakage pass through the same zone in order to simplify the problem.

D. Case of leakages between pole shoe and armature (M4)

In the case of leakage between pole shoe and armature, the (0xy) plane is no longer a symmetry plane for the surfaces under study (e.g. Fig. 8). The system of equations can only be solved by a unique solution, or no solution at all. The effective leakage surface is determined by the position of the contour points of the surface that allow a solution to the problem under study. For a given couple of surfaces, an average length of flux tube is determined and the cross-section of the tube is assumed to vary linearly between the effective surface area of the leakage from the pole shoe to that of the armature.

III. VALIDATION OF ANALYTICAL MODEL

The different analytical models are validated by comparing their results with the results obtained by JMAG ® for a given geometry. It should be noted that the experimental results obtained on the test bench for the whole system agree well with the results of the finite element software.

For the actuator under study, the magnetic induction remains below saturation limit according to JMAG ® simulation. Regarding the used ferromagnetic material, it can be deduced, from the maximum induction reached, that the relative permeability remains very high (>1e4). The impact of the iron reluctance is therefore low on the actuator magnetic performance and the assumption for modelling the leakage lines remains consistent. For an optimization process, however, magnetic induction modeling is mandatory to avoid magnetic saturation issues. In a such geometric construction, magnetic induction cannot be approximated to its normal component which is related to the magnetic flux. The estimation of tangential component is required to accurately model the magnetic induction in the actuator. This issue will be addressed in further work.

The improvement in actuator performance modeling accuracy is assessed for the four successive models. Model 4 provides the most accurate depiction of the magnetic fluxes passing through the magnets and air gaps as shown in Table I. However, it is noted that the deviation for air gaps magnetic fluxes (and in particular from air gap 2) remain significantly even at zero angle. This difference can be explained by the approximate calculation of the leakage reluctances in 3-D, but also by the position of the reluctances in the equivalent reluctance network. Model 1 1.23e -4 9.82e -5 1.11e -4 1.67e -4 6.83e -5 1.18e -4 Model 2

1.05e -4 8.02e -5 9.27e -5 1.44e -4 5.63e -5 1.00e -4 Model 3

5.67e -5 3.18e -5 1.54e -4 7.96e -5 2.27e -5 1.56e -4 Model 4

5.37e -5 2.88e -5 1.57e -4 7.55e -5 2.06e -5 1.59e -4 JMAG ® 4.92e -5 2.18e -5 1.57e -4 6.64e -5 1.65e -5 1.58e -4 Two methods of computing the electromagnetic torque are used: the Maxwell stress tensor (A) in agreement with the expression used in [START_REF] Merritt | Hydraulic Control Systems[END_REF]- [START_REF] Zhang | High Torque Density Torque Motor With Hybrid Magnetization Pole Arrays for Jet Pipe Servo Valve[END_REF] and the virtual work method (B). The results given in TABLE II. show the torque modelling deviation comparing to JMAG ® using both methods and following the expression: 𝑒𝑟𝑟 = (Γ Model (𝐼, 𝜃) -Γ JMAG (𝐼, 𝜃)) Γ JMAG (𝐼 𝑛𝑜𝑚 , 3°) ⁄ ( 5)

Here, coils are supplied with the rated current (0.3A). As TABLE II. depicted, the accuracy of the estimated torque has been improved progressively with model enhancement. However, the use of Maxwell stress tensor, using the assumptions of previous studies described in I, can be questioned by the presence of leakage between the pole shoes and the armature, which contributes to the generation of torque, as well as the presence of the magnetic field tangential component in the air gap for non-zero armature pivot angles (the flux lines being modelled by ellipses).

Using magnetic coenergy to calculate the torque shows an improvement in the accuracy of the modelling, especially at high angles with the increase of leakage dissymmetry and the occurrence of a tangential component in the air gaps. TABLE III. describes the evolution of the accuracy achieved by successive improvements of the model using magnetic coenergy to estimate the torque. The armature movement is between -3° and +3°. The characterization time (CPU time) corresponds to the time required to calculate the torque at a given current, from -3° to +3° by increments of 1°, whose evolution can be evaluated according to the model used. The needed time to characterize the actuator (e.g. for 7 points) using JMAG ® software is 315s. This finite element analysis is done with the largest possible mesh size in order to limit the calculation time without any loss of accuracy. As TABLE III. depicted, the developed model M4 allows to converge to the JMAG ® results with a much lower simulation time (up to 11 times faster for characterization time).

IV. CONCLUSION

In this paper, an enhanced analytical model is presented to predict the torque motor electromagnetic performance, whatever its geometry and without any need of reluctance recalibration with Finite Element Method (FEM) software, which is the major improvement regarding the literature. The purpose of this model is to be used in the torque motor design optimization process, which allows to reduce the characterization time. An incremental improvement of the analytical model is presented to show the enhancement of estimated torque. The results of these different analytical models are established by comparing them to a 3-D FEM software. This methodology enables fast pre-optimization of the structure to meet defined performance requirements. However, in an optimization process, it is worth to model the magnetization state of soft ferromagnetic parts to avoid their saturation at nominal operation. This issue is the topic of ongoing work.

The applied methodology can be used whenever elliptical flux tubes in air need to be accurately modeled to assess magnetic performance, regardless of actuator design as long as the iron is not saturated.

Fig. 2 .

 2 Fig. 2. 2-D diagram of the actuator with different magnetic fluxes

Fig. 3 .

 3 Fig. 3. Developed reluctance network model

Fig. 4 .

 4 Fig. 4. Magnet modeling with several layers (example with 4 layers)

Fig. 5 .

 5 Fig. 5. Visualization of flux tube between two different surfaces

Fig. 6 .

 6 Fig. 6. Visualization of a set of possible ellipses that represent the leakage

Fig. 7 .

 7 Fig. 7. Maximum cross-section area of the magnetic flux tube in (0xy) plane

Fig. 8 .

 8 Fig. 8. Visualization of a leakage line between pole shoe and armature

TABLE I

 I

	.	MAGNETIC FLUX EVOLUTION BASED ON THE MODEL USED
		𝒊 = 𝟎. 𝟑𝑨 / 𝜽 = 𝟎°	𝒊 = 𝟎. 𝟑𝑨 / 𝜽 = 𝟑°
		𝜱 𝟏	𝜱 𝟐	𝜱 𝒂	𝜱 𝟏	𝜱 𝟐	𝜱 𝒂
		(Wb)	(Wb)	(Wb)	(Wb)	(Wb)	(Wb)

TABLE III

 III

	.	ACCURACY AND CPU TIME OF THE DIFFERENT METHODS
		max(err) at	max(err) at	Characterization
		𝑰 = 𝑰 𝒏𝒐𝒎 (%)	𝑰 = 𝟎𝑨 (%)	Time-7 points (s)
	Model 1	348%	284%	0.8
	Model 2	238%	190%	0.8
	Model 3	17%	16%	1.1
	Model 4	14%	10%	28-35