Keywords: Centaurea corymbosa, Climate change, Conservation, Extinction risk, Long-term survey, Matrix projection models, Stochastic projections

Climate change is a growing threat for global biodiversity, in particular for narrow endemic species. The Mediterranean region, which harbors an exceptional biodiversity, has been identified as one of the most sensitive regions to climate change. Based on a 22-year monitoring period, we analyzed the dynamic and viability of the six extant populations of a narrow endemic plant species of the Mediterranean area, Centaurea corymbosa, to predict their fate under two climatic scenarios. We constructed matrix projection models to calculate current asymptotic growth rates and to perform stochastic projections including both demographic and environmental stochasticity. Neither asymptotic growth rates nor their temporal variance were linked to population size and age at flowering. Randomization tests showed that asymptotic growth rates were significantly different among years but not among populations. An increase in temperature and a decrease in the number of wet days had a negative impact on the whole life-cycle, particularly in the summer period, and thus reduced asymptotic growth rates. Stochastic projections showed that an increased frequency of extreme climatic events increased population extinction risk and decreased mean time to extinction. The warm scenario had a more dramatic impact on population viability than the dry scenario. Management recommendations are proposed to increase population viability of endangered plant species such as C. corymbosa that face climate change.

Introduction

Ongoing climate change is expected to result in increase in drought and warm periods in several regions due to both an increasing frequency of extreme climatic events and long-term gradual changes (IPCC, 2007). Compared to some other taxonomic groups, plants are more vulnerable to climate change since they have relatively low migration capacity [START_REF] Malcolm | Global warming and extinctions of endemic species from biodiversity hotspots[END_REF][START_REF] Thomas | Comparative losses of British butterflies, birds, and plants and the global extinction crisis[END_REF]. Gradual climate change and extreme events have already caused range shifts for some plant species (Chen et al., 2011), including range contraction with long-term population declines and extinctions [START_REF] Lennartsson | Demographic variation and population viability in Gentianella campestris: effects of grassland management and environmental stochasticity[END_REF][START_REF] Doak | Demographic compensation and tipping points in climateinduced range shifts[END_REF][START_REF] Selwood | The effects of climate change and landuse change on demographic rates and population viability[END_REF][START_REF] Wiens | Climate-Related Local Extinctions Are Already Widespread among Plant and Animal Species[END_REF], or range expansion [START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF][START_REF] Williams | Life-history evolution under climate change and its influence on the population dynamics of a long-lived plant[END_REF]. This change is expected to become an even more important driver of global biodiversity loss over the next century [START_REF] Heller | Biodiversity management in the face of climate change: A review of 22 years of recommendations[END_REF]. Thus, population dynamics of many plant species are expected to be significantly affected by such changes [START_REF] Breda | Temperate forest trees and stands under severe drought: a review of ecophysiological responses, adaptation processes and long-term consequences[END_REF][START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF][START_REF] Nicolè | Interdependent effects of habitat quality and climate on population growth of an endangered plant[END_REF][START_REF] Ulrey | Life at the top: Long-term demography, microclimatic refugia, and responses to climate change for a highelevation southern Appalachian endemic plant[END_REF]. For conservation purposes, identifying climatic factors that impact population dynamics and predicting the consequences of climate change on species persistence have therefore become an important prerequisite to develop effective conservation strategies aimed at limiting population decline and extinction risk [START_REF] Cotto | A dynamic eco-evolutionary model predicts slow response of alpine plants to climate warming[END_REF][START_REF] Malcolm | Global warming and extinctions of endemic species from biodiversity hotspots[END_REF][START_REF] Thomas | Comparative losses of British butterflies, birds, and plants and the global extinction crisis[END_REF].

Climate change can either positively or negatively affect population dynamics [START_REF] Lawson | Environmental variation and population responses to global change[END_REF][START_REF] Marrero-Gómez | Population viability of the narrow endemic Helianthemum juliae (Cistaceae) in relation to climate variability[END_REF]. For instance, climate change proved beneficial to populations of the widespread orchid species (Orchis purpurea) and is predicted to induce range expansion in the near future [START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF][START_REF] Williams | Life-history evolution under climate change and its influence on the population dynamics of a long-lived plant[END_REF]. In contrast, several studies have documented negative effects of summer temperature on growth rates for other plant species [START_REF] Aragón-Gastélum | Seedling survival of three endemic and threatened Mexican cacti under induced climate change[END_REF][START_REF] Riba | Effects of regional climate and small-scale habitat quality on performance in the relict species Ramonda myconi[END_REF][START_REF] Shryock | Population viability of pediocactus bradyi (cactaceae) in a changing climate[END_REF]. In some cases, a given climatic variable has been shown to have diverging effects on separate vital rates within a species life-history [START_REF] Csergő | Less favourable climates constrain demographic strategies in plants[END_REF][START_REF] Jolls | Population viability analysis and the effects of light availability and litter on populations of Cirsium pitcheri, a rare, monocarpic perennial of Great Lakes shorelines[END_REF][START_REF] Nicolè | Interdependent effects of habitat quality and climate on population growth of an endangered plant[END_REF][START_REF] Peñuelas | Nonintrusive field experiments show different plant responses to warming and drought among sites, seasons, and species in a north-south european gradient[END_REF][START_REF] Treurnicht | Environmental drivers of demographic variation across the global geographical range of 26 plant species[END_REF]. For example, a warm summer increased the flowering probability of Dracocephalum austriacum, whereas it decreased plant survival, leading to reduced effects of climate on population growth rates [START_REF] Nicolè | Interdependent effects of habitat quality and climate on population growth of an endangered plant[END_REF]. Such opposite effects on population dynamics and demographic processes make it difficult to predict the net consequences of climate change on population viability [START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF][START_REF] Nicolè | Interdependent effects of habitat quality and climate on population growth of an endangered plant[END_REF]. The Mediterranean region has been identified as one of the most sensitive regions to climate change [START_REF] Ducrocq | Climate change in the Mediterranean region[END_REF]. Climate model projections indicate that warming and drying will likely continue (AllEnvi, 2016). At the end of the century (2080-2099 period), annual mean temperature is predicted to increase (+3.5°C) with more extreme dry event frequency (+46%) in this region [START_REF] Christensen | Regional climate projections[END_REF]. Moreover, in most areas of the Mediterranean region, precipitation is predicted to decrease (-12%), particularly in summer (-24%, [START_REF] Christensen | Regional climate projections[END_REF]. The Mediterranean region harbors an exceptional biodiversity, including a large number of rare and endemic plant species. Among them, 36% are narrow endemics, i.e., they grow only in a single area or have a narrow geographic range [START_REF] Thompson | Ecological differentiation of Mediterranean endemic plants[END_REF]. Most of these species occur in rocky habitats, on steep slopes and in open vegetation with low species richness [START_REF] Lavergne | The biology and ecology of narrow endemic and widespread plants: a comparative study of trait variation in 20 congeneric pairs[END_REF][START_REF] Thompson | Ecological differentiation of Mediterranean endemic plants[END_REF]. Such habitats are characterized by their stability both in relation to vegetation succession and human disturbance [START_REF] Lavergne | Environmental and human factors influencing rare plant local occurrence, extinction and persistence: a 115-year study in the Mediterranean region[END_REF]. Narrow endemic species are expected to be particularly sensitive to environmental shifts, as several of these species are extremely specialized and have evolved low dispersal ability [START_REF] Isik | Rare and endemic species: why are they prone to extinction? Turk[END_REF][START_REF] Lavergne | The biology and ecology of narrow endemic and widespread plants: a comparative study of trait variation in 20 congeneric pairs[END_REF][START_REF] Lavergne | Environmental and human factors influencing rare plant local occurrence, extinction and persistence: a 115-year study in the Mediterranean region[END_REF][START_REF] Thuiller | Niche properties and geographical extent as predictors of species sensitivity to climate change[END_REF].

Projection models that incorporate temporal environmental stochasticity are a powerful tool for predicting long-term population dynamics under climate change (Andrello et al., 2012;[START_REF] Crone | How do plant ecologists use matrix population models?[END_REF][START_REF] Crone | Ability of matrix models to explain the past and predict the future of plant populations[END_REF][START_REF] Garcıá | An evaluation of the status of five threatened plant species in the Pyrenees[END_REF]. An increase in variability in vital rates due to climatic variations will usually decrease the long-term population growth [START_REF] Morris | Longevity can buffer plant and animal populations against changing climatic variability[END_REF]. Thus, temporal variations may amplify population fluctuations and thus increase extinction risk, an effect being more pronounced for small populations [START_REF] Isik | Rare and endemic species: why are they prone to extinction? Turk[END_REF][START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF][START_REF] Morris | Quantitative Conservation Biology: Theory and Practice of Population Viability Analysis[END_REF]. Species life-histories can however buffer the effects of climate variation. For instance, long duration of the life-cycle has been shown to decrease the impact of environmental stochasticity, making long-lived species facing climate change less prone to extinction [START_REF] Morris | Longevity can buffer plant and animal populations against changing climatic variability[END_REF]. Demographic stochasticity can also be an important driver of population dynamics, whenever populations are of a small size [START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF]. Such processes refer to chance events of individual survival and reproduction causing random variations in population growth rates [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF][START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF]. These variations are expected to be more pronounced in small populations and thus lead to random fluctuations in population size that increase extinction risks [START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF][START_REF] Zeigler | Predicting responses to climate change requires all life-history stages[END_REF]. Including both environmental and demographic stochasticity in population viability analyses is thus crucial to forecast the fate of populations under climate change [START_REF] Crone | How do plant ecologists use matrix population models?[END_REF][START_REF] Crone | Ability of matrix models to explain the past and predict the future of plant populations[END_REF][START_REF] Kaye | The effect of stochastic technique on estimates of population viability from transition matrix models[END_REF][START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF]. Demographic studies investigating the long-term effect of climate change on narrow endemic Mediterranean plants remain rare. Such studies have mostly relied on short-term surveys, thus exploring limited year-to-year climatic variations [START_REF] Crone | How do plant ecologists use matrix population models?[END_REF] and consider only a subset of extant populations (e.g. Centaurea hyssopifolia; Matesanz et al., 2009, Brassica insularis;Noel et al., 2010, Ramonda myconi;[START_REF] Riba | Effects of regional climate and small-scale habitat quality on performance in the relict species Ramonda myconi[END_REF], making it difficult to predict species responses to climate change [START_REF] Crone | How do plant ecologists use matrix population models?[END_REF][START_REF] Ehrlén | Advancing environmentally explicit structured population models of plants[END_REF]. Long-term demographic surveys are needed to understand how past climatic variations have affected population dynamics, and to predict population viability under climate change (Andrello et al., 2012;[START_REF] Franklin | Big data for forecasting the impacts of global change on plant communities[END_REF][START_REF] Hunter | Climate change threatens polar bear populations: a stochastic demographic analysis[END_REF][START_REF] Menges | Population viability analyses in plants: challenges and opportunities[END_REF]. Typically, at least 15-20 years of observations are necessary to predict population growth rate or extinction risk adequately [START_REF] Zeigler | Predicting responses to climate change requires all life-history stages[END_REF]. This is particularly true for perennial species, which may have a stage in their life-cycle that allows for species persistence under unfavorable conditions [START_REF] Blume-Werry | Short-term climate change manipulation effects do not scale up to long-term legacies: effects of an absent snow cover on boreal forest plants[END_REF][START_REF] Huelber | Uncertainty in predicting range dynamics of endemic alpine plants under climate warming[END_REF][START_REF] Morris | Longevity can buffer plant and animal populations against changing climatic variability[END_REF].

Centaurea corymbosa is a narrow monocarpic short-lived species endemic to the Mediterranean region. The species grows in the Massif de la Clape in Southern France, on the top of cliffs and in nearby rocky areas of open vegetation [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF]. Only six populations are known, which have been surveyed since June 1994 using permanent plots. In this paper, we used this unique demographic dataset collected during a 22-year period (totaling 6112 individual lifehistories) to predict species dynamics under different climate scenarios. To do so, using matrix population models, we estimated asymptotic growth rates and vital rates for each population and each pair-of-years over the 22-year period. First, we tested for spatial and temporal variations in population dynamics and also tested whether small populations had lower asymptotic growth rate and showed higher variance in asymptotic growth rate than large populations. In addition, we tested whether age at flowering could buffer the effects of environmental stochasticity. Next, we assessed the relationship of climatic variables with demographic parameters over the 22-year period. More specifically, we tested whether any climatic situation contributing to drought negatively affected asymptotic growth rates and vital rates, since water availability is essential for the establishment of newly recruited plants and the survival of vegetative plants. Moreover, because flowering is closely linked with size and thus with plant growth in monocarpic plants [START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF][START_REF] Metcalf | Evolutionary demography of monocarpic perennials[END_REF][START_REF] Williams | Life-history evolution under climate change and its influence on the population dynamics of a long-lived plant[END_REF], we tested whether probability of flowering depended on climatic conditions prevailing only the months before flowering or varied according to a cumulative multiyear climate effect. Finally, by incorporating both environmental and demographic stochasticity, we simulated the fate of populations under two climatic scenarios, a warm scenario reflecting an increase in the frequency of hot years, and a dry scenario corresponding to an increase in the frequency of drought events, and thus tested which of the two scenarios would have the largest impact on population persistence. [START_REF] Colas | Statut démographique de Centaurea corymbosa Pourret (Asteraceae), Hormatophylla pyrenaica (Lapeyr.) Cullen & Dudley (Brassicaceae) et Marsilea strigosa Willd. (Marsileaceae-Pteridophyta), trois plantes rares du sud de la France[END_REF][START_REF] Imbert | Dispersal by ants in Centaurea corymbosa (Asteraceae): What is the elaiosome for?[END_REF][START_REF] Imbert | Do endemic species always have a low competitive ability? A test for two Mediterranean plant species under controlled conditions[END_REF][START_REF] Kirchner | Effects of local density on insect visitation and fertilization success in the narrow-endemic Centaurea corymbosa (Asteraceae)[END_REF], population dynamics [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Kirchner | Modelling the dynamics of introduced populations in the narrow-endemic Centaurea corymbosa: a demo-genetic integration[END_REF] and population genetics [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF][START_REF] Colas | Spatio-temporal variation of reproductive success and conservation of the narrow-endemic Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Fréville | Comparative allozyme and microsatellite population structure in a narrow endemic plant species, Centaurea corymbosa Pourret (Asteraceae)[END_REF][START_REF] Fréville | Predicting endemism from population structure of a widespread species: case study in Centaurea maculosa Lam. (Asteraceae)[END_REF][START_REF] Hardy | Fine-scale genetic structure and gene dispersal in Centaurea corymbosa (Asteraceae). II. Correlated paternity within and among sibships[END_REF][START_REF] Riba | Variation in dispersal traits in a narrow-endemic plant species, Centaurea corymbosa Pourret. (Asteraceae)[END_REF]. Here, we only summarize the most important features and results related to the current study. C. corymbosa is restricted to a 3-km 2 area and is known from only six populations found on different cliffs 0.3-2.3 km apart (mean elevation 132 m, SD = 37.4, n=41 permanent plots, see below). Individuals grow on the top of cliffs and in rocky areas. These populations are currently not directly threatened by human activities. However, habitat closure due to the abandonment of grazing contributes to the isolation of open and suitable habitats, and thus increased fragmentation among patches. The species has a monocarpic perennial life-cycle. Individuals stay as a rosette for 2 to 13 years before flowering. The flowering period extends from May to mid-August, and most of the seeds germinate between September and December. Seeds are dispersed over short distances by wind and ants. Despite suitable habitats in the vicinity of the six populations, it has been shown that C. corymbosa is very unlikely to colonize new sites [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF][START_REF] Olivieri | Why evolution matters for species conservation: perspectives from three case studies of plant metapopulations[END_REF][START_REF] Riba | Variation in dispersal traits in a narrow-endemic plant species, Centaurea corymbosa Pourret. (Asteraceae)[END_REF].

Materials & Methods

Demographic dataset

Data have been collected since June 1994 in the six populations. However, in the first census, the status of non-flowering plants (seedling or rosette) could not be assessed. Thus, matrix population models were based on 22 years of demographic survey , except for one population (Portes, 1996(Portes, -2016)). Data collection is described in details in [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF]. Every 3 months (June, September, December and March), we recorded the presence and the status of each individual within 41 permanent plots and new seedlings were added to the datasets (see Appendix A1). During the flowering period (mid-June), we also surveyed the whole distribution area of the species to count the total number of flowering plants per population (exhaustive count, Table 1 and Appendix A2). A total of 6112 individual life-histories were used to construct population projection models (Table 1 andAppendix A1).

Climatic dataset

Climatic data were obtained from the closest meteorological station located at INRA Pech Rouge (43°1444 N, 3°1338 E, elevation = 40 m) less than 5 km from the populations of C. corymbosa. We used daily mean temperature, daily minimum and maximum temperatures and daily precipitation. Over the 22 years of this study, the Massif de la Clape experienced a classical warm Mediterranean climate (average mean temperature 15.1 ± 0.6 °C from June t to May t+1, Fig. 1) with a warm summer period. Precipitation regime was also characteristic of the Mediterranean climate with an average of 540.8 mm per year from June t to May t+1 with large inter-annual variation (SD = 175.46, Fig. 1). The number of days with precipitation > 1 mm ranged from 35 days to 68 days with an average of 48.5 days per pair-of-years (SD = 8.9, Fig. 1).

Deterministic analyses 2.2.1. Matrix projection models

As described in [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF], the life-cycle is based on a prebreeding census performed before the germination pulse, with a one-year interval from June t to May t+1. Three stages were defined: seedlings (individuals less than 1 year old), rosettes (vegetative plants older than 1 year), and flowering plants. We estimated lower-level vital rates (survival s i , flowering probability α i conditional to survival and fecundity f) to construct a stage structured Matrix Projection Model (MPM). For each population and each pair-of-years, the projection matrix representing the life-cycle is given by:

                  3 3 2 2 1 1 3 3 2 2 1 1 0 1 1 ) 1 ( 0 0    species,
and had negligible effects on population dynamics (see appendix B). Using a deterministic MPM (linear, time-invariant), we calculated the asymptotic growth rate λ a as the dominant eigenvalue of the matrix A of the corresponding pair-of-years. The model did not include seed bank and migration. Indeed, less than 5% of seeds may remain dormant for one year and seed dispersal among populations is highly limited [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF][START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Imbert | Dispersal by ants in Centaurea corymbosa (Asteraceae): What is the elaiosome for?[END_REF]. The net fecundity fs 0 was estimated as the ratio of the number of seedlings observed in June of a given year over the number of flowering plants observed within permanent plots in June of the previous year. Our three-month survey made it possible to decompose the net fecundity fs 0 into a fecundity term f (number of just-emerged seedlings assessed every three months over number of flowering plants in June t-1) and survival of just-emerged seedlings s 0 (number of seedlings observed in June t over total number of just-emerged seedlings observed from June t-1 to May t, [START_REF] Colas | Restoration demography: a 10-year demographic comparison between introduced and natural populations of endemic Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF]. We obtained four null values in asymptotic growth rates (three for Cruzade and one for Peyral), which corresponded to years with no flowering plant within plots and null values in rosette survival s 2 . In such case, the persistence of the plot depended on new rosettes coming from the seedling stage (s 1 never equaled zero), and seed dispersal from the closest flowering plants outside plots. When calculating the geometric mean of asymptotic growth rates over years for each population, we excluded these null values.

Patterns of spatio-temporal variations in asymptotic growth rates

To test for temporal and spatial variation in asymptotic growth rates, we used nonparametric randomisation tests [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF]. Temporal variation was tested by randomly permuting individual life-histories (status at t and fate at t+1) among pair-of-years when considering all data as a single population. For spatial variation, we permuted the whole individual life-history (from germination to death) between populations (see [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF]. Each lifehistory appeared exactly once in each randomized dataset, maintaining the original sample sizes. For each set of permuted data, we calculated asymptotic growth rate for each group (year or population) and then computed the standard deviation of λ a between groups. The probability that σ(λ a ) ≥ σ obs under the null hypothesis H 0 (no group effect) was computed based on 2000 random permutations. We rejected H 0 when this probability was smaller than 0.05 [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF]. We used linear models to test for the effect of population size on asymptotic growth rates and their temporal variance, using the total number of flowering plants obtained by the exhaustive survey as a proxy of population size (Table 1). We used also a linear model to test for the effect of mean age at flowering (Table 1) on the variance in population growth rate.

Retrospective and prospective analyses

We performed a life-table response analysis (LTRE) with a random design to assess how lower-level vital rates contributed to the variance in asymptotic growth rates var(λ a ) observed among years. This analysis was performed on the pooled data over populations, provided that we did not detect significant spatial variation in population growth rates. The decomposition of var(λ a ) in term of lower-level vital rates p is expressed as:

𝑣𝑎𝑟(𝜆 𝑎 ) ≈ ∑ 𝑖,𝑗 𝑐𝑜𝑣(𝑝 𝑖 ,𝑝 𝑗 ) ∂𝜆 𝑎 ∂𝑝 𝑖 ∂𝜆 𝑎 ∂𝑝 𝑗
where is the covariance of p i and p j [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF]. A prospective analysis was also 𝑐𝑜𝑣(𝑝 𝑖 ,𝑝 𝑗 ) conducted to assess the population dynamic response to changes in lower-level vital rates. We calculated the elasticity of λ a to lower-level vital rates p as:

𝑒 = 𝑝 𝜆 𝑎 ∂𝜆 𝑎 ∂𝑝 2.

Relationships between demographic parameters and climate

From the raw climatic dataset, we extracted seven relevant variables assumed to have an influence on demographic parameters, given the existing literature [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Riba | Effects of regional climate and small-scale habitat quality on performance in the relict species Ramonda myconi[END_REF] and our knowledge of Mediterranean plant species. We thus calculated the average of daily mean temperature and daily maximum temperature, the number of days with mean T° > 25°C (corresponding to mean + SD) and the number of freezing days (minimum T° < 0°C). In addition, we calculated both the average cumulative precipitation and the number of days with precipitation > 1 mm that hereafter will be referred as the number of wet days. We also calculated the number of days with precipitation > 20 mm. This threshold value was equal to mean + SD. We thus considered this variable as reflecting the number of heavy precipitation events in the Massif de la Clape.

We tested for the effect of climate on demographic parameters using generalized linear models with identity link for asymptotic growth rates and fecundity f (both log-transformed), and logit link for survival and flowering probabilities with the binomial distribution. First, we tested for relationships between annual averages of each climatic variable calculated from June t to May t+1 and asymptotic growth rates λ a , and also tested for quadratic effects and multiyear cumulative effect with a time lag of one year of each climatic variable (Appendix E.1). We then tested for the effect of the variables that we found significant in the single-variable GLMs, using a forward stepwise selection approach. Starting from the null model, we added at each step the variable having the highest R 2 value in the single-variable models (Appendix E.1). For logistic models, we used R 2 LR , the pseudo-coefficient of determination based on the likelihood-ratio test and given by R 2 LR = 1-(L null /L full ) 2/n where L null and L full are the likelihoods of the null and the fitted models, respectively (Shinichi [START_REF] Nakagawa | A general and simple method for obtaining R2 from generalized linear mixed-effects models[END_REF]. The significance of each added variable in the model was tested using an analysis of deviance (Appendix E.2). From the best model, we reported the estimated coefficient (β) and its significance for each variable in the model after scaling climatic variables. Second, to understand biological processes at a finer scale, we tested for the impact of the seasonal climate on lower-level vital rates by considering only climatic variables that significantly impacted asymptotic growth rates, using the same approach as described above. We included the seasonality effect in GLMs by averaging variables over three-month periods separating two successive censuses (Appendix E.3). GLMs were fitted using R Stats-package (version 3.3.1). The R 2 LR was calculated using the R-command 'r.squaredLR' from the MuMIn-package (version 1.15.6).

Stochastic projections under climate change

Based on the IPCC Fourth Assessment Report, the frequency of extreme warm and dry events is expected to increase [START_REF] Christensen | Regional climate projections[END_REF]. We thus built stochastic matrix models to simulate the fate of C. corymbosa populations under two scenarios of climatic variations, representing an increased frequency of extreme events either for temperature or precipitation (Bucharovà et al., 2012;[START_REF] Shryock | Population viability of pediocactus bradyi (cactaceae) in a changing climate[END_REF]. To simulate the fate of populations, we incorporated environmental stochasticity by drawing an entire matrix at each time step (sampling with replacement) among the 21 available matrices [START_REF] Crone | How do plant ecologists use matrix population models?[END_REF][START_REF] Crone | Ability of matrix models to explain the past and predict the future of plant populations[END_REF][START_REF] Kaye | The effect of stochastic technique on estimates of population viability from transition matrix models[END_REF]. Because C. corymbosa has small population sizes, we also incorporated demographic stochasticity in the models. Such stochasticity is due to the independent chance of transition and reproduction among individuals [START_REF] Engen | Demographic and Environmental Stochasticity-Concepts and Definitions[END_REF][START_REF] Lande | Risks of Population Extinction from Demographic and Environmental Stochasticity and Random Catastrophes[END_REF]Melbourne and Hasting, 2008). From the existing individuals at year t, we drew the number of individuals making each transition from stage j to stage i including death at each time step in a multinomial distribution characterized by the a ij transition probabilities of the matrix that we randomly drew (Andrieu et al., 2017;[START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF]Melbourne and Hasting, 2008;[START_REF] Morris | Quantitative Conservation Biology: Theory and Practice of Population Viability Analysis[END_REF]. The number of recruited seedlings was drawn in a Poisson distribution with mean equal to the fs 0 value of the drawn matrix. In the case of C. corymbosa, demographic stochasticity indeed increased fluctuations in population size over time and thus increased extinction risk (see Appendix F), in agreement with both theoretical studies (e.g. [START_REF] Engen | Extinction in relation to demographic and environmental stochasticity in age-structured models[END_REF][START_REF] Lande | Risks of Population Extinction from Demographic and Environmental Stochasticity and Random Catastrophes[END_REF][START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF] and empirical ones (e.g. [START_REF] Fujiwara | Demography of the endangered North Atlantic right whale[END_REF]Melbourne and Hasting, 2008;Jepsonn and Forslund, 2012). Therefore, we combined both demographic and environmental stochasticity to simulate the fate of populations under climate scenarios.

The initial population size N 0 used to start our simulations was calculated from the stable stage distribution obtained from the arithmetic mean of the total number of flowering plants recorded in the population (exhaustive survey) from 2010 to 2016, and the scaled eigenvector W from the arithmetic mean matrix over years (2010)(2011)(2012)(2013)(2014)(2015)(2016) when pooling data over populations. N 0 is given by where w 3 , the third element of the scaled eigenvector, represents the 𝑁 0 = 𝑊 𝑁𝑓𝑟 𝑤 3 proportion of the number of flowering plants. We simulated population dynamics using 1000 stochastic iterations, each iteration representing a trajectory of population size over 100 years. For a given trajectory we calculated the growth rate λ j as:

𝜆 𝑗 = 𝑒𝑥𝑝 [ 𝑙𝑛(𝑛 𝑗 (𝑇)) -𝑙𝑛(𝑛 𝑗 (0)) 𝑇 ]
where n j (T) and n j (0) were the population size of the j th trajectory at t = T and t = 0, respectively, and T being the last year at which population size was non-null. The stochastic growth rate λ s was obtained as the median of λ j over 1000 trajectories. Extinction probability at a given time t corresponds to the number of trajectories for which N(t) = 0 over the total number of trajectories; we calculated extinction probability at t = 50 years (P 50 ), t = 100 years (P 100 ) and the mean time to extinction (T ext ) based on 100 trajectories over 100 years. We carried out 1000 such simulations, each of 100 trajectories, to obtain a sampling distribution of P 50 , P 100 and T ext . We calculated mean value of those parameters over the 1000 simulations and 95% confidence intervals by taking the 2.5 th and the 97.5 th percentile of the simulated distribution [START_REF] Shryock | Population viability of pediocactus bradyi (cactaceae) in a changing climate[END_REF]. All stochastic analyses were implemented using the R popbio-package, version 2.4.3 [START_REF] Stubben | Estimating and analyzing demographic models using the popbio package in R[END_REF].

Climatic scenarios were first simulated by increasing the frequency of drawing matrices corresponding to extreme warm and extreme dry years (Andrello et al. 2012;[START_REF] Hunter | Climate change threatens polar bear populations: a stochastic demographic analysis[END_REF][START_REF] Shryock | Population viability of pediocactus bradyi (cactaceae) in a changing climate[END_REF] from the observed frequency over the 22-year period q* to q = 0.8, that represent, respectively, the warm and the dry scenario. Based on the results of the GLM analyses testing for the effects of climatic variables on asymptotic growth rates, we used the daily mean temperature and the number of wet days to characterize warm and dry scenarios, respectively. Following [START_REF] Williams | Life-history evolution under climate change and its influence on the population dynamics of a long-lived plant[END_REF], extreme years corresponded to years with climate values larger than one standard deviation above the daily mean for temperature and lower than one standard deviation below the mean for precipitation. In the warm scenario, warm years were represented by pair-of-years with high daily mean temperature [START_REF] Imbert | Dispersal by ants in Centaurea corymbosa (Asteraceae): What is the elaiosome for?[END_REF]( -2007[START_REF] Zeigler | Predicting responses to climate change requires all life-history stages[END_REF]( -2014( , 2014( -2015( and 2015( -2016, Fig. 1, Fig. 1, q*=4/21=0.19). In the dry scenario, dry years were represented by pair-of-years with low numbers of wet days [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF]( -2002[START_REF] Imbert | Dispersal by ants in Centaurea corymbosa (Asteraceae): What is the elaiosome for?[END_REF]( -2007( and 2010( -2011, Fig. 1, Fig. 1, q*=3/21=0.14).

Second, following the approach described in Salguero-Gomez et al. ( 2012), we focused on a particular climate scenario resulting from the Regional Climate Model (RCM) proposed by the Euro-Cordex 2014 project [START_REF] Jacob | EURO-CORDEX: new high-resolution climate change projections for European impact research[END_REF]. We used the RCP4.5 scenario representing an intermediate scenario of increasing CO 2 concentration until 2060. The regional climate model simulations provided predicted values of daily precipitation and mean temperature. We then extracted these data for the closest geographic location, which is less than 1 km apart from the centroid of the species distribution (43.13710 N, 3.07327 E) and less than 6 km apart from the meteorological station we used to get observed climatic data. Overestimation of precipitation in RCMs is a well-known problem, in particular in areas close to the Mediterranean Sea [START_REF] Gao | Increased aridity in the Mediterranean region under greenhouse gas forcing estimated from high resolution simulations with a regional climate model[END_REF][START_REF] Ruffault | Projecting future drought in Mediterranean forests: bias correction of climate models matters![END_REF]. Therefore, to estimate the frequency of extreme years in the future (2020-2099), we used threshold values from the back-projected model values [START_REF] Salguero-Gomez | A demographic approach to study effects of climate change in desert plants[END_REF]. Using the back-projected data from the reference period , we defined warm years as those with mean temperature from May t to June t+1 greater than 14.5 + 1.23 (mean + SD) and dry years as those with number of wet days lower than 68.6 -14.8 (mean -SD). Comparing the observed data and the back-projected data for the overlapping period 1995-2005, we indeed document an overestimation of the number of wet days predicted by the regional model compared to the observed one (paired t-test = 4.57, df = 8, p < 0.002). In contrast, predicted and observed daily mean temperatures and total precipitation were not significantly different (p > 0.10). Simulated data are available on the Drias platform (www.drias-climat.fr).

Results

Patterns of spatio-temporal variations in asymptotic growth rates

Asymptotic growth rates λ a calculated per population and per pair-of-years showed a decreasing trend over time (β = -0.01, p = 0.02, Fig. 2A). The slope of the trend for each population separately was significant only for the smallest population (Cruzade: β = -0.04; p = 0.01, Fig. 2D). When pooling populations within each pair-of-years, the geometric mean of asymptotic growth rates over years was 0.880 (Table 1), with λ a ranging from 0.432 (2006-2007) to 1.815 (2012-2013). Geometric means of λ a over years per population were all lower than 1 and varied between 0.710 (Peyral) and 0.836 (Enferret1, Table 1). No significant relationship was detected between asymptotic growth rate and population size (β = 0.0003; p = 0.46). Randomization tests showed that the difference in λ a among years was highly significant (p < 0.001, see appendix C). In contrast, asymptotic growth rates λ a were not significantly different among populations (p = 0.16, see appendix C). The variance in λ a per population was not significantly correlated with mean age at flowering (β = -0.02; p = 0.68) nor with population size (β = 0.0003; p = 0.55).

Retrospective and prospective analyses

As λ a did not differ significantly among populations, retrospective and prospective analyses were performed after pooling individuals across populations. The observed variance in asymptotic growth rates among years was equal to 0.083, and was mainly explained by the variation in both the fecundity term (f), the survival of seedlings (s 1 ) and the survival of rosettes (s 2 ) having the highest contributions (0.025, 0.023 and 0.021, respectively, Fig. 3), followed by the survival of justemerged seedling (s 0 ) and the flowering probability of rosettes (α 2 ) having a similar contribution (0.008, Fig. 3). Rare events in the life-cycle (flowering probabilities α 1 and α 3 , and survival of flowering plants s 3 ) had negligible contributions (sum to 0.0009, Fig. 3 and appendix D). The largest elasticity of asymptotic growth rate was associated with rosette survival (s 2 ) followed by the fecundity term f (mean number of just-emerged seedlings per flowering plant), the survival of justemerged seedling (s 0 ), seedling survival (s 1 ) and the flowering probability of rosettes (α 2 , Fig. 3). Elasticity of λ a to the remaining lower-level vital rates representing rare events (s 3 , α 1 and α 3 ) summed only to 1% (Fig. 3). These latter rates were thus not considered for the GLMs investigating the relationship between lower-level vital rates and climatic variables.

Relationship between asymptotic growth rates and climate

Asymptotic growth rate, when pooling populations, was best explained by a model including only two climatic variables (R 2 = 0.58, Appendix E.2), with a high positive effect of the annual number of wet days (β = 0.15, p = 0.004) and a negative effect of the annual average daily mean temperature (β = -0.11, p = 0.03). The number of freezing days, the cumulative precipitation and the number of days with precipitation > 20 mm proved non-significant (p>0.05, Appendix E.1). No multiyear cumulative effect and no quadratic effect of climatic variables were significant (p > 0.05, Appendices E.1).

Relationships between lower-level vital rates and climate

Overall, warming and drought negatively affected vital rates. The mean number of justemerged seedlings per flowering plants f was negatively impacted by the daily mean temperature during both summer and fall periods (Table 2). The survival of just-emerged seedlings s 0 was positively impacted by the number of wet days during the germination period (both fall and the winter periods), and negatively impacted by the daily mean temperature during both summer and fall periods (Table 2). Seedling survival s 1 was negatively affected by an increase in the daily mean temperature during the summer and fall periods, and positively affected by an increase in the number of wet days during the same periods (Table 2). Rosette survival s 2 decreased with increasing summer daily mean temperature (Table 2). Finally, the flowering probability of rosettes α 2 was negatively impacted by the summer daily mean temperature and positively impacted by the number of wet days during the fall period (Table 2).

Stochastic projections under climate change

In absence of climate change, when randomly drawing each matrix with equal probability (1/21), stochastic growth rates λ s were significantly lower than 1 when pooling populations (λ s = 0.881, CI = [0.877, 0.889], Table 1). Stochastic growth rates were also lower than 1 in each population, ranging from 0.742 for the smallest population Cruzade (CI = [0.732, 0.770]) to 0.836 for the largest one Enferret2 (CI = [0.829, 0.851], Table 1).

To assess the combined effect of environmental and demographic stochasticity on population viability, we compared the fate of the smallest population Cruzade and the largest one Enferret2, using their respective population size at t = 0 (see details above and Fig. 4). As we did not detect any significant variation in asymptotic growth rates among populations, we pooled data across population for each pair-of-years to obtain the 21 matrices used to simulate climate change. In the warm scenario, when increasing the frequency of extreme years from q* (4/21 = 0.19) to 0.8, stochastic growth rates decreased from 0.882 to 0.761 for the largest population (Enferret2, Fig. 4A) and from 0.879 to 0.765 for the smallest one (Cruzade, Fig. 4C). Under the RCP4.5 scenario, the predicted frequency of warm year q pred characterized by a daily mean temperature > 15.63 °C for the period 2020-2099 equaled 0.78 and led to the extinction of both populations at 100 years in most simulations (Figs. 4A and 4C,solid black lines). Comparing this scenario with those of no climate change (Figs. 4A and4C, dashed green lines), extinction probability at 50 years (P 50 ) increased from 0.37 to 0.99 for Enferret2 and from 0.82 to 0.99 for Cruzade, while extinction probabilities at 100 years (P 100 ) were close or equal to 1 for both populations. The mean time to extinction decreased from 57 to 26.4 years for Enferret2, and from 32 to 16 for Cruzade.

The dry scenario had less dramatic consequences on the viability of populations than the warm scenario (Fig. 4B and4D). The increase in dry year frequency from q* (3/21=0.14) to 0.8 led to a small reduction in λ s from 0.882 to 0.855 for Enferret2 and from 0.879 to 0.851 for Cruzade (Figs. 4B and 4D,respectively). The RCP4.5 scenario predicted a frequency of dry years q pred , corresponding to a number of wet days lower than 53.8 days, equaled to 0.15, a value very close to the one observed during the 22-year period of our demographic survey. The RCP4.5 scenario did not result in any significant change in extinction probabilities and mean time to extinction compared to the scenario of no climate change, whatever the size of populations (Figs. 4B and4D, solid black lines).

Discussion

Long term population dynamics of Centaurea corymbosa

Our results demonstrate the need for long-term demographic surveys to better understand the spatio-temporal demographic variability on plants [START_REF] Blume-Werry | Short-term climate change manipulation effects do not scale up to long-term legacies: effects of an absent snow cover on boreal forest plants[END_REF][START_REF] Compagnoni | The effect of demographic correlations on the stochastic population dynamics of perennial plants[END_REF][START_REF] Crone | How do plant ecologists use matrix population models?[END_REF][START_REF] Huelber | Uncertainty in predicting range dynamics of endemic alpine plants under climate warming[END_REF]. Indeed, our study based on 22 years of data provided a more accurate picture of the determinants of population dynamics in C. corymbosa than the study of [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF] based on 8 years of data. In [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF], the standard deviation in asymptotic growth rates σ(λ) was 0.116 among years and 0.099 among populations, with significant differences among years and populations. In the current study, variation among populations became non-significant and temporal variation in asymptotic growth rates was rather synchronized among populations (Fig. 2). This pattern is not surprising given the narrow distribution of C. corymbosa and is largely consistent with studies of other narrowly distributed species (Buckley et al., 2010;[START_REF] Kiviniemi | Spatial (a) synchrony in population fluctuations of five plant species in fragmented habitats[END_REF][START_REF] Ramula | Management actions are required to improve the viability of the rare grassland herb Carlina biebersteinii[END_REF]. More importantly, the standard deviation σ(λ) observed over the 22-year period was much larger (0.230) than the one (0.116) reported in [START_REF] Fréville | Spatial and temporal demographic variability in the endemic plant species Centaurea corymbosa (Asteraceae)[END_REF]. In agreement with the results of the permutation tests, we did not find any significant correlation between population dynamics and population size. Indeed, we did not find any evidence that small populations had lower population growth rates. Moreover, the variance in asymptotic growth rates among years did not increase with decreasing population size. In contrast to other demographic studies [START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF][START_REF] Zeigler | Predicting responses to climate change requires all life-history stages[END_REF], we thus did not find any evidence of a greater sensitivity of the smallest populations to both environmental and demographic stochasticity in our species. In addition, the variance in asymptotic growth rate was not correlated with mean age at flowering per population. We thus did not find evidence for demographic buffering in populations with greater longevity, in contrast to what has been found at an inter-specific scale in [START_REF] Morris | Longevity can buffer plant and animal populations against changing climatic variability[END_REF]. Population dynamics of C. corymbosa is thus mainly affected by environmental factors that vary over time rather than by spatial factors.

Average and most of the yearly asymptotic growth rates were lower than 1 for each population and when pooling individuals across populations. Overall, we observed a declining trend in yearly asymptotic growth rates over the 22-year period (Fig. 2), this trend being significant for the smallest population and when considering all populations (Figs. 2A and2D, respectively). Asymptotic growth rates were negatively impacted by an increase in temperature and positively impacted by the number of wet days. Consistent with these results, asymptotic growth rates attained extreme values in years corresponding to extreme climatic events. Indeed, when pooling all individuals in a single population, the lowest value (0.432, Fig. 2A) was observed in 2006-2007 when the mean temperature was the highest (16.3° C, Fig. 1) and the number of wet days was the lowest (34 days, Fig. 1). The highest growth rate (1.815, Fig. 2A) corresponded to 2012-2013, when we observed the second lowest mean temperature (14.2° C, Fig. 1) and the highest number of wet days (66 days, Fig. 1). Thus, population dynamics of C. corymbosa was mainly affected by extreme climatic events, which are predicted to increase in the future. Climatic variables investigated here did not show any marked temporal trend over the 22-year period of our demographic survey (Fig. 1), which could explain the weak temporal trend observed in asymptotic growth rates. Overall, our study confirms that extreme climatic events are an important driver of plant population dynamics as reported in other studies (Andrello et al., 2012;[START_REF] Davison | Demographic effects of extreme weather events on a short-lived calcareous grassland species: stochastic life table response experiments[END_REF][START_REF] Mcdowell | Mechanisms of plant survival and mortality during drought: why do some plants survive while others succumb to drought?[END_REF][START_REF] Shryock | Population viability of pediocactus bradyi (cactaceae) in a changing climate[END_REF][START_REF] Ulrey | Life at the top: Long-term demography, microclimatic refugia, and responses to climate change for a highelevation southern Appalachian endemic plant[END_REF].

In contrast to some studies reporting both negative and positive effects of climate change on vital rates [START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF][START_REF] Meisner | Temperature effects on long-term population dynamics in a parasitoid-host system[END_REF][START_REF] Nicolè | Interdependent effects of habitat quality and climate on population growth of an endangered plant[END_REF], our study documents a consistent negative effect of increased temperature and drought on lower-level vital rates in C. corymbosa. Regarding temperature, the most critical season that impacted lower-level vital rates was the summer period, since increasing summer daily mean temperature negatively affected both fecundity, survival probability and flowering probability. In contrast to our study, flowering probability was found to be positively impacted by high temperature in some temperate plant species (e.g. Himantoglossum hircinum, [START_REF] Meer | Recent range expansion of a terrestrial orchid corresponds with climate-driven variation in its population dynamics[END_REF]Dracocephalum austriacum, Nicolè et al., 2011). While a negative effect of increasing temperature on flowering probability was also reported in [START_REF] Peñuelas | Nonintrusive field experiments show different plant responses to warming and drought among sites, seasons, and species in a north-south european gradient[END_REF], this pattern was only observed in the southern Mediterranean sites but not in the northern ones. A likely explanation of this negative effect is that warm summers increase evapo-transpiration and decrease soil moisture [START_REF] Christensen | Vulnerability of the Asian typical steppe to grazing and climate change[END_REF][START_REF] Peñuelas | Nonintrusive field experiments show different plant responses to warming and drought among sites, seasons, and species in a north-south european gradient[END_REF]. This could lead to a decrease in plant biomass as a consequence of rosette shrinking that in turn reduces the amount of resources that a plant can allocate to reproduction [START_REF] Manders | Effects of watering regime on growth and competitive ability of nursery-grown Cape fynbos and forest plants[END_REF]. This is particularly true for monocarpic perennial species, where flowering probability has been shown to increase with plant size [START_REF] Metcalf | Evolutionary demography of monocarpic perennials[END_REF]. Regarding precipitation, the youngest plants were the most impacted by the number of wet days. This factor had a positive effect on both survival of just-emerged seedlings (s 0 ) during the germination period (fall and winter) and seedling survival (s 1 ) during the summer and fall periods, which is consistent with several studies showing a positive impact of the frequency of wet days [START_REF] Riba | Effects of regional climate and small-scale habitat quality on performance in the relict species Ramonda myconi[END_REF][START_REF] Shriver | Quantifying how short-term environmental variation leads to long-term demographic responses to climate change[END_REF]. Both temperature and precipitation effects may be explained by the fact that recurrent warm and dry days can induce a seasonal water deficit, which may negatively impact the whole life-cycle [START_REF] Manders | Effects of watering regime on growth and competitive ability of nursery-grown Cape fynbos and forest plants[END_REF][START_REF] Peñuelas | Nonintrusive field experiments show different plant responses to warming and drought among sites, seasons, and species in a north-south european gradient[END_REF][START_REF] Ruffault | Projecting future drought in Mediterranean forests: bias correction of climate models matters![END_REF]. Species growing in rocky habitats, such as C. corymbosa, are likely to be the most affected, since these soils have a very low water capacity [START_REF] Ruiz-Sinoga | Analysis of dry periods along a pluviometric gradient in Mediterranean southern Spain[END_REF][START_REF] Silva | Demographic vulnerability in cliff-dwelling Sonchus species endemic to the western Mediterranean[END_REF]. Moreover, herbaceous species such as C. corymbosa do not have a deep root system ensuring access to water over dry periods in such rocky habitats.

Predictions from climate scenarios

Both demographic and environmental stochasticity are important factors in determining the viability of species with small population sizes [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF][START_REF] Crone | Ability of matrix models to explain the past and predict the future of plant populations[END_REF][START_REF] Garcıá | An evaluation of the status of five threatened plant species in the Pyrenees[END_REF]. We here showed that climate variation explained a large amount of variation in vital rates and growth rates in C. corymbosa. We thus used stochastic models to predict the fate of C. corymbosa populations with both environmental stochasticity arising from climate change and demographic stochasticity. We did not include density-dependence in our model, which could make quantitative predictions overly pessimistic [START_REF] Ehrlén | Advancing environmentally explicit structured population models of plants[END_REF]. Population viability analyses are acknowledged to be relevant tools to quantitatively compare the impact of different scenarios rather than quantitatively predict the future status of populations [START_REF] Coulson | The use and abuse of population viability analysis[END_REF]. We thus used population viability analyses to compare how the warm and the dry scenarios impacted population viability of C. corymbosa rather than attempting to explicitly predict future population size.

Stochastic projections predicted population declines (λ s < 1) under no climate change. Populations were predicted to decline even faster in the future in response to an increased frequency of extreme years in both the warm and dry scenarios. Such results are consistent with the GLMs showing negative impacts of warming and drought, and several studies investigating population viability in response to climate change (Andrello et al., 2012;[START_REF] Lawson | Environmental variation and population responses to global change[END_REF][START_REF] Marrero-Gómez | Population viability of the narrow endemic Helianthemum juliae (Cistaceae) in relation to climate variability[END_REF][START_REF] Shryock | Population viability of pediocactus bradyi (cactaceae) in a changing climate[END_REF]. More importantly, the warm scenario had more severe consequences on population viability than the dry scenario whatever the initial population size. However in our simulations, the initial population size impacted both extinction probability and the mean time to extinction, a pattern consistent with theoretical studies since small populations are more vulnerable to extinction than largest ones [START_REF] Groom | Principles of conservation biology[END_REF][START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF]. Using the RCP4.5 scenario, the change in temperature projected for the period 2071-2099 was of the same order of magnitude than the one predicted by other regional climatic models [START_REF] Christensen | Regional climate projections[END_REF][START_REF] Ruffault | Projecting future drought in Mediterranean forests: bias correction of climate models matters![END_REF]. In contrast, our dry scenario predicted an annual mean number of wet days equal to 66.41 (-3.2%), while other models predicted a much severe decrease (-19.6%, Ruffault et al., 2014). This suggests that uncertainty in climate models should be carefully considered in management plans and when evaluating extinction risk. Such uncertainty is particularly pronounced for the Mediterranean region [START_REF] Gao | Increased aridity in the Mediterranean region under greenhouse gas forcing estimated from high resolution simulations with a regional climate model[END_REF][START_REF] Ruffault | Projecting future drought in Mediterranean forests: bias correction of climate models matters![END_REF].

Faced with climate change, populations can either migrate or adapt by microevolution or phenotypic plasticity to avoid extinction [START_REF] Groom | Principles of conservation biology[END_REF][START_REF] Lande | Stochastic population dynamics in ecology and conservation[END_REF]. In the Mediterranean region, many narrow endemic species are characterized by a low ability to colonize existing suitable habitats due to several biological traits such as low dispersal ability [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF][START_REF] Lavergne | The biology and ecology of narrow endemic and widespread plants: a comparative study of trait variation in 20 congeneric pairs[END_REF][START_REF] Thompson | Ecological differentiation of Mediterranean endemic plants[END_REF]. In C. corymbosa, self-incompatibility and monocarpy make successful colonization even less likely [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF]. Thus, for such species, persistence mainly depends upon their ability to respond plastically to climate change or to become locally adapted to new environmental conditions [START_REF] Cotto | A dynamic eco-evolutionary model predicts slow response of alpine plants to climate warming[END_REF]Knight et al. 2008;[START_REF] Menges | Population viability analyses in plants: challenges and opportunities[END_REF]. Theoretical work suggests that perennial species should display slower evolutionary responses than annual plants, since higher adult survival limits the adaptive capacity of local populations [START_REF] Cotto | A dynamic eco-evolutionary model predicts slow response of alpine plants to climate warming[END_REF]. Moreover, small populations have a lower evolutionary potential, since they become genetically impoverished due to the impact of genetic drift. Using a shorter survey (1995)(1996)(1997)(1998)(1999)(2000)(2001), [START_REF] Fréville | Comparative allozyme and microsatellite population structure in a narrow endemic plant species, Centaurea corymbosa Pourret (Asteraceae)[END_REF] showed that demographic rates were not correlated with intra-population genetic diversity in Centaurea corymbosa [START_REF] Fréville | Comparative allozyme and microsatellite population structure in a narrow endemic plant species, Centaurea corymbosa Pourret (Asteraceae)[END_REF]. Using the same genetic data (Table 1 in [START_REF] Fréville | Comparative allozyme and microsatellite population structure in a narrow endemic plant species, Centaurea corymbosa Pourret (Asteraceae)[END_REF], we likewise confirmed that genetic diversity was correlated with neither λ s (Spearman correlation rho = 0.26, p = 0.61) nor the slope of regression in asymptotic growth rates over years (rho = 0.09, p = 0.89). Thus, our results do not support evidence for accumulation of deleterious mutations [START_REF] Lynch | Mutation Accumulation and the Extinction of Small Populations[END_REF], nor a loss of standing variation due to genetic drift in small populations of C. corymbosa. 4.3 Management recommendations Persistence of narrow endemic species in the Mediterranean region may have been favored by the capacity of those species to grow in rocky habitats [START_REF] Thompson | Ecological differentiation of Mediterranean endemic plants[END_REF], since these remote habitats do not face direct threats due to human activities. Demographic studies of cliffdwelling species have shown unusual demographic stability and resilience of these plants [START_REF] García | Demographic viability of a relict population of the critically endangered plant Borderea chouardii[END_REF][START_REF] Lavergne | Environmental and human factors influencing rare plant local occurrence, extinction and persistence: a 115-year study in the Mediterranean region[END_REF][START_REF] Picó | Regional-scale demography of Ramonda myconi: Remnant population dynamics in a preglacial relict species[END_REF][START_REF] Thompson | Ecological differentiation of Mediterranean endemic plants[END_REF], likely arising from much higher importance of rosette survival than recruitment for the maintenance of populations [START_REF] Cotto | A dynamic eco-evolutionary model predicts slow response of alpine plants to climate warming[END_REF][START_REF] Larson | Evidence for the widespread occurrence of ancient forests on cliffs[END_REF][START_REF] Silva | Demographic vulnerability in cliff-dwelling Sonchus species endemic to the western Mediterranean[END_REF]. In a scenario of no climate change, rosette survival (that had the highest elasticity on growth rates) should be increased from 0.71 to 0.92 to ensure the persistence of C. corymbosa in a deterministic model (Appendix G). However, management actions aimed at increasing specific vital rates, such as watering plants are not feasible in such rocky habitats. Implementing management actions that reduce the negative effects of demographic, genetic and environmental stochasticity by manipulating population size and population number appear much more promising.

Previous studies have shown the importance of biological features such as selfincompatibility and low dispersal ability on the population persistence of C. corymbosa [START_REF] Colas | Centaurea corymbosa, a cliff-dwelling species tottering on the brink of extinction: A demographic and genetic study[END_REF]. Two new populations were introduced in 1994 on the top of unoccupied cliffs thus confirming the existence of suitable but empty habitats nearby existing natural populations [START_REF] Colas | Restoration demography: a 10-year demographic comparison between introduced and natural populations of endemic Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Kirchner | Modelling the dynamics of introduced populations in the narrow-endemic Centaurea corymbosa: a demo-genetic integration[END_REF]. Introduced populations exhibited on average higher plant survival than natural populations, mainly because seeds had been introduced manually in a priori suitable microsites [START_REF] Colas | Restoration demography: a 10-year demographic comparison between introduced and natural populations of endemic Centaurea corymbosa (Asteraceae)[END_REF]). However, fecundity in introduced populations was lower than in natural ones, likely as a consequence of low local density of flowering plants, leading to strong pollen limitation [START_REF] Colas | Restoration demography: a 10-year demographic comparison between introduced and natural populations of endemic Centaurea corymbosa (Asteraceae)[END_REF]. One introduced population has been extinct for 5 years, while the other population only harbored 35 plants in the last census performed in December 2017. In the future, protocols should be carefully designed to ensure successful introductions. For instance, introducing a large number of seeds (>1000) distributed over a few sites and at yearly intervals should allow population persistence on the short term [START_REF] Colas | Restoration demography: a 10-year demographic comparison between introduced and natural populations of endemic Centaurea corymbosa (Asteraceae)[END_REF][START_REF] Kirchner | Modelling the dynamics of introduced populations in the narrow-endemic Centaurea corymbosa: a demo-genetic integration[END_REF].

Faced with accelerated climate change, conservation policies for endangered species, such as C. corymbosa, should be reconsidered. As our study demonstrated the importance of climate on the fate of populations, introductions in distant habitats should be considered, although such assisted colonization event requires suitable and available habitats at a larger scale. At present, prioritization actions of conservation rely on the climatic niche of the species (Jones et al., 2016;Oliver et al., 2016), and depend on our ability to define new protected areas in Northern localities [START_REF] Araújo | Climate change threatens European conservation areas[END_REF][START_REF] Bellard | Impacts of climate change on the future of biodiversity[END_REF] or microrefugia (i.e. location with a low impact of climate change, Jones et al., 2016;[START_REF] Ulrey | Life at the top: Long-term demography, microclimatic refugia, and responses to climate change for a highelevation southern Appalachian endemic plant[END_REF]. Recently, an increasing amount of empirical evidence has been gathered to document that organisms, including plants, can show microevolutionary responses in natural populations [START_REF] Franks | Evolutionary and plastic responses to climate change in terrestrial plant populations[END_REF][START_REF] Thomann | Flowering plants under global pollinator decline[END_REF], suggesting plants can adapt to new environmental conditions [START_REF] Thompson | Ecological differentiation of Mediterranean endemic plants[END_REF]. Therefore, reinforcement, which allows increasing population size and thus reducing extinction risk arising mainly from demographic and genetic stochasticity, is also potentially an efficient and reasonable management strategy for species facing climate change. A. Duncan for English corrections, as well as the editor-in-chief V. Devictor and three anynomous reviewers for their helpful suggestions. We also thank the INRA Pech Rouge for providing us the climatic data and F. Mouillot for his advice on climate predictions. This research was supported by a fellowship from the Algerian Ministry of Higher Education (Ph-D grant to A H-B), and by the OSU-OREME.

Bibliography

AllEnvi (2016). The Mediterranean region under climate change (Paris: IRD Editions). Andrello, M., Bizoux, J.-P., Barbet-Massin, M., Gaudeul, M., Nicole, F., and Till-Bottraud, I. (2012). Effects of management regimes and extreme climatic events on plant population viability in Eryngium alpinum. Biol. Cons. 147, 99-106.

Andrieu, E., Besnard, A., Fréville, H., Vaudey, V., Gauthier, P., Thompson, J.D., and Debussche, M. (2017). Population dynamics of Paeonia officinalis in relation to forest closure: From model predictions to practical conservation management. Biol. Cons 215, 51-60.

1 Tables Table 1: Demographic characteristics of each of the six populations of Centaurea corymbosa, and the pooled population after pooling data across populations: number of individual life-histories recorded over 22 years (see also Appendix A1), number of surviving plants older than 1 year, mean life-time of individuals that reached the rosette stage, mean age at flowering, geometric mean of asymptotic growth rates (λ a ) over 22 years, minimum and maximum of λ a , stochastic growth rate (λ s ) when including both demographic and environmental stochasticity (uniform frequency, see text) with the confidence intervals (2.5 th and 97.5 th percentiles of the simulated distributions) and the rounded mean of total numbers of flowering plants (Nb.FP) recorded by the exhaustive count over the 22 years (see appendix A2 andE1), the medium size populations (Au and Po) and the smallest ones (Pe and Cr). R 2 corresponds to the r-squared value obtained from the linear regression and "ns" indicates that the slope was not significantly different from 0. Year on the x-axis corresponds to June (t). 2006-2007, 2013-2014, 2014-2015), and the dry scenario an increase in the frequency of years with an extremely low number of wet days (3 extreme years out of 21: 2001-2002, 2006-2007 and 2010-2011). 'q' indicates the frequency of extreme years with q* the observed frequency over the 22-year of our demographic survey and q pred the frequency predicted by the RCP4.5 climatic model for the warm and dry scenarios (q pred are 0.78 and 0.15, respectively, see text). The initial population size was estimated as the stable stage distribution predicted by the mean matrix over years after pooling data across populations, using the mean total number of flowering plants recorded from 2010 to 2016 in Enferret2 and Cruzade, respectively.

Figure 1 :

 1 Figure 1: Climatic variation observed in the Massif de la Clape between 1995 and 2016. Climatic variables were calculated per pair-of-years from June(t) to May(t+1). The solid line represents the average daily mean temperature (C°), and the dashed line represents the number of wet days. Values in the vertical bars indicate the cumulative precipitation per pair-of-years. Year on the x-axis corresponds to June(t). Data were obtained from INRA Pech Rouge.

Figure 2 :

 2 Figure 2: Asymptotic growth rates (λ a ) per pair-of-years. Panel A represents yearly λ a values (circles) with their trend over years (dashed line) calculated from 125 matrices. The solid line represents the values of λ a (triangles) pooling individuals in a single population. Panels B, C and D represent the trend in λ a over years for the largest populations (E2 and E1), the medium size

Figure 3 :

 3 Figure3: Contribution of lower-level vital rates to the variance of λ a observed among years, and elasticity of growth rates to these vital rates calculated on the arithmetic mean matrix calculated over 21 pair-of-years matrices.

Figure 4 :

 4 Figure4: Predicted consequences of the warm and dry scenarios on the viability of populations using two different initial population sizes representing the largest (Enferret2) and the smallest (Cruzade) populations when incorporating both demographic and environmental stochasticity (see text for details). Panels (A) and (C) represent the warm scenario for the largest population size (N 0 = {685 796 118}, Enferret2), and the smallest one (N 0 = {41 47 7}, Cruzade). Panels (B) and (D) represent the dry scenario for Enferret2 and Cruzade, respectively. The warm scenario consisted of an increase in the frequency of years with extremely high temperatures (4 extreme years out of 21:

  Pourret (Asteraceae) is a narrow Mediterranean species endemic to the Massif de la Clape near Narbonne (southern France, 43°13 N, 3°08 E). It is listed in the French Red Book of endangered species and in the European Habitat Directive list of priority species, thus precise coordinates of the populations are not given. C. corymbosa has been extensively studied since 1994 and several papers have already reported on its ecological characteristics

	2.1. Biological model, demographic and climatic datasets
	2.1.1. Study species
	Centaurea corymbosa

Table 2 :

 2 ). Populations are ranked by decreasing order of total number of flowering plants. The null values of asymptotic growth rates corresponded to pair-of-years with no flowering plant within plots and null values for rosette survival s 2 . Summary of GLMs explaining the effect of two climatic variables (averaged over three-month periods) on lower-level vital rates. Only climatic variables affecting population growth rate were included in the GLMs. From the best model explaining a given lower-level vital rate, we report the estimated coefficient (β) and its significance (t-test) after scaling the climatic variables. 'ns' indicates that the variable tested was not significant in the model, based on a forward stepwise selection approach (see text and appendix E.3 for details).

	Population	Nb. individuals	Nb. one year plants	Mean longevity of the rosettes	Mean age at flowering	Mean λ a [λ a min, λ a max]	λ s	CI(λ s )	Nb.FP
				(years)	(years)
	Pooled pop.	6112	1579	3.5	5.0	0.880	[0.432, 1.815]	0.881	[0.877, 0.889]	478
	Enferret2	1712	286	3.3	4.5	0.749	[0.240, 1.585]	0.836	[0.829, 0.851]	173
	Enferret1	1165	324	3.6	5.1	0.836	[0.425, 2.325]	0.824	[0.819, 0.837]	147
	Auzils	1175	415	3.5	5.7	0.783	[0.331, 1.391]	0.792	[0.786, 0.807]	81
	Portes	1064	268	3.7	6.2	0.819	[0.364, 1.610]	0.824	[0.818, 0.842]	34
	Peyral	540	192	3.2	4.7	0.710	[0.00 (a) , 1.264]	0.804	[0.799, 0.827]	28
	Cruzade	456	94	3.4	4.2	0.720	[0.00 (a) , 1.444]	0.742	[0.732, 0.770]	13
	(a)

Acknowledgments

This work is part of the 'Centaurea corymbosa project' initiated by I. Olivieri, B. Colas and M. Riba in 1994. We are indebted to the many contributors to this project, and in particular to B. Colas and M. Riba who provided constructive comments on the manuscript. We would like to thank

Appendices:

Appendix A1: Number of individuals recorded within the 41 permanent plots over the 22-year period. Number of new seedlings represents all new plants recorded in the permanent plots every 3 months during year t to t+1, out of which some did not survive until June t+1 with a probability 1-s 0 , and thus did not reach the seedling stage. The number of individuals recorded in June (t) in each life-stage of the life-cycle used to construct population projection models, is given for: Seedling, the one-year plants that survive until June (t), Rosette, vegetative plants older than one year, and Flowering plants that represent reproductive plants. The total number of individuals per year did not include the new seedlings. The total individual life-histories (6112) corresponds to the sum of number of new seedlings (5779) and number of rosettes and flowering plants at the first census (315+18).

Year New seedlings

Seedling stage Appendix D: Arithmetic mean and coefficient of variation of lower-level vital rates over years after pooling data over populations, sensitivity and elasticity of growth rates to these vital rates calculated at the arithmetic mean matrix over 21 matrices, and contributions of lower-level vital rates to the variation of growth rates among years. Appendix G: Required values of lower-level vital rates for achieving population persistence in a deterministic model (λ a =1), using two different methods: lower-level vital rates values were either predicted from the elasticity values, or directly from the global mean matrix after pooling individuals over populations and years to obtain a single population. For both methods, we varied only one vital rate at a time, while keeping others at their observed value in the mean matrix.

Rosette stage

Flowering plant stage

Required value


A Ayadi

M-F Lacrampe

P Krawczak

A comprehensive study of bubble inflation in vacuumassisted thermoforming based on whole-field strain measurements

This paper focuses on the potential use of stereo-DIC in thermoforming conditions to monitor large deformations of softened thermoplastic sheets posteriori to the sagging phenomenon. The study concerns HIPS sheets which are softened by the radiative heat-transfer mode then stretched by inflation of compressed-air for 1.5 s to form a large and quasi-spherical dome of 250 mm in diameter. While the bubble-inflation operation leads to large deformations of the softened sheet, it shows transitional geometrical instabilities due to the initial surface sagging. When the temperature-induced surface deformations are inaccessible by the stereoscopic system during the heating operation, the geometrical instabilities limit the identification of the reference of displacements which affects the accuracy of results based on image-correlation computations. To compare between the principal strains assessed from bubble-inflation tests conducted at different thermal conditions, a method for filtering these instabilities is developed in this study.

INTRODUCTION

Experimental characterization of the mechanical behavior of thermoplastics in the range of thermoforming temperatures is classically performed based on unidirectional and multi-axial tests performed at controlled deformation speeds at isothermal conditions. However, the optimization of thermoforming simulation packages requires the development of original mechanical tests which are more representative of the material behavior under processing conditions. One potential solution is the use of open-mold thermoforming equipment and full-field strain computation techniques to characterize the thermomechanical behavior of thermoplastics. In the field of experimental mechanics, the whole-field optical techniques have an advantage of being fast and contact-less [1,2]. In particular, stereo-DIC is more and more applied in thermoforming to compute whole-field strains [3], investigate thermoformability limits [4] and compute full-field thickness maps [5]. During a vacuum-assisted thermoforming cycle, thermoplastic sheets are thermally softened then pre-stretched by forming a quasi-spherical bubble to homogenize their thicknesses. During the bubble-inflation operation, the interaction between the non-conditioned blown air and the softened sheet does not only exhibit large deformations but also causes geometrical non-linearity due to the stochastic thermal deformations taking place during the heating operation. In fact, the sheets are mainly softened based on the radiative heat-transfer mode which inevitably induces warpage of the surface due to relaxation of the residual stresses and descent of the softened surface under the effect of its own weight. The latter phenomenon is known by sag in thermoforming. When the access to the sheet's surface during heating by the stereoscopic system is not possible, an appropriate protocol is required to avoid erroneous interpretations of the stereo-DIC results relative to the large deformations taking place during the bubble formation. This study provides an overview of the potential use of stereo-DIC measurements in thermoforming conditions, particularly during a pre-stretching operation of 1.5s by bubble inflation of High-Impact Polystyrene sheets (HIPS). This case focuses on the applicability of stereo-DIC measurements when the monitoring of large deformations can only be possible after thermal sagging of the surface. First, the constraints imposed by the equipment that affect the accuracy of stereo-DIC computations are presented. Second, an objective stereo-DIC post-processing method is presented to evaluate the principle strains from stereo-DIC measurements during bubble-inflation tests which are performed after two heating temperatures. With access to the temperature distributions of the HIPS sheet after heating, full-field principle strains will be used to understand the effect of sagging on the resulting wall-thickness distributions.

EXPERIMENTAL PROTOCOL

In this study, bubble-inflation tests are performed by using a thermoforming machine of the type Illig UA-100Ed equipped with a clamp-frame which is designed to form a semi-spherical bubble of 250 mm in diameter. The used HIPS sheets are commercialized under the trade name Athpol P91® and have an average thickness of 1.5 mm. Before each thermoforming cycle, a sheet of 600×330 mm 2 is cut to meet the dimensions of the forming plateau and a black and white speckle is applied on its surface as required for stereo-DIC experiments. The thermoforming cycle starts automatically by clamping the sheet to seal a cavity located below the forming plateau of the machine. Then, a sliding radiative heater entirely covers the forming plateau to heat the fixed sheet at a defined regulation temperature and for a fixed duration of 70 s. The heater comprises a battery of infra-red (IR) ceramic elements of the type Acim Jouanin® which are located at a distance of 150 mm from the surface of the clamped HIPS sheet. After the complete retraction of the heater (within less than 2 s), compressed-air is inflated inside the sealed cavity covered by the softened sheet for 1.5 s to force the softened thermoplastic sheet through the circular opening in the clamping-frame. The realized experimental plan comprised a total of 10 bubble-inflation tests which are carried out at the regulation temperatures of 310 °C and 330 °C. Because the used machine does not provide continuous time monitoring of the process parameters such as the material's temperature and air pressure, external acquisition devices are installed. K-type thermocouples are used to measure temperatures during the heating step on the surface of thermoplastic sheets. A pressure sensor of the type JUMO AP-30R is used to measure the relative pressure inside the bubble during the air-inflation operation. The evolution of the surface speckle is observed from different angles by using two identical CMOS cameras (Imager MX 4M) which are calibrated to make stereoscopic measurements within a common coordinates system. During the bubble-inflation operation, the pressure signal is synchronized with a high-speed-controller of a stereo-DIC system commercialized by LaVision. Pairs of images are recorded at a full-resolution of 2048 × 2048 pixels and a frequency of 180 fps. An overview of the experimental setup is illustrated in Fig. 1.

CONSTRAINTS IMPOSED BY THE THERMOFORMING EQUIPMENT

Existence of Temperature Gradients after the Material Heating

During the radiative heating operation, a circular domain of the HIPS sheet of 250 mm in diameter is exposed to the IR ceramic elements. Because the machine does not provide direct measurements of the thermoplastic temperature at the end of the heating operations, an experimental procedure based on the Kriging interpolation method is used to evaluate the temperature distributions within a surface domain covering 100×120 mm 2 . As illustrated in Fig. 2, the obtained temperature maps indicate the existence of significant in-plane thermal gradients. Such observation reflects a non-uniform heating despite the automatic regulation of the heater's temperature. These temperature distributions confirm the existence of local changes of the surface flatness and notably sagging which are both known to affect the thermal performance of the IR heaters.

Large Deformations and Speckle Uniformity Considerations

The used thermoforming cycle gives the possibility to deform a softened thermoplastic sheet at two successive stages. First, the pressure build-up during the inflation of compressed-air inside the sealed cabinet forces the softened HIPS sheet through the relatively large circular opening of the clamp-frame. The height of the thermoplastic sheet at the end of this step is limited to few centimeters and it can be calibrated with variation of the heater's regulation temperature (domain between B and C in Fig. 3.a). To avoid exceeding the depth-of-field of the used cameras, the control temperatures were limited to 330 °C and 310 °C resulting in bubble heights lower than 1/4 of the diameter. This height threshold is defined based on trial tests and visual inspections of deformed surfaces to check the uniformity of the speckle and absence of paint cracking at the end of the bubble formation. In the second deformation stage, a forming table comprising a positive punch of 200 mm in diameter moves upwards starting from the bottom of the forming cabinet to deform the sheet for a maximum height of 250 mm (domain between C and E in Fig. 3 With consideration of the large extent of the analyzed surface and the calibration constraints of the used stereoscopic system (such as the variations of luminosity as illustrated in Fig. 3.b), only deformations taking place by the interaction between compressed-air and the softened thermoplastic sheet are considered in this study.

Limited Access to the Speckle During Sagging

During the heating operation, the sliding IR heater totally covers the thermoplastic sheet (Fig. 3.b). Consequently, the potential deformations of the speckle due the collapse of the softened sheet under the effect of its weight and due to the relaxation of residual stresses cannot be observed by the used stereoscopic system. Thus, the initially flat surface of the HIPS sheet should no longer be considered as a reference for stereo-DIC computations. Moreover, the temperature-induced deformations are dependent of the thermal history of the used thermoplastic sheet (extrusion direction, calendaring …) and the homogeneity of the temperature distributions on its surface. Thus, different sag levels are obtainable even posteriori to similar heating conditions. Consequently, the identification of a reference state of the speckle (which is required for the computation of displacements and strains) requires an objective method to avoid errors and misinterpretations caused by the random and uncontrolled aspects of the thermally activated deformations.

POST-PROCESSING METHOD FOR STEREO-DIC COMPUTATIONS

To filter the out-of-plane undulations that affect the accuracy of the stereo-DIC computations, the developed approach consists on using a two-step computation. The first step is based on the assumption that the speckle of the sagged surface represents the reference for the displacement vectors. To identify the moment of attenuation of the undulations, the exploitation of the computed displacements is limited to qualitative analyses of their temporal evolution. The second step consists on correcting the reference of the displacement and deformation fields after filtering the surface instabilities. The aim of this study is to propose an objective and repetitive method to filter surface undulations regardless of the temperature of the test. The commercial software DaVis 8.4 is used to compute the displacements using the sum of the differential algorithms. The stereo-DIC computations are limited to the polar domain of the bubbles defined by a virtual gauge of 50 mm radius concentric to the circular opening of the used clamping frame. The size of the subset and the step size were respectively 81 * 81 pixels and 21 * 21 pixels.

Qualitative Analysis of the Geometrical Instability

In the first step, the reference state of the speckle is attributed to the onset of bubble inflation based on the measured pressure signals. The assessed average out-of-plane displacements of two representative tests performed at 310 °C and 330 °C and their corresponding measured pressures are presented in Fig. 4.a. The temporal evolution of the out-ofplane components of displacements indicates the existence of two regimes. An initial transient regime of nearly 300 ms which presents local maxima and minima (between 100 ms and 250 ms) corresponding to the transition from the sagged to the strained geometries of the softened sheet. The second regime is more uniform and represents the effective extension of the thermoplastic sheet. The repetitiveness of formation of the irregular out-of-plane displacement transitions is verified by comparing the out-of-plane displacements from five repetitive bubble-inflation tests. Similar aspects are observed in all the carried-out bubble-inflation tests. However, the amplitude and the surface transition are more significant at the regulation temperature of 330 °C. As an illustrative example, the out-of-plane components of displacement vectors relative to the inflation tests that were conducted after a heating temperature of 330 (°C) are provided in Fig. 4.b.

Identification of the Effective Reference for Stereo-DIC computations

With consideration of the used image recording frequency of 180 fps, the out-of-plane displacements at the pole of the bubble are assumed as differentiable functions of time. The time derivatives of displacements are expressed in discretized form as expressed in Eq.1:

  i 1 i i 1 i i 1 i t t w w 180 t t w w t w lim i 1 i                      (1)
Where: w and t indicate respectively the out-of-plane components of the displacement vectors and time increment corresponding to the two successive stereoscopic-image pairs of the incremental orders i and i+1. The inflation test conducted at 330°C and referenced by Exp.n°5 is considered as a representative case. The corresponding out-of-plane and computed derivatives as function of time are reported in Fig. 4.c. The transitional domain of displacements (during about 300 ms) corresponds to an alternation between local maxima and minima of the derivative. Such variations can be explained by the opposing effects of gravity and the fast pressure increase inside the cavity of the forming cabinet. A decayed exponential analytical form as expressed in (Eq.2) is selected to fit the time decrease of local maxima of the derivatives:

             τ t t exp a a 0 0  (2)
Where: t0, a0,  and τ respectively referred to a time offset, an amplitude offset, a pre-exponential coefficient and characteristic time duration. The results of the fitting operations are presented in Table .1. The characteristic time-durations (τ) relative to bubble-inflation tests conducted at identical heating temperatures present similar order of magnitudes. The advantage of using of a decayed exponential form is related to the significance of the characteristic time duration which corresponds to the duration required to the attenuation of 63% of the maximal velocity amplitude at the origin of times. Consequently this parameter can be considered as an objective indicator to select the reference of the stereo-DIC computations, regardless of the initial sagging level.

Conversion of Principle Strains to Thickness Maps

Under the assumption of material incompressibility, which is reported in different studies in the case of HIPS, the maximum and minimum deformation fields on the surface can be used to compute full-fields to thicknesses. Such information is helpful especially to adjust the heating levels and to evaluate the influence of any temperature heterogeneous distribution on the resulting quality of the pre-stretching phase. According to the formalism detailed in [5] the thickness descriptions are not influenced with the adopted definition of the strain fields. We note that the stereo-DIC technique was successfully used to image the whole circular region of the plate that was exposed to heater. However, to be coherent with the evaluated temperature maps based on the Kriging interpolation method, only thinning distributions within the mapped zones are presented in Fig. 5. Comparison between Fig. 2 and Fig. 5 indicates similarities between the distributions of temperatures and thinning levels. The highest thinning levels are measured at the zones of highest surface temperatures.

CONCLUSIONS

By using the Kriging interpolation method during the radiative heating of thin HIPS sheets, the existence of surface sagging was detected from the presence of temperature gradients. To quantify the actual stretching of the sagged surface that takes place during bubble-inflation by stereo-DIC, an objective identification of the reference of deformations was required. Stereoscopic measurements performed at a frequency of 180 fps during bubble-inflation posteriori to two different heating conditions showed that the geometrical instabilities corresponded to out-of-plane damping undulations of the sheet. The time derivatives of the out-of-plane displacements confirmed that the transition moment from the sagged to the initially strained geometries of the sheet corresponded to the attenuation of 63% of their maximal velocity amplitude, regardless of the initial sagging level. The developed method allowed comparison between the principal strains that were assessed from different thermal conditions and correlation of the surface thinning with the initial temperature distributions of the softened HIPS sheets. This work constitutes a first step towards using open-mold thermoforming equipment and stereo-DIC based strain measurements to identify the thermomechanical parameters of thermoplastic sheets.

FIGURE 1 .

 1 FIGURE 1. Overview of the used experimental configuration combining stereo-DIC and thermoforming.

FIGURE 2 .

 2 FIGURE 2. a-b) Temperature maps at the surface of HIPS sheets based on Kriging interpolation method. c) Temperature profiles measured during IR heating at the position of coordinates (X=29cm; Y=15cm).

  .a). A latency duration of almost two seconds marks the time required for the punch to reach the surface of the inflated bubble (domain between C and D in Fig.3.a).

FIGURE 3 .

 3 FIGURE 3. a) Bubble-height variations during large deformations obtained by the used thermoforming equipment. b) Stereo images of the observed field-of-view at the representative moments corresponding to the moments A, B, C and E.

FIGURE 4 .

 4 FIGURE 4. a) Pressure and average out-of-plane displacement evolution during bubble inflation. b) Reproducibility of the displacement instabilities of the tests conducted at 330°C. c) Suggested undulation filtering method.

FIGURE 5 .

 5 FIGURE 5. Thinning distributions at the end of a 1.5s bubble inflation operation. a) Command temperature of the heater at 310°C. b) Command temperature of the heater at 330°C.

TABLE 1 .

 1 Identified parameters using the expression in Eq.2

	Test		310°C				330°C
	Ref.	a0 (mm/s) (mm/s) t0 (s)	 (s)	a0 (mm/s) (mm/s) t0 (s)	 (s)
	1	13.636	127.912 0.738 0.215	27.586	370.775 0.750 0.050
	2	12.031	125.812 0.789 0.217	24.948	305.372 0.714 0.074
	3	14.025	137.466 0.560 0.201	28.473	388.115 0.702 0.050
	4	15.121	126.166 0.561 0.214	27.772	347.009 0.684 0.051
	5	14.123	132.499 0.550 0.213	27.224	334.630 0.827 0.051
	Average	13.787	129.971 0.640 0.212	27.201	349.180 0.735 0.055
	Std	1.006	4.440	0.103 0.006	1.197	28.698	0.051 0.009

ACKNOWLEDGEMENTS

The authors acknowledge the European Regional Development Fund FEDER, the French state and the Hauts-de-France Region council for co-funding the ELSAT 2020 by CISIT project (POPCOM action).