Keywords: MapReduce, Hadoop, Speculation, Stragglers, Heartbeat

Speculative execution can significantly improve the performance of Big Data applications by launching other copies of stragglers (slow tasks). Stragglers detection plays an important role in the effectiveness of speculative execution. The methods employed to detect stragglers use the information extracted from the last received heartbeats which may be outdated when triggering detection. This, in turn, can mislead Big Data analytic systems to make wrong detection with high inaccuracy. To shed the light on this issue, we carry out extensive simulations to identify how heartbeat arrival, task starting times, and detection methods impact the accuracy of stragglers detection in Big Data analytic systems. We reveal that the asynchrony in heartbeat arrivals not only lead to marking normal tasks as stragglers (false positives) but can also result in overlooking real stragglers (false negatives).

I. INTRODUCTION

The past decade has witnessed a tremendous increase in the deployment of Big Data analytic systems in large-scale environments to meet the ever growing size and velocity of data. Nevertheless, a major challenge -when running Big Data applications at large-scale infrastructures -is performance variability. Performance variability in large-scale environments causes stragglers (tasks performing relatively slower than other tasks) which result in a severe degradation in performance. Traces from production clusters at Facebook and Microsoft pointed out that stragglers can be 7-8 X slower than the median task and can increase the average job duration by up to 47% [START_REF] Ananthanarayanan | Effective straggler mitigation: Attack of the clones[END_REF]. Big Data analytic systems (Google MapReduce [START_REF] Dean | Mapreduce: Simplified data processing on large clusters[END_REF], [START_REF] Jin | The MapReduce Programming Model and Implementations[END_REF], Apache Hadoop [START_REF][END_REF], Apache Spark [START_REF] Zaharia | Spark: Cluster computing with working sets[END_REF]) employ speculative execution to mitigate stragglers at large scale. Specifically, new copies of detected stragglers are launched on available machines. Over the past years, there have been several efforts to improve the effectiveness of speculative execution, such as launching speculative copies on lightly loaded machines [START_REF] Zaharia | Improving mapreduce performance in heterogeneous environments[END_REF], making speculative execution resourceaware [START_REF] Ananthanarayanan | Reining in the outliers in map-reduce clusters using mantri[END_REF], [START_REF] Zhou | Energy-efficient speculative execution using advanced reservation for heterogeneous clusters[END_REF], or integrating speculation and job scheduling decisions [START_REF] Ren | Hopper: Decentralized speculation-aware cluster scheduling at scale[END_REF].

Stragglers detection plays an important role in the effectiveness of speculative execution. Thus, many studies have focused on improving stragglers detection. The basic philosophy of existing methods to detect stragglers is to compare (evaluate) Fig. 1: CDF of the freshness of heartbeats for Sort applications with different input sizes tasks, using the information extracted from the last received heartbeats 1 . These methods can broadly be categorized as either ScoreBased or RateBased. While ScoreBased detection methods [START_REF] Dean | Mapreduce: Simplified data processing on large clusters[END_REF], [START_REF][END_REF] use the progress score of tasks (i.e., the fraction of the data has been processed) to judge if a task is a straggler or not (e.g., in Hadoop [START_REF][END_REF], a task is marked as a straggler if its progress score is smaller than the average progress score of all running tasks minus 20%), RateBased detection methods [START_REF] Zaharia | Improving mapreduce performance in heterogeneous environments[END_REF], [START_REF] Ananthanarayanan | Reining in the outliers in map-reduce clusters using mantri[END_REF] mark tasks which are 20% slower than the average progress rate (or tasks which have 20% longer "estimated" elapsed times compared to the mean or median). Recent studies have shown that current detection methods (ScoreBased and RateBased) achieve very low precision and recall (i.e., the precision of ScoreBased detection, used in Hadoop, can be only 12% in heterogeneous infrastructures [START_REF] Phan | A new framework for evaluating straggler detection mechanisms in mapreduce[END_REF]). Unfortunately, only a few efforts [START_REF] Phan | A new framework for evaluating straggler detection mechanisms in mapreduce[END_REF], [START_REF] Phan | Energydriven straggler mitigation in mapreduce[END_REF], [START_REF] Phan | On understanding the energy impact of speculative execution in hadoop[END_REF] have touched on how to explain the (in)accuracy of detection methods.

The main contribution of this study is to identify and to quantify the impact of heartbeat arrival on the inaccuracy of stragglers detection in large-scale infrastructures. To the best of our knowledge, no previous studies have worked on identifying and analyzing the impact of heartbeat arrival on stragglers detection. Our study is motivated by two observations. First, having up-to-date information about the running tasks is essential when detecting stragglers. The second observation is that throughout the execution of Big Data applications, the freshness of the information reported by last received heartbeats vary between tasks, depending on the heartbeat window (e.g., 6s in Hadoop) and the task starting times. To demonstrate that, we deployed Hadoop 3.2.0 (Yarn) on 23 virtual machines cluster using EnosLib [START_REF] Cherrueau | Enoslib: A library for experiment-driven research in distributed computing[END_REF]. We use 6 nodes from Ecotype cluster (each machine is equipped with two Intel Xeon E5-2630L v4 10-cores processors, 128 GB of main memory, and 400 GB of SSD) at Nantes site of Grid'5000 [START_REF] Balouek | Adding virtualization capabilities to the Grid'5000 testbed[END_REF]. In this cluster, 1 VM (configured with 6 CPU cores and 6 GB of RAM) is running alone on 1 node and acts as ResourceManager. The remaining 22 VMs (each configured with 1 CPU core and 6 GB of RAM) are distributed among the remaining 5 nodes (we deployed 2 VMs, 2 VMs, 4 VMs, 4 VMs, and 10 VMs per node, respectively) and act as NodeManagers and DataNodes. We limit the network bandwidth to 500 Mbps and run 6 applications including three Sort applications with 10, 20, and 30 GB inputs and three Wordcount applications with 10, 20, and 30 GB inputs. We modified Hadoop to log the progress scores and time-stamps of each task whenever a heartbeat arrives and reported the age of the most outdated progress score per second (the age of the heartbeat with the oldest value) for each application. In Fig. 1, we can clearly see that the freshness of heartbeats information for the three Sort applications varies from 0 -6.7 seconds.

Unfortunately, current detection methods do not take into account the freshness of heartbeats information. To this end, we have developed a simulator written in Java and implemented two state-of-the-art ScoreBased and RateBased detection methods used in Hadoop [START_REF][END_REF] and in Bing [START_REF] Ananthanarayanan | Reining in the outliers in map-reduce clusters using mantri[END_REF]. Using synthetic data sets, we carry out extensive simulations over a large number of scenarios to identify how heartbeat arrival, task starting times, and detection methods impact the accuracy of stragglers detection in Big Data analytic systems. We reveal that the asynchrony of heartbeat arrivals (and the resulting stale information) not only result in marking normal tasks as stragglers (false positive) but also lead to the non-detection of real stragglers (false negative).

The rest of this paper is organized as follows. The problem alongside two detection methods are formalized in Section II. The experimental methodology is explained in Section III. Section IV discusses the experimental results. Finally, Section V concludes this study.

II. MODEL AND DETECTION METHODS

A. Model

We present the model to describe the problem and on which we build the simulator for experiments in Sections III and IV.

Let J be a job (a set of tasks). For each task T i ∈ J , we define its duration as d i and its starting time as s i . We consider both to be independent from the placement of T i . We use d J to denote the average duration of the tasks of J . A task T i is considered as a straggler if and only if

d i ≥ 1.2 × d J , i.e.,
if its duration is 20% higher than the average duration of all tasks. Heartbeats are signals sent by tasks to the master node (e.g., from the tasks to ApplicationMaster) at a fixed interval; we use hw to denote the heartbeat window (the default hw is 6s in Hadoop and 10s in Spark). In addition, heartbeats are also sent when the task starts or ends. Let st i,j be the sending time of the j th heartbeat of T i . By definition st i,0 = s i and st i,j = s i + j × hw. We use j end (i) to denote the index of the last heartbeat (end of the task). Then j end (i) = d i /hw and st i,j end (i) = s i + d i . Each heartbeat has also a heartbeat latency l i,j (usually l i,j varies from few hundreds of milliseconds -up to 500ms in the experiments in Section I). Thus, a heartbeat is received by the master node at time rt i,j = st i,j + l i,j . At time t, we use j last (t, i) to denote the index of the last received heartbeat from task T i , more precisely j last (t, i) = max{j, rt i,j ≤ t}. The information transmitted through heartbeats include the progress scores of running tasks (for example, the progress score of a map task is the fraction of input data read). To simplify the problem, we assume the computation speed to be constant. Thus, the progress score of a task T i at time t is P S i (t) = (t -s i )/d i . As the master node can only follow the progress of a task through heartbeats, the perceived progress score may differ from the read onethis strongly depends on the heartbeat window. We refer to perceived one as approximated progress score. It is donated by P S i (t), more precisely, P S i (t) = P S i (st i,j last (t,i) ).

B. Stragglers Detection Methods

We present two state-of-the-art methods to detect stragglers: 1. ScoreBased: This method relies on the progress scores (precisely, the approximated ones which were received in the last heartbeats) of the running and completed tasks. Specifically, at a time t, the average progress score is computed as:

P S(t) = T i , r i,0 ≤t P Si(t)
|{Ti, ri,0≤t}| . Accordingly, ScoreBased method marks a task T i as a straggler if P S i (t) ≤ P S(t) -0.2 [START_REF][END_REF]. 2. RateBased: In order to be less sensitive to delayed tasks (task with skewed starting times), some detection methods use the notion of progress rate, that is the rate at which progress score increases. More precisely, the progress rate of a task T i at time t is P R i (t) = P S i (t)/(t -s i ). However, P S i (t) represents the approximate progress score, thus the perceived progress rate is an approximate one and can be referred to as approximated progress rate. It is denoted by P R i (t) and is computed as P R i (t) = P S i (t)/(t -s i ).

Usually, the progress rate based method (will refer to as RateBased) computes the expected duration ET D i (t) of a task T i at time t, more precisely:

ET D i (t) = t -s i + (1 -P S i (t))/ P R i (t).
A task T i is considered as a straggler if and only if ET D i (t) ≥ 1.2 × ET D(t). ET D(t) denotes the average expected duration for all started tasks. Similar method is used by LATE [START_REF] Zaharia | Improving mapreduce performance in heterogeneous environments[END_REF] and Mantri (with the factor 1.5) [START_REF] Ananthanarayanan | Reining in the outliers in map-reduce clusters using mantri[END_REF].

III. EXPERIMENTAL SETUP

We build a Java simulator based on the model described in the previous section. We describe the different parameters and the synthetic jobs sets we use in our experiments below. We use the values 10s, 20s, 50s and 100s for d avg and for f we use 0 (homogeneous, all tasks with same duration), 0.1 (slightly heterogeneous, no stragglers), and 0.25 (more heterogeneous durations with some stragglers). In the last case, the amount of stragglers is expected to be 10%. Indeed, with a uniform draw on

A. Synthetic Jobs

[(1 -f )d avg , (1 + f )d avg ], the probability of being in [1.2d avg , (1 + f )d avg ] is (1 + f -1.
2)/2f = 0.1 when f = 0.25. We will use "d min -d max " to distinguish the different sets of jobs (e.g., 18 -22s or 75 -125s) in Section IV.

Starting Times:

For starting times, we use two main possible modes. In the first one, all starting times are equal to 0. We refer to this mode as UNIFORMSTARTS. In the second one, we use ending times from UNIFORMSTARTS as starting times, this mode is referred to as SKEWEDSTARTS.

B. Hosts

First, we propose two modes for heartbeat latencies, one without (NOLATENCY) where all heartbeats arrive as soon as they are sent, and one with (WITHLATENCY) where latency follows a Pareto Distribution of scale 1s and shape 5 (our objective is to mimic the heartbeat latency in real Hadoop cluster -up to 500ms in our experiment -and to have few "extreme" events). In 90% of the send events, latency is a few dozen to a few hundreds of milliseconds, but for the last 10%, latency is above 1s and more (which is then above the stragglers detection window, i.e., 1s). A CDF of latencies is shown in Fig. 2 (1 million draw, note that we forced the latency to stay below 2s by redrawing any higher value). We always consider the number of hosts to be equal to the number of tasks in the simulated job. For each configuration, if the simulation uses randomness (i.e., WITHLATENCY or for SKEWEDSTARTS in synthetic jobs) then we run the experiments 50 times. The other two parameters we are tuning are the heartbeat window (hw) and the speculative lag, i.e., the time since the starting of a job after which the stragglers detection starts. We use a Hadoop-like configuration, i.e., with 6s heartbeat window. In every case, stragglers are detected every second after the speculative lag (i.e., detection starts as soon as one task is finished). In our experiments, the detection considers tasks with at least one received heartbeat.

D. Metrics

A task is said to be detected if there is a moment during the execution where the detection method marked it as a straggler. For example, if for the task T i there is a time t after the speculative lag where P S i (t) ≤ P S(t) -0.2, then this task is detected by the ScoreBased detection method. Note that a task can be detected at time t, but might not be detected anymore later (because of more or less up-to-date information). In such a case, the task is still considered as detected in the results (the task has been reported at least once as a straggler). If a detected task is a straggler (its duration exceeds 1.2 times the average duration of all tasks), then it is a true positive. If a detected task is not a straggler, it is then a false positive. Finally, a straggler that is never detected is a false negative.

IV. EXPERIMENTAL RESULTS

We present all our results with box plots. As a reminder, the horizontal line inside the box is the median of the values for a given setup, frontiers of the box are the limits of first and last quartiles, and the vertical bars at the end are the limits of first and last deciles. Outliers are represented by dots. We will precise the average value in the text when needed.

The false positive rates (respectively, the false negative rates) are depicted in Fig. 3 (respectively, in Fig. 4). In all sub-figures of Fig. 3 and4, the left figure depicts the case where heartbeats have no latency (they are received as soon as they are sent), the right figure shows the case where heartbeat latency is added (up to 2s, but most of the time less than 1s which is equivalent to the detection window interval). We normalize false positives and false negatives using the number of normal tasks (i.e., all non-stragglers including the false positives) and the number of real stragglers. We refer to them as false positive rate and false negative rate, respectively. In the case of false negatives, we only show results for duration ranges where there are stragglers (i.e., 7.5 -12.5s, 15 -25s, 37.5 -62.5s, and 75 -125s).

1. False Positives (Figure 3): For Homogeneous task durations, as expected, no false positives are detected under NOLATENCY mode. However, adding latency leads to a few false positives when task durations are set to 10s under both ScoreBased and RateBased (e.g., false positive rates when the job size is set to 10 tasks are 4% and 2%, respectively). For heterogeneous task durations, more false positives appear under both ScoreBased and RateBased detection methods. In the case of ScoreBased, for jobs with 10s average task In the case of RateBased detection method, the number of false positives is larger in general. For example, for the 15-25s task durations range and 50-task jobs, the average false positive rate under SKEWEDSTARTS mode is 53.27% under NOLATENCY mode (median 53.26%) and 54.07% under WITHLATENCY mode (median 54.17%). This indicates that for more than half of the runs, more than 50% of the non-stragglers are tagged as stragglers at least once during the run (however, it is important to note that they are not handled as stragglers at the same time, some of them may lose this tag and may have it back later). This issue occurs under UNIFORMSTARTS mode too, but with smaller amplitude (the average false positive rate is 24.31% for the same task durations range and job size). Regardless of the detection method, we find that the false positive rate is impacted by the average duration of tasks. In general, the false positive rate decreases (sometimes significantly) when the average task duration increases. For example, for 50-task jobs and under NOLATENCY mode, the false positive rate is reduced from 53.27% to 13.27% on average when average task duration increases from 20s to 100s. This is more obvious with RateBased than ScoreBased. The size of the job seems to have little impact. The presence of heartbeats latency increases the size of the box plots and the outliers count but does not alter the general trend.

2. False Negatives (Figure 4): In all cases, RateBased does not result in false negatives. On the other hand, ScoreBased results in false negatives. This method does not detect any stragglers for jobs with average task durations of 50s and 100s under UNIFORMSTARTS-NOLATENCY mode (rates inferior to 1 are outliers). For jobs with 20s average task duration, ScoreBased detects most of the stragglers (on average, only 6.43% of stragglers are not detected under NOLATENCY mode for 50task jobs). A surprising result is that heartbeat latency seems to improve (to decrease) the number of missed stragglers for ScoreBased detection method. Maybe as surprisingly, the false negative rate decreases under SKEWEDSTARTS mode.

V. CONCLUSION

We study how heartbeat arrivals and using outdated information when detecting stragglers impact the accuracy of stragglers detection. Our findings can be summarized as follows:

(1) Not considering the freshness of information during stragglers detection can lead to a large amount of false positives (mainly noticeable with RateBased). (2) This phenomenon is greatly amplified when the heartbeats are asynchronous, either because of skewed starting times or because of late heartbeats (due to latency). (3) False negative is mainly an issue with ScoreBased methods. As future work, we plan to evaluate detection methods when considering the time-stamp of heartbeats (e.g., they use the time-stamps of heartbeats to estimate the progress scores/rates when the detection is triggered) using synthetic and production data sets.

1 .

 1 Durations: We generate random task durations. More precisely, we follow uniform distribution with range [d min , d max ]. d min and d max are fixed from the expected average d avg . More precisely d min = (1 -f ) × d avg and d max = (1 + f ) × d avg .

Fig. 2 :

 2 Fig. 2: CDF for the latency of Heartbeat arrivals

Fig. 3 :

 3 Fig. 3: False positive rates in Hadoop-like configuration on synthetic jobs

Not to be confused with the heartbeats that report the resource availability (from the NodeManager to the ResourceManager in Yarn) or the ones that report the job status (from the ApplicationMaster to the ResourceManager in Yarn), in this paper, heartbeats refer to the periodic signals that report the progress/status of running tasks: from the tasks to the ApplicationMaster in Yarn (known as HearbeatTask in Spark).

ACKNOWLEDGMENTS

This work is supported by the ANR KerStream project (ANR-16-CE25-0014-01). The work of the third author was partially supported by National Science Foundation Grant OAC-1740218. Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RE-NATER, and several Universities as well as other organizations (see http://www.grid5000.fr/).


est destinée au dépôt

ASTERICS: A Tool for the ExploRation and Integration of omiCS data

Élise Maigné, Céline Noirot, Jérôme J. Mariette, Yaa Adu Kesewaah, Sébastien Dejean, Camille Guilmineau, Julien Henry, Arielle Krebs, Laurence Liaubet, Fanny Mathevet, et al.

To cite this version:

Élise A Tool for the ExploRation and Integration of omiCS data

Élise Maigné 1, * , Céline Noirot 1,2, * , Jérôme Mariette 1,2 , Yaa Adu Kesewaah 1,3 , Sébastien Déjean 3,4 , Camille Guilmineau 1,3 , Julien Henry 1,3 , Arielle Krebs 1,2 , Laurence Liaubet 5 , Fanny Mathevet 1,3 , Hyphen-Stat 6 , Christine Gaspin 1,2 , Nathalie Vialaneix A Tool for the ExploRation and Integration of omiCS data

Élise Maigné 1, * , Céline Noirot 1,2, * , Jérôme Mariette 1,2 , Yaa Adu Kesewaah 1,3 , Sébastien Déjean 3,4 , Camille Guilmineau 1,3 , Julien Henry 1,3 , Arielle Krebs 1,2 , Laurence Liaubet 5 , Fanny Mathevet 1,3 , Hyphen-Stat 6 , Christine Gaspin 1,2 , Nathalie Vialaneix

ASTERICS web application

Explore, integrate and visualize your omics data.

backend frontend

Rserve {renv}

PyRServe

• Available at: http://asterics.miat.inrae.fr

• Source code (GPLv2): https://forgemia.inra.fr/asterics/asterics

Available analyses

Generated datasets and analyses are navigable using a DAG:

Help pages

On each analysis page, focused help pages are available to help the user choose analysis settings or interpret the results. These pages include images and GIF animations.

Full documentation: https://asterics.pages.mia.inra.fr/user_documentation/

Outputs of analyses

• Interactive and easily customizable plots:

• Sortable and filtrable tables:

Additional features

• Use your UUID to come back later to your analyses.

• Export the results of your analysis in HTML reports (using Rmarkdown).

• Export complex objects as RDS data or data.frame as CSV files.

• Use the online tool or use Docker for a local use or for deployment.

Contributors

Partners Funding

  Maigné, Céline Noirot, Jérôme J. Mariette, Yaa Adu Kesewaah, Sébastien Dejean, et al.. AS-TERICS: A Tool for the ExploRation and Integration of omiCS data. ECCB: 21st European Conference on Computational Biology, Sep 2022, Sitges, Spain. , 2022, 10.7490/f1000research.1119290.1. hal-03890660

  1,3 1 Université de Toulouse, INRAE, UR MIAT, 31326, Castanet-Tolosan, France -2 Université Fédérale de Toulouse, INRAE, BioinfOmics, GenoToul Bioinformatics facility, 31326, Castanet-Tolosan, France -3 Plateforme Biostatistique, Genotoul, Toulouse, France -4 IMT, UMR5219, Université de Toulouse, CNRS, UPS, 31062, Toulouse, France -5 GenPhySE, Université de Toulouse, INRAE, ENVT, F-31326, Castanet-Tolosan France -6 Hyphen-stat, https://hyphen-stat.com/, Toulouse, France ( * ) These authors contributed equally to the work.

  1,3 1 Université de Toulouse, INRAE, UR MIAT, 31326, Castanet-Tolosan, France -2 Université Fédérale de Toulouse, INRAE, BioinfOmics, GenoToul Bioinformatics facility, 31326, Castanet-Tolosan, France -3 Plateforme Biostatistique, Genotoul, Toulouse, France -4 IMT, UMR5219, Université de Toulouse, CNRS, UPS, 31062, Toulouse, France -5 GenPhySE, Université de Toulouse, INRAE, ENVT, F-31326, Castanet-Tolosan France -6 Hyphen-stat, https://hyphen-stat.com/, Toulouse, France ( * ) These authors contributed equally to the work.


Antoine Perasso

Quentin Richard
email: quentin.richard@math.cnrs.fr

Ezio Irene Azzali

Ezio Venturino

Well-posedness, positivity and time asymptotics properties for a reaction-diffusion model of plankton communities, involving a rational nonlinearity with singularity

Keywords: Reaction-diffusion models, positivity, well-posedness, plankton modeling, predator-prey models AMS classification scheme numbers: 35A01, 35B09, 35K57, 47D06, 92D25

Well-posedness, positivity, and time asymptotics properties for a reaction-diffusion model of plankton communities, involving a rational nonlinearity with singularity

because the phytoplankton-zooplankton trophic interactions are at the base of the food chain on our planet, [START_REF] Edwards | Oscillatory behaviour in a three-component plankton population model[END_REF] and disturbances to this basic ecosystem such as those mentioned above may have serious consequences that are far beyond the nutrition chain and may involve the worldwide oxygen production, see [START_REF] Petrovskii | Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change[END_REF] and the references quoted there. This is mainly the consequence of the unregulated human activities, [START_REF] Anderson | Harmful algal blooms and eutrophication: nutrient sources, composition, and consequences[END_REF], e.g. utilization of chemical pesticides in agriculture and the release of untreated wastewaters, [START_REF] Davidson | Anthropogenic nutrients and harmful algae in costal waters[END_REF], that ultimately flow into the shallow waters near the coastlines and thus contribute to the raise in the organic nutrients concentration in the ocean, [START_REF] Friligos | Eutrophication and red tide in aegean coastal waters[END_REF]. The harmful algal blooms deplete the water from its oxygen content and thereby threaten the life of aquatic creatures. For these reasons it is important to be able to predict them and mathematical models are fundamental tool to achieve that goal, [START_REF] Franks | Models of harmful algal blooms[END_REF][START_REF] Steele | A simple plankton model[END_REF][START_REF] Steele | A simple model for plankton patchiness[END_REF]. Most of these models have been formulated by explicitly avoiding space, assuming that the ocean environment properties are independent of time or position in space, [START_REF] Medvinsky | Biological factors underlying regularity and chaos in aquatic ecosystems: simple models of complex dynamics[END_REF]. But this is unrealistic as hydrodynamics plays an important role in the shaping of an aquatic community, as well as factors as temperature, salinity, turbulent mixing intensity. A consequence is the fact that spatial structures become possible in this context, both induced by the heterogeneities in the aquatic medium and by the trophic interactions, [START_REF] Steele | Spatial heterogeneity and populations stability[END_REF][START_REF] Steele | Some comments on plankton patches[END_REF]. Thus multi-habitat and multi-patch formation is possible, [START_REF] Medvinsky | Chaos and regular dynamics in a model communities[END_REF].

Large amplitude oscillations of plankton populations are predicted by theoretical analyses, when nutrients abound in the ocean, [START_REF] Rosenzweig | Paradox of enrichment: destabilization of exploitation ecosystems in ecological time[END_REF][START_REF] Gilpin | Enriched predator prey systems: Theoretical stability[END_REF][START_REF] Edwards | Oscillatory behaviour in a three-component plankton population model[END_REF], but are not confirmed by empirical data, [START_REF] Slaughter | Grazing impact of mesozooplankton in an upwelling region off northern california[END_REF][START_REF] Goericke | Top-down control of phytoplankton biomass and community structure in the monsoonal arabian sea[END_REF][START_REF] Benndorf | Top-down control of phytoplankton: the role of time scale, lake depth and trophic state[END_REF], originating thus the paradox of enrichment, [START_REF] Rosenzweig | Paradox of enrichment: destabilization of exploitation ecosystems in ecological time[END_REF][START_REF] Gilpin | Enriched predator prey systems: Theoretical stability[END_REF]. The original Rosenzweig model has been modified to improve it, in particular accounting for the zooplankton vertical movement, following phytoplankton for feeding, [START_REF] Farkas | Revisiting the stability of spatially heterogeneous predator-prey systems under eutrophication[END_REF]. The latter indeed distributes inhomogeneously in view of the diminishing light in the water with depth, due to absorption in the upper layers, [START_REF] Raymont | Plankton and Productivity in the Oceans[END_REF]. The properties of the combined above mechanisms leading oscillations to settle to a stable coexistence equilibrium have been elucidated in [START_REF] Morozov | Nutrient-rich plankton communities stabilized via predator-prey interactions: revisiting the role of vertical heterogeneity[END_REF][START_REF] Farkas | Revisiting the stability of spatially heterogeneous predator-prey systems under eutrophication[END_REF].

Based in part on these results, further explorations have been carried out in [START_REF] Azzali | Exploring the role of vertical heterogeneity in the stabilization of planktonic ecosystems under eutrophication[END_REF], including a depth-dependent vertical turbulent diffusion, providing a more realistic scenario. In this paper, we study the following model equations, where t > 0 is time, h ∈ [0, H] is the depth and others parameters are given in Table 1 (see [START_REF] Azzali | Exploring the role of vertical heterogeneity in the stabilization of planktonic ecosystems under eutrophication[END_REF] for details). Moreover p, n and z respectively represent the phytoplankton and nutrients densities and the average density of zooplankton:

           ∂n ∂t = D ∂ 2 n ∂h 2 -L h (p) n 1+χn , ∂p ∂t = D ∂ 2 p ∂h 2 + L h (p) n 1+χn -zg(p) -m p p, dz dt = kz(t) H H 0 g(p)(t, h) dh -mz(t). (1)
Due to the functional response g, i.e. the predator ingestion rate of the zooplankton as a function of phytoplankton density, the latter model is a generalization of the one proposed in [START_REF] Azzali | Exploring the role of vertical heterogeneity in the stabilization of planktonic ecosystems under eutrophication[END_REF]. The operator L h is given either by or by

L h (p) = r exp(-γh)p (2)
L h (p) = r exp -ν h 0 p(t, x) dx p (3)
assuming, as the case may be, an exponential decay of light with increasing depth or a light attenuation due to phytoplankton self-shading. The parameters r, γ and ν introduced in the the latter equations are also defined in Table 1. Moreover, system (1) is equipped with the following boundary conditions:

∂n ∂h (t, 0) = 0, n(t, H) = n H , ∂p ∂h (t, 0) = 0, ∂p ∂h (t, H) = 0,
for every t > 0, where n H ≥ 0 is constant. We also add some initial conditions:

n(0, h) = n 0 (h), p(0, h) = p 0 (h), z(0) = z 0 .
In the following, we will consider the model 1) and prove its well-posedness. To this end, tools from functional analysis are employed.

In this paper, we want to prove existence and uniqueness of a nonnegative solution for Problem (1) for both cases of operator L h given in (2)-(3), in a L 2 framework, and where g is a positive functional satisfying some Lipschitz property and is positive up to a translation (i.e. (g + λI) is positive for some λ). To achieve that goal, we follow a standard line of proof, sketched next with an outline of the changes and difficulties encountered. We rewrite the model as a Cauchy problem, we prove that the linear part generates a positive C 0 -semigroup, we check that the nonlinear part verifies a Lipschitz property and is positive up to a translation. These latter points then allow us to use a fixed point theorem to get the desired result. Such kind of mathematical developments have already been published for PDE structured models ( [START_REF] Magal | On semilinear cauchy problems with non-dense domain[END_REF], [START_REF] Perasso | Infection load structured SI model with exponential velocity and external source of contamination[END_REF], [START_REF] Perasso | Implication of age-structure on the dynamics of Lotka-Volterra equations[END_REF]), reaction diffusion systems ( [START_REF] Amann | Local and global strong solutions to continuous coagulationfragmentation equations with diffusion[END_REF], [START_REF] Apreutesei | On a prey-predator reaction-diffusion system with Holling type III functional response[END_REF], [START_REF] Apreutesei | An optimal control problem for a two-prey and one-predator model with diffusion[END_REF], [START_REF] Duprez | Criterion of positivity for semilinear problems with applications in biology[END_REF]) and a case mixing diffusion and age-structure [START_REF] Walker | Global existence for an age and spatially structured haptotaxis model with nonlinear age-boundary conditions[END_REF].

In the present model, some new technical difficulties appear, due to the shape of the system. First, there is a nonhomogeneous Dirichlet boundary condition for n, so we need to make the change of variables:

ñ = n -n H (4)
in order to get a Cauchy problem. Consequently, in addition to the proof that the linear part generates a positive C 0 -semigroup, we also need to prove a lower bound property, implying that {f ∈ L 2 (0, H) : f (x) ≥ -n H a.e. x ∈ [0, H]} is invariant under the semigroup. Moreover, another critical point in the mathematical analysis stands in a singularity of the nonlinear part at: n = -1/χ so we need to restrict the space to a subset where the denominator is nonzero. A final difficulty is that the nonlinear part does not satisfy the required Lipschitz property in L 2 , but does in L ∞ . Consequently, we need some L ∞ estimates, that are proved by using the truncation method of Stampacchia (see e.g. [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF]).

The paper is structured as follows: in the next section, we make explicit the framework used in the sequel, taking into account the model specificities as previously described. Section 3, dealing with well-posedness, is dedicated to the main results of the article; we first prove that the linear part generates a C 0 -semigroup that satisfies some lower and upper bounds; we then handle the nonlinear part showing it satisfies a Lipschitz property and checking that it is positive up to a translation, implying the existence and uniqueness of a nonnegative solution; we then show that the solution is global since it cannot explode in finite time, prove that n is bounded and give a sufficient condition to get extinction for p and z. All these results are obtained for the two cases of operator L h as defined in (2)-(3).

Framework

For sake of simplicity and without loss of generality, we assume in all that follows that the diffusion coefficient is D = 1. Recalling (4), it follows that the model (1) becomes

           ∂ ñ ∂t = ∂ 2 ñ ∂h 2 -L h (p) ñ+n H 1+χ(ñ+n H ) , ∂p ∂t = ∂ 2 p ∂h 2 + L h (p) ñ+n H 1+χ(ñ+n H ) -zg(p) -m p p, dz dt = kz(t) H H 0 g(p)(t, h) dh -mz(t), (5)
for every t ≥ 0, h ∈ [0, H], with the boundary conditions:

∂p ∂h (t, 0) = 0, ∂p ∂h (t, H) = 0, ∂ ñ ∂h (t, 0) = 0, ñ(t, H) = 0.
Since n = ñ + n H , it suffices to prove that the problem (5) is well-posed in a suitable Banach space, in the semigroups setting. We will then drop the tilde in the following and write n instead of ñ, for a better reading. We work in the Hilbert space

X = (L 2 (0, H) × L 2 (0, H) × R, • X ), endowed with the norm (n, p, z) X = n L 2 (0,H) + p L 2 (0,H) + |z|
and the scalar product

(n 1 , p 1 , z 1 ), (n 2 , p 2 , z 2 ) X = n 1 , n 2 L 2 (0,H) + p 1 , p 2 L 2 (0,H) + z 1 z 2 .
We define the linear operator A : D(A) ⊂ X → X by:

A    n p z    =    n p -m p p -mz    , with domain D(A) given by {(n, p, z) ∈ H 2 (0, H) × H 2 (0, H) × R : n (0) = n(H) = p (0) = p (H) = 0}.
Note here that (n, p, z) belong to D(A) ⊂ X and are time-independent, while it was a function of time (and space) in the model [START_REF] Amann | Local and global strong solutions to continuous coagulationfragmentation equations with diffusion[END_REF]. The derivatives are consequently taken with respect to h ∈ [0, H], e.g.

n = dn dh , p = dp dh .
For sake of simplicity we keep the same notations, though the space will always be specified to avoid some possible confusion.

Since we are interested in the positivity of the solutions, we denote by X + the positive cone of X . Actually, because of the change of variable (4), we have

n ≥ 0 ⇐⇒ ñ ≥ -n H ,
where n and ñ are respectively the solutions of (1) and [START_REF] Azzali | Exploring the role of vertical heterogeneity in the stabilization of planktonic ecosystems under eutrophication[END_REF]. To this end we define, for every ε ≥ 0, the space

X ε := {(n, p, z) ∈ X : (n + ε1 [0,H] , p, z) ∈ X + }.
We see that X 0 = X + and the sequence of spaces {X ε } ε≥0 is increasing in the sense that

X + ⊂ X ε 1 ⊂ X ε 2 , ∀ε 2 ≥ ε 1 ≥ 0.
We will then obtain the positivity when considering ε = n H . We now suppose, and in all that follows, that the nonlinear functional g satisfies the assumption below.

Assumption 1. We suppose that g :

L ∞ + (0, H) → L ∞ + (0, H) and there exists λ > 0 such that λp -g(p) ∈ L ∞ + (0, H) for every p ∈ L ∞ + (0, H)
, and for every m > 0 there exists some constant l m ≥ 0 such that for every

(p 1 , p 2 ) ∈ {p ∈ L ∞ + (0, H) : p L ∞ (0,H) ≤ m} 2 , we have g(p 1 ) -g(p 2 ) L ∞ (0,H) ≤ l m p 1 -p 2 L ∞ (0,H)
Remark 2. We can note that the classical functional responses Holling types I,II II, Ivlev and the one given in [START_REF] Azzali | Exploring the role of vertical heterogeneity in the stabilization of planktonic ecosystems under eutrophication[END_REF]:

g(p) = p, g(p) = p 1 + p , g(p) = p 2 1 + p 2 , g(p) = (1 -e -p ), g(p) = p 2 (1 + p) satisfy the Assumption 1.
Since the functional g is defined in L ∞ (0, H), we need to define the Banach space

X ∞ = (L ∞ (0, H) × L ∞ (0, H) × R, • X ∞ ) ⊂ X endowed with the norm (n, p, z) X ∞ = n L ∞ (0,H) + p L ∞ (0,H) + |z|
and we also define X ∞ + the positive cone of X ∞ , as well as the spaces

X ∞ ε := {(n, p, z) ∈ X ∞ : (n + ε1 [0,H] , p, z) ∈ X ∞ + } ⊂ X ∞ ,
for every ε ≥ 0. Because of the singularity of the nonlinear part in (5) at

-n H - 1 χ
we define, according to the two cases of operator L h given in ( 2)-(3), the functions

f i : X ∞ n H +(2χ) -1 → X ∞ by: f 1 (n, p, z) =       -r exp(-γ•)p n+n H 1+χ(n+n H ) r exp(-γ•)p n+n H 1+χ(n+n H ) -g(p) kz H H 0 g(p)(t, h) dh       , f 2 (n, p, z) =       -r exp(-ν h 0 p(x) dx)p n+n H 1+χ(n+n H ) r exp(-ν h 0 p(x) dx)p n+n H 1+χ(n+n H ) -g(p) kz H H 0 g(p)(t, h) dh       for each i ∈ {1, 2}. Lemma 3. The range of f i is included in X ∞ for each i ∈ {1, 2}. Proof. Let (n, p, z) ∈ X ∞ n H +(2χ) -1 , then f 1 (n, p, z) X ∞ ≤ 2r χ p L ∞ + g 2 (p) L ∞ + k|z| g 2 (p) L ∞ < ∞.
and the same inequality holds for f 2 .

When focusing on (5), we will consequently study thereafter the following abstract Cauchy problems:

U (t) = AU (t) + f i (U (t)), ∀t > 0, in X ∞ n H , U (0) = U 0 ∈ X ∞ n H ⊂ X ∞ n H +(2χ) -1 , (6)
for every i ∈ {1, 2}, where U (t) = (n(t), p(t), z(t)) T . The approach used to prove existence and uniqueness of a solution of ( 6) is classical (see e.g. [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]). The techniques used for both models are the same: we first show that A generates a C 0 -semigroup in X , then we prove some Lipschitz property for f i . Now that the framework is clear, we can deal with the well-posedness of the Cauchy problem (6).

Well-posedness

Linear part

We start this section by handling the linear part.

Theorem 4. For every ν ≥ 0, the operator A -νI generates a C 0 -semigroup {T A-νI (t)} t≥0 on X . Moreover it satisfies

∀u 0 ∈ X , t -→ T A-νI (t)u 0 ∈ C([0, ∞), X ) ∩ C 1 ((0, ∞), X ), (7)
T A-νI (t)u 0 X ∞ ≤ u 0 X ∞ , ∀t ≥ 0, ∀u 0 ∈ X ∞ , (8)
whence {T A-νI (t)} t≥0 ⊂ L(X ∞ ) and

T A-νI (t)u 0 ∈ X ε , ∀t ≥ 0, ∀ε ≥ 0, ∀u 0 ∈ X ε . (9)
Note that (9) implies the positivity of {T A-νI (t)} t≥0 .

Proof. The sketch of the proof is the following: we first prove that A-νI generates a C 0semigroup by verifying the surjectivity and the dissipativity properties. We deduce that for every initial condition (n 0 , p 0 , z 0 ) ∈ X , the solution of the linear problem verifies [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF].

We then show that this solution (denoted by (n, p, z)) verifies the following inequalities:

min{0, inf h∈[0,H] n 0 (h)} ≤ n(t, h) ≤ max{0, sup h∈[0,H] n 0 (h)}, ( 10
) min{0, inf h∈[0,H] p 0 (h)} ≤ p(t, h) ≤ max{0, sup h∈[0,H] p 0 (h)}, (11)
-|z 0 | ≤ z(t) ≤ |z 0 |, (12)
for every t ≥ 0, a.e. h ∈ [0, H] and (8) follows. We then check that {T A-νI (t)} t≥0 is positive. Finally we prove [START_REF] Davidson | Anthropogenic nutrients and harmful algae in costal waters[END_REF].

(i) Clearly, D(A) is dense into X . Moreover, for every (n, p, z) ∈ D(A), we have

A(n, p, z), (n, p, z) X = Dn , n L 2 + Dp -m p p, p L 2 -mz 2 = D H 0 n(h) ∂ 2 n ∂h 2 dh + D H 0 p(h) ∂ 2 p ∂h 2 dh -m p H 0 p(h) 2 dh -mz 2 = -D H 0 ∂n ∂h 2 dh -D H 0 ∂p ∂h 2 dh -m p H 0 p(h) 2 dh -mz 2 ≤ 0.
Consequently, A is dissipative in X . Let us show now that λI -A : D(A) → X is surjective for any λ > 0. Let H = (h n , h p , h z ) ∈ X and λ > 0. We look for

U := (n, p, z) T ∈ D(A) such that (λI -A)U = H, i.e. λn -n = h n , (13)
λp -p + m p p = h p , (14)
λz

+ mz = h z , so z = h z λ + m .
We multiply ( 13) and ( 14) respectively by u ∈ H 1 (0, H) and v ∈ H 1 (0, H), then integrate between 0 and H to get

       λ H 0 nu - H 0 n u = H 0 h n u, λ H 0 pv - H 0 p v + m p H 0 pv = H 0 h p v.
An integration by parts gives

λ H 0 nu + H 0 n u = H 0 h n u, (15)
λ H 0 pv + H 0 p v + m p H 0 pv = H 0 h p v, (16)
whence

a 1 (n, u) = L 1 (u), a 2 (p, v) = L 2 (v),
where the bilinear forms a 1 :

V × V → R, a 2 : H 1 (0, H) × H 1 (0, H) → R and the linear forms L 1 : V → R, L 2 : H 1 (0, H) → R are defined by: a 1 (n, u) = λ H 0 nu + H 0 n u , a 2 (p, v) = λ H 0 pv + H 0 p v + m p H 0 pv, L 1 (u) = H 0 h n u, L 2 (v) = H 0 h p v, where V := {u ∈ H 1 (0, H) : u(H) = 0}.
A simple application of Lax-Milgram theorem implies that for every (h n , h p ) ∈ (L 2 (0, H)) 2 , there exists a unique (n, p) ∈ V × H 1 (0, H) such that:

a 1 (n, u) = L 1 (u), a 2 (p, v) = L 2 (v),
for every (u, v) ∈ V × H 1 (0, H). Now, we verify that U belongs to D(A). For this, we use ( 15) and ( 16) with

u ∈ C ∞ c ([0, H]) and v ∈ C ∞ c ([0, H]) respectively, where C ∞ c (0, H) refers to C ∞ functions with compact support. Then, we get H 0 n u ≤ [|λ| n L 2 (0,H) + h n L 2 (0,H) ] u L 2 (0,H) ≤ c 1 u L 2 , H 0 p v ≤ [(|λ| + |m p |) p L 2 (0,H) + h p L 2 (0,H) ] v L 2 (0,H) ≤ c 2 v L 2 ,
for some constant c 1 and c 2 . Consequently n ∈ H 1 (0, H) and p ∈ H 1 (0, H), so n ∈ H 2 (0, H) and p ∈ H 2 (0, H). Finally, to prove the surjectivity, an integration by parts of ( 15)-( 16) with u ∈ C c (0, H) and v ∈ C c (0, H) implies ( 13) and [START_REF] Friligos | Eutrophication and red tide in aegean coastal waters[END_REF]. Moreover, an integration by parts of ( 15) with u ∈ C(0, H), u(0) = 1, u(H) = 1 implies that n (0) = 0. Similarly, we get p (0) = 0 and p (H) = 0 after an integration by parts of ( 16) with v ∈ C(0, H) and respectively v(0) = 1, v(H) = 0 and v(0) = 0, v(H) = 1. Thus A generates a C 0 -semigroup {T A (t)} t≥0 by Lumer-Phillips theorem, and A -νI also generates a C 0 -semigroup {T A-νI (t)} t≥0 for every ν ≥ 0 by bounded perturbation arguments.

(ii) Let ν ≥ 0. We readily see that A -νI is a symmetric operator. It is actually a self-adjoint operator since it is m-dissipative (with [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF], Proposition VII.6, p. 113). Using [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF], Theorem VII.7, p. 113, we obtain that the solution of

U (t) = (A -νI)U (t) U (0) = u 0 ∈ X (17)
verifies [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF].

(iii) Let ν ≥ 0. We want to prove that the solution U (t) := (n(t, •), p(t, •), z(t)) of

U (t) = (A -νI)U (t) U (0) = (n 0 , p 0 , z 0 ) ∈ X (18)
verifies ( 10)-( 11)-( 12), for every t ≥ 0. It is clear that

z(t) = z 0 e -(ν+m)t
so that ( 12) is satisfied for every t ≥ 0. To get the result on n and p, we use the truncation method of Stampacchia (see e.g. [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF], Theorem X.3, p. 211). In all the following, we will use the notation

K σ := max{0, sup h∈[0,H] σ(h)} ≥ 0, K σ := -min{0, inf h∈[0,H] σ(h)} ≥ 0 for every function σ ∈ L ∞ (0, H). Define the function G ∈ C 1 (R) such that (a) |G (x)| ≤ M, ∀x ∈ R, (b) G is strictly increasing on (0, ∞), (c) G(x) = 0, ∀x ≤ 0.
We introduce the functions

κ : x → x 0 G(σ)dσ, ∀x ∈ R, (19)
ϕ 1 : t → H 0 κ(p(t, h) -K p 0 )dh, ϕ 2 : t → H 0 κ(p(t, h) -K p 0 )dh, ∀t ≥ 0, ϕ 3 : t → H 0 κ(n(t, h) -K n 0 )dh, ϕ 4 : t → H 0 κ(n(t, h) -K n 0 )dh, ∀t ≥ 0, where p := -p, n := -n.
Define the set

Y := {ϕ ∈ C([0, ∞), R), ϕ(0) = 0, ϕ ≥ 0 on [0, ∞), ϕ ∈ C 1 ((0, ∞), R)}.
We can show that ϕ i ∈ Y for every i ∈ 1, 4 , using [START_REF] Brézis | Analyse Fonctionnelle : Théorie et Applications[END_REF]. Moreover, we have

ϕ 1 (t) = H 0 G(p(t, h) -K p 0 ) ∂p ∂t (t, h)dh = H 0 G(p(t, h) -K p 0 ) ∂ 2 p ∂h 2 (t, h) -(ν + m p )p(t, h) dh = - H 0 G (p(t, h) -K p 0 ) ∂p ∂h (t, h) 2 dh - H 0 G(p(t, h) -K p 0 )(ν + m p )p(t, h)dh ≤ 0, ∀t > 0,
since G ≥ 0. Finally ϕ 1 ≤ 0 on (0, ∞) and consequently ϕ 1 ≡ 0, so

p(t, h) ≤ K p 0 ≤ max{0, sup h∈[0,H] p 0 (h)}, ∀t ≥ 0, a.e. h ∈ [0, H].
The same computations lead to

ϕ 2 (t) = - H 0 G (p(t, h) -K p 0 ) ∂p ∂h (t, h) 2 dh - H 0 G(p(t, h) -K p 0 )(ν + m p )p(t, h)dh ≤ 0
for every t > 0 and ϕ 2 ≡ 0 on (0, ∞), so

p(t, h) ≥ -K p 0 ≥ min{0, inf h∈[0,H] p 0 (h)}, ∀t ≥ 0, a.e. h ∈ [0, H]
and ( 11) is satisfied. Similarly, we have

ϕ 3 (t) = H 0 G(n(t, h) -K n 0 ) ∂n ∂t (t, h)dh = H 0 G(n(t, h) -K n 0 ) ∂ 2 n ∂h 2 (t, h) -νn(t, h) dh = - H 0 G (n(t, h) -K n 0 ) ∂n ∂h (t, h) 2 dh - H 0 G(n(t, h) -K n 0 )νn(t, h)dh ≤ 0, ∀t > 0, since G(n(t, H) -K n 0 ) = G(-K n 0 ) = 0.
We can also show that ϕ 4 (t) ≤ 0, ∀t > 0 whence [START_REF] Duprez | Criterion of positivity for semilinear problems with applications in biology[END_REF] holds. Considering an initial condition (n 0 , p 0 , z 0 ) ∈ X ∞ leads easily to [START_REF] Clément | One-Parameter Semigroups[END_REF].

(iv) Let us prove now that {T A-ν (t)} t≥0 is positive for every ν ≥ 0, that is, the resolvent

R λ (A -νI) := ((λ + ν)I -A) -1
is positive for λ large enough (see e.g. [START_REF] Clément | One-Parameter Semigroups[END_REF], p. 165). Let ν ≥ 0, λ ≥ 0, H := (h n , h p , h z ) ∈ X + . As point 1. above, one can consider

U := (n, p, z) = (R λ (A -νI))H ∈ D(A).
We have to prove that U ∈ X + . Since C([0, H]) is dense in L 2 (0, H), we may assume without loss of generality (using the dissipativity and the closedness of A) that

h n ∈ C([0, H]), h p ∈ C([0, H]).
Thus, we have

-p + (λ + ν + m p )p = h p , with p ∈ H 2 (0, H) ⊂ C([0, H]).
Since h p is continuous, then the latter equation implies that p is also continuous and then p ∈ C 2 ([0, H]). The absolute minimum of p is achieved at some h ∈ [0, H]. Suppose that p(h) < 0. The function

q := -p verifies the equation q -(λ + ν + m p )q = h p ≥ 0,
and its absolute maximum is reached at h. If h = 0, then by Hopf's maximum principle (see [START_REF] Protter | Maximum Principles in Differential Equations[END_REF], Theorem 4, p. 7), we would have -p (0) = q (0) > 0, which contradicts the Neumann boundary condition. If h = H then by Hopf's maximum principle, we would have

-p (H) = q (H) < 0, which is absurd. Finally, if h ∈ (0, H) then 0 ≥ -p (h) = h p (h) -(λ + ν + m p )p(h) > 0 which is not possible. Consequently p(h) ≥ p(h) ≥ 0, ∀h ∈ [0, H].
Similarly, n ∈ C 2 ([0, H]) verifies the equation

-n + (λ + ν)n = h n ≥ 0.
Moreover, n reaches its absolute minimum at h ∈ [0, H]. If n(h) < 0, then the same arguments than before lead to h = H, which contradicts the fact that n(H) = 0. Consequently

n(h) ≥ n(h) ≥ 0, ∀h ∈ [0, H].
Finally, it is clear that

z = h z λ + ν + m p ≥ 0,
which proves that R λ (A + νI) is positive and consequently that the C 0 -semigroup {T A-νI (t)} t≥0 is positive for every ν ≥ 0.

(v) Now we want to prove [START_REF] Davidson | Anthropogenic nutrients and harmful algae in costal waters[END_REF]. Let ε ≥ 0, ν ≥ 0, (n 0 , p 0 , z 0 ) ∈ X ε and (n, p, z) the solution of [START_REF] Medvinsky | Biological factors underlying regularity and chaos in aquatic ecosystems: simple models of complex dynamics[END_REF]. Because of the positivity of {T A-νI (t)} t≥0 , it only remains to prove that

n(t, h) ≥ -ε, ∀t ≥ 0, a.e. h ∈ [0, H]
which arises from (10).

Nonlinear part

In this section we handle the nonlinear part by showing a Lipschitz and a positivity properties of f i for each i ∈ {1, 2}. Let m > 0, then define the set

B m := {(n, p, z) ∈ X ∞ : (n, p, z) X ∞ ≤ m}.
Proposition 5. For every m > 0, there exists some constant k m ≥ 0 such that for every

((n 1 , p 1 , z 1 ), (n 2 , p 2 , z 2 )) ∈ X ∞ n H +(2χ) -1 ∩ B ∞ m 2
, we have

f i    n 2 p 2 z 2    -f i    n 1 p 1 z 1    X ∞ ≤ k m    n 2 p 2 z 2    -    n 1 p 1 z 1    X ∞ .
Proof. We first prove the result for f 1 , the case f 2 being similar.

Let ((n 1 , p 1 , z 1 ), (n 2 , p 2 , z 2 )) ∈ X n H +(2χ) -1 ∩ B m 2 . Some computations give f 1 n 2 , p 2 , z 2 T -f 1 n 1 , p 1 , z 1 T X ∞ ≤ 2r p 2 (n 2 +n H ) 1+χ(n 2 +n H ) -p 1 (n 1 +n H ) 1+χ(n 1 +n H ) L ∞ + z 1 g(p 1 ) -z 2 g(p 2 ) L ∞ + k H H 0 z 1 g(p 1 ) -z 2 g(p 2 ) L ∞ dh ≤ 2r (m n 2 -n 1 L ∞ (1 + χ(m + n H )) + (m + n H ) p 2 -p 1 L ∞ ) + (ml m p 1 -p 2 L ∞ + ml m |z 1 -z 2 |) (1 + k)
by Assumption 1, which proves the result.

Proposition 6. For every m > 0, there exists λ m ≥ 0 and η m ≥ 0 such that for every

(n, p, z) ∈ X ∞ n H +(2χ) -1 ∩ B m , we have f i (n, p, z) + λ m (n, p, z) ∈ X ∞ ηm . Proof. Let (n, p, z) ∈ X ∞ n H +(2χ) -1 ∩ B m , then f 1 (n, p, z) + λ m (n, p, z) =       n λ m -r exp(-γ•) p 1+χ(n+n H ) -r exp(-γ•) pn H 1+χ(n+n H ) p λ m + r exp(-γ•) n+n H 1+χ(n+n H ) -zg(p) z λ m + k H H 0 g(p)(t, h)dh      
.

Note that by Assumption 1, there exists λ > 0 such that λp -g(p) ≥ 0, so choosing λ m ≥ mλ induces that pλ m -zg(p) ≥ m(λp -g(p)) ≥ 0. Consequently, it suffices to consider

λ m = mλ (20)
and

η m = mλ m + rm 2 + rmn H (21)
which ends the proof.

Local existence and positivity

We are now able to show existence and uniqueness of a solution.

Theorem 7. Suppose that operator L h has one of the shapes given in [START_REF] Anderson | Harmful algal blooms and eutrophication: nutrient sources, composition, and consequences[END_REF] or in [START_REF] Apreutesei | On a prey-predator reaction-diffusion system with Holling type III functional response[END_REF]. Then for every initial condition (n 0 , p 0 , z 0 ) ∈ X ∞ n H , there exists a unique solution (n, p, z) ∈ C [0, t max ), X ∞ n H for the system (5), where t max ≤ ∞.

Proof. Let (n 0 , p 0 , z 0 ) ∈ X ∞ n H and m = 2 (n 0 , p 0 , z 0 ) X ∞ .
Define the constants λ m ≥ 0, η m ≥ 0 respectively by ( 20) and ( 21), the linear operator

A m = A -λ m I : D(A) ⊂ X → X ,
and for i = 1, 2 the nonlinear function

f m = f i + λ m I : X ∞ n H +(2χ) -1 → X .
We readily see that A m is the infinitesimal generator of a C 0 -semigroup {T Am (t)} t≥0 on X . Let

τ = min 1 2(k m + λ m ) , 1 2χη m > 0.
A consequence of Theorem 4 and Proposition 5 is that the nonlinear operator

G : C [0, τ ], X ∞ n H +(2χ) -1 → C([0, τ ], X ) defined by G    n(t, •) p(t, •) z(t)    = T Am (t)    n 0 p 0 z 0    + t 0 T Am (t -s)f m    n(s, •) p(s, •) z(s)    ds (22)
is a 1/2-shrinking operator on

Z := C [0, τ ], X ∞ n H +(2χ) -1 ∩ B m with G(Z) ⊂ B m , since t ≤ τ ≤ 1 2(k m + λ m ) .
Moreover, using Theorem 4, the fact that

τ ≤ 1 2χη m ,
and Proposition 6, then

G    n(t, •) p(t, •) z(t)    ∈ X n H +(2χ) -1 ∀t ∈ [0, τ ].
Consequently G preserves the space Z. The Banach-Picard theorem then implies the existence and uniqueness of a local solution

(n, p, z) ∈ C [0, τ ], X ∞ n H +(2χ) -1 ∩ B m .
It remains to prove that

n(t, h) ≥ -n H , ∀t ∈ [0, τ ], ∀h ∈ [0, H]. (23)
First, suppose that

(n 0 , p 0 , z 0 ) ∈ D(A) ∩ X ∞ n H . (24)
Using [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Theorem 6.1.7,p. 190], the solution (n, p, z) of ( 5) is classical. Consequently, the function n := -n satisfies the equation

∂n ∂t (t, h) = ∂ 2 n ∂h 2 (t, h) + L h (p)(t, h) n H -n(t, h) 1 + χ(n + n H ) ,
for every t ∈ (0, τ ] and a.e. h ∈ [0, H]. Define the function

ϕ n (t) = H 0 κ(n(t, h) -n H )dh,
where κ is given by [START_REF] Medvinsky | Chaos and regular dynamics in a model communities[END_REF], for every t ∈ (0, τ ]. We can check that

ϕ n ∈ C([0, τ ], R), ϕ n (0) = 0, ϕ n ≥ 0 on [0, τ ], ϕ n ∈ C 1 ((0, τ ], R),
then some computations lead to

ϕ n (t) = H 0 G(n(t, h) -n H ) ∂n ∂t (t, h)dh = H 0 G(n(t, h) -n H ) ∂ 2 n ∂h 2 (t, h) + L h (p)(t, h) n H -n(t, h) 1 + χ(n(t, h) + n H ) dh = - H 0 G (n(t, h) -n H ) ∂n ∂h (t, h) 2 dh + H 0 G(n(t, h) -n H )L h (p)(t, h) n H -n(t, h) 1 + χ(n(t, h) + n H ) dh ≤ 0 since G(n(t, H) -n H ) = 0, 1 + χ(n(t, h) + n H ) ≥ 1/
, z 0 ) ∈ X ∞ n H . Since D(A)∩X ∞ n H is dense into X ∞ n H , there exists a sequence (n k 0 , p k 0 , z k 0 ) k≥0 ∈ D(A)∩X ∞ n H such that lim k→∞ (n 0 , p 0 , z 0 ) -(n k 0 , p k 0 , z k 0 ) X ∞ = 0.
For every k ≥ 0, there exists a unique solution (n k , p k , z k ) ∈ C([0, τ ], X ∞ n H ) for the system (5) with initial condition (n k 0 , p k 0 , z k 0 ). Using [START_REF] Perasso | Infection load structured SI model with exponential velocity and external source of contamination[END_REF], for every k ≥ 0, we get

   n(t, •) p(t, •) z(t)    -    n k (t, •) p k (t, •) z k (t)    = T Am (t)    n 0 -n k 0 p 0 -p k 0 z 0 -z k 0    + t 0 T Am (t -s)    f m    n(s, •) p(s, •) z(s)    -f m    n k (s, •) p k (s, •) z k (s)       ds for every t ∈ [0, τ ], so    n(t, •) p(t, •) z(t)    -    n k (t, •) p k (t, •) z k (t)    X ∞ ≤    n 0 -n k 0 p 0 -p k 0 z 0 -z k 0    X ∞ + t 0 (k m + λ m )    n(s, •) p(s, •) z(s)    -    n k (s, •) p k (s, •) z k (s)    X ∞ ds ≤    n 0 -n k 0 p 0 -p k 0 z 0 -z k 0    X ∞ + τ (k m + λ m ) max s∈[0,τ ]    n(s, •) p(s, •) z(s)    -    n k (s, •) p k (s, •) z k (s)    X ∞ for every t ∈ [0, τ ], since ((n, p, z), (n k , p k , z k )) ∈ X ∞ n H +(2χ) -1 ∩ B m 2
and using [START_REF] Clément | One-Parameter Semigroups[END_REF]. Thus, we have max

t∈[0,τ ]    n(t, •) p(t, •) z(t)    -    n k (t, •) p k (t, •) z k (t)    X ∞ ≤    n 0 -n k 0 p 0 -p k 0 z 0 -z k 0    X ∞ + 1 2 max t∈[0,τ ]    n(t, •) p(t, •) z(t)    -    n k (t, •) p k (t, •) z k (t)    X ∞ whence max t∈[0,τ ]    n(t, •) p(t, •) z(t)    -    n k (t, •) p k (t, •) z k (t)    X ∞ ≤ 2    n 0 -n k 0 p 0 -p k 0 z 0 -z k 0    X ∞ → 0
as k goes to infinity, for every t ∈ [0, τ ]. Consequently ( 23) holds and we have

(n, p, z) ∈ C [0, τ ], X ∞ n H ∩ B m .
Some standard time extending properties of the solution allow to extend the solution (n, p, z) over a maximal interval [0, t max ).

Global existence and boundedness

We now prove that the solution of ( 5) is global in time and that n is bounded. We also give an example where p and z are bounded and go to extinction. We then deduce the result for (1).

Theorem 8. Suppose that operator L h has one of the shapes given in [START_REF] Anderson | Harmful algal blooms and eutrophication: nutrient sources, composition, and consequences[END_REF] or in [START_REF] Apreutesei | On a prey-predator reaction-diffusion system with Holling type III functional response[END_REF]. Then for every initial condition (n 0 , p 0 , z 0 ) ∈ X ∞ n H , there exists a unique solution

(n, p, z) ∈ C [0, ∞), X ∞ n H for the system (5), that satisfies n(t, h) ≤ max{0, sup h∈[0,H] n 0 (h)} for every t ≥ 0 and h ∈ [0, H]. Moreover, if m p > r χ (25)
holds true, then

lim t→∞ p(t, •) L ∞ (0,H) = 0, lim t→∞ z(t) = 0. Proof. Let (n 0 , p 0 , z 0 ) ∈ X ∞ n H and (n, p, z) ∈ C [0, t max ), X ∞ n H
be the solution of (5). Using the same argument of density as in the proof of Theorem 7, we only need to consider the case where the initial condition satisfies [START_REF] Petrovskii | Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change[END_REF]. Because of the positivity of the solution, we have ∂n ∂t (t, h) ≤ ∂ 2 n ∂h 2 (t, h). We define the function

ϕ n (t) = H 0 κ(n(t, h) -K n 0 )dh.
We can show that

ϕ n ∈ C([0, t max ), R), ϕ n (0) = 0, ϕ n ≥ 0 on [0, t max ), ϕ n ∈ C 1 ((0, t max ), R), and
ϕ n (t) = H 0 G(n(t, h) -K n 0 ) ∂n ∂t (t, h)dh ≤ - H 0 G (n(t, h) -K n 0 ) ∂n ∂h (t, h) 2 dh ≤ 0, ∀t > 0 so n(t, h) ≤ K n 0 , ∀t ≥ 0, a.e. h ∈ [0, H].
) 27
Consequently t max = ∞ and the solution is global in time. Suppose now that (25) holds and consider an initial consider that satisfies [START_REF] Petrovskii | Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change[END_REF]. Since the solution is classical, we get the inequality [START_REF] Raymont | Plankton and Productivity in the Oceans[END_REF]. An integration leads to Using the change of variable (4), we deduce the same result for the initial problem.

Corollary 9. Suppose that operator L h has one of the shapes given in [START_REF] Anderson | Harmful algal blooms and eutrophication: nutrient sources, composition, and consequences[END_REF] or in [START_REF] Apreutesei | On a prey-predator reaction-diffusion system with Holling type III functional response[END_REF]. Then for every initial condition (n 0 , p 0 , z 0 ) ∈ X ∞ + , there exists a unique solution (n, p, z) ∈ C [0, ∞), X ∞ + for the system (1), that satisfies n(t, h) ≤ max{n H , n 0 L ∞ } for every t ≥ 0 and a.e. h ∈ [0, H]. Moreover, if (25) holds, then lim t→∞ p(t, •) L ∞ (0,H) = 0, lim t→∞ z(t) = 0.

Open questions and perspectives

The well-posedness, positivity and asymptotic results that we proved in this article have wide range of applicability to reaction-diffusion model of plankton communities since the functional response g covers several types of predation, such as Holling types I, II, III as well as Ivlev.

The asymptotic results of extinction are obtained under a threshold condition related to phytoplankton population, stating that the mortality rate is bigger than the maximum growth rate.

The case where this threshold condition [START_REF] Protter | Maximum Principles in Differential Equations[END_REF] is not satisfied is an open question that will be investigated in a future work.

Another research direction concerns existence of steady states. The trivial equilibrium (n H 1 [0,H] , 0, 0) ∈ X clearly always exist. However the existence of non trivial steady states need deeper analysis. In [START_REF] Azzali | Exploring the role of vertical heterogeneity in the stabilization of planktonic ecosystems under eutrophication[END_REF], the authors proved numerically the existence of such non trivial equilibria for a slightly different model than the one presented in this paper.

Finally, re-cycling of the nutrient is contemplated in the boundary condition on function n as a constant inflow of nutrient at position H. It could also be alternatively considered as a flux in the n-equation, but this would lead to different cases of studies in terms of modelling as well mathematical analysis.

  ). Since p(t, •) ∈ H 2 (0, H) ⊂ C 1 ([0, H]) for every t > 0, then lim t→∞ p(t, •) L ∞ (0,H) = 0which concludes the proof.

Table 1 .

 1 Parameters involved in the model

	Notations	Definitions
	D	Vertical turbulent diffusion
	H	Depth of water column
	χ	Inverse half-saturation density of nutrient intake
	m, m p	Zooplankton and phytoplankton mortality rates
	k	Food utilization coefficient
	r/χ	Maximum phytoplankton growth rate
	γ	Light attenuation coefficient
	ν	Self-shading coefficient

  To prove that the solution is global, suppose by contradiction that t max < ∞. Since n is bounded, classical results (see e.g.[START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], Theorem 6.1.4, p. 185) imply that, either lim t→tmax p(t, •) L ∞ (0,H) = ∞

	or
			lim t→tmax	z(t) = ∞.
	However, the former cannot hold since
	∂p ∂t	(t, h) ≤	∂ 2 p ∂h 2 (t, h) +	r χ	-m p p(t, h), ∀t > 0, a.e. h ∈ [0, H]	(26)
	and the latter contradicts the fact that
	z (t) ≤ z(t)	k H

H 0 g(p)(t, h)dh -m , ∀t > 0.

(

Acknowledgement This research was undertaken within the framework of the Epimath project, funded by Region Bourgogne Franche-Comté.


M Gille

H Proudhon

J Oddershede

T F Morgeneyer

Projection-DIC analysis of projected 3D data for a 6xxx aluminium alloy under plane strain tension imaged by in situ absorption contrast tomography

This work focuses on experiments developed to better understand the deformation and damage mechanisms leading to failure when stamping 6xxx aluminium alloy sheets. A miniaturized plane strain tensile specimen was developed and validated using Digital Image Correlation (DIC) on the specimen surface to verify the plane strain condition. An in situ tensile test imaged with absorption contrast lab-tomography was then performed on the miniaturized specimen up to failure. Finally, advantage was taken of the plane strain condition to perform two-dimensional image correlation in the material bulk, so called projection-DIC (p-DIC). Intermetallic particles provided image contrast that was registered during the correlation analyses. The p-DIC method was used to compute internal strain fields that were consistent with fracture location observed on the post-mortem scan.

Introduction

Aluminium alloys are widely considered by car manufacturers to replace steel and meet the light weighting demand to reduce CO2 emissions. For automotive body panels, formability is crucial to obtain complex geometries while guaranteeing a good surface aspect and a sufficient ductility in service. Most of the stamping failures are found to occur in a state close to plane strain tension [START_REF] Bottema | Recent Developments in AA6016 Aluminium Type Body Sheet[END_REF]. This strain state, critical for formability, is thus investigated in this work aiming at assessing internal strain heterogeneity up to fracture in an aluminium alloy sheet. The experimental technique of 4D imaging enables the in situ investigation of the deformation and damage mechanisms by acquiring 3D scans at several time steps of a mechanical test with non-destructive imaging techniques like absorption contrast tomography [START_REF] Proudhon | In situ 4D mechanical testing of structural materials: The data challenge[END_REF]. The natural grey level contrast of scans coming from differently absorbing elements in the material (particles, voids, …) is used by image correlation techniques such as Digital Volume Correlation (DVC) or the more robust p-DIC to compute the displacement field between a reference image and a deformed image and deduce the internal strain field [START_REF] Buljac | On deformation and damage micromechanisms in strong work hardening 2198 T3 aluminium alloy[END_REF][START_REF] Morgeneyer | On crystallographic aspects of heterogeneous plastic flow during ductile tearing: 3D measurements and crystal plasticity simulations for AA7075-T651[END_REF].

Material and specimen geometry

The material is a 6016 aluminium alloy in T4 temper typically used for automotive body panels and provided as a metal sheet with nominal thickness of 1.15mm. To create the plane strain condition, the specimen design was inspired by flat grooved specimens presented by Bai et al. [START_REF] Bai | On the Application of Stress Triaxiality Formula for Plane Strain Fracture Testing[END_REF] and Park et al. [START_REF] Park | A new approach for fracture prediction considering general anisotropy of metal sheets[END_REF]. The geometry was then optimized by Finite Element method to have miniaturized specimens suitable for tomography with a wide enough central region of interest satisfying the plane strain condition defined in Eq. 1.

|εyy/ εxx| > 10 (1) εyy corresponds to the strain in the loading direction Y and εxx is the strain in the horizontal direction X. Axes are defined on Fig. 1 presenting the specimen geometry.

To validate the geometry, a tensile test was performed with camera instrumentation to perform DIC on the surface of the specimen where a paint speckle was applied. It was thus verified that the plane strain condition was satisfied on a 2 mm-wide region of interest and that the fracture initiated in this plane strain region. Loading curve as well as |εyy/ εxx| field at maximal load are displayed on Fig. 2 to illustrate this.

Projection-DIC to assess internal strain field evolution during in-situ plane strain tensile test

An in situ tensile test was performed on the validated plane strain specimen and 12 loading steps as well as the post-mortem one were imaged with the lab tomograph ZEISS Xradia 520 Versa with a resolution of 0.8 µm. For the selected 6016 alloy, image contrast in the reconstructed 3D images (mainly provided by intermetallic particles) was low, and it would be difficult to reliably perform DVC measurements. For this reason, contrast was increased by projecting the maximum grey levels (white) found in a stack of 40 slices along the X direction (i.e. the direction without strain) onto a single 2D section. In other words, contrast provided by intermetallic (white) particles and aluminium matrix (grey) in a 32 µm thick slice was projected onto a 2D image and subsequently used for the 2D registrations. This approach was deemed valid since plane strain conditions prevailed. From the natural speckle provided by the projected intermetallic particles, it was possible to compute the internal strain field using the previously introduced p-DIC technique. As shown on Fig. 3, an heterogeneous εyy field was revealed with the localization of strain corresponding to the location of the final fracture.

Conclusions

A miniaturized plane strain tensile specimen was designed and validated to be in plane strain condition thanks to DIC performed on the surface. An in situ tensile test later performed on this specimen provided 4D data from which an analysis by the p-DIC method revealed an internal heterogeneous strain field with localization bands corresponding to the fracture location.


S Erard

B Cec- Coni

P Le Sidaner

A P Rossi

T Capria

B Schmitt

N André

A C Vandaele

M Scherf

R Hueso

A Määttänen

B Carry

N Achilleos

C Marmo

O Santolik

K Benson

P Fernique 15

VESPA: ENLARGING THE VIRTUAL OBSERVATORY TO PLANETARY SCIENCE

Introduction: Modern space borne instruments produce huge datasets, especially on long-lived missions. This calls for new ways to handle the data, not only to perform mass processing, but also more basically to access them easily and efficiently. Virtual Observatory (VO) techniques developed in Astronomy during the past 15 years can be adapted to address this problem, provided they are enlarged to handle specificities of Solar System studies. The VESPA data access system focuses on applying VO techniques and tools to Planetary Science data, and supports all aspects of Solar System science [1]. VESPA (Virtual European Solar and Planetary Access) is developed in the framework of the EU-funded Europlanet-2020 program started Sept 1 st , 2015 for 4 years. The objective of this activity is to facilitate searches in big archives as well as in sparse databases, to provide simple data access and on-line visualization tools, and to allow small data providers to make their data available in an interoperable environment with minimum effort. This system makes intensive use of studies and developments led in Astronomy (International Virtual Observatory Alliance, IVOA [2]), Solar Physics (HELIO), and space archive services (International Planetary Data Alliance, IPDA [3]).

Data services: the VESPA architecture [1] is based on a new data access protocol, a specific user interface to query the available services, and intensive usage of tools and standards developed for the Astronomy VO [4]. The Europlanet data access protocol, EPN-TAP, relies on the general TAP (Table Access Protocol) mechanism associated to a set of parameters that describe the contents of a data service [1] [5]. These parameters are defined to enable queries on quantities relevant to the scientific user, including observational and instrumental conditions. Data services are required to return answers formatted as VOtables, which are handled by all standard VO tools.

Data services are installed at their respective provider institutes and are declared in the standard IVOA registries, so that they are always visible and reachable from query interfaces. At the time of writting, 37 data services are publicly open, and about 20 more are be-ing finalized (see, e. g., [6]). They encompass a wide scope, including surfaces, atmospheres, magnetospheres and plasmas, small bodies, experimental data such as spectroscopy in solid phase, heliophysics, exoplanets, and selected amateur data services. ESA's Planetary Science Archive (PSA) will get an EPN-TAP interface in 2018, and bridges with PDS4 are being studied.

Data access: Although accessible in many ways, EPN-TAP data services are best queried from the optimized VESPA user interface, or portal (Fig. 1) In the frame of TAP, all data services present a list of granules (usually data files) described by a series of parameters. The Europlanet data access protocol, EPN-TAP, defines a set of mandatory parameters introducing metadata that describe all granules; this is similar to the ObsTAP protocol from IVOA, which describes observational datasets in Astronomy. EPN-TAP metadata introduce both observational and instrumental conditions and are defined to handle the specific diversity and complexity of Planetary Science: ranges on several axes (spatial, temporal, spectral, photometric), measurement type, origin of data, and various references. Location is provided in the most appropriate coordinate system (e.g., sky or planetary coordinates); target-related time (local time and season, through Ls) can be provided when relevant. The VESPA portal uses the mandatory parameters to search for individual granules in all registered data services at once, allow-

6033.pdf Informatics and Data Analytics (2018)

ing for discovery of data content unknown to the user. In addition, specific parameters may also be used to describe individual services in more details, and can be used to identify granules more precisely when querying a single service.

All granules provide a link to a data file, or include the data itself in the table when possible (e.g., for scalar quantities). Data description parameters are used to identify adequate VO tools to access, plot and handle the data correctly. They not only provide a description of the file format, but also specify the dimensions, units, and physical quantities, relying on IVOA data models extended for VESPA. For instance, spectra and images are handled in different tools, and spectra measured in radiance or in reflectance are handled differently by the spectral tools.

Tools: Metadata are smoothly transferred from the VESPA portal to VO tools according to the IVOA SAMP protocol, with no user intervention. Standard VO tools are connected to the VESPA portal so that they readily display metadata, e. g., spatial footprints are plotted on a 3D sphere in Aladin or Mizar; other metadata such as local time, instrument modes, etc… can be plotted in 2D or 3D with TOPCAT.

The data themselves can be transferred in a similar way for display and standard analyses. Data description is used to select appropriate tools, e.g. TOPCAT will handle all types of tables, Aladin most images and spectral cubes, CASSIS and SPLAT-VO spectra in general (Fig. 2), 3Dview can plot measurements along a spacecraft trajectory, Autoplot is dedicated to extracting data from long time series, etc. Both Aladin and TOPCAT can produce multiresolution maps from large datasets. Most of these tools have been updated to support Planetary Science and to handle specificities of Solar System data, e. g., measurements in reflected light, coordinate systems on surfaces and in magnetospheres, etc. Other, non-VO tools are provided with a SAMP interface so that they can be included in pipe-lines (e. g., ImageJ which now provides conversions and image processing functions to the VO) and in some case new applications have been developed for VESPA (e. g., to handle fits images of planetary surfaces [7], or PDS3 spectral cubes). A significant on-going activity is the development of a connection between the VO world and Geographic Information Systems (GIS), so that EPN-TAP data services can provide links to WMS services which are then handled in QGIS; the intermediate VO layer allows for powerful search functions in the data, and cross-examinations with other datasets.

Computation services: another goal is to connect on-line computation services with interfaces similar to the data services, so as to compare observations and simulations more routinely. This activity has obvious applications, e. g., for radiative transfer in planetary atmospheres. A complementary aspect is to provide low level computation functions on-line, e.g., averages, resampling, deconvolution, etc. This is currently supported only to some extent by standard VO tools and ImageJ; in addition, higher level processing such as retrieval of Hapke parameters, multivariate analyses, etc, would also be beneficial and are being studied.

External contributions: A procedure has been identified to install data services with little resources, and hands-on sessions are organized twice a year at EGU and EPSC conferences in Europe (see VESPA web site). This is expected to favor the installation of services by individual research teams, e. g. to distribute derived data related to a published study. In complement, regular discussions are held with big data providers, starting with space agencies (IPDA). In parallel, a Solar System Interest Group has just been started in the IVOA, where several VESPA partners contribute; the goal is here to adapt existing astronomy standards to Planetary Science.

Fig 1 :

 1 Fig 1: The VESPA search interface: http://vespa.obspm.fr

Fig 2 :

 2 Fig 2: telescopic spectra of Vesta compared to basaltic meteorites from the PDS spectral library in CASSIS.

6033.pdf Informatics and Data Analytics (2018)

Acknowledgements:

The Europlanet 2020 Research Infrastructure project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 654208. VESPA web site: http://www.europlanet-vespa.eu


Paul Temple
email: paul.temple@irisa.fr

Explicit or Implicit? On Feature Engineering for ML-based Variability-intensive Systems

Keywords: feature, machine learning, software variability

Software variability engineering benefits from Machine Learning (ML) to learn e.g., variability-aware performance models, explore variants of interest and minimize their energy impact. As the number of applications of combining variability with ML grows, we would like to reflect on what is the core to the configuration process in software variability and inference in ML: feature engineering. These disciplines previously managed features explicitly, easing graceful combinations. Now, deep learning techniques derive automatically obscure but efficient features from data. Shall we give up explicit feature management in variability-intensive systems to embrace machine learning advances?

INTRODUCTION

To deal with the inherent engineering complexity of variabilityintensive systems, machine learning (ML) techniques are helpful. Applications include specifying software product lines [START_REF] Acher | Varylatex: Learning paper variants that meet constraints[END_REF][START_REF] Temple | Using machine learning to infer constraints for product lines[END_REF] and predicting systems' performance [START_REF] Siegmund | Performance-influence models for highly configurable systems[END_REF][START_REF] Siegmund | Predicting performance via automated feature-interaction detection[END_REF]. Conversely, variability management techniques can support the design of machine learning models [START_REF] Ghamizi | Automated Search for Configurations of Convolutional Neural Network Architectures[END_REF].

Core to the cross-fertilization of software variability and ML is the process of feature engineering. While the term "feature" has many meanings [START_REF] Berger | What is a Feature? A Qualitative Study of Features in Industrial Software Product Lines[END_REF][START_REF] Classen | What's in a Feature: A Requirements Engineering Perspective[END_REF] in the software variability and relates to various types of data processed by ML algorithms, feature engineering is essential for relevant configurations (variability) and accurate decisions (ML). In this paper, we show that though variability management relies on explicit feature engineering, ML transitioned from explicit to implicit feature engineering, notably enhancing the performance of deep learning models. Embracing these advances via implicit feature engineering may imply giving up the essence of variability management by losing control over the features. Should we go there?

Section 2 presents explicit feature engineering in ML and variability. Section 3 describes implicit feature management in deep learning. Finally, Section 4 discusses the implications of implicit feature management for software variability practice.

EXPLICIT FEATURE ENGINEERING

Software variability features. From a user-oriented perspective, we find multiple definitions of "feature" in software product lines textbooks: "a feature is a characteristic or end-user-visible behavior of a software system" [START_REF] Apel | Featureoriented software product lines[END_REF]; "a set of software-intensive systems that share a common, managed set of features satisfying the specific needs of a particular market segment or mission and that are developed from a common set of core assets in a prescribed way. " [START_REF] Clements | Software Product Lines : Practices and Patterns[END_REF]. Rashid et al. define a feature as "characteristic of a product that is visible to the end user in some way" [START_REF] Rashid | Aspect-Oriented, Model-Driven Software Product Lines The AMPLE Way[END_REF]. Thus, from a user perspective, a feature can be anything (requirements, design element, software artifact) as long as it helps distinguish one product from another.

ML features for image processing. On the other hand, ML models also try to distinguish products (i.e., data) based on their feature descriptions. ML features are supposed observable characteristics that, if appropriately selected, should help decide which group data belong to. Their definition remains difficult and historically relies on domain expertise. In image processing, Harris et al. [START_REF] Harris | A combined corner and edge detector[END_REF] defined corners as the most relevant parts of images helping to differentiate them. Schmid et al. [START_REF] Schmid | Evaluation of interest point detectors[END_REF] improved this approach before the introduction of the Scale Invariant Feature Transform (SIFT) descriptors [START_REF] David G Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. In this context, the literature defines features as visually explicit elements: corners and boundaries.

ML features for tabular data. The Iris dataset 1 is a toy example for pedagogical purposes. It contains only 150 instances separated into three classes, all described by four features (or attributes). They are the length and width of the sepals and petals. The ML model task is to analyze the attributes of the sepals and petals to understand what makes one instance belongs to a class or another. According to domain knowledge, these four characteristics should be sufficient to classify irises. Taking a different example, the Postoperative dataset 2 tries to determine where patients in a postoperative recovery area should go next (home, intensive care unit, or general hospital floor). Domain knowledge informs us that hypothermia is an important risk after surgery. Thus, the identified features that describe the situation (i.e., overall state of patients) focus on body temperature.

All these examples have defined their features according to domain knowledge that documents how experts analyze data to decide. Because of this human and somewhat manual aspect, features carry some explicit, explainable semantics. It allows direct mapping from the decisions to a combination of these features.

Using software variability features as ML features. Regarding software variability and feature models, concrete features (abstract features help structure information) also relate to the decisions users make when building their products. Here again, the semantics of features is explicit, allowing users to understand the impact of their choices on the resulting products. However, despite using software variability features as directly as ML features (e.g., [START_REF] Acher | Varylatex: Learning paper variants that meet constraints[END_REF][START_REF] Benoit Amand | Towards learning-aided configuration in 3D printing: Feasibility study and application to defect prediction[END_REF][START_REF] Guo | Variability-aware performance prediction: A statistical learning approach[END_REF][START_REF] Temple | Using machine learning to infer constraints for product lines[END_REF][START_REF] Temple | Empirical assessment of generating adversarial configurations for software product lines[END_REF]), some difficulties arise. For instance, ML models often require homogeneous features (i.e., over the same range of values and preventing the use of a list of elements in a single string). It demands changing the representation of enumerations. Usually, one turns literals into integers. Yet, ML models consider a spurious order between literals. One thus flattens enumerations into a set of Boolean features, for example: "is choice X selected?". The Pandas framework, in Python, calls this decomposition "dummification" 3 . While this causes the number of features artificially increase (which may cause other problems for ML, e.g., the curse of dimensionality [START_REF] Indyk | Approximate nearest neighbors: towards removing the curse of dimensionality[END_REF]) and makes them independent, the enumeration semantics remains.

IMPLICIT FEATURE ENGINEERING

The rise of deep learning (DL) has forced models to learn themselves which features are relevant to their tasks. Doing so, DL models have shown incredible performances that continue to be improved every day.

Embedding in Deep Learning. A first step (called embedding) transforms data into an intermediate homogeneous representation that accounts for relations between elements of input data (e.g., close meaning or frequent co-occurrences). One turns raw input data into vectors of real-value numbers. This vector defines a high-dimension space that groups data points of close meaning or frequent co-occurrence. Conversely, there is a greater distance among points having different meanings or infrequent occurrences. While this transformation may take as input the previously mentioned characteristics, the fact that dimensions are now a non-trivial combination of these different elements renders the interpretation of each dimension difficult, if not impossible. However, the new representation is prone to define a continuous space allowing the use of, for instance, gradient-based methods for optimization (and especially to converge when learning the task of the DL model). The creation and selection of such dimensions is fully automatic. Hence, embedding and feature extraction do not have to rely on the knowledge of domain experts to guide the selection or definition of such features. Yet, there is no guarantee of meeting all the expected properties of the resulting encoding space. 3 https://pandas.pydata.org/docs/reference/api/pandas.get_dummies.html Interpretation and control over DL features. Automated embedding and feature extraction techniques raise the problem of building humanly understandable models (i.e., explainable AI), in which we seek to understand why the model took a decision. Also, multiple software engineering tasks rely on a deep understanding of the system's behavior. For instance, trying to debug a system because of unexpected behavior is facilitated by clear code structures. With DL models, understanding the model decision or why it did not activate the expected neuron is much more difficult because the features do not have concrete meaning. As an example of an application, Pierazzi et al. [START_REF] Pierazzi | Intriguing Properties of Adversarial ML Attacks in the Problem Space[END_REF] used some gradient-based techniques to modify programs such that they become malicious. Yet, the goal of the transformation is to insert a malicious behavior in the program while retaining its basic functionality (i.e., the program still behaves correctly, in addition, it retrieves and leaks information about the system). This work first transforms the code of the application into an intermediate representation. This representation supports an optimal introduction of malicious behavior. Yet, the transformation needs to be bidirectional since we need to produce back the modified source code to run tests. Such a property is desirable for domain expert intervention and control.

Balancing performance and control. Deep learning can achieve far better performances in various tasks than older ML models and relies on techniques that demand fewer human actions to optimize the data representation. The current trend is to use embedding to provide an efficient intermediate representation to employ gradientbased optimization techniques. Unfortunately, this prevents domain experts from controlling how the model behaves and, if necessary, modifying it accordingly. On the other hand, their intuitions on which characteristics to consider to solve a problem may be wrong, leading to ML models of lower performance. In the end, it seems that there is a trade-off to find between the level of control and understanding experts need to have to pursue their different activities, such as model debugging, and the performance of the models.

EXPLICIT OR IMPLICIT?

So the dilemma in this paper is the following: either we acknowledge that explicit feature engineering promoted by the variability modeling community is crucial, implying non-deep models of lower performance, or we do fully embrace the deep learning advances, allowing ML models to transform the variability space of our systems automatically in ways that domain experts may not understand.

Giving up on three decades of feature management research since the definition of feature models [START_REF] Kyo | Feature-oriented domain analysis (FODA) feasibility study[END_REF] for performance is not easy. First, it means denying in part the achievements members of the variability management community contributed to, in this conference or other venues. They successfully offered techniques to model, design, test, and verify variability-intensive systems with myriad options. Second, there are categories of critical systems where features need to be interpretable by humans, should they be involved in medical decisions or justice ones. On the other hand, it leads to missing automation and performance opportunities that only deep learning can bring. Thus, our community have to explore trade-offs between explicit and implicit feature engineering. We also think that this exploration could lead to a new notion of synthetic feature, issued from reversible transformations of the original feature space.

Such a graceful combination is required to address complex challenges such as predicting the performance or energy consumption of variability-intensive systems [START_REF] Siegmund | Green Configuration: Can Artificial Intelligence Help Reduce Energy Consumption of Configurable Software Systems[END_REF]. We think now is the time to do it, as prompt-enabled language models generate code and documentation for features automatically [START_REF] Chen | Evaluating Large Language Models Trained on Code[END_REF].

https://archive.ics.uci.edu/ml/datasets/iris

https://archive.ics.uci.edu/ml/datasets/Post-Operative+Patient Conference'17, July 2017, Washington, DC, USA Paul Temple and Gilles Perrouin

ACKNOWLEDGMENTS

Gilles Perrouin is an FNRS Research Associate. This work was partly funded by the EOS-VeriLearn, project number 30992574 of the Fonds de la Recherche Scientifique (F.R.S-FNRS) in Belgium.