Meghan Shaughnessy

Rosalie Defino
email: rdefino@umich.edu

Erin Pfaff
email: erpfaff@umich.edu

Merrie Blunk
email: mblunk@umich.edu

Timothy Boerst
email: tboerst@umich.edu

Impacts of a mathematical mistake on preservice teachers' eliciting of student thinking

Keywords:

We report on a study of preservice teachers' eliciting performances in a scenario in which a student has made a mistake and, if sufficiently probed, is able to recognize the mistake and revise their work. Our findings reveal the skills and capabilities of one group of preservice teachers at the start of a teacher education program.

Student responses in teaching

Mr. Chinn, an experienced sixth grade teacher, is circulating in his sixth-grade mathematics class as students work independently on a set of problems involving operations with fractions, including mixed numbers. He pauses by Chloe and notices that she has arrived at an answer for a subtraction problem with "mixed numbers" that looks wrong. By mixed numbers, we mean numbers that combine integers and proper fractions. Chloe's work, shown in Figure 1, has resulted in an incorrect answer. Mr. Chinn decides that he does not want to assume what Chloe was doing and so he asks her some questions about her work: "Chloe, what did you notice when you started to work on this problem?" Chloe responds, "I was trying to take three-fifths away from two-fifths but I didn't have enough so I needed to borrow." Mr. Chinn nods and follows up, "Where did you borrow from?" Cloe responds, "I borrowed from the 3 and then the two-fifths become twelve-fifths." Mr. Chinn wonders whether Chloe is overgeneralizing "borrowing" from work with subtracting multi-digit numbers or if she has made a bookkeeping error. He asks some more questions and presses Chloe to explain the value of the "little one" in fraction notation and Chloe pauses, and then says softly, "I think I made a mistake." Mr. Chinn asks her what she thinks is a mistake and to articulate why she thought she did it. Through this interaction, he learns that Chloe believes she was just "working too quickly" and that she forgot that she was borrowing "five-fifths." Then, Mr. Chinn asks Chloe to solve the problem again. When Chloe changes the "two-fifths" to "seven-fifths," Mr. Chinn asks Chloe how she knew to write seven-fifths and why the mixed number can be re-written as "three and sevenfifths."

In this episode, Mr. Chinn encounters a regular problem of practice in mathematics teaching. A student is doing something when solving a mathematics problem that misaligns with the lesson's mathematics goal of accurately computing the difference. The processes of teaching and learning often involve such mismatches between goals and the current status of students' performances. For example, students might use an approach that is not familiar to a teacher. Or a student may make what Bass (personal communication) terms a "bookkeeping" error. [START_REF] Radatz | Students' errors in the mathematical learning process: A survey[END_REF] refers to such an error as a mistake, an isolated and unrepresentative mis-executions of an algorithm (a careless move) as opposed a systematic and persistent mis-execution that reflect a conceptual or procedural misunderstanding. Chloe's response reflects a book-keeping error. Another possibility is that student's work might reflect a misunderstanding that results in an incorrect answer. Of course, other sorts of mismatches happen in teaching that are crucial, such as when teachers favors/expects a particular approach or route of reasoning and a student treads an alternative path. In a broader sense these mismatches illustrate a broader categorization of students' actions/productions that are "not what the teacher is looking for." This is not about anticipation as teachers may very well anticipate that students might produce some of these responses. This is more about the need for teachers to have generative responses when there is a mismatch between what the teacher prefers/hopes to see and the variability in what students produce in the course of learning mathematics.

Interactions around student responses that are not what the teacher is looking for can be powerful sites of learning for both teachers and students [START_REF] Borasi | Capitalizing on errors as "springboards for inquiry": A teaching experiment[END_REF][START_REF] Hiebert | Making sense: Teaching and learning mathematics with understanding[END_REF]. When the answer is incorrect, the joint sense-making required to interpret student thinking goes beyond the identification and correction of mistakes/errors into the conceptual analysis of why the mistake/error was made [START_REF] Borasi | Capitalizing on errors as "springboards for inquiry": A teaching experiment[END_REF][START_REF] Kazemi | Promoting conceptual thinking in four upper-elementary mathematics classrooms[END_REF]. The information that teachers can uncover by eliciting students' thinking can then guide their pedagogical response. This is not to say that all student responses that are different from the responses that teachers are looking for have the same potential for being a site for learning. For example, one could argue that a bookkeeping error has less potential to support learning for the teacher and the student compared to an instance in which a student's work reflects a misunderstanding that results in an incorrect answer. While the importance of interacting with students around incorrect answers is well-documented, research suggests that practicing teachers vary greatly in their attention to and treatment of errors and mistakes in classroom discussion [START_REF] Bray | A collective case study of the influence of teachers' beliefs and the knowledge on error-handling practice during class discussion of mathematics[END_REF][START_REF] Santagata | Practices and beliefs in mistake-handling activities: A video study of Italian and U.S. mathematics lessons[END_REF][START_REF] Silver | Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom[END_REF]. Less is known about the ways in which preservice teachers (PSTs) elicit student thinking when they see answers that are not the answers that they want students to produce. This matters for several reasons. For one, when students do something that the teacher is not looking for, this indicates a place where the teacher could learn something about the student and/or about mathematics. Both of which merit being able to leverage the opportunity. For another, when students make what seem to be mistakes, asking questions may reveals understandings that are not evident from their written work.

In this study, we examine PSTs' skills around asking questions to learn about student thinking (eliciting student thinking) when a student has an incorrect answer. We explored a case in which the student makes a bookkeeping error, a mistake, in their arithmetic process and will recognize the mistake if asked specific questions about their reasoning. The context is challenging for many PSTs. It has significant implications for students because teachers' understanding the basis of student's mistakes is necessary for responsive instruction. In other words, teachers need to know the degree to which a mistake reflects underlying understandings in order to ascertain how to move a student's understanding forward. Specifically, we sought to analyze the extent to which PSTs' (a) elicited the full process used; (b) elicited the student's understanding of the process, and (c) elicited the student's mistake, including the reason for the mistake and their revised process. We illustrate how knowledge of this kind of skill can be gathered through the use of a simulation in which a PST interacts with a teacher educator who is taking on the role of a student, what such findings reveal about PSTs in one teacher education program, and the implications of these findings.

Focusing on PSTs' skills at the beginning of teacher preparation

There has been increasing focus on preparing PSTs for the work of teaching by focusing on specific instructional practices (e.g., [START_REF] Ball | The work of teaching and the challenge for teacher education[END_REF][START_REF] Ball | Combining the development of practice and the practice of development in teacher education[END_REF][START_REF] Mcdonald | Core practices and pedagogies of teacher education: A call for a common language and collective activity[END_REF]. In "practice-based teacher education," learning goals for PSTs are tied to developing skills in carrying out specific teaching tasks. To help PSTs develop skill, it would be useful to know what they bring with them to teacher education, so as to be able to design the program in ways that capitalizes to their prior ways of acting and interacting. Prior research provides knowledge of some of the orientations and assumptions that PSTs bring to teacher preparation and their knowledge of subject-matter content. Research on particular teaching practices often focuses on the utility of the approach, composite parts of the teaching practice, and/or challenges in learning to enact particular teaching practices. Such knowledge is useful for the design of teacher education; however, it is insufficient. Needed is knowledge of the skill with which PSTs can enact particular teaching practices upon entry to a teacher education program.

We build on a prior study which examined the skills with eliciting student thinking at the start of a teacher education program (Shaughnessy, & Boerst, 2018a). PSTs elicited the thinking of a student who arrived at a correct answer to a multi-digit addition problem using a method that was likely to be unfamiliar to PSTs. That study revealed skills with eliciting student thinking that could be built on (e.g., eliciting the student's process), needed to be learned (e.g., eliciting the student's understanding), and needed to be unlearned (e.g., directing the student to use a different process). However, the case of eliciting when a student has made a mistake is different. We wondered whether PSTs would learn that the student had made a mistake and what understandings underlie the mistake.

The practice of eliciting student thinking

In teaching, "teachers pose questions or tasks that provoke or allow students to share their thinking about specific academic content in order to evaluate student understanding, guide instructional decisions, and surface ideas that will benefit other students" [START_REF] Teachingworks | High leverage teaching practices[END_REF]. Eliciting student thinking makes the nature of students' current knowledge available to the teacher. Such information is essential for engaging students' preconceptions and building on their existing knowledge (Bransford, Brown, & Cocking, 2000). In actual practice, eliciting student thinking is often done in conjunction with interpreting student thinking and responding to student thinking in ways that support students in building on their current understandings [START_REF] Jacobs | Professional noticing of children's mathematical thinking[END_REF].

Because of the crucial nature of the practice and the need to teach PSTs to do this work, it is necessary to specify the work involved in eliciting student thinking. We conceive of the work of eliciting student thinking in the context of a mathematics problem as asking questions to bring out the student's process and the student's understanding of key ideas underlying the process [START_REF] Shaughnessy | Designing simulations to learn about preservice teachers' capabilities with eliciting and interpreting student thinking[END_REF]. Students are at the center of this work. It is their thinking which is sought and intended to be understand, and the work is situated in mathematical context that focus dialog, shape interaction, and influence follow-up questions. Throughout this paper, we use "skill with eliciting student thinking" to refer to the degree to which PSTs are able to engage in this work.

Studying skill with eliciting student thinking

Since 2011, we have used teaching simulations to study PSTs' skill with eliciting student thinking. A simulation serves as an "approximation of practice" [START_REF] Grossman | Redefining teaching, reimagining teacher education[END_REF]. Simulations have been used in many professional fields such as medicine and more recently in the preparation of teachers [START_REF] Self | Designing and using clinical simulations to prepare teachers for culturally responsive teaching[END_REF]. In these simulations, a PST interacts with a "standardized student" (a teacher educator taking on the role of a student using a well-defined set of rules for responding) around a specific piece of written work. This form of assessment has advantages over field-based interviews to assess skill with eliciting student thinking [START_REF] Shaughnessy | Complementary assessments of preservice teachers' skill with eliciting student thinking[END_REF]. Field-based interviews leave the particular student thinking being elicited to chance and can be highly variable. The teaching simulation uses highly specified protocols that enable us to control key aspects of the student's process, understanding, and demeanor. This leads to comparable eliciting contexts for all of our PSTs and facilitates fair and more nuanced assessment of PSTs' eliciting skill.

We design teaching simulations to have a consistent three-part format [START_REF] Shaughnessy | Designing simulations to learn about preservice teachers' capabilities with eliciting and interpreting student thinking[END_REF]. First, PSTs are provided with student work on a problem and given 10 minutes to prepare for an interaction. Second, PSTs have five minutes to interact with the standardized student, eliciting the student's thinking to understand the steps they took, why they performed particular steps, and their understanding of the key mathematical ideas involved. Third, PSTs are interviewed about their interpretations of the student's thinking. In total, the assessment takes approximately 25 minutes.

We designed this simulation to be one in which the student uses non-standard process (meaning a process other than the standard US algorithm) for solving multi-digit subtraction problems and makes a mistake. The process we selected is sometimes referred to as "Expand and Trade" (see the student work in Figure 2). The process involves writing the value of the minuend and subtrahend in expanded form and making any necessary trades. When used correctly, the user would then subtract the numbers place-by-place in expanded form. This student correctly applies the expanding and trading process, but mistakenly adds values by place instead of subtracting. This student has conceptual understanding of expanded form, the meanings of addition and subtraction, and when, how, and why to make trades. In this instance, however, the student loses sight of the fact that they are solving a subtraction problem due to the addition symbols in the expanded form. This is a bookkeeping error. During the interaction, the student will change their mind when pressed to: make and evaluate an estimate for the original problem, represent their process with a picture, talk about the meaning of the operation, explain why trades were made, solve another multi-digit subtraction problem, or re-solve the original problem. The student is trained to only reveal the mistake if pressed by the PST on one of these specific points. In this paper, we focus our analysis on: (a) eliciting the student's original process, (b) eliciting the student's understanding of the process, (c) eliciting the student's mistake, including the reason for the mistake and the revised process and solution. For each of these components, we identified specific "moves" and tracked their presence or absence in each performance. We used the software package Studiocode© to parse the video into talk turns. Then, we identified "instances," which contain a question posed by a PST and the student's response to that question. Two independent coders applied all of the relevant codes to each instance. Disagreements were resolved through discussion and by referencing a code book.

Findings Eliciting the student's original process

The student's process had five steps: expanding both the minuend and subtrahend, comparing the numbers in each place, trading, adding (rather than subtracting) numbers by place, and adding the partial sums to arrive at the answer. The highest rates of eliciting occurred around the expansion (70%), the comparison of the numbers in each place (90%), and the trading steps (80%). In fact, 65% of PSTs elicited all three of these steps and 90% of the PSTs elicited two or more. However, only 53% of the PSTs elicited that the student added numbers by place in the expanded for after trading. This was surprising given that it was the point where the mistake occurred. The smallest percentage of eliciting occurred around the adding of the partial sums (10%). This was not surprising given that (a) it occurred after the point where the student made the mistake and may have not been relevant if the PST had asked questions that resulted in the student revealing that they had made a mistake, and (b) it could be easily inferred from the written work. Given that these PSTs elicited some, but not all, of the student's steps, we concluded that they brought skills relevant to eliciting a student's process that may be leveraged and built upon in the teacher education program; however, there was also a need for these PSTs to work on identifying and actively asking about steps that are particularly important to understand in the context of the problem.

Eliciting the student's understanding of the original process

We coded whether PSTs elicited the student's understanding of six mathematical ideas underlying the process. The highest percentages of eliciting of understanding occurred around the operation in the problem (27%), why the student expanded (37%), and why the student traded (27%). The lowest percentages occurred around the equivalence of expanded form and the "original" number (7%), the reasonableness of the answer (7%) and what trading means (0%). This suggests that limited eliciting of understanding of mathematical ideas occurred. However, when we looked across the set of ideas, we found that 67% of PSTs elicited the student's understanding of one or more idea. Thus, we concluded that this 67% of PSTs brought skills relevant to eliciting the student's understanding in this scenario, but note that their eliciting does not show discernment of which understandings are more important to focus on and/or more relevant to subsequent instruction.

Eliciting the student's mistake, including the reason for the mistake and the revised process We coded the extent to which the PSTs elicited the student's mistake, including their understanding of why they made the mistake. We found three foci in PSTs' performances: eliciting the student's realization that a mistake had been made (47% of PSTs), pointing out the mistake and getting the student to admit a mistake without first eliciting the mistake from the student (20% of PSTs), and only asking questions that were not focused on the mistake (33% of PSTs). In other words, 67% of the PSTs uncovered the mistake either by eliciting it or asking the student to confirm that they made a mistake; 33% percent did not learn about the mistake through questioning.

We now turn to the 47% of the PSTs who elicited the student's realization that a mistake had been made. These 14 PSTs elicited the mistake in different ways. The most common way, used by 11 PSTs, was to ask about the operation involved in the problem and to press on how the operation in the problem (subtraction) differed from the operation that the student had used (addition). Another two PSTs posed another problem for the student to solve or asked the student to redo the original problem. The student profile specifies that the student should correctly apply the Expand and Trade method if prompted to solve another problem. A third approach to eliciting the mistake was to ask about why the student had made the trades. One PST leveraged this to ask why the student had added.

We next turn briefly to the 20% of the PSTs uncovered the student's mistake by pointing it out themselves and getting the student to admit the mistake. Often, PSTs made statements about what it appeared the student had done based on the written work. For example, one PST stated her impression that that expansion and trades were executed correctly and that the student had added instead of subtracting. This exchange took place after the PST had elicited the steps of the student's original process through the trading step. In the exchange, the idea that the student mistakenly added instead of subtracted is introduced by the PST, not elicited from the student. The student merely confirms what the PST has stated. We see this approach to uncovering the mistake as qualitatively distinct from eliciting the mistake through asking questions.

To explore what PSTs did after learning the student had made a mistake, we considered the 20 cases in which PSTs uncovered the student's mistake either by eliciting the student's realization that a mistake had been made or pointing out the mistake. Twelve of the 20 PSTs did not ask any further questions about the mistake. These PSTs need to learn to ask additional questions once a student recognizes a mistake. Of the 8 PSTs who followed up about the mistake, 5 made statements based on their own inferences about how and/or why the student made the mistake and got the student to agree. The remaining 3 PSTs asked questions to learn about how and/or why the mistake was made. These findings suggest that these 8 PSTs bring skills relevant for eliciting a student's mistake, but may need to learn reasons and strategies for eliciting how and why a mistake was made.

Discussion

Understanding the skills with teaching practices that PSTs bring to teacher education is key to designing experiences that are responsive to the needs of PSTs (Shaughnessy, & Boerst, 2018a). This study examined the ways in which PSTs at the beginning of a teacher preparation program elicited the thinking of a student who made a mistake when solving a subtraction problem. Learning about the reason for a mistake is important for teachers to accurately assess and respond to student thinking. For example, a student who has made a mistake due to misunderstanding about ideas underlying the process would need different instruction than a student who has made a bookkeeping error when carrying out a well-understood process. However, at the start of their teacher education program, these PSTs focused more on eliciting the revised method and/or solution than asking about why the mistake was made. We hypothesize that there might be multiple reasons that this pattern occurred. Some PSTs might refrain from directly asking the student why they made the mistake because they are fearful of embarrassing the student by asking them to talk about it. For other PSTs, limitations in their content knowledge for teaching might impede being able to identify important mathematical ideas related to the process. Still other PSTs might not have been able to formulate a question that could be used to elicit evidence of the student's understanding.

The data suggest several directions for continuing research. First, these PSTs were in their first week of a teacher education program. In what ways do their skills with eliciting student thinking develop over time? Second, what course activities might effectively cultivate an inclination to elicit how and why students make mistakes? Third, the assessment itself involved a mistake where the student was using a "non-standard" approach to solve the problem. Anecdotally, we have reason to think that some PSTs were discounting the student's reasoning because they believed that the student should be using a different method to solve the subtraction problem. A future study could compare skills in eliciting around a mistake with a "standard" and an "alternative" algorithm.

Figure 1 :

 1 Figure 1: Student work, the student makes a mistake

Figure 2 :

 2 Figure 2: Student work, the student makes a mistake

Acknowledgment

The research reported here was supported by the National Science Foundation under Award No. 1535389. Any opinions, findings, and recommendations expressed are those of the authors and do not reflect the views of the National Science Foundation.


Elise De La Rochebrochard

Heather Joshi

Why are there more unplanned births in the UK than in France? A comparative analysis between the French ELFE and the UK Millennium birth cohorts

Context. Whether the pregnancy was planned is one of the factors that could have long-term consequences for the child. Despite widespread use of contraception and abortion, the proportion of births following an unplanned pregnancy remains high but variable between developed countries.

Objective. To compare the proportion of unplanned births in the UK and in France and to explore if the socioeconomic or demographic characteristics of the parents could explain the much higher level of unplanned pregnancies in the UK. Data. Two large birth cohorts with nationally representative samples: the French ELFE cohort having recruited 18,322 families with children born in 2011 and the UK Millennium Cohort Study (MCS) covering 18,552 families with children born in 2000-2002.

Results and discussion. The MCS has already exhibited a high level of unplanned births (41%). ELFE data will be available imminently and the rate of unplanned births is expected to be below 20%. In the UK, unplanned births occurred mainly among low educated mothers, large families, young mothers

and mothers without partners. We will test if the differences in unplanned pregnancies between France and the UK could be explained by differentials in such characteristics of the families recruited into the two cohorts.


Carlos Martínez
email: carlos.martinez@inria.fr

Francis Mairet
email: francis.mairet@ifremer.fr

Olivier Bernard
email: olivier.bernard@inria.fr

Dynamics of the periodically forced light-limited Droop model

Keywords: Variable quota model, Positive periodic solution, Global stability, Microalgae, Cooperative System

The periodically forced light-limited Droop model represents microalgae growth under co-limitation by light and a single substrate, accounting for periodic fluctuations of factors such as light and temperature. In this paper, we describe the global dynamics of this model, considering general monotone growth and uptake rate functions. Our main result gives necessary and sufficient conditions for the existence of a positive periodic solution (i.e. a periodic solution characterized by the presence of microalgae) which is globally attractive. In our approach, we reduce the model to a cooperative planar periodic system. Using results on periodic Kolmogorov equations and on monotone sub-homogeneous dynamical systems, we describe the global dynamics of the reduced system. Then, using the theory of asymptotically periodic semiflows, we extend the results on the reduced system to the original model. To illustrate the applicability of the main result, we include an example considering a standard microalgae population model.

Introduction

Microalgae are photosynthetic microorganisms, converting light energy into chemical energy [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF]. Microalgae have many applications, among them biomass production for food and fine chemicals, biodiesel production, and wastewater treatment [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF][START_REF] Abdel-Raouf | Microalgae and wastewater treatment[END_REF]. For industrial applications, microalgae are grown in open ponds or photobioreactors [START_REF] Grobbelaar | Factors governing algal growth in photobioreactors: the "open" versus "closed" debate[END_REF]. In these systems, algae growth is mainly limited by the amount of nutrients and light availability. Different mathematical models have been developed to describe microalgae growth under these limitations. Under nutrient limitation, we find the Monod model and the Droop (or Cell Quota) model [START_REF] Droop | Vitamin b 12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF]. The former relates the growth rate to the nutrient concentration in the medium, while the latter relates the growth rate to an intracellular pool of nutrient known as cell quota. The applicability of the Monod model is limited to steady state condition [START_REF] Sommer | A comparison of the droop and the monod models of nutrient limited growth applied to natural populations of phytoplankton[END_REF]. The applicability of the Droop model is more widespread and has successfully described the growth rate even under fluctuations of the environmental conditions [START_REF] Droop | Vitamin b 12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF][START_REF] Bougaran | Modeling continuous cultures of microalgae colimited by nitrogen and phosphorus[END_REF][START_REF] Geider | A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature[END_REF]. On the other hand, to describe the growth under light-limitation, Huisman and collaborators [START_REF] Huisman | Principles of the light-limited chemostat: theory and ecological applications[END_REF] introduced the theory of light-limited chemostat. Light-limitation differs considerably from nutrient-limitation. Light rapidly decreases as it passes through the microalgae culture due to absorption and scattering by algal cells. This results in a light gradient whose pattern depends on the microalgae concentration.

As a consequence, the growth rate depends on the microalgae concentration. On top of that, the light source in microalgae cultures is not always constant along time. Outdoor cultures are subject to a light phase (day) and a dark phase (night) following a periodic pattern. Thus, the growth rate, that depends on light availability, becomes a periodic function in time. Periodicity on the models can also be induced by water temperature or nutrient supply fluctuations. Many theoretical works analyze single microalgae population growth with the Droop model [START_REF] Lange | The attractiveness of the droop equations[END_REF][START_REF] Bernard | Transient behavior of biological loop models with application to the droop model[END_REF], or light limitation [START_REF] Hsu | Dynamics of phytoplankton communities under photoinhibition[END_REF][START_REF] Gerla | Photoinhibition and the assembly of light-limited phytoplankton communities[END_REF][START_REF] Huisman | Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model[END_REF]. Models with both substrate and light limitations are studied in [START_REF] Huisman | Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model[END_REF][START_REF] Khonthapagdee | Crowding and light limitation affect phytoplankton competition for nitrogen[END_REF][START_REF] Wolkowicz | Effect of light on the growth of non-nitrogenfixing and nitrogen-fixing phytoplankton in an aquatic system[END_REF] with Monod approach, and in [START_REF] Hsu | Dynamics of two phytoplankton species competing for light and nutrient with internal storage[END_REF] with variable quota. Finally, a few studies deal with periodic forcing. Microal-gae cultures under light limitation with a periodic light source are analyzed in [START_REF] Grognard | Optimal strategies for biomass productivity maximization in a photobioreactor using natural light[END_REF][START_REF] Bayen | Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion[END_REF], and the Droop model with periodic nutrient supply is studied in [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF].

But, to our knowledge, nothing has yet been done for both light and substrate limitations under periodic forcing.

In this work we study the asymptotic behavior of the periodically forced light-limited Droop model i.e. a model that results from combining the modelling approaches of Droop [START_REF] Droop | Vitamin b 12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF] and Huisman [START_REF] Huisman | Principles of the light-limited chemostat: theory and ecological applications[END_REF], when the growth rate, the uptake rate, the nutrient supply, and the dilution rate are periodic functions of time. We consider general monotone growth and uptake rate functions. In our approach, we reduce the model to a cooperative two-dimensional system to show that any solution approaches asymptotically to a periodic solution. Following results on Kolmogorov periodic equations [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF], we find conditions such that any solution of the reduced system is asymptotic to a positive periodic solution i.e. a solution characterized by the presence of microalgae. This proves the existence of positive periodic solutions for the original system. Using results of the theory of subhomogeneous (or sublinear) dynamical systems [START_REF] Zhao | Dynamical systems in population biology[END_REF], we give conditions for the uniqueness of positive periodic solutions. Finally, using the theory of asymptotically periodic semiflows [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF] and classical results of the theory of differential equations such as the comparison method [START_REF] Coppel | Stability and asymptotic behavior of differential equations[END_REF], we find a result on the global dynamics of the original model. This article is organized as follows. In Section 2, we introduce the periodically forced light-limited Droop model and we state some basic results on the existence, uniqueness, and boundedness of solutions. In Section 3, we study a limiting two-dimensional periodic system of the model. We prove that any solution of this system is asymptotic to a periodic solution (Proposition 3.3), and we give conditions for the extinction (Proposition 3.5) and persistence (Theorem 3.6) of the population. We also determine conditions for the uniqueness of positive periodic solutions (Theorem 3.8). In Section 4, we present the main result (Theorem 4.1), a result on the global dynamics of the model. In Section 5, we apply our results to study a model describing microalgae growth under limitation by phosphorus and light. In Section 6, we discuss our results and some possible extensions. Finally, we include two appendices. In Appendix A we present some results on the asymptotic of scalar differential equations. In Appendix B we prove some properties of a growth rate function.

Model description and basic properties

Model description

Let us consider a well-mixed culture system with a biomass x(t) of microalgae. Microalgae growth is only limited by light and a nutrient at concentration s(t) in the medium. The light is provided by an external light source (artificial or natural) and its intensity can vary with time. The nutrient is supplied at variable concentration s in (t), from an external reservoir at the variable volumetric flow rate Q in (t). The dilution rate is the ratio

D(t) := F in (t)/V (t) with V (t)
the volume of the culture. Following the Droop model [START_REF] Droop | Vitamin b 12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF], microalgae growth depends on the internal quota of nutrient q(t). The quota increases with nutrient uptake and decreases with cell growth (by the effect of intracellular dilution).

Following the theory of light-limited chemostats [START_REF] Huisman | Principles of the light-limited chemostat: theory and ecological applications[END_REF], the growth of microalgae affects their own light environment (self-shading). Then, the cell growth rate depends on the biomass concentration x(t). Since the incident light may vary over time, the growth rate depends on time. The light-limited Droop model takes the following form:

dx dt = [µ(t, x, q) -D(t)]x, dq dt = ρ(t, q, s) -µ(t, x, q)q, ds dt = D(t)(s in (t) -s) -ρ(t, q, s)x. (1)
The functions µ and ρ represent the growth rate of microalgae and the nutrient uptake rate respectively. Let J = [q 0 , ∞) with q 0 > 0. We assume that µ : R 2 + × J -→ R, ρ : R + × J × R -→ R, and D, s in : R + -→ R + are continuous functions and satisfy the following set of assumptions:

H 2.1. µ, ρ, D, and s in are ω-periodic in t with ω > 0.

H 2.2. q -→ ρ(t, q, s) is decreasing, s ∈ [0, ∞) -→ ρ(t, q, s) is increasing, and ρ(t, q, s) = 0 for all s ≤ 0.

H 2.3. µ(t, x, q 0 ) ≤ 0 for any t, x ≥ 0, and q -→ µ(t, x, q) is increasing.

H 2.4. For any q > q 0 , x -→ µ(t, x, q) is decreasing.

H 2.5. lim q→∞ ρ(t, q, s) = 0 and lim x→∞ µ(t, x, q) ∈ (-∞, 0], both uniformly

for t ∈ [0, ω]. H 2.6. ω 0 D(t)dt > 0 and ω 0 D(t)s in (t)dt > 0.
H 2.7. µ and ρ are locally Lipschitz uniformly for t in [0, ω].

H 2.8. There exists q > q 0 such that ω 0 µ(t, 0, q )dt > 0.

Remark 2.9. (Subsistence quota) The parameter q 0 is known as the subsistence quota and represents the value of q at which growth ceases. H2.3 shows that there cannot be growth when q = q 0 . In particular, this implies that the quota cannot be smaller than q 0 . Indeed, the derivative of q(t) is non-negative when q = q 0 (see the second equation in (1)).

Remark 2.10. In H2.5, the existence of the limits is given by the monotonicity of µ and ρ. The limit for µ is allowed to be -∞.

Remark 2.11. (Respiration rate) In hypothesis H2.3, the growth rate is allowed to be negative. When microalgae is measured in terms of carbon biomass, µ corresponds to the carbon gain rate i.e. µ = p -m, with p the photosynthesis (carbon uptake) rate and m the specific carbon loss rate. Thus, µ may be negative, especially in absence of light when p = 0.

Remark 2.12. From a biological point of view, H2.8 states that there is a quota such that a very small population can grow. Hypothesis H2.8 is necessary to avoid the extinction of the population and unbounded values of the cell quota (see Remarks 2.15 and 2.18).

Existence, uniqueness, and boundedness of solutions

We define the total amount of limiting nutrient both in the substrate and in the biomass by means of S = s + xq. A simple calculation shows that S satisfies the differential equation:

dS dt = D(t)(s in (t) -S). (2)
With respect to the solutions of (2), we have the following lemma.

Lemma 2.13. Equation (2) admits a unique ω-periodic solution s * (t) which is positive and globally attractive.

Proof. From a direct calculation we have that:

S(t) = (S(0) + f (t))e -d(t) , (3)
and that s * (t) is given by:

s * (t) = e -d(t) f (ω) e d(ω) -1 + f (t) , (4)
dition on R + × J × R + .
Proof. The existence and uniqueness of solutions is given by hypothesis H2.7.

Let (x, q, s) be a solution of (1) such that x(0), s(0) ≥ 0 and q(0) ≥ q 0 , with ∆ the maximal interval of existence. We have (x(t), q(t), s(t)) ∈ R + × J × R + for any t ∈ ∆. Since the variable S = xq + z satisfies the differential equation [START_REF] Abdel-Raouf | Microalgae and wastewater treatment[END_REF] and (x, q, s) is non-negative, by Lemma 2.13, xq and s cannot be unbounded in a finite interval of time. Now we note that x(t)q(t) ≥ x(t)q 0 , then x(t) ≤ S(t)/q 0 for all t ∈ ∆. Finally, since dq/dt ≤ ρ(t, q 0 , S(t))-µ(t, S(t)/q 0 , q 0 )q, we conclude that q cannot be unbounded in a finite interval of time. Thus, ∆ = [0, ∞).

Remark 2.15. Let (x, q, s) be a solution of (1). If H2.8 does not hold, then x(t) → 0 as t → ∞. Indeed, since µ is decreasing in x we have that:

dx(t) dt ≤ x(t)[µ(t, 0, q(t)) -D(t)]. (5)
Since H2.8 does not hold, nω 0 µ(t, 0, q(t))dt < 0 for any integer n ≥ 1. Thus, applying Gronwall's inequality to [START_REF] Sommer | A comparison of the droop and the monod models of nutrient limited growth applied to natural populations of phytoplankton[END_REF] on the interval [t -ω[t/ω], t] we obtain:

x(t) ≤ x(t -ω[t/ω])e -α[t/ω] , (6)
where [t/ω] is the greatest integer less than or equal to t/ω and α = ω 0 D(τ )dτ > 0. Letting t → ∞ in [START_REF] Bougaran | Modeling continuous cultures of microalgae colimited by nitrogen and phosphorus[END_REF] we obtain that x(t) → 0.

The following lemma will be repeatedly used in the rest of the paper. Lemma 2.16. For any non-negative continuous function σ there is Q > 0 such that:

ω 0 ρ(t, Q, σ(t)) Q -µ(t, 0, Q) dt < 0.
Proof. From H2.5 we have that lim q→∞ ω 0 ρ(t, q, s)dt = 0 for any s ≥ 0. Then there exists Q > q , with q given by H2.8, such that:

ω 0 ρ(t, Q, max t∈[0,ω] σ(t))dt < := q ω 0 µ(t, 0, q )dt. ( 7
)
From the monotonicity of µ and ρ as functions of q (see H2.2 and H2.3), we have that:

Q ω 0 µ(t, 0, Q)dt > q ω 0 µ(t, 0, q )dt = ≥ ω 0 ρ(t, Q, σ(t))dt,
from where we complete the proof.

Lemma 2.17. Solutions of (1) starting on R + × J × R + are uniformly bounded.

Proof. From Theorem 8.5 in [START_REF] Yoshizawa | Stability theory and the existence of periodic solutions and almost periodic solutions[END_REF], the ultimate boundedness of solutions of a periodic system implies the uniform boundedness of solutions. Thus, we prove that solutions of (1) starting on R + × J × R + are ultimately bounded. Let (x, q, s) be a solution of (1) with x(0), s(0), q(0) -q 0 ≥ 0. We have that S(t) =

x(t)q(t) + s(t) satisfies the differential equation ( 2). From Lemma 2.13, there is t > 0 such that S(t) ≤ s for all t ≥ t , with s := 1 + max s * (t). By similar arguments as in Proof of Lemma 2.14, we have x(t) ≤ s /q 0 and s(t) ≤ s for all t > t . It remains to prove the existence of a constant β, not depending on initial conditions, such that lim sup t→∞ q(t) ≤ β . For this purpose, let us define h(t, q) := ρ(t,q,s ) q -µ(t, 0, q) and g(t, q) = ρ(t,q,s ) q -D(t). From Lemma 2.16 and H2.5, there exists Q > q(0) such that:

ω 0 h(t, Q)dt < 0 and ω 0 g(t, Q)dt < 0. (8)
Now, if q(t) ≤ Q for all t ≥ t , then the proof is ready. Then, let us assume that q(t 1 ) = Q for some t 1 > t and that q(t) ≥ Q for all t ≥ t 1 . Then we have that xs (t) := x(t)q(t) satisfies the following equation for all t ≥ t 1 :

dx s dt = ρ(t, q(t), s(t)) q(t) -D(t) x s ≤ g(t, Q)x s . (9)
Using Gronwall's inequality on the interval [t 1 , t 1 + t], t > 0 gives:

xs (t 1 + t) ≤ xs (t 1 + t -ω[t/ω])e -α[t/ω] ,
where [t/ω] is the greatest integer less than or equal to t/ω and α = -ω 0 g(τ, Q)dτ > 0. Since s is an upper bound for xs and q 0 is a lower bound for q, we obtain:

x(t 1 + t) ≤ s q 0 e -α[t/ω] . (10)
Now, from H2.7, there exists δ 0 > 0 such that:

|µ(t, x, Q) -µ(t, 0, Q)| ≤ l|x|, (11)
for all t ∈ [0, ω] and x ∈ [-δ 0 , δ 0 ], with l the Lipschitz constant of µ. Let := -1 2 ω 0 h(t, Q)dt and let us choose t 2 > t 1 such that s q0 e -[(t2-t1)/ω]α < min{δ 0 , /l}. Thus, from ( 10) and ( 11), we obtain that |µ(t,

x(t), Q)-µ(t, 0, Q)| <
for all t ≥ t 2 , and consequently:

dq(t) dt ≤ q(t) (h(t, Q) + ) . ( 12
)
using Gronwall's inequality on the interval [t 2 , t 2 + nω] gives q(t 2 + nω) ≤

q(t 2 )e -n . Let γ := max t∈[0,ω] ρ(t,q0,s ) q0
-µ(t, s /q 0 , q 0 ). Then dq(t)/dt ≤ γ q(t).

Applying Gronwall's inequality on the interval [t 1 , t 2 ] gives q(t 2 ) ≤ Qe γ(t2-t1) .

Consequently q(t 2 + nω) ≤ Qe γ(t2-t1)+n . Thus, for n > γ(t 2 -t 1 )/ , we have that q(t 2 + nω) < Q. Therefore q must return to Q in a finite time smaller than

T := t 2 -t 1 + nω.
Since T does not depends on initial conditions, we conclude that q is ultimately bounded by Qe T γ .

Remark 2.18. If H2.8 does not hold, then solutions of (1) are not bounded.

Indeed, let (x, q, s) be a solution of (1) with x(0), s(0) ≥ 0 and q(0) ≥ q 0 . Let us assume that q is bounded from above by Q > 0. Since ρ is non-negative and µ is decreasing in x, we have dq(t) dt ≥ -µ(t, 0, Q)q. Applying Gronwall's inequality on the interval [0, nω] with n ≥ 1 an integer, we obtain:

q(nω) = q(0)e -n ω 0 µ(t,0,Q)dt . ( 13
)
If H2.8 does not hold, then ω 0 µ(t, 0, Q)dt < 0. Thus, letting n → ∞ in (13), we conclude that q is not bounded which is a contradiction.

A solution (x, q, s) of (1) will be called an ω-periodic solution provided each component is ω-periodic. An ω-periodic solution with absence of microalgae is called washout periodic solution. The following proposition shows that (1) has a washout periodic solution.

Proposition 2.19. The system (1) has at least one washout periodic solution.

Proof. It is not difficult to see that any washout periodic solution must be of the form (0, q(t), s * (t)) with s * (t) the periodic solution of (2). Thus, putting x = 0 and s = s * (t) in the second equation of (1) results in:

dq dt = ρ(t, q, s * (t)) -µ(t, 0, q)q. ( 14
)
Let us define:

F 0 (t, q) = ρ(t, q, s * (t)) q -µ(t, 0, q). (15)
From Lemma 2.16, there exists Q > 0 such that ω 0 F 0 (t, Q)dt < 0. From H2.3 we have that F 0 (t, q 0 ) ≥ 0 for all t ≥ 0. Thus, the proof follows from a direct application of Proposition 6.4 in Appendix A.

Remark 2.20. (Uniqueness of the washout) The uniqueness of the washout can be stated under additional assumptions over the monotonicity of ρ and µ. For 155 example, consider F 0 defined in [START_REF] Wolkowicz | Effect of light on the growth of non-nitrogenfixing and nitrogen-fixing phytoplankton in an aquatic system[END_REF]. If for some t the function q -→ F 0 (t, q) is strictly decreasing, then we have the uniqueness of the washout.

Reduced system

Dropping the equation for s and replacing s in (1) by s = s * (t) -xq results in the following reduced ω-periodic system for (x, q):

dx dt = [µ(t, x, q) -D(t)]x, dq dt = ρ(t, q, s * (t) -xq) -µ(t, x, q)q. . (16)
In the following we study the asymptotic behavior of the reduced system [START_REF] Hsu | Dynamics of two phytoplankton species competing for light and nutrient with internal storage[END_REF].

We are interested in solutions of ( 16) starting with a positive initial microalgae concentration and an internal quota not lower than q 0 i.e. solutions with initial conditions on the set:

P := {(x, q) ; x > 0, q ≥ q 0 }.
Our first lemma states a basic property of solutions of ( 16).

Lemma 3.1. For any solution (x, q) of ( 16) starting on P we have that x(t) > 0, 160 q(t) ≥ q 0 for all t > 0. Moreover, there is t ≥ 0 such that s * (t) ≥ x(t)q(t) for all t ≥ t .

Proof. Since dq dt | q=q0 ≥ 0, if q(0) ≥ q 0 then q(t) ≥ q 0 for all t ≥ 0. If x(0) > 0, x cannot reach x = 0 in a finite time by the uniqueness of solutions of initial value problems. Then x(t) > 0 for all t ≥ 0. The variable x s := xq satisfies the differential equation:

dx s dt = x s ρ(t, q, s * (t) -x s ) q -D(t) . (17)
Thus, the variable y(t) = s * (t) -x s (t) satisfies:

dy dt = D(t)(s in (t) -y) + (y -s * (t)) ρ(t, q, y) q . ( 18
)
We note that dy dt | y=0 = s in (t)D(t) ≥ 0, therefore if y(t ) ≥ 0 for some t ≥ 0 then y(t) ≥ 0 for all t ≥ t and the proof is completed. Then we have to prove the existence of t > 0 such that y(t ) ≥ 0. By contradiction, let us assume that y(t) < 0 for all t ≥ 0. From (18) and H2.2 we have dy/dt = D(t)(s in (t) -y).

From Lemma 2.13, y approaches asymptotically to s * , which is a contradiction because s * is positive.

The following convergence results for the reduced system need the uniqueness of the washout periodic solution. Proposition 3.2. Let us assume that (16) admits a unique washout periodic solution (0, q * ). Then, for any solution (x, q) of ( 16) satisfying lim t→∞ x(t) = 0, we have that lim t→∞ |q(t) -q * (t)| = 0.

Proof. Let (x, q) a solution of [START_REF] Hsu | Dynamics of two phytoplankton species competing for light and nutrient with internal storage[END_REF]. Following the proof of Proposition 2.19, we define F (t, q) = ρ(t, q, s * (t) -qx(t))/q -µ(t, x(t), q). From H2.3 we have that F (t, q 0 ) ≥ 0 for all t ≥ 0. Since lim t→∞ x(t) = 0, we have that lim t→∞ |F 0 (t, q)-F (t, q)| = 0. Thus, the proof follows from a direct application of Proposition 6.4b) in Appendix A. Now we prove that any solution of ( 16) is asymptotic to an ω-periodic solution. The heart of the proof lies in the fact that the change of variables x s = xq leads the reduced system to a cooperative system. Proposition 3.3. If (16) admits a unique washout periodic solution, then any solution of ( 16) starting on P approaches asymptotically to an ω-periodic solution.

Proof. Let (x, q) be a solution of ( 16) with x(0) > 0 and q(0) ≥ q 0 . Let xs (t) := x(t)q(t). From Lemma 3.1, it easily follows that xs (t) and x(t) are bounded.

Considering the change of variables x s := qx, we have that (x(t), xs (t)) is a solution of the following system:

dx/dt = f 1 (t, x, x s ) := [µ(t, x, x s /x) -D(t)]x, dx s /dt = f 2 (t, x, x s ) := ρ(t, x s /x, s * (t) -x s )x -D(t)x s . ( 19
)
The system ( 19) is cooperative i.e. f 1 and f 2 are increasing in x s and x respectively. Following the proof of Theorem 4.2 in Chapter 7 in the Book [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF], we have that the sequences xn := x(nω) and xsn := xs (nω) are convergent. Let l := lim n→∞ xn and l := lim n→∞ xsn . If l > 0, then l > 0 and consequently qn := q(nω) = xsn /x n → l/l as n → ∞. Thus, (x, q) approaches asymptotically an ω-periodic solution of ( 16) with initial conditions (l, l /l). Let us assume now that l = 0 and let g(t) := µ(t, x(t), q(t)) -D(t). We can write x(t) = x(0)e α(t)+β(t) , with:

α(t) = ω[t/ω] 0 g(τ )dτ, and β(t) = t ω[t/ω] g(τ )dτ.
Let Q be an upper bound for q given by Lemma 2.17, then we have g(t) ≤ µ(t, 0, Q). Thus, β(t) ≤ b := ω max t∈[0,ω] µ(t, 0, Q). We have that xn = x(0)e α(nω) . Since xn → 0, we conclude that α(nω) → -∞. Then, it is trivial that α(t) → -∞ as t → ∞. Thus, we conclude that x(t) ≤ x(0)e α(t)+b → 0 as t → ∞. From Proposition 2.19, we conclude that (x, q) is asymptotic to the washout periodic solution.

Remark 3.4. The monotonicity of µ as a function of x is not essential in the proof of Proposition 3.3. Indeed, the system [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF] does not lose the property of being cooperative.

The following proposition states conditions for the extinction of the population.

Proposition 3.5. Let us assume that (16) admits a unique washout periodic solution (0, q * ). If one of the following conditions holds:

a) ω 0 [µ(t, 0, q * (t)) -D(t)]dt < 0; b) ω 0
[µ(t, 0, q * (t)) -D(t)]dt = 0 and the function x -→ µ(t, x, q * (t)) is strictly decreasing for some t ∈ [0, ω]; 200 then, any solution of ( 16) starting on P approaches asymptotically (0, q * (t)).

Proof. Let (x, q) be a solution of ( 16) starting on P , and let x s = xq. Following the proof of Lemma 2.3 in [START_REF] Hsu | Competition for two essential resources with internal storage and periodic input[END_REF], we define x(t) to be the unique solution of:

dx dt = [µ(t, x, q * (t)) -D(t)]x, (20)
with x(0) := max{x(0), x s (0)/q * (0)}. We also define xs (t) := x(t)q * (t). It is easy to verify that:

dx s dt = ρ(t, xs /x, s * (t))x -D(t)x s .
Let us consider the functions f i , i = 1, 2 defined in [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF]. We have the following inequality:

dx dt ≥ f 1 (t, x, xs ), dx s dt ≥ f 2 (t, x, xs ), (21)
with x(0) ≥ x(0) and xs (0) ≥ x s (0). Applying Theorem B.1 from Appendix B in [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF], we conclude that x(t) ≤ x(t) and x s (t) ≤ xs (t).

Now, let us define the sequence xn = x(nω), n ∈ N. Since µ is decreasing in andb)), we have:

x and ω 0 [µ(t, 0, q * (t)) -D(t)]dt ≤ 0 (in a)
xn+1 = xn exp ω 0 [µ(t, x, q * (t)) -D(t)]dt ≤ xn exp ω 0 [µ(t, 0, q * (t)) -D(t)]dt ≤ xn ,
hence xn is a decreasing sequence. Since xn ≥ 0 for all n ∈ N, we conclude that xn is convergent and therefore x approaches asymptotically an ω-periodic solution of [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF]. We prove now that in both cases, a) and b), x = 0 is the unique periodic solution. By contradiction, let xp be a positive periodic solution. Then we have

ω 0 [µ(t, xp (t), q * (t)) -D(t)]dt = 0. ( 22
)
However, in case a):

ω 0 [µ(t, xp (t), q * (t)) -D(t)]dt ≤ ω 0 [µ(t, 0, q * (t)) -D(t)]dt < 0,
which contradicts [START_REF] Coppel | Stability and asymptotic behavior of differential equations[END_REF]. In case b):

ω 0 [µ(t, xp (t), q * (t)) -D(t)]dt < ω 0 [µ(t, 0, q * (t)) -D(t)]dt = 0,
which again contradicts [START_REF] Coppel | Stability and asymptotic behavior of differential equations[END_REF]. Hence x = 0 is the unique periodic solution of [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF] and we have that lim t→0 x(t) = 0. This implies lim t→∞ x(t) = 0. By Lemma 3.2, we conclude that lim t→∞ |q(t) -q * (t)| = 0.

An ω-periodic solution (x, q) of ( 16) will be called positive ω-periodic solution, if x(t) > 0, q(t) ≥ q 0 , and x(t)q(t) ≤ s * (t) for all t ∈ [0, ω]. The following theorem gives conditions to ensure that any solution of ( 16) approaches a positive ω-periodic solution.

Theorem 3.6. Let us assume that (16) admits a unique washout periodic solution (0, q * ) and that ω 0

[µ(t, 0, q * (t)) -D(t)]dt > 0. Then, ( 16) admits at least one positive ω-periodic solution and any solution of ( 16) starting in P approaches asymptotically a positive ω-periodic solution.

Proof. Along the proof we will write u = (x, q). Let us define G = (G 1 , G 2 ) :

R 2 + × J × R + -→ R 2 by: G 1 (t, u, v) = µ(t, u)-D(t) and G 2 (t, u, v) = ρ(t, u 2 , v-u 1 u 2 )/u 2 -µ(t, u), (23)
with ρ a continuous extension of ρ on R + × J × R to R 2 + × R such that ρ is ωperiodic in t and locally Lipschitz in u uniformly in t. Consider the Kolmogorov periodic system:

du i dt = u i G i (t, u, s * (t)), i = 1, 2, (24)
For initial conditions on R + × [0, q 0 ], solutions of (24) stay on R + × [0, q 0 ] or they intersect the set R + × J for some t > 0. Thus, solutions of (24) exist for any initial condition on R 2 + and they are uniformly bounded. Let φ 0 (t, u) be the unique solution of ( 24) with φ 0 (0, u) = u ∈ R 2 + and let ϕ := φ(ω,

•) : R 2 + -→ R 2 +
be the Poincaré map associated to [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF]. From Lemma 1 in the appendix of [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF],

we conclude that there is δ > 0 such that lim n→∞ d(ϕ n (u), (0, q * (0))) ≥ δ for all u ∈ int(R 2 + ). This implies that for any u ∈ (0, ∞)×J, φ 0 (t, u) is not asymptotic to the washout periodic solution. From Proposition 3.3, we conclude that φ(t, u) approaches an ω-periodic solution (x * , q * ) different from the washout periodic solution. From Lemma 3.1, we have that x * (t)q * (t) ≤ s * (t) for all t ≥ 0. Thus (x * , q * ) is a positive periodic solution and the proof is completed.

The following result that states an order of the positive periodic solutions of ( 16). Lemma 3.7. For any two periodic solutions (x * i , q * i ), i = 1, 2 of ( 16) with x * i (0) > 0, we have that either

• x * 1 (t) ≤ x * 2 (t) and x * 1 (t)q * 1 (t) ≤ x * 2 (t)q * 2 (t) for all t ∈ [0, ω], or • x * 1 (t) ≥ x * 2 (t) and x * 1 (t)q * 1 (t) ≥ x * 2 (t)q * 2 (t) for all t ∈ [0, ω].
Proof. We write x * si = x * i (t)q * i (t), i = 1, 2. Then, we have that (x * i , x * si ), i = 1, 2 are periodic solutions of [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF]. We claim that either (a) we conclude that (a) holds. In the same way, if (II) holds then (b) holds. Thus, the claim is proved. Now, since f 1 (see [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF]) is increasing in x s , we conclude that (a) implies x * 1 (t) ≤ x * 2 (t), and (b) implies x * 1 (t) ≥ x * 2 (t). This completes the proof.

x * 1s (t) ≤ x * 2s (t) for all t ∈ [0, ω] or (b) x * 1s (t) ≥ x * 2s (t) for all t ∈ [0, ω]. Indeed, let us assume that there is t 0 ∈ [0, ω] such that x * 1s (t 0 ) = x * 2s (t 0 ), otherwise the claim is trivial. Then either (I) x * 1 (t 0 ) < x * 2 (t 0 ) or (II) x * 2 (t 0 ) > x * 1 (t 0 ),
We end this section with a theorem that gives conditions for the uniqueness of positive ω-periodic solutions of [START_REF] Hsu | Dynamics of two phytoplankton species competing for light and nutrient with internal storage[END_REF]. For an interpretation of the hypotheses in the following theorem, see the remarks at the end of this section. Theorem 3.8. We recall the subsistence quota q 0 introduced in Section 2. Assume that: I) ρ(t, q 0 , s) > 0 for all t ∈ [0, ω], s > 0, and that for any continuous function q on [0, ω], satisfying q(t) > q 0 for all t ∈ [0, ω], we have: II) the function x -→ µ(t, x, q(t)) is strictly decreasing for some t ∈ [0, ω], and III) the function s -→ ρ(t, q(t), s) is either strictly increasing or equal to zero for all s ≥ 0.

Then, ( 16) admits at most one ω-periodic solution (x * , q * ) with x * (0) > 0 and q * (0) ≥ q 0 . Proof. Let φ(t, v) be the unique solution of (19) satisfying φ(0, v) = v, let K := int(R 2 + ) and ϕ = φ(ω, •) : K -→ K be the Poincaré map associated to [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF]. Let u be a positive fixed point of ϕ and let α ∈ (0, 1). We define the variables y(t) := αφ(t, u) and z(t) := φ(t, αu). Let us consider the functions f i , i = 1, 2 defined in [START_REF] Smith | The periodically forced droop model for phytoplankton growth in a chemostat[END_REF]. We can easily verify that for all t ∈ [0, ω]:

dy i (t) dt = αf i (t, y 1 (t)/α, y 2 (t)/α) ≤ f i (t, y 1 (t), y 2 (t)), i = 1, 2, y(0) = αu, (25)
and that:

dz i (t) dt = f i (t, z 1 (t), z 2 (t)), i = 1, 2, z(0) = αu. (26)
Applying Theorem B.1 from Appendix B in [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF], we conclude that y i (t) ≤ z i (t) for all t ∈ [0, ω], i = 1, 2.

Let q y (t) := y 2 (t)/y 1 (t). Since y(t)/α corresponds to an ω-periodic solution of ( 19), (y 1 (t)/α, q y (t)) corresponds to an ω-periodic solution of ( 16). We claim that q y (t) > q 0 for all t ∈ [0, ω]. Indeed, from Lemma 3.1, we know that q y (t) cannot be lower than q 0 . Thus, by contradiction, if q y (t ) = q 0 for some t , then q y reaches a minimum at t = t . Hence, dq y (t )/dt = 0. However, from hypothesis I), we have that dq y (t )/dt > 0 which is a contradiction. Therefore our claim is true. From hypothesis II), we conclude that x -→ µ(t , x, q y (t )) is strictly decreasing for some t ∈ [0, ω]. Consequently, for i = 1, the inequality in ( 25) is strict for t . Again, since y(t)/α is an ω-periodic solution of ( 19), we have:

ω 0 ρ(t, y 2 (t)/y 1 (t), s * (t) -y 2 (t)/α) = ω 0 D(t)dt > 0,
from where there exists an interval of time t such that: ρ(t , y 2 (t )/y 1 (t ), s * (t ) -y 2 (t )/α) > 0

From hypothesis III), we conclude that for i = 2, the inequality in ( 25) is strict for some t . Since inequalities in ( 25) are strict at some moment and f 1 and f 2 are continuous, we obtain that for i = 1, 2:

0 = α ω 0 f i (t, y 1 (t)/α, y 2 (t)/α)dt < ω 0 f i (t, y 1 (t), y 2 (t))dt. ( 27
)
We prove now that y i (t) < z i (t), i = 1, 2 for some t ∈ [0, ω]. Without loss of generality, we do it for i = 1. By contradiction, if z 1 (t) = y 1 (t) for all t ∈ [0, ω],

then we have:

z 1 (ω) -z 1 (0) = ω 0 f 1 (t, z 1 (t), z 2 (t))dt = ω 0 f 1 (t, y 1 (t), z 2 (t))dt ≥ ω 0 f 1 (t, y 1 (t), y 2 (t))dt.
> 0 (see [START_REF] Passarge | Competition for nutrients and light: stable coexistence, alternative stable states, or competitive exclusion?[END_REF]).

Thus, we conclude that z 1 (0) = z 1 (ω) which is a contradiction because y 1 (0) = y 1 (ω). Therefore there exists t 0 such that z 1 (t 0 ) > y 1 (t 0 ). Let us consider z 1 defined by:

dz 1 dt = f 1 (t, z 1 , z 2 (t)), z 1 (t 0 ) = y 1 (t 0 ). ( 28
)
Since dy1 dt ≤ f 1 (t, y 1 , z 2 (t)), by a comparison argument, we have that y 1 (ω) ≤ z 1 (ω). By an uniqueness argument, we have that z 1 (ω) < z 1 (ω). Hence, we conclude that y 1 (ω) < z 1 (ω). Similarly, we can argue that y 2 (ω) < z 2 (ω).

Since αϕ(u) = y(ω) and z(ω) = ϕ(αu), and due to arbitrary choice of α and u, we conclude that for any α ∈ (0, 1) and u ∈ K: αϕ(u) < ϕ(αu).

(29)

Now, let us assume that ϕ admits two different fixed points u, u ∈ K. From a Kamke's Theorem argument, it follows that ϕ is monotone. Thus, following the same arguments as in the proof of Lemma 2.3.1 in Chapter 2 in [START_REF] Zhao | Dynamical systems in population biology[END_REF], we obtain the existence of σ > 0 such that u = σu . From Lemma 3.7 we can assume that u ≤ u (component-wise inequality). Therefore, σ ∈ (0, 1). Thus,

u = ϕ(u) = ϕ(σu ) > σϕ(u ) = σu = u, which is a contradiction. Therefore, ϕ
(subsistence quota q 0 ), they will absorb nutrients from the medium. To interpret hypothesis II), first we must consider that any increase of the microalgae population is expected to reduce the light availability in the medium (self-shading).

Thus, hypothesis II) states the existence of a moment at which any decrease of the light availability reduces the specific growth rate, in other words, there is a moment of the day at which the culture is light limited. Hypothesis III) is inspired by the fact that for high values of the quota microalgae stop absorbing nutrients, independent of the concentration of nutrients in the medium. Thus, hypothesis III) says that at any moment of the day, if there is absorption of nutrients (low values of quota), then increasing the nutrient concentration in the medium will increase the absorption rate. However, if there is no absorption of nutrients (high values of quota), then it is impossible to initiate the consumption of nutrients by increasing their concentration in the medium.

Main result

An ω-periodic solution (x * , q * , s * ) of ( 1) is known as positive ω-periodic solution if x * (t) > 0, q * (t) ≥ q 0 , and s * (t) ≥ 0. The following theorem states a threshold type result on the global asymptotics of (1). In particular, it gives necessary and sufficient conditions for the existence of a globally attractive positive periodic solution.

Theorem 4.1. Let us assume that (1) admits a unique washout periodic solution (0, q * , s * ) and that assumptions of Theorem 3.8 hold. Let (x, q, s) be a solution of (1) with x(0) > 0, q(0) ≥ q 0 , and s(0) ≥ 0. We have:

a) If ω 0 [µ(t, 0, q * (t)) -D(t)]dt > 0, (1)
admits a unique positive ω-periodic solution (x * , q * , s * ), and

lim t→∞ |(x(t), q(t), s(t)) -(x * (t), q * (t), s * (t))| = 0. b) If ω 0 [µ(t, 0, q * (t)) -D(t)]dt ≤ 0, then lim t→∞ |(x(t), q(t), s(t)) -(0, q * (t), s * (t))| = 0.
Proof. As in the proof of Theorem 3.6, we write u = (x, q) and we consider the functions G i , i = 1, 2 defined in [START_REF] Yoshizawa | Stability theory and the existence of periodic solutions and almost periodic solutions[END_REF]. Consider the Kolmogorov non-autonomous system:

du i dt = u i G i (t, u, S(t)), i = 1, 2, (30)
where S(t) is the unique solution of (2) with S(0) ≥ u 1 (0)u 2 (0). Recalling the proof of Theorem 3.6, solutions of ( 24) and (30) exist for any initial condition on R 2 + and they are uniformly bounded. Let φ 0 (t, s, u) and φ(t, s, u) be the unique solutions of ( 24) and (30) respectively with φ(s, s, u) = φ 0 (s, s, u) = u ∈ R 2 + . We note that for initial conditions on R + × J, (1) is equivalent to (30) (take s(0) = S(0) -u 1 (0)u 2 (0)). From Theorems 3.6 and 3.8 and Proposition 3.5 we obtain the following result on the global asymptotics of [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF]. 24) admits a unique positive ω-periodic solution (x * , q * ), and for any u ∈ (0, ∞) × J we have lim t→∞ |φ 0 (t, 0, u) -

I) If ω 0 [µ(t, 0, q * (t)) -D(t)]dt > 0, (
(x * (t), q * (t))| = 0. II) If ω 0 [µ(t, 0, q * (t)) -D(t)]dt ≤ 0, for any u ∈ [0, ∞)×J, lim t→∞ |φ 0 (t, 0, u)- (0, q * (t))| = 0.
From Theorem 3.6, x * (t)q * (t) ≤ s * (t) for all t ∈ [0, ω]. Then, (x * , q * , s *x * q * ) is the unique positive ω-periodic solution of [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF]. Given the equivalence between (30) and (1), we have to prove that I) and II) remain valid when replacing φ 0 by φ.

From Lemma 2.13, lim t→∞ |S(t)-s * (t)| = 0, and hence lim t→∞ |G(t, u, S(t))-G(t, u, s * (t))| = 0. By Proposition 3.2 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], φ(t, s, u) is asymptotic to the ω-

periodic semiflow T (t) := φ 0 (t, 0, •) : R 2 + -→ R 2 + , and hence T n (u) = φ(nω, 0, u),
n ≥ 0, is an asymptotically autonomous discrete dynamical process with limit discrete semiflow ϕ n : R 2 + -→ R 2 + , n ≥ 0, where ϕ = T (ω) is the Poincaré map associated to [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF]. By Theorem 3.1 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], it suffices to prove in case a) that lim n→∞ T n (u) = u * := (x * (0), q * (0)) for any u ∈ (0, ∞) × J, and in case b) that lim n→∞ T n (u) = u * := (0, q * (0)) for any u ∈ R + × J In case a), by conclusion I), u * is a globally attractive fixed point of ϕ in (0, ∞) × R + . Thus, the only fixed points of ϕ are u * and the washout u * . By Theorem 2.4 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], the ω-limit of u is a fixed point of ϕ. By Lemma 2 (with n = 2) in [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF], we have:

{u ∈ R 2 + ; lim n→∞ T n (u) = (0, q * (0))} ∩ int(R 2 + ) = φ.
Thus, lim n→∞ T n (u) = u * for any u ∈ int(R 2 + ), which proves a).

In case b), by conclusion II), u * is a globally attractive fixed point of ϕ in R + × R + . Thus, the only fixed point of ϕ is u * . By Theorem 2.4 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], the ω-limit of u is a fixed point of ϕ, hence u * . This proves b).

5. Application: Microalgae growth under phosphorus and light limitation.

Here we consider a periodic version of the light-limited Droop model proposed by Passarge and collaborators in [START_REF] Passarge | Competition for nutrients and light: stable coexistence, alternative stable states, or competitive exclusion?[END_REF] for describing microalgae growth under light and phosphorus limitation. The model reads:

dx/dt = [min {µ I (t, x), µ P (q)} -D]x, dq/dt = ρ(q, s) -min {µ I (t, x), µ P (q)} q, ds/dt = D(s in -s) -ρ(q, s)x,

with s in and D constant and positive, and the functions µ I , µ P defined as follows. µ P (q) = µ max 1 -q0 q is the specific growth rate as described by Droop with I in (t) the incident light intensity, k > 0 the specific light extinction coefficient of microalgae, and K bg ≥ 0 the background turbidity. A direct integration shows that:

I(t, x, z) = I in (t)e -(kx+K bg )z , z ∈ [0, L], (32)
µ I (t, x) = µ max (kx + K bg )L ln K I + I in (t) K I + I out (t, x) , (33)
with I out (t, x) = I(t, x, L) the light intensity at the bottom of the culture. The incident light intensity varies periodically according to

I in (t) = I max max{0, sin(2πt/ω)} 2 , (34)
with ω > 0 the length of a day and I max the maximal incident light (at midday).

Figure 1 illustrates the function I in .

330

The uptake rate function is:

ρ(q, s) =    ρ max s Ks+s q L -q q L -q0 if q ≤ q L , 0 if q > q L , (35)
where ρ max is the maximal uptake rate of phosphorus, q L is the hypothetical maximal quota, and K s is a half-saturation constant.

It is not difficult to see that (31) satisfies the hypotheses H2.1-H2.8 presented in section 2 (see Appendix B for the properties of µ I ). Thus, we can apply Theorem 4.1 to obtain the following result. two periodic solutions, the ω periodic solution represented by x = 0 and q * , and a positive ω-periodic solution represented by x * > 0 and q * . Any solution starting with a positive microalgae concentration approaches the positive ω-periodic solution. In this case, x 1 , q 1 and x 2 , q 2 correspond to two different solutions of (31) with x 1 (0), x 2 (0) > 0 and q 1 (0) = q 2 (0).

We note that the cell quota remains between q 0 and q L . A. Microalgae population density.

B. Cell quota.

Theorem 5.1. System (31) admits a unique washout ω-periodic solution, (0, q * (t), s in ).

Moreover, for

∆ := 1 ω ω 0 min{µ I (t, 0), µ P (q * (t))}dt -D,

we have: a) if ∆ > 0, then (31) admits a unique positive ω-periodic solution (x * (t), q * (t), s * (t))

and any solution to (31) with a positive initial population density approaches it asymptotically, 340 b) if ∆ ≤ 0, then any solution to (31) asymptotically approaches the washout ω-periodic solution.

Proof. We recall equation [START_REF] Khonthapagdee | Crowding and light limitation affect phytoplankton competition for nitrogen[END_REF] to study the uniqueness of the washout periodic solution. We note that ω 0 F 0 (t, q(t))dt < 0 for any function q(t) ∈ [q L , ∞). Thus, the quota associated to any washout must intersect the set [q 0 , q L ]. Since 345 µ(t, x, q) := min{µ I (t, x), µ P (q)} ≥ 0, we have that [q 0 , q L ] is positively invariant with respect to [START_REF] Khonthapagdee | Crowding and light limitation affect phytoplankton competition for nitrogen[END_REF]. Thus, the quota associated to any washout stays on [q 0 , q L ]. Since q -→ ρ(q, s in ) is strictly decreasing on [q 0 , q L ], we have that q -→ F 0 (t, q)/q is also strictly decreasing on [q 0 , q L ]. This implies the uniqueness of the washout and part a) is proved.

Let q : [0, ω] -→ (q 0 , ∞) be a continuous function and q m := min t∈[0,ω] q(t).

Since q m > q 0 , we have µ P (q m ) > 0. Now, we note that

µ I (t, x) ≤ ν(t) :=    µ max Iin(t) K I +Iin(t) if K bg = 0, µmax K bg L ln 1 + Iin(t) K I if K bg > 0.
Thus, from the definition of I in (t), it is clear that we can choose t ∈ [0, ω] such that I in (t ) > 0 and µ I (t , x) ≤ ν(t ) ≤ µ P (q m ) ≤ µ P (q(t )) for all x ≥ 0 i.e. µ(t , x, q(t )) = µ I (t , x) for all x ≥ 0. Then we have that x -→ µ(t , x, q(t )) is strictly decreasing (see Proposition 6.5 in Appendix B). We note now that s -→ ρ(q, s) is strictly increasing for any q ∈ [q 0 , q L ] and that ρ(q 0 , s) > 0 for any s > 0. Thus, applying Theorem 4.1 we conclude the proof. Remark 5.2. A crucial fact to ensure the uniqueness of positive periodic solutions of (31) is that the incident light intensity I in (t) is continuous, and zero during some time (night period). Indeed, this implies that I in (t) can take values as close to zero as we want. Hence, for any evolution of the quota q(t) (greater than q 0 ), it is possible to find a time t at which µ I (t , x) ≤ µ P (q(t )) for all x ≥ 0 (details are in the proof of Theorem 5.1). In other words, there is a moment during the day at which limitation by light is predominant. This implies hypothesis II) in Theorem 3.8 (see Remark 3.9).

To illustrate Theorem 5.1, let us consider the kinetic parameters for Chlorella vulgaris provided in [START_REF] Passarge | Competition for nutrients and light: stable coexistence, alternative stable states, or competitive exclusion?[END_REF]. The rest of parameters are chosen as D = 0.02 h -1 , K bg = 6 m -1 , s in = 15 µmol /L, L = 0.4 m, and I max = 2000 µmol m -2 s -1 .

Figure 2 illustrates the microalgae population density and the cell quota associated to the periodic solutions of (31) and their attractiveness property. Figure 3 illustrates the positive periodic solution (x * , q * , s * ) and its evolution during one day. The shaded area corresponds to the night (i.e. I in (t) = 0). Figure 3D shows that during the day (t ∈ [0, 0.5]) microalgae growth is mainly limited by phosphorus, while during the night (t ∈ [0.5, 1]), there is no growth due to the absence of light. Thus, microalgae population only grows during the day (see Figure 3A), and consequently the internal cell quota and external nutrient concentration decrease during the day (see Figures 3B and3C).

Discussion and conclusions

In this work, we studied the asymptotic behavior of a single microalgae model accounting for nutrient and light limitation. We found conditions such that prolonged continuous periodic culture operation (periodic dilution rate and nutrient supply) under periodic fluctuations of environmental conditions (such as the light source or the medium temperature) allows periodic concentrations to be maintained in the culture. More precisely, if (1) admits only one washout periodic solution (0, q * , s * ), then the following condition: As an application of our results, we gave necessary and sufficient conditions for the existence of a unique positive globally attractive periodic solution for a periodic version of the model proposed by Passarge and collaborators [START_REF] Passarge | Competition for nutrients and light: stable coexistence, alternative stable states, or competitive exclusion?[END_REF] (see Theorem 5.1). In this model, the specific growth rate is represented by the law of minimum. If in (31) the specific growth rate is a multiplicative function i.e.

µ I (t, x)(1 -q 0 /q), a new version of Theorem 5.1 can be readily stated.

A possible extension of this work consist in allowing the function µ not to be monotone as a function of x. In [START_REF] Martínez | Theory of turbid microalgae cultures[END_REF] it is shown that when microalgae suffer from photoinhibiton (i.e. a decrease of the photosynthetic rate due to an excess of light), then an Allee effect may occurs i.e. µ in (1) is increasing as a function

From these inequalities we conclude that F (t, u) is bounded from above, and consequently φ(t, s, u) exists for all t ≥ s ≥ 0.

Let R > 0 be according to 6.2 and let > 0 be such that < -1 ω ω 0 F 0 (t, R)dt. From 6.1, there is t * such that |F (t, R) -F 0 (t, R)| < for all t ≥ t * . Then, for all t ≥ t * we have (39)

If u = 0 then φ(t, 0, u) = 0 for all t ≥ 0, hence suppose that u > 0. In that case φ(t, 0, u) > 0 for all t ∈ R + . For the rest of the proof we need the following claim:

Claim 1: If there is t 1 ≥ t * such that φ(t, 0, u) ≥ R for all t ∈ [t 1 , t 1 + ω] then φ(t 1 + ω, 0, u) < φ(t 1 , 0, u)e -1 .

The proof of the claim follows directly from the following inequality: Let us assume that φ(t, 0, u) ≥ R for all t ≥ t * . Using Claim 1 we obtain that φ(t * + kω, 0, u) < φ(t * , 0, u)exp(-k 1 ), for any k ∈ N and a contradiction is achieved letting k → ∞. We may therefore assume without loss of generality that φ(t * , 0, u) < R.

Now suppose that there is t 1 > t * such that φ(t 1 , 0, u) = R. Let us define ∆ := max{δ ≥ 0 ; φ(t 1 + δ, 0, u) ≥ R} and I := [t 1 , t 1 + ∆]. From the Claim 1 we have that φ(t 1 + ω, u) < Re -k 1 < R, therefore ∆ is well defined and smaller than ω. For each t ∈ I we have: ln φ(t, 0, u) φ(t 1 , 0, u) = t t1 F (t, φ(τ, 0, u))dτ ≤ (t -t 1 )M ≤ ωM, (40)

all t ∈ [ω/2, ω].
c) µ I is Lipschitz in x uniformly in t.

Proof. We recall that µ I (t, x) = L 0 p(I(t, x, z))dz. By doing the change of variable I = I(t, x, z), we rewrite µ I as:

µ I (t, x) = g(q) (kx + K bg )L Iin(t) Iout(t,x) p(I) I dI (41)
where I out (t, x) = I(t, x, L). We can easily verify that:

Since p is strictly increasing and I out (t, x) < I for all I ∈ (I out (t, x), I in (t)] and

x > 0, we conclude that ∂µ(t,x,q) ∂x < 0 for all x > 0, and consequently µ is strictly decreasing in x. For c), let us define θ = (kx + K bg )L. Let l be a Lipschitz constant of p, then we have:

∂µ I (t,

[ 4 ]

 4 under nutrient limitation, and µ I (t, x) = 1 L L 0 p(I(t, x, z))dz is the vertical average of the local specific growth rate p(I) = µ max I K I +I when microalgae is only limited by light. I(t, x, z) is the light intensity perceived by microalgae at a distance z from the surface of the culture vessel and is determined from the Lambert-Beer law:

Figure 1 :

 1 Figure 1: I in as a function of t.

Figure 2 :

 2 Figure 2: Periodic solutions of (31) and their asymptotic behavior. System (31) admits only

Figure 3 :

 3 Figure 3: Unique positive periodic solution of (31). A. Population density. B. Intracellular phosphorus content. C. External phosphorus concentration. D. Light and phosphorus limitation.

  , 0, q * (t))dt, (36) is sufficient and necessary for the existence of a unique positive periodic solution. This solution is globally attractive (Theorem 4.1).

F 0 F 0

 00 (τ, R)dτ < -1 := ω + ω (t, R)dt < 0.

ln φ(t 1 +FF

 1 ω, 0, u) φ(t 1 , 0, u) = t1+ω t1 (t, φ(t, 0, u))dt ≤ t1+ω t1 (t, R)dt < -1 .

0

  ≤ µ I (t, x) ≤ 1 kxL I max µ max .(42)Letting x → ∞ in (42), we prove a). For b), if I in (t) > 0 we have:∂µ I (t, x) ∂x = g(q)kL (kx + K bg ) 2 L 2 Iin(t) Iout(t,x)p(I out (t, x)) -p(I) I dI.

  Thus µ I is Lipschitz in x uniformly in t and c) is proved.

	∂x	x)	≤	kL θ 2	Iin(t) Iout(t,x)	|p(I) -p(I out (t, x))| I	dI
			≤	klL θ 2	Iin(t) Iout(t,x)	|I -I out (t, x)| I	dI
			≤ lkLI max	1 -e -θ θ	≤ lkLI max .

Acknowledgments

This work was supported by the CONICYT doctoral grant (Carlos Martínez), and by the Phycover (ANR-14-CE04-0011) and IPL Algae in silico (INRIA) projects. We would like to thank the anonymous referee for the helpful suggestions which led to improvements of our original manuscript.

admits at most one fixed point and the proof is completed. Remark 3.9. (Interpretation of hypotheses in Theorem 3.8) Hypothesis I) simply says that at any moment of the day, if microalgae reach their minimal quota of x for small values of x. In such a case, the cooperativity of the reduced system [START_REF] Hsu | Dynamics of two phytoplankton species competing for light and nutrient with internal storage[END_REF] is not lost (see Remark 3.4). Thus, a similar result to Proposition 3.3 could be obtain for this new model.

Appendix A

Consider the non-autonomous Kolmogorov equation:

and the ω-periodic Kolmogorov equation:

where F (t, u) : R 2 + -→ R is continuous, decreasing in u and locally Lipschitz in u, and A 6.2.

ω 0 F 0 (t, R)dt < 0 for some R > 0.

Lemma 6.3. Assume that A6.1 and A6.2 hold. Then, solutions of (37) are ultimately bounded.

Proof. Let φ(t, s, u), t ≥ s ≥ 0, be the unique solution of (37) with φ(s, s, u) = u.

From 6.1, there is t 0 > 0 such that |F (t, 0) -F 0 (t, 0)| < 1 for all t ≥ t 0 . Since F is decreasing in u, we have that

with M an upper bound for F (t, u). From 40, we conclude that φ(t, 0, u) ≤ Re M ω for all t ∈ I. This implies that φ(t, 0, u) ≤ β = Re M ω for all t ≥ t * , and consequently lim sup t→∞ φ(t, 0, u) ≤ β.

The following proposition is inspired by part b) of Theorem 2.1 in [START_REF] Wolkowicz | n-species competition in a periodic chemostat[END_REF]. b) Assume that F (t, a) ≥ 0 for all t ≥ 0. If (38) admits a unique ω-periodic solution u * satisfying u * (t) ≥ a, then any solution to (37) with initial condition on J approaches asymptotically to u * .

Proof. Let φ(t, s, u) and φ 0 (t, s, u) be the unique solutions of ( 37) and ( 38) respectively with φ(s, s, u) = φ 0 (s, s, u) = u ∈ R + . From Lemma 6.3, solutions of (38) and (37) are ultimately bounded, and hence, uniformly bounded. Let S : J -→ J be the Poincaré map associated to (38). We note that J is positively invariant with respect to (38), then S is well defined. Let u ∈ J. Since S n (u) is monotone and bounded, S n (u) is convergent. Since J is positively invariant with respect to (38), u 0 = lim n→∞ S n (u) ∈ J. Thus, u * (t) = φ 0 (t, s, u 0 ) is an ω-periodic solution satisfying u * (t) ∈ J, and the part a) is proved.

For the part b), let u * be the unique ω-periodic solution with u * (0) ∈ J.

By Proposition 3.2 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], φ(t, s, u) is asymptotic to the ω-periodic semiflow T (t) := φ 0 (t, 0, •) : R + -→ R + , and hence T n (u) = φ(nω, 0, u), n ≥ 0, is an asymptotically autonomous discrete dynamical process with limit discrete semiflow S n : R + -→ R + , n ≥ 0. Since u * (0) is the unique globally stable fixed point of S, by Theorem 2.4 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], we conclude that lim n→∞ T n (u) = u * (0) for any u ∈ J. Applying Theorem 3.1 in [START_REF] Zhao | Asymptotic behavior for asymptotically periodic semiflows with applications[END_REF], we conclude the proof.

Appendix B

Here, we state some properties of the function µ I defined in (33). b) x -→ µ I (t, x) is strictly decreasing for all t ∈ (0, ω/2) and µ I (t, x) = 0 for