T. Anning, H. L. Macintyre, S. M. Pratt, P. J. Sammes, S. Gibb et al., Photoacclimation in the marine diatom skeletonema costatum, Limnology and Oceanography, vol.45, issue.8, pp.1807-1817, 2000.

W. Berger, Global maps of ocean productivity, productivity of the ocean: Present and past wh berger, vs smetacek, g. wefer, pp.429-455, 1989.

G. Bougaran, O. Bernard, and A. Sciandra, Modeling continuous cultures of microalgae colimited by nitrogen and phosphorus, Journal of theoretical biology, vol.265, issue.3, pp.443-454, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00608413

J. Caperon, Population growth response of isochrysis galbana to nitrate variation at limiting concentrations, Ecology, vol.49, issue.5, pp.866-872, 1968.

M. Cuaresma, M. Janssen, C. Vílchez, and R. H. Wijffels, Productivity of chlorella sorokiniana in a short light-path (slp) panel photobioreactor

F. Grognard, A. R. Akhmetzhanov, and O. Bernard, Optimal strategies for biomass productivity maximization in a photobioreactor using natural light, Automatica, vol.50, issue.2, pp.359-368, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00871341

B. Han, A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii, Journal of theoretical biology, vol.214, issue.4, pp.519-527, 2002.

J. Huisman, R. R. Jonker, C. Zonneveld, and F. J. Weissing, Competition for light between phytoplankton species: experimental tests of mechanistic theory, Ecology, vol.80, issue.1, pp.211-222, 1999.

J. Huisman, H. C. Matthijs, P. M. Visser, H. Balke, C. A. Sigon et al., Principles of the light-limited chemostat: theory and ecological applications, Antonie van Leeuwenhoek, vol.81, issue.1, pp.117-133, 2002.

J. Huisman and F. J. Weissing, Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model, Ecology, vol.75, issue.2, pp.507-520, 1994.

M. Iehana, Kinetic analysis of the growth of spirulina sp. on continuous culture, Journal of fermentation technology, vol.61, issue.5, pp.457-466, 1983.

W. E. Kardinaal, L. Tonk, I. Janse, S. Hol, P. Slot et al., Competition for light between toxic and nontoxic strains of the harmful cyanobacterium microcystis, Appl. Environ. Microbiol, vol.73, issue.9, pp.2939-2946, 2007.

Y. Lee and C. Soh, Accumulation of astaxanthin in haematococcus lacustris (chlorophyta) 1, Journal of phycology, vol.27, issue.5, pp.575-577, 1991.

S. Long, S. Humphries, and P. G. Falkowski, Photoinhibition of photosynthesis in nature. Annual review of plant biology, vol.45, pp.633-662, 1994.

J. J. Milledge, Commercial application of microalgae other than as biofuels: a brief review, Reviews in Environmental Science and Bio/Technology, vol.10, issue.1, pp.31-41, 2011.

J. Monod, La technique de culture continue: theorie et applications, 1950.

N. Norsker, M. J. Barbosa, M. H. Vermuë, and R. H. Wijffels, Microalgal productiona close look at the economics, Biotechnology advances, vol.29, issue.1, pp.24-27, 2011.

A. Novick and L. Szilard, Description of the chemostat, Science, vol.112, pp.715-716, 1950.

S. J. Pirt, Y. Lee, A. Richmond, and M. W. Pirt, The photosynthetic efficiency of chlorella biomass growth with reference to solar energy utilisation, Journal of Chemical Technology and Biotechnology, vol.30, issue.1, pp.25-34, 1980.

S. A. Razzak, M. M. Hossain, R. A. Lucky, A. S. Bassi, and H. De-lasa,

H. Haario, L. Kalachev, and M. Laine, Reduced models of algae growth, Bulletin of mathematical biology, vol.71, issue.7, pp.1626-1648, 2009.

S. Hsu and C. Lin, Dynamics of two phytoplankton species competing for light and nutrient with internal storage, Discrete & Continuous Dynamical Systems-S, vol.7, issue.6, pp.1259-1285, 2014.

M. Huesemann, B. Crowe, P. Waller, A. Chavis, S. Hobbs et al., A validated model to predict microalgae growth in outdoor pond cultures subjected to fluctuating light intensities and water temperatures, Algal Research, vol.13, pp.195-206, 2016.

J. Huisman and F. J. Weissing, Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model, Ecology, vol.75, issue.2, pp.507-520, 1994.

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: a theoretical analysis, The American Naturalist, vol.146, issue.4, pp.536-564, 1995.

Y. Jeon, C. Cho, and Y. Yun, Measurement of microalgal photosynthetic activity depending on light intensity and quality, Biochemical Engineering Journal, vol.27, issue.2, pp.127-131, 2005.

J. T. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 1994.

C. A. Klausmeier, E. Litchman, and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnology and Oceanography, vol.49, issue.4part2, pp.1463-1470, 2004.

E. Lee, M. Jalalizadeh, and Q. Zhang, Growth kinetic models for microalgae cultivation: a review, Algal Research, vol.12, pp.497-512, 2015.

E. Lee and Q. Zhang, Integrated co-limitation kinetic model for microalgae growth in anaerobically digested municipal sludge centrate, Algal research, vol.18, pp.15-24, 2016.

H. L. Macintyre, T. M. Kana, T. Anning, and R. J. Geider, Photoacclimation of photosynthesis irradiance response curves and photosynthetic pigments in microalgae and cyanobacteria 1, Journal of phycology, vol.38, issue.1, pp.17-38, 2002.

F. Mairet, O. Bernard, T. Lacour, and A. Sciandra, Modelling microalgae growth in nitrogen limited photobiorector for estimating biomass, carbohydrate and neutral lipid productivities, IFAC Proceedings Volumes, vol.44, pp.10591-10596, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00848427

J. Monod, The growth of bacterial cultures, Annual Reviews in Microbiology, vol.3, issue.1, pp.371-394, 1949.

A. Nikolaou, P. Hartmann, A. Sciandra, B. Chachuat, and O. Bernard, Dynamic coupling of photoacclimation and photoinhibition in a model of microalgae growth, Journal of theoretical biology, vol.390, pp.61-72, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01247049

F. J. Oyarzun and K. Lange, The attractiveness of the droop equations ii. generic uptake and growth functions, Mathematical biosciences, vol.121, issue.2, pp.127-139, 1994.

A. Packer, Y. Li, T. Andersen, Q. Hu, Y. Kuang et al., Growth and neutral lipid synthesis in green microalgae: a mathematical model, Bioresource technology, vol.102, issue.1, pp.111-117, 2011.

J. Passarge, S. Hol, M. Escher, and J. Huisman, Competition for nutrients and light: stable coexistence, alternative stable states, or competitive exclusion?, Ecological Monographs, vol.76, issue.1, pp.57-72, 2006.

J. Quinn, L. D. Winter, and T. Bradley, Microalgae bulk growth model with application to industrial scale systems, Bioresource technology, vol.102, issue.8, pp.5083-5092, 2011.

O. N. Ross and R. J. Geider, New cell-based model of photosynthesis and photo-acclimation: accumulation and mobilisation of energy reserves in phytoplankton. Marine ecology progress series, vol.383, pp.53-71, 2009.

H. Smith, The periodically forced droop model for phytoplankton growth in a chemostat, Journal of Mathematical Biology, vol.35, issue.5, pp.545-556, 1997.

H. L. Smith and P. Waltman, The theory of the chemostat: dynamics of microbial competition, vol.13, 1995.

J. H. Steele, Environmental control of photosynthesis in the sea, Limnology and Oceanography, vol.7, pp.137-150, 1962.

. .. Chapter-presentation,

, 48 a flat-plate photobioreactor, Applied Microbiology and Biotechnology, vol.49, issue.6, pp.655-662, 1998.

J. Huisman and F. J. Weissing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model, Ecology, vol.75, issue.2, pp.507-520, 1994.

J. T. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 1994.

C. Martínez, F. Mairet, and O. Bernard, Theory of turbid microalgae cultures, Journal of theoretical biology, vol.456, pp.190-200, 2018.

J. Neidhardt, J. R. Benemann, L. Zhang, and A. Melis, Photosystemii repair and chloroplast recovery from irradiance stress: relationship between chronic photoinhibition, light-harvesting chlorophyll antenna size and photosynthetic productivity in dunaliella salina

, Photosynthesis Research, vol.56, issue.2, pp.175-184, 1998.

H. Qiang, H. Guterman, and A. Richmond, Physiological characteristics of spirulina platensis (cyanobacteria) cultured at ultrahigh cell densities 1, Journal of Phycology, vol.32, issue.6, pp.1066-1073, 1996.

H. Qiang, H. Guterman, and A. Richmond, Physiological characteristics of spirulina platensis (cyanobacteria) cultured at ultrahigh cell densities1, Journal of Phycology, vol.32, issue.6, pp.1066-1073, 1996.

H. Qiang and A. Richmond, Productivity and photosynthetic efficiency ofspirulina platensis as affected by light intensity, algal density and rate

W. A. Coppel, Stability and asymptotic behavior of differential equations, 1965.

M. R. Droop, Vitamin B 12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri, Journal of the Marine Biological Association of the United Kingdom, vol.48, issue.3, pp.689-733, 1968.

C. Martínez, F. Mairet, and O. Bernard, Theory of turbid microalgae cultures, Journal of theoretical biology, vol.456, pp.190-200, 2018.

H. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat, Journal of Mathematical Biology, vol.35, issue.5, pp.545-556, 1997.

H. L. Smith and P. Waltman, The theory of the chemostat: dynamics of microbial competition, vol.13, 1995.

G. S. Wolkowicz and X. Zhao, n-species competition in a periodic chemostat, Differential and Integral Equations, vol.11, issue.3, pp.465-491, 1998.

T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, vol.14, 1975.

X. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Communications on Applied Nonlinear Analysis, vol.3, issue.4, pp.43-66, 1996.

X. Zhao, Dynamical systems in population biology, CMS Books in Mathematics, vol.16, 2003.

D. J. Gerla, W. M. Mooij, and J. Huisman, Photoinhibition and the assembly of light-limited phytoplankton communities, Oikos, vol.120, issue.3, pp.359-368, 2011.

M. W. Hirsch and H. Smith, Monotone dynamical systems. Handbook of differential equations: ordinary differential equations, vol.2, pp.239-357, 2006.

S. Hsu and C. Lin, Dynamics of two phytoplankton species competing for light and nutrient with internal storage, Discrete Contin. Dyn

, Syst. Ser. S, vol.7, pp.1259-1285, 2014.

H. R. Thieme, Convergence results and a poincaré-bendixson trichotomy for asymptotically autonomous differential equations, Journal of mathematical biology, vol.30, issue.7, pp.755-763, 1992.

K. Yeh, J. Chang, and W. Chen, Effect of light supply and carbon source on cell growth and cellular composition of a newly isolated microalga chlorella vulgaris esp-31, Engineering in Life Sciences, vol.10, issue.3, pp.201-208, 2010.

, Modeling light-limited growth of microalgae, p.142

, Maximizing microalgae productivity: Indoor case, vol.144

. .. Chemostat, 145 6.4.2 Maximizing biomass productivity, p.148

, Maximizing microalgae productivity by shading outdoor cultures

. .. Control-strategy, , p.154

. .. , 2 Dynamics of the controlled culture, p.158

. .. Simulations, , p.160

. .. Conclusions, , p.163

. Bibliography,

J. S. Burlew, Current status of the large-scale culture of algae, vol.600, pp.3-23, 1953.

A. P. Carvalho, S. O. Silva, J. M. Baptista, and F. X. Malcata, Light requirements in microalgal photobioreactors: an overview of biophotonic aspects, Applied Microbiology and Biotechnology, vol.89, issue.5, pp.1275-1288, 2011.

J. Deschnes and A. V. Wouwer, Dynamic optimization of biomass productivity in continuous cultures of microalgae isochrysis galbana through modulation of the light intensity, IFAC-PapersOnLine, vol.48, issue.8, pp.1093-1099, 2015.

S. P. Long, S. Humphries, and P. G. Falkowski, Photoinhibition of photosynthesis in nature, Annual Review of Plant Physiology and Plant Molecular Biology, vol.45, issue.1, pp.633-662, 1994.

F. Mairet and O. Bernard, The photoinhibistat: Operating microalgae culture under photoinhibition for strain selection, IFAC-PapersOnLine, vol.49, issue.7, pp.1068-1073, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01410835

F. Mairet, R. Muñoz-tamayo, and O. Bernard, Driving species competition in a light-limited chemostat, IFAC Proceedings Volumes, vol.46, pp.175-180, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00852258

C. Martínez, O. Bernard, and F. Mairet, Maximizing microalgae productivity by shading outdoor cultures. IFAC-PapersOnLine, vol.50, pp.8734-8739, 2017.

C. Martínez, O. Bernard, and F. Mairet, Maximizing microalgae productivity in a light-limited chemostat, IFAC-PapersOnLine, vol.51, issue.2, pp.735-740, 2017.

P. Masci, F. Grognard, and O. Bernard, 11th ifac symposium on computer applications in biotechnology microalgal biomass surface productivity optimization based on a photobioreactor model. IFAC Proceedings Volumes, vol.43, pp.180-185, 2010.

H. Qiang, H. Guterman, and A. Richmond, Physiological characteristics of spirulina platensis (cyanobacteria) cultured at ultrahigh cell densities1, Journal of Phycology, vol.32, issue.6, pp.1066-1073, 1996.

A. Vonshak and R. Guy, Photoadaptation, photoinhibition and productivity in the blue-green alga, spirulina platensis grown outdoors, Plant, Cell Environment, vol.15, issue.5, pp.613-616, 1992.

F. Zanolin, Continuation theorems for the periodic problem via the translation operator, Rend. Sem. Mat. Univ. Politec. Torino, vol.54, issue.1, pp.1-23, 1996.

. .. Maximizing-microalgae-productivity,

N. Abdel-raouf, A. Al-homaidan, and I. Ibraheem, Microalgae and wastewater treatment, Saudi journal of biological sciences, vol.19, issue.3, pp.257-275, 2012.

Q. Béchet, A. Shilton, and B. Guieysse, Modeling the effects of light and temperature on algae growth: state of the art and critical assessment for productivity prediction during outdoor cultivation, Biotechnology advances, vol.31, issue.8, pp.1648-1663, 2013.

O. Bernard, Hurdles and challenges for modelling and control of microalgae for co2 mitigation and biofuel production, Journal of Process Control, vol.21, issue.10, pp.1378-1389, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00847274

J. Bonnans, V. Grelard, and P. Martinon, Bocop, the optimal control solver, open source toolbox for optimal control problems, 2011.

M. A. Borowitzka, Limits to growth, Wastewater treatment with algae, pp.203-226, 1998.

L. Christenson and R. Sims, Production and harvesting of microalgae for wastewater treatment, biofuels, and bioproducts, Biotechnology advances, vol.29, issue.6, pp.686-702, 2011.

D. Dochain, Automatic control of bioprocesses, 2013.

D. J. Gerla, W. M. Mooij, and J. Huisman, Photoinhibition and the assembly of light-limited phytoplankton communities, Oikos, vol.120, issue.3, pp.359-368, 2011.

K. Larsdotter, Wastewater treatment with microalgae-a literature review, vol.62, p.31, 2006.

C. Martínez, O. Bernard, and F. Mairet, Maximizing microalgae productivity by shading outdoor cultures. IFAC-PapersOnLine, vol.50, pp.8734-8739, 2017.

C. Martínez, F. Mairet, and O. Bernard, Theory of turbid microalgae cultures, Journal of theoretical biology, vol.456, pp.190-200, 2018.

A. Morel and A. Bricaud, Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton, Deep-Sea Res, vol.28, issue.11, pp.1375-1393, 1981.

F. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton 1, Journal of phycology, vol.23, issue.1, pp.137-150, 1987.

J. Park, R. Craggs, and A. Shilton, Wastewater treatment high rate algal ponds for biofuel production, Bioresource technology, vol.102, issue.1, pp.35-42, 2011.

, 208 pilot high rate algal ponds for simultaneous nutrient removal and lipids production, Science of the Total Environment, vol.589, pp.66-72, 2017.

O. Bernard, F. Mairet, and B. Chachuat, Modelling of microalgae culture systems with applications to control and optimization, Microalgae Biotechnology, pp.59-87, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01245875

O. Bernard, F. Mairet, and B. Chachuat, Modelling of Microalgae Culture Systems with Applications to Control and Optimization, p.59
URL : https://hal.archives-ouvertes.fr/hal-01245875

P. S. Davies, The biological basis of wastewater treatment, Strathkelvin Instruments Ltd, vol.3, 2005.

Q. Dortch, The interaction between ammonium and nitrate uptake in phytoplankton. Marine ecology progress series, Oldendorf, vol.61, issue.1, pp.183-201, 1990.

J. S. Guest, M. C. Van-loosdrecht, S. J. Skerlos, and N. G. Love, Lumped pathway metabolic model of organic carbon accumulation and mobilization by the alga chlamydomonas reinhardtii, Environmental science & technology, vol.47, issue.7, pp.3258-3267, 2013.

B. Kim, J. Choi, K. Cho, Z. Kang, R. Ramanan et al., Influence of water depth on microalgal production, biomass harvest, and energy consumption in high rate algal pond using municipal wastewater, J. Microbiol. Biotechnol, vol.28, issue.4, pp.630-637, 2018.

F. Mairet, O. Bernard, P. Masci, T. Lacour, and A. Sciandra, Modelling neutral lipid production by the microalga isochrysis aff. galbana under nitrogen limitation, Bioresource Technology, vol.102, issue.1, pp.142-149, 2011.

B. S. Sturm and S. L. Lamer, An energy evaluation of coupling nutrient removal from wastewater with algal biomass production, Applied Energy, vol.88, issue.10, pp.3499-3506, 2011.

D. S. Wágner, B. Valverde-pérez, and B. G. Plósz, Light attenuation in photobioreactors and algal pigmentation under different growth conditions-model identification and complexity assessment. Algal research, vol.35, pp.488-499, 2018.

D. S. Wágner, B. Valverde-pérez, M. Saebø, M. B. De-la-sotilla, J. Van-wagenen et al., Towards a consensusbased biokinetic model for green microalgae-the asm-a, Water research, vol.103, pp.485-499, 2016.

Z. Arbib, I. De-godos, J. Ruiz, and J. A. Perales, Optimization of pilot high rate algal ponds for simultaneous nutrient removal and lipids production, Science of the Total Environment, vol.589, pp.66-72, 2017.

Q. Béchet, M. Laviale, N. Arsapin, H. Bonnefond, and O. Bernard, Modeling the impact of high temperatures on microalgal viability and photosynthetic activity, Biotechnology for biofuels, vol.10, issue.1, p.136, 2017.

J. M. Beman, K. R. Arrigo, and P. A. Matson, Agricultural runoff fuels large phytoplankton blooms in vulnerable areas of the ocean, Nature, vol.434, issue.7030, p.211, 2005.

O. Bernard and B. Rémond, Validation of a simple model accounting for light and temperature effect on microalgal growth, Bioresource technology, vol.123, pp.520-527, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00848389

H. Bonnefond, N. Moelants, A. Talec, O. Bernard, and A. Sciandra, Concomitant effects of light and temperature diel variations on the growth rate and lipid production of dunaliella salina, Algal research, vol.14, pp.72-78, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01266717

C. Combe, P. Hartmann, S. Rabouille, A. Talec, O. Bernard et al., Long-term adaptive response to high-frequency light signals in the unicellular photosynthetic eukaryote dunaliella salina, Biotechnology and bioengineering, vol.112, issue.6, pp.1111-1121, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01102019

J. Cornet, Calculation of optimal design and ideal productivities of volumetrically lightened photobioreactors using the constructal approach, Chemical Engineering Science, vol.65, issue.2, pp.985-998, 2010.
URL : https://hal.archives-ouvertes.fr/hal-02614188

R. Craggs, D. Sutherland, and H. Campbell, Hectare-scale demonstration of high rate algal ponds for enhanced wastewater treatment and biofuel production, Journal of Applied Phycology, vol.24, issue.3, pp.329-337, 2012.

R. De-luca, F. Bezzo, Q. Béchet, and O. Bernard, Exploiting meteorological forecasts for the optimal operation of algal ponds, Journal of Process Control, vol.55, pp.55-65, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01666404

M. P. Gent, Effect of degree and duration of shade on quality of greenhouse tomato, HortScience, vol.42, issue.3, pp.514-520, 2007.

D. J. Gerla, W. M. Mooij, and J. Huisman, Photoinhibition and the assembly of light-limited phytoplankton communities, Oikos, vol.120, issue.3, pp.359-368, 2011.

S. Lee and D. Lee, Improved prediction of harmful algal blooms in four major south koreas rivers using deep learning models, International journal of environmental research and public health, vol.15, issue.7, p.1322, 2018.

S. Long, S. Humphries, and P. G. Falkowski, Photoinhibition of photosynthesis in nature. Annual review of plant biology, vol.45, pp.633-662, 1994.

P. Masci, F. Grognard, and O. Bernard, Microalgal biomass surface productivity optimization based on a photobioreactor model. IFAC Proceedings Volumes, vol.43, pp.180-185, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00847297

W. Mulbry, S. Kondrad, C. Pizarro, and E. Kebede-westhead, Treatment of dairy manure effluent using freshwater algae: algal productivity and recovery of manure nutrients using pilot-scale algal turf scrubbers, Bioresource technology, vol.99, issue.17, pp.8137-8142, 2008.

E. J. Olgu?n, Phycoremediation: key issues for cost-effective nutrient removal processes, Biotechnology advances, vol.22, issue.1-2, pp.81-91, 2003.

W. Oswald, Waste water reclamation through the production of algae, 1959.

J. Park, R. Craggs, and A. Shilton, Wastewater treatment high rate algal ponds for biofuel production, Bioresource technology, vol.102, issue.1, pp.35-42, 2011.

M. Ras, J. Steyer, and O. Bernard, Temperature effect on microalgae: a crucial factor for outdoor production, Reviews in environmental science and bio/technology, vol.12, issue.2, pp.153-164, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00852286

E. Sforza, E. Barbera, and A. Bertucco, Improving the photoconversion efficiency: An integrated photovoltaic-photobioreactor system for microalgal cultivation, Algal research, vol.10, pp.202-209, 2015.

H. Takache, G. Christophe, J. Cornet, and J. Pruvost, Experimental and theoretical assessment of maximum productivities for the microalgae chlamydomonas reinhardtii in two different geometries of photobioreactors, Biotechnology progress, vol.26, issue.2, pp.431-440, 2010.
URL : https://hal.archives-ouvertes.fr/hal-02534166

A. Vonshak and R. Guy, Photoadaptation, photoinhibition and productivity in the blue-green alga, spirulina platensis grown outdoors, Plant, Cell & Environment, vol.15, issue.5, pp.613-616, 1992.

G. S. Wolkowicz and X. Zhao, n-species competition in a periodic chemostat, Differential and Integral Equations, vol.11, issue.3, pp.465-491, 1998.

X. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, MR 97i: 58150. Citeseer, 1996.

X. Zhao, J. Borwein, and P. Borwein, Dynamical systems in population biology, vol.16, 2003.

, For any I out ? [?(I in ), I * ), the function g(I out , ·) is strictly decreasing on [?(I out ), ?). Since I out < I * , we have that ?(I out ) ? (I * , ?(I out )). We also note that I out > ?(I in ) > ?(I in ), therefore ?(I out ) < I in . The last inequality implies that the function g(I out , ·) is strictly decreasing on [I in , ?), Proof. Let I in , I in > I * such that I in < I in

, This shows that g(I in , ·) cannot intersect p on [?(I in ), I * ), therefore ?(I in ) < ?(I in ), Consequently, ? is strictly decreasing on (I * , ?)

, Since ? is strictly decreasing, it is enough to prove that for any y ? (0, I * ) there exist I y ? (I * , ?) such that g(I y , y) = p(y)

/. , ) ? p(y) < 0, we conclude that lim I in ?? G(I in ) =

?. ?. , Thus, since G is continuous, there is I y > I * such that G(I y ) = 0

, 0 for all t ? 0. If (11) admits a unique ?-periodic solution u * satisfying u * (t) ? a, then any solution to (10) with

, ) be the unique solutions of (10) and (11) respectively with ?(s, s, u) = ? 0 (s, s, u) = u ? R + . From Lemma B8, solutions of (11) and (10) are ultimately bounded, and hence

S. Let and . The, We note that J is positively invariant with respect to (11), 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, issue.11

G. S. Wolkowicz and X. Zhao, n-species competition in a periodic chemostat, Differential and Integral Equations, vol.11, issue.3, pp.465-491, 1998.

X. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, MR 97i: 58150. Citeseer, 1996.

?. Martínez, F. Mairet, and O. Bernard, Dynamics of the periodically forced light-limited Droop model, Journal of Differential Equations, 2019.

?. Martínez, F. Mairet, L. Plaza, A. Sciandra, and O. Bernard, Quantifying the potential of microalgae culture systems to remove nutrients from wastewater, Proceedings of the FOSBE conference, 2019.

?. Martínez, F. Mairet, and O. Bernard, Effects of photoinhibition on the dynamics of the lightlimited Droop model, Journal of Differential Equations, 2019.

?. Martínez, O. Bernard, P. Martinon, and F. Mairet, Dynamics and Control of a Periodically Forced Microalgae Culture, Proceedings of the DYCOPS conference, 2019.

?. Martínez, O. Bernard, and F. Mairet, Theory of turbid microalgae cultures, Journal of Theoretical Biology, vol.456, pp.190-200, 2018.

?. Martínez, O. Bernard, and F. Mairet, Maximizing Microalgae Productivity in a Light-Limited Chemostat, IFAC-PapersOnLine, vol.51, issue.2, pp.735-740, 2018.

C. Martínez, O. Bernard, and F. Mairet, Maximizing microalgae productivity by shading outdoor cultures, IFAC-PapersOnLine, vol.50, pp.8734-8739, 2017.

, Dans une dernière partie, un modèle est proposé et validé pour rendre compte de la limitation par la lumière, l'azote et le phosphore

, Mots-clés: microalgues, modélisation mathématique, productivité, photoinhibition, modèleà quota, système dynamique