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1. Theory of turbid microalgae cultures

Author

Francis Mairet, Olivier Bernard, Carlos Martínez, Université Côte d'Azur (UCA), Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA)-Laboratoire d'océanographie de Villefranche (LOV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Institut Français de Recherche pour l'Exploitation de la Mer - Nantes (IFREMER Nantes), Université de Nantes (UN), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Institut Français de Recherche pour l'Exploitation de la Mer - Atlantique (IFREMER Atlantique), Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER), CONICYT doctoral grant, Phycover [ANR-14-CE04-0011], and IPL Algae in silico (INRIA) projects

Subjects
0106 biological sciences, 0301 basic medicine, Statistics and Probability, Photoinhibition, Materials science, Light, Light limitation, Photobioreactor, 01 natural sciences, Optical depth, Modelling, General Biochemistry, Genetics and Molecular Biology, Turbidity, Photobioreactors, 03 medical and health sciences, Game Theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 010608 biotechnology, Microalgae, Animals, Microalgae growth, Growth rate, Photosynthesis, Population Density, General Immunology and Microbiology, Applied Mathematics, General Medicine, Ray, Culture Media, 030104 developmental biology, Modeling and Simulation, General Agricultural and Biological Sciences, Biological system, Intensity (heat transfer)
Abstract

International audience; Microalgae can be cultivated in closed or open photobioreactors (PBR). In these systems, light rapidly decreases as it passes through the culture due to the turbidity of the medium. Thus, microalgae experiment different light intensities depending on their position in the medium. In this paper, we study theoretically how the growth rate of microalgae is affected by different factors; incident light intensity, form of the PBR, microalgae population density, turbidity of non-microalgae components, and light path-length of the reactor. We show that for different types of PBR the average growth rate is completely determined by the incident light intensity and the optical depth. In the case of vertical cylindrical PBRs illuminated from above (e.g. race-way or panel-type reactors), we described (and we prove under general assumptions) in details the dependence of the AGR on the aforementioned factors. Finally, we discuss some implications of our analysis; the occurrence of the Allee effect, if light ostensibly limits or inhibits the growth rate in outdoor cultures, and how the geometry of the PBR affects microalgae growth rate and productivity.

Published
2018
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2. The algal trophic mode affects the interaction and oil production of a synergistic microalga-yeast consortium

Author

R. van Lis, G. Zuccaro, Jean-Philippe Steyer, Laboratoire de Biotechnologie de l'Environnement [Narbonne] (LBE), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Department of Chemical, Materials and Production Engineering, University of Naples Federico II, ANR PHYCOVER (ANR-14-CE04-0011), and ANR-14-CE04-0011,Phycover,Durabilité des productions microalgales par recyclage du phosphore et de l'azote des eaux résiduaires : vers la station d'épuration du futur(2014)

Subjects
0106 biological sciences, Environmental Engineering, [SDV]Life Sciences [q-bio], Heterotroph, Biomass, Bioengineering, consortium, 010501 environmental sciences, Raw material, 01 natural sciences, complex mixtures, oleaginous yeast, fatty acid profile, 010608 biotechnology, Microalgae, Lipomyces, Waste Management and Disposal, 0105 earth and related environmental sciences, 2. Zero hunger, Biodiesel, Renewable Energy, Sustainability and the Environment, business.industry, Chemistry, Fossil fuel, food and beverages, trophic mode, General Medicine, Pulp and paper industry, microalga, Yeast, oil production, 13. Climate action, Biofuels, Biodiesel production, [SDE]Environmental Sciences, business, Oils, Mixotroph
Abstract

International audience; The use of non-food feedstocks to produce renewable microbial resources can limit our dependence on fossil fuels and lower CO2 emissions. Since microalgae display a virtuous CO2 and O2 exchange with heterotrophs, the microalga Chlamydomonas reinhardtii was combined with the oleaginous yeast Lipomyces starkeyi, known for their production of oil, base material for biodiesel. The coupled growth was shown to be synergistic for biomass and lipid production. The species were truly symbiotic since synergistic growth occurred even when the alga cannot use the organic carbon in the feedstock and in absence of air, thus depending entirely on CO2-O2 exchange. Since addition of acetate as the algal carbon source lowered the performance of the consortium, the microbial system design should take into account algal mixotrophy. The mixed biomass was found be suitable for biodiesel production, and whereas lipid production increased in the consortium, yields should be improved in future studies.

Published
2019
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3. Comparative assessment of evaporation models in algal ponds

Author

Bruno Sialve, Jean-Philippe Steyer, Andy Shilton, Quentin Béchet, Benoit Guieysse, Biological control of artificial ecosystems (BIOCORE), Laboratoire d'océanographie de Villefranche (LOV), Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), School of Engineering and Advanced Technology, Laboratoire de Biotechnologie de l'Environnement [Narbonne] (LBE), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Déchets Eaux Environnement Pollutions (DEEP), Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA), Phycover (ANR-14-CE04-0011), ANR-14-CE04-0011,Phycover,Durabilité des productions microalgales par recyclage du phosphore et de l'azote des eaux résiduaires : vers la station d'épuration du futur(2014), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Hydrology, Desert climate, 020209 energy, [SDV]Life Sciences [q-bio], Evaporation, modeling, 02 engineering and technology, 010501 environmental sciences, 01 natural sciences, 6. Clean water, Water demand, evaporation, 13. Climate action, water demand, Air temperature, outdoor pond, [SDE]Environmental Sciences, 0202 electrical engineering, electronic engineering, information engineering, Environmental science, Raceway, Agronomy and Crop Science, 0105 earth and related environmental sciences
Abstract

International audience; The ability to predict water evaporation from shallow ponds is needed to accurately assess the water demand and costs of microalgae farming. This study assessed the accuracy of seven evaporation models available in the literature against experimental data collected in a raceway algal pond located in Narbonne, France. A theoretical ‘flat-plate’ evaporation model and the “Sartori model” were identified as the most accurate models (errors of 14.2% and 9.2%, respectively, over a period of 274 days). As these two models require the mathematical determination of pond temperature, simulations were performed to determine if pond temperature could be substituted for air temperature to compute yearly evaporation estimates. Unfortunately, assuming that pond temperature was equal to air temperature caused significant inaccuracies on the yearly evaporation (e.g. up to 68% in an arid climate with the Sartori model). High-resolution co-modeling of evaporation and temperature is therefore required for accurate evaporation predictions.

Published
2018
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4. Maximizing microalgae productivity in a light-limited chemostat

Author

Martínez, Carlos, Bernard, Olivier, Mairet, Francis, Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'océanographie de Villefranche (LOV), Institut National de la Recherche Agronomique (INRA)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Universités (COMUE), Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER), CONICYT, Phycover [ANR-14-CE04-0011], IPL Algae in silico (INRIA) projects, Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA)

Subjects
Chlorophyll, Photoacclimation, [SDV]Life Sciences [q-bio], Biomass productivity, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Photoinhibition, Microalgae, Dynamics and control, Industrial biotechnology
Abstract

International audience; Light supply is one of the most important parameters to be considered for enhancing microalgae growth in photobioreactors (PBR) with artificial light. However, most of the mathematical works do not consider incident light as a parameter to be optimized. In this work based on a simple model of light-limited growth, we determine optimal values for the dilution rate and the incident light intensity in order to maximize the steady-state microalgal surface productivity in a continuous culture. We also show that in optimal conditions there is a minimal initial microalgal concentration (and we give a simple expression to determine it) to guarantee the persistence of the population. Finally, in the context of enhancing microalgae productivity by reducing light absorption by microalgae, we conclude our work by studying the influence of the chlorophyll-carbon quota on the maximal productivity. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Published
2018
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5. Modelling an Artificial Microalgae-Cyanobacteria Ecosystem

Author

Olivier Bernard, Margaux Caia, Jean-Philippe Steyer, Laboratoire de Biotechnologie de l'Environnement [Narbonne] (LBE), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'océanographie de Villefranche (LOV), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Phycover ANR-14-CE04-0011, and ANR-14-CE04-0011,Phycover,Durabilité des productions microalgales par recyclage du phosphore et de l'azote des eaux résiduaires : vers la station d'épuration du futur(2014)

Subjects
0301 basic medicine, Cyanobacteria, Microorganism, Crocosphaera watsonii, Interactions, Environmental engineering, Diazotrophy, Photosynthesis, Modelling, 03 medical and health sciences, chemistry.chemical_compound, Nitrate, Nitrogen fixation, Ecosystem, 14. Life underwater, biology, Ecology, biology.organism_classification, 030104 developmental biology, chemistry, Control and Systems Engineering, Environmental chemistry, [SDE]Environmental Sciences, Dunaliella salina, Environmental science, Biotechnology, Allelopathy
Abstract

International audience; Microalgae based processes have been actively studied in the last decades with perspective for food, feed, and source of chemicals such as biofuels. Most of the developments focused on monospecific culture of microalgae, with dedicated practices to avoid any contaminations. However, interactions between microalgae and bacteria are likely to enhance microalgae growth, provide more resilience to external changes and eventually limit external contaminations. But interactions within these natural ecosystems are still poorly understood and are affected by the environment. A photosynthetic marine ecosystem composed of the microalgae Dunaliella salina and the nitrogen fixing cyanobacteria Crocosphaera watsonii was therefore studied. A model was designed to represent the competition for light and the interactions with nitrogen between these two microorganisms. An allelopathic effect was noticed and a toxin production by C. watsonii was assumed and included in the model. Calibration was carried out with experimental data where cell densities and nitrate concentrations were measured. The predictions of the mathematical model accurately represented the experimental data. The model therefore highlighted the interactions within this artificial ecosystem. The model confirms that D. salina growth was limited by nitrate concentration and did not consume dissolved organic nitrogen produced by C. watsonii from its diazotrophic activity. D. salina and C. watsonii were competing for light, which favored D. salina and limited C. watsonii when grown in cocultures. The model supports the hypothesis that C. watsonii produced toxins enhancing D. salina mortality in the cocultures.

Published
2018
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6. Maximizing microalgae productivity by shading outdoor cultures

Author

Martínez, Carlos, Bernard, Olivier, Mairet, Francis, Biological control of artificial ecosystems (BIOCORE), Institut National de la Recherche Agronomique (INRA)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Station Zoologique, Centre National de la Recherche Scientifique (CNRS), Sorbonne Universités, Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA), This work was supported by the CONICYT doctoral grant (Carlos Martínez), and by the Phycover (ANR-14-CE04-0011) and Purple Sun (ANR-13-BIME-0004) projects. F. Mairet is grateful to 'FMJH Program Gaspard Monge in optimization and operation research', Laboratoire d'océanographie de Villefranche (LOV), Observatoire océanologique de Villefranche-sur-mer (OOVM), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), Station Zoologique de Villefranche, Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), International Federation of Automatic Control (IFAC). AUT., Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)

Subjects
[SPI]Engineering Sciences [physics], Biomass productivity, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, [SDV]Life Sciences [q-bio], fungi, Photoinhibition, Microalgae, food and beverages, Dynamics and control, [MATH]Mathematics [math], Industrial biotechnology
Abstract

International audience; Outdoor microalgae cultures can undergo a photoinhibitory process that can result in a loss in biomass productivity. This loss can be reduced by shading the culture such that the incident photon flux decreases. Based on a simple model of light-limited growth, we look for a control strategy to shadow the culture in order to maximize the biomass productivity. The strategy results in a feedback control that depends on the microalgae strain, the microalgae concentration, and the incident light. In the case that the incident light and the loss rate vary periodically in time, we give conditions for the existence of a positive periodic solution that is globally stable. We show the performance of the feedback control by means of numerical simulations.

Published
2017
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