# American Institute of Mathematical Sciences

February  2021, 14(2): 615-633. doi: 10.3934/dcdss.2020424

## A mathematical model for marine dinoflagellates blooms

 1 Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, Pau, France 2 Université de Pau et des Pays de l'Adour, UPPA, CNRS, LMAP, Pau, France

Received  December 2019 Published  February 2021 Early access  September 2020

We present a model for the life cycle of a dinoflagellate in order to describe blooms. We prove the mathematical well-posedness of the model and the possibility of extinction in finite time of the alga form meaning that the full population is under the cysts from.

Citation: M. Dambrine, B. Puig, G. Vallet. A mathematical model for marine dinoflagellates blooms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 615-633. doi: 10.3934/dcdss.2020424
##### References:
 [1] D. Anderson, Alexandrium fundyense cyst dynamics in the Gulf of Maine, Deep Sea Research Part Ⅱ: Topical Studies in Oceanography, 52 (2005), 2522-2542. [2] F. Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations, 18 (2005), 891-934. [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 5, Evolution problems. I, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [4] K. Flynn and D. McGillicuddy, Modelling Marine Harmful Algal Blooms: Current Status and Future Prospects, Harmful Algal Blooms: A Compendium Desk References, 3, John Wiley & Sons Ltd, 2018. [5] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modèles Non Linéaires de L'ingénierie Pétrolière, Mathématiques & Applications (Berlin) [Mathematics & Applications], 22, Springer-Verlag, Berlin, 1996. [6] O. Guibé, A. Mokrane, Y. Tahraoui and G. Vallet, Lewy-Stampacchia's inequality for a pseudomonotone parabolic problem, Adv. Nonlinear Anal., to appear. doi: 10.1515/anona-2020-0015. [7] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer Science, 2017. doi: 10.1007/978-981-10-0188-8. [8] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, The Macmillan Co., New York, 1964. [9] D. J. McGillicuddy, Models of harmful algal blooms: Conceptual, empirical, and numerical approaches, Journal of Marine Systems, 83 (2010), 105-105. [10] H. Ruoying, D. J. McGillicuddy, B. Keafer and D. Anderson, Historic 2005 toxic bloom of Alexandrium fundyense in the western Gulf of Maine: 2. Coupled biophysical numerical modeling, Journal of Geophysical Research: Oceans, 113 (2008).

show all references

##### References:
 [1] D. Anderson, Alexandrium fundyense cyst dynamics in the Gulf of Maine, Deep Sea Research Part Ⅱ: Topical Studies in Oceanography, 52 (2005), 2522-2542. [2] F. Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations, 18 (2005), 891-934. [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 5, Evolution problems. I, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [4] K. Flynn and D. McGillicuddy, Modelling Marine Harmful Algal Blooms: Current Status and Future Prospects, Harmful Algal Blooms: A Compendium Desk References, 3, John Wiley & Sons Ltd, 2018. [5] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modèles Non Linéaires de L'ingénierie Pétrolière, Mathématiques & Applications (Berlin) [Mathematics & Applications], 22, Springer-Verlag, Berlin, 1996. [6] O. Guibé, A. Mokrane, Y. Tahraoui and G. Vallet, Lewy-Stampacchia's inequality for a pseudomonotone parabolic problem, Adv. Nonlinear Anal., to appear. doi: 10.1515/anona-2020-0015. [7] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer Science, 2017. doi: 10.1007/978-981-10-0188-8. [8] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, The Macmillan Co., New York, 1964. [9] D. J. McGillicuddy, Models of harmful algal blooms: Conceptual, empirical, and numerical approaches, Journal of Marine Systems, 83 (2010), 105-105. [10] H. Ruoying, D. J. McGillicuddy, B. Keafer and D. Anderson, Historic 2005 toxic bloom of Alexandrium fundyense in the western Gulf of Maine: 2. Coupled biophysical numerical modeling, Journal of Geophysical Research: Oceans, 113 (2008).
Sketch of the parts of Alexandrium catenella life we are interested in
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