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  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 & 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.   Google Scholar

[2]

F. Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations, 18 (2005), 891-934.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer Science, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[8]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, The Macmillan Co., New York, 1964.  Google Scholar

[9]

D. J. McGillicuddy, Models of harmful algal blooms: Conceptual, empirical, and numerical approaches, Journal of Marine Systems, 83 (2010), 105-105.   Google Scholar

[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). Google Scholar

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.   Google Scholar

[2]

F. Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations, 18 (2005), 891-934.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer Science, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[8]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, The Macmillan Co., New York, 1964.  Google Scholar

[9]

D. J. McGillicuddy, Models of harmful algal blooms: Conceptual, empirical, and numerical approaches, Journal of Marine Systems, 83 (2010), 105-105.   Google Scholar

[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). Google Scholar

Figure 1.  Sketch of the parts of Alexandrium catenella life we are interested in
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