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Shadow system approach to a plankton model generating harmful algal bloom
A dynamical approach to phytoplankton blooms
1. | Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA |
2. | Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA |
Algae in the ocean absorb carbon dioxide from the atmosphere and thus play an important role in the carbon cycle. An algal bloom occurs when there is a rapid increase in an algae population. We consider a reaction-advection-diffusion model for algal bloom density and present new proofs of existence and uniqueness results for the steady state solutions using techniques from dynamical systems. On the question of stability of the bloom profiles, we show that the only possible bifurcation would be due to an oscillatory instability.
References:
[1] |
M. J. Behrenfeld,
Climate-driven trends in contemporary ocean productivity, Nature, 444 (2006), 752-755.
doi: 10.1038/nature05317. |
[2] |
D. Bleecker and G. Csordas, Basic Partial Differential Equations, International Press, Cambridge, MA, 1966.
![]() ![]() |
[3] |
J. E. Cloern,
Tidal stirring and phytoplankton bloom dynamics in an estuary, J. Mar. Res., 49 (1991), 203-221.
doi: 10.1357/002224091784968611. |
[4] |
Y. Du and L. Mei,
On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.
doi: 10.1088/0951-7715/24/1/016. |
[5] |
Y. Du and S. Hsu,
On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333.
doi: 10.1137/090775105. |
[6] |
U. Ebert, M. Arrayás, N. Temme and B. Sommeijer,
Critical conditions for phytoplankton blooms, Bulletin of Mathematical Biology, 63 (2001), 1095-1124.
doi: 10.1006/bulm.2001.0261. |
[7] |
J. Huisman,
Population dynamics of light-limited phytoplankton: Microcosm experiments, Ecology, 80 (1999), 202-210.
|
[8] |
J. Huisman, M. Arrayás, U. Ebert and B. Sommeijer,
How do sinking phytoplankton species manage to persist?, The American Naturalist, 159 (2002), 245-254.
doi: 10.1086/338511. |
[9] |
J. Huisman, P. van Oostveen and F. J. Weissing,
Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms, Limnology and Oceanography, 44 (1999), 1781-1787.
doi: 10.4319/lo.1999.44.7.1781. |
[10] |
S. Hsu and Y. Lou,
Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.
doi: 10.1137/100782358. |
[11] |
H. Ishii and I. Takagi,
Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, Journal of Mathematical Biology, 16 (1982), 1-24.
doi: 10.1007/BF00275157. |
[12] |
H. Kaper and H. Engler,
Mathematics and Climate, Society for Industrial & Applied Mathematics, US, 2013. |
[13] |
C. A. Klausmeier and E. Litchman,
Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnology and Oceanography, 46 (2001), 1998-2007.
doi: 10.4319/lo.2001.46.8.1998. |
[14] |
T. Kolonikov, C. H. Ou and Y. Yuan,
Phytoplankton depth profiles and their transitions near the critical sinking velocity, J. Math. Biol., 59 (2009), 105-122.
doi: 10.1007/s00285-008-0221-z. |
[15] |
L. Mei and X. Zhang,
Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics, J. Differential Equations, 253 (2012), 2025-2063.
doi: 10.1016/j.jde.2012.06.011. |
[16] |
T. Platt, C. L. Gallegos and W. G. Harrison,
Photoinhibition of photosynthesis in natural assemblages of marine phytoplankton, J. Mar. Res, 38 (2011), 687-701.
|
[17] |
G. A. Riley, H. Stommel and D. F. Bumpus,
Quantitative ecology of the plankton of the
western North Atlantic, Bull. Bingham Oceanogr. Collection, 12 (1949), article 3.
|
[18] |
N. Shigesada and A. Okubo,
Analysis of the self-shading effect on algal vertical distribution in natural waters, J. Math. Biol., 12 (1981), 311-326.
doi: 10.1007/BF00276919. |
[19] |
W. L. Webb, M. Newton and D. Starr,
Carbon dioxide exchange of Alnus rubra: A mathematical model, Oecologia, 17 (1974), 281-291.
doi: 10.1007/BF00345747. |
[20] |
A. Zagaris, A. Doelman, N. N. Pham Thi and B. P. Sommeijer,
Blooming in a nonlocal, coupled phytoplankton-nutrient model, SIAM J. Appl. Math., 69 (2009), 1174-1204.
doi: 10.1137/070693692. |
show all references
References:
[1] |
M. J. Behrenfeld,
Climate-driven trends in contemporary ocean productivity, Nature, 444 (2006), 752-755.
doi: 10.1038/nature05317. |
[2] |
D. Bleecker and G. Csordas, Basic Partial Differential Equations, International Press, Cambridge, MA, 1966.
![]() ![]() |
[3] |
J. E. Cloern,
Tidal stirring and phytoplankton bloom dynamics in an estuary, J. Mar. Res., 49 (1991), 203-221.
doi: 10.1357/002224091784968611. |
[4] |
Y. Du and L. Mei,
On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.
doi: 10.1088/0951-7715/24/1/016. |
[5] |
Y. Du and S. Hsu,
On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333.
doi: 10.1137/090775105. |
[6] |
U. Ebert, M. Arrayás, N. Temme and B. Sommeijer,
Critical conditions for phytoplankton blooms, Bulletin of Mathematical Biology, 63 (2001), 1095-1124.
doi: 10.1006/bulm.2001.0261. |
[7] |
J. Huisman,
Population dynamics of light-limited phytoplankton: Microcosm experiments, Ecology, 80 (1999), 202-210.
|
[8] |
J. Huisman, M. Arrayás, U. Ebert and B. Sommeijer,
How do sinking phytoplankton species manage to persist?, The American Naturalist, 159 (2002), 245-254.
doi: 10.1086/338511. |
[9] |
J. Huisman, P. van Oostveen and F. J. Weissing,
Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms, Limnology and Oceanography, 44 (1999), 1781-1787.
doi: 10.4319/lo.1999.44.7.1781. |
[10] |
S. Hsu and Y. Lou,
Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.
doi: 10.1137/100782358. |
[11] |
H. Ishii and I. Takagi,
Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, Journal of Mathematical Biology, 16 (1982), 1-24.
doi: 10.1007/BF00275157. |
[12] |
H. Kaper and H. Engler,
Mathematics and Climate, Society for Industrial & Applied Mathematics, US, 2013. |
[13] |
C. A. Klausmeier and E. Litchman,
Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnology and Oceanography, 46 (2001), 1998-2007.
doi: 10.4319/lo.2001.46.8.1998. |
[14] |
T. Kolonikov, C. H. Ou and Y. Yuan,
Phytoplankton depth profiles and their transitions near the critical sinking velocity, J. Math. Biol., 59 (2009), 105-122.
doi: 10.1007/s00285-008-0221-z. |
[15] |
L. Mei and X. Zhang,
Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics, J. Differential Equations, 253 (2012), 2025-2063.
doi: 10.1016/j.jde.2012.06.011. |
[16] |
T. Platt, C. L. Gallegos and W. G. Harrison,
Photoinhibition of photosynthesis in natural assemblages of marine phytoplankton, J. Mar. Res, 38 (2011), 687-701.
|
[17] |
G. A. Riley, H. Stommel and D. F. Bumpus,
Quantitative ecology of the plankton of the
western North Atlantic, Bull. Bingham Oceanogr. Collection, 12 (1949), article 3.
|
[18] |
N. Shigesada and A. Okubo,
Analysis of the self-shading effect on algal vertical distribution in natural waters, J. Math. Biol., 12 (1981), 311-326.
doi: 10.1007/BF00276919. |
[19] |
W. L. Webb, M. Newton and D. Starr,
Carbon dioxide exchange of Alnus rubra: A mathematical model, Oecologia, 17 (1974), 281-291.
doi: 10.1007/BF00345747. |
[20] |
A. Zagaris, A. Doelman, N. N. Pham Thi and B. P. Sommeijer,
Blooming in a nonlocal, coupled phytoplankton-nutrient model, SIAM J. Appl. Math., 69 (2009), 1174-1204.
doi: 10.1137/070693692. |





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