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January  2012, 17(1): 221-243. doi: 10.3934/dcdsb.2012.17.221

On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients

Linfeng Mei 1, and  Xiaoyan Zhang 2,

1. 

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
2. 

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

Received  October 2010 Revised  February 2011 Published  October 2011

We investigate the steady state solutions of a phytoplankton-nutrient system proposed by Huisman et al. in [14] that models the dynamics of a single phytoplankton species whose growth is limited by light and nutrients in a vertical water column. We first study the existence and nonexistence problem of the model and prove there is at least one positive solution to the system when the parameters involved are in a suitable range. We then analyze the limiting profiles of the positive solutions as the specific phytoplankton loss rate approaches zero and as the diffusion coefficient of the system tends to zero, respectively. In the small diffusion case, we show that the phytoplankton species all die out regardless of how large the nutrient supply is, and the nutrients distribution approaches a linear function determined by the parameters of the system. This phenomenon is in sharp contrast to that of the model studied by Du and Hsu [3, 4], where the phytoplankton species can concentrate at the bottom, at the surface or at a specific depth of the water column decided by the amplitude of the nutrient supply. We also study the asymptotic profile of the positive solution when the diffusion coefficient is very large. Our results reveal that in such a case the phytoplankton and the nutrients distribute evenly in the water column.
    Our concentration results also reveal that passive diffusion and active movement (sinking or floating) should be in proportion to the oscillation phenomena showed in [14, 24] to occur.
Keywords: Nutrient, Phytoplankton dynamics, Reaction-diffusion-advection system..
Mathematics Subject Classification: Primary: 35J55, 35J65; Secondary: 92D2.
Citation: Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-729. doi: 10.1137/1018114.  Google Scholar

[2]

F. Belgacem, "Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications," Pitman Res, Notes Math. Ser., 368, Longman Sci., 1997. Google Scholar

[3]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton I: Existence, SIAM J. Math. Anal., 40 (2008), 1419-1440. doi: 10.1137/07070663X.  Google Scholar

[4]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton II: Limiting profile, SIAM J. Math. Anal., 40 (2008), 1441-1470. doi: 10.1137/070706641.  Google Scholar

[5]

Y. Du and S. B. 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.  Google Scholar

[6]

Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[7]

U. Ebert, M. Arrays, N. Temme, B. Sommeijer and J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. doi: 10.1006/bulm.2001.0261.  Google Scholar

[8]

K. Fennel and E. Boss, Surface maxima of phytoplankton and chlorophyll: Steady-state solutions from a simple model, Limnol. Oceanogr., 48 (2003), 1521-1534. doi: 10.4319/lo.2003.48.4.1521.  Google Scholar

[9]

S. Ghosal and S. Mandre, A simple model illustrating the role of turbulence on phytoplankton blooms, J. Math. Biol., 46 (2003), 333-346. doi: 10.1007/s00285-002-0184-4.  Google Scholar

[10]

J. Huisman, M. Arrayas, N. Temme and B. Sommeijer, How do sinking phytoplankton species manage to persist?, American Naturalist, 159 (2002), 245-254. doi: 10.1086/338511.  Google Scholar

[11]

S. B. 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.  Google Scholar

[12]

J. Huisman, P. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, American Naturalist, 154 (1999), 46-67. doi: 10.1086/303220.  Google Scholar

[13]

J. Huisman, P. van Oostveen and F. J. Weissing, Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms, Limnol. Oceanogr., 44 (1999), 1781-1787. doi: 10.4319/lo.1999.44.7.1781.  Google Scholar

[14]

J. Huisman, N. N. Pham Thi, D. K. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic deep chlorophyll, Nature, 439 (2006), 322-325. doi: 10.1038/nature04245.  Google Scholar

[15]

J. Huisman and B. Sommeijer, Population dynamics of sinking phytoplankton in light limited environments: Simulation techniques and critical parameters, J. Sea Res., 48 (2002), 83-96. doi: 10.1016/S1385-1101(02)00137-5.  Google Scholar

[16]

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: A theoretical analysis, American Naturalist, 146 (1995), 536-564. doi: 10.1086/285814.  Google Scholar

[17]

H. Ishii and I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. Math. Biology, 16 (1982), 1-24. doi: 10.1007/BF00275157.  Google Scholar

[18]

C. A. Clausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. doi: 10.4319/lo.2001.46.8.1998.  Google Scholar

[19]

C. A. Clausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. doi: 10.4319/lo.2004.49.4_part_2.1463.  Google Scholar

[20]

T. Kolokolnikov, 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.  Google Scholar

[21]

E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield and P. G. Falkowski, Multinutrient, multi-group model of present and future oceanic phytoplankton communities, Biogeosciences, 3 (2006), 585-606. doi: 10.5194/bg-3-585-2006.  Google Scholar

[22]

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.  Google Scholar

[23]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, American Naturalist, 174 (2009), 190-203. doi: 10.1086/600113.  Google Scholar

[24]

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.  Google Scholar

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-729. doi: 10.1137/1018114.  Google Scholar

[2]

F. Belgacem, "Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications," Pitman Res, Notes Math. Ser., 368, Longman Sci., 1997. Google Scholar

[3]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton I: Existence, SIAM J. Math. Anal., 40 (2008), 1419-1440. doi: 10.1137/07070663X.  Google Scholar

[4]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton II: Limiting profile, SIAM J. Math. Anal., 40 (2008), 1441-1470. doi: 10.1137/070706641.  Google Scholar

[5]

Y. Du and S. B. 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.  Google Scholar

[6]

Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[7]

U. Ebert, M. Arrays, N. Temme, B. Sommeijer and J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. doi: 10.1006/bulm.2001.0261.  Google Scholar

[8]

K. Fennel and E. Boss, Surface maxima of phytoplankton and chlorophyll: Steady-state solutions from a simple model, Limnol. Oceanogr., 48 (2003), 1521-1534. doi: 10.4319/lo.2003.48.4.1521.  Google Scholar

[9]

S. Ghosal and S. Mandre, A simple model illustrating the role of turbulence on phytoplankton blooms, J. Math. Biol., 46 (2003), 333-346. doi: 10.1007/s00285-002-0184-4.  Google Scholar

[10]

J. Huisman, M. Arrayas, N. Temme and B. Sommeijer, How do sinking phytoplankton species manage to persist?, American Naturalist, 159 (2002), 245-254. doi: 10.1086/338511.  Google Scholar

[11]

S. B. 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.  Google Scholar

[12]

J. Huisman, P. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, American Naturalist, 154 (1999), 46-67. doi: 10.1086/303220.  Google Scholar

[13]

J. Huisman, P. van Oostveen and F. J. Weissing, Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms, Limnol. Oceanogr., 44 (1999), 1781-1787. doi: 10.4319/lo.1999.44.7.1781.  Google Scholar

[14]

J. Huisman, N. N. Pham Thi, D. K. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic deep chlorophyll, Nature, 439 (2006), 322-325. doi: 10.1038/nature04245.  Google Scholar

[15]

J. Huisman and B. Sommeijer, Population dynamics of sinking phytoplankton in light limited environments: Simulation techniques and critical parameters, J. Sea Res., 48 (2002), 83-96. doi: 10.1016/S1385-1101(02)00137-5.  Google Scholar

[16]

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: A theoretical analysis, American Naturalist, 146 (1995), 536-564. doi: 10.1086/285814.  Google Scholar

[17]

H. Ishii and I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. Math. Biology, 16 (1982), 1-24. doi: 10.1007/BF00275157.  Google Scholar

[18]

C. A. Clausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. doi: 10.4319/lo.2001.46.8.1998.  Google Scholar

[19]

C. A. Clausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. doi: 10.4319/lo.2004.49.4_part_2.1463.  Google Scholar

[20]

T. Kolokolnikov, 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.  Google Scholar

[21]

E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield and P. G. Falkowski, Multinutrient, multi-group model of present and future oceanic phytoplankton communities, Biogeosciences, 3 (2006), 585-606. doi: 10.5194/bg-3-585-2006.  Google Scholar

[22]

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.  Google Scholar

[23]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, American Naturalist, 174 (2009), 190-203. doi: 10.1086/600113.  Google Scholar

[24]

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.  Google Scholar

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