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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

 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.
Citation: Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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