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Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension
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 |
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.
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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