# American Institute of Mathematical Sciences

March  2015, 20(2): 587-597. doi: 10.3934/dcdsb.2015.20.587

## Concentration phenomenon in a nonlocal equation modeling phytoplankton growth

 1 Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007, China 2 Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056021, China, China

Received  June 2013 Revised  November 2014 Published  January 2015

We study a nonlocal reaction-diffusion-advection equation arising from the study of a single phytoplankton species competing for light in a poorly mixed water column. When the diffusion coefficient is very small, the phytoplankton population concentrates around certain zeros of the advection function. The corresponding phytoplankton distribution approaches a $\delta$-like function centered at those zeros.
Citation: Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587
##### References:
 [1] 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. [2] X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204. [3] Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling 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 modelling 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 modelling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333. doi: 10.1137/090775105. [6] Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modeling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016. [7] W. M. Durham, J. O. Kessler and R. Stoker, Disruptive of vertical motility by shear triggers formation of phytoplankton layers, Science, 323 (2009), 1067-1070. doi: 10.1126/science.1167334. [8] J. Huisman, M. Arrayas, U. Ebert and B.Sommeijer, How do sinking phytoplankton species manage to persist? Amer. Naturalist, 159 (2002), 245-254. doi: 10.1086/338511. [9] U. Ebert, M. Arrays, N. Temme, B. Sommeijer and J. Huisman, Critical conditions for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. doi: 10.1006/bulm.2001.0261. [10] J. Huisman, P. Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, Amer. Naturalist, 154 (1999), 46-68. doi: 10.1086/303220. [11] 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. [12] J. Huisman and F. J. Wessing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model, Ecology, 75 (1994), 507-520. doi: 10.2307/1939554. [13] J. Huisman and F. J. Wessing, Competition for nutrients and light in a mixed water column: A theoretical analysis, Amer. Naturalist, 146 (1995), 536-564. doi: 10.1086/285814. [14] 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. [15] 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. [16] C. A. Klausmeier and E. Litchman, Algae games: The vertical distribution of pytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. [17] C. A. Klausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichimetry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. doi: 10.4319/lo.2004.49.4_part_2.1463. [18] E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield and P. G. Falkowski, Multi-nutrient, multi-group model of present and future oceanic phytoplankton communities, Biogeosci., 3 (2006), 585-606. doi: 10.5194/bg-3-585-2006. [19] 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. [20] K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Amer. Naturalist, 174 (2009), 190-203. doi: 10.1086/600113. [21] F. J. Weissing and J. Huisman, Growth and competition in a lighted gradient, J. Theoret. Biol., 168 (1994), 323-336. doi: 10.1006/jtbi.1994.1113.

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##### References:
 [1] 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. [2] X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204. [3] Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling 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 modelling 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 modelling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333. doi: 10.1137/090775105. [6] Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modeling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016. [7] W. M. Durham, J. O. Kessler and R. Stoker, Disruptive of vertical motility by shear triggers formation of phytoplankton layers, Science, 323 (2009), 1067-1070. doi: 10.1126/science.1167334. [8] J. Huisman, M. Arrayas, U. Ebert and B.Sommeijer, How do sinking phytoplankton species manage to persist? Amer. Naturalist, 159 (2002), 245-254. doi: 10.1086/338511. [9] U. Ebert, M. Arrays, N. Temme, B. Sommeijer and J. Huisman, Critical conditions for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. doi: 10.1006/bulm.2001.0261. [10] J. Huisman, P. Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, Amer. Naturalist, 154 (1999), 46-68. doi: 10.1086/303220. [11] 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. [12] J. Huisman and F. J. Wessing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model, Ecology, 75 (1994), 507-520. doi: 10.2307/1939554. [13] J. Huisman and F. J. Wessing, Competition for nutrients and light in a mixed water column: A theoretical analysis, Amer. Naturalist, 146 (1995), 536-564. doi: 10.1086/285814. [14] 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. [15] 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. [16] C. A. Klausmeier and E. Litchman, Algae games: The vertical distribution of pytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. [17] C. A. Klausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichimetry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. doi: 10.4319/lo.2004.49.4_part_2.1463. [18] E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield and P. G. Falkowski, Multi-nutrient, multi-group model of present and future oceanic phytoplankton communities, Biogeosci., 3 (2006), 585-606. doi: 10.5194/bg-3-585-2006. [19] 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. [20] K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Amer. Naturalist, 174 (2009), 190-203. doi: 10.1086/600113. [21] F. J. Weissing and J. Huisman, Growth and competition in a lighted gradient, J. Theoret. Biol., 168 (1994), 323-336. doi: 10.1006/jtbi.1994.1113.
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