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On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect
Competitive exclusion in phytoplankton communities in a eutrophic water column
1. | Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA |
2. | Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA |
We analyze a reaction-diffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the two-species model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of $ N $ similar phytoplankton species, for any number $ N $, generically leads to competitive exclusion. The main tool is the theory of a normalized principal bundle for linear parabolic equations.
References:
[1] |
R. S. Cantrell and K.-Y. Lam, On the evolution of slow dispersal in multi-species communities, 2020, arXiv: 2008.08498 [math.AP] Google Scholar |
[2] |
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), 1305-333.
doi: 10.1137/090775105. |
[3] |
U. Ebert, M. Arrayas, 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. |
[4] |
J. Huisman and F.J. Weissing, Light-limited growth and competition for light in well-mixed acquatic environments: an elementary model, Ecology, 75 (1994), 507-520. Google Scholar |
[5] |
J. Huisman and F. J. Weissing,
Competition for nutrients and light in a mixed water column: A theoretical analysis, Am. Nat., 146 (1995), 536-564.
doi: 10.1086/285814. |
[6] |
J. Huisman, P. van Oostveen and F. J. Weissing,
Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, Am. Nat., 154 (1999), 46-67.
doi: 10.1086/303220. |
[7] |
J. Húska and P. Poláčik,
The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations, 16 (2004), 347-375.
doi: 10.1007/s10884-004-2784-8. |
[8] |
J. Húska,
Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557.
doi: 10.1016/j.jde.2006.02.008. |
[9] |
J. Húska, P. Poláčik and M. V. Safonov,
Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 711-739.
doi: 10.1016/j.anihpc.2006.04.006. |
[10] |
G. E. Hutchinson,
The paradox of the plankton, Am. Nat., 95 (1961), 137-145.
doi: 10.1086/282171. |
[11] |
S.-B. Hsu and Y. Lou,
Single species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.
doi: 10.1137/100782358. |
[12] |
H. Ishii and I. Takagi,
Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. Math. Biol., 16 (1982/83), 1-24.
doi: 10.1007/BF00275157. |
[13] |
D. Jiang, Y. Lou, K.-Y. Lam and Z. Wang,
Monotonicity and global dynamics of a nonlocal two-species phytoplankton model, SIAM J. Appl. Math., 79 (2019), 716-742.
doi: 10.1137/18M1221588. |
[14] |
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. |
[15] |
M. J. Ma and C. H. Ou,
Existence, uniqueness, stability and bifurcation of periodic patterns for a seasonal single phytoplankton model with self-shading effect, J. Differential Equations, 263 (2017), 5630-5655.
doi: 10.1016/j.jde.2017.06.029. |
[16] |
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. |
[17] |
J. Mierczyński,
Globally positive solutions of linear parabolic PDEs of second order with robin boundary conditions, J. Math. Anal. Appl., 209 (1997), 47-59.
doi: 10.1006/jmaa.1997.5323. |
[18] |
R. Peng and X.-Q. Zhao,
A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species, J. Math. Biol., 72 (2016), 755-791.
doi: 10.1007/s00285-015-0904-1. |
[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. |
show all references
References:
[1] |
R. S. Cantrell and K.-Y. Lam, On the evolution of slow dispersal in multi-species communities, 2020, arXiv: 2008.08498 [math.AP] Google Scholar |
[2] |
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), 1305-333.
doi: 10.1137/090775105. |
[3] |
U. Ebert, M. Arrayas, 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. |
[4] |
J. Huisman and F.J. Weissing, Light-limited growth and competition for light in well-mixed acquatic environments: an elementary model, Ecology, 75 (1994), 507-520. Google Scholar |
[5] |
J. Huisman and F. J. Weissing,
Competition for nutrients and light in a mixed water column: A theoretical analysis, Am. Nat., 146 (1995), 536-564.
doi: 10.1086/285814. |
[6] |
J. Huisman, P. van Oostveen and F. J. Weissing,
Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, Am. Nat., 154 (1999), 46-67.
doi: 10.1086/303220. |
[7] |
J. Húska and P. Poláčik,
The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations, 16 (2004), 347-375.
doi: 10.1007/s10884-004-2784-8. |
[8] |
J. Húska,
Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557.
doi: 10.1016/j.jde.2006.02.008. |
[9] |
J. Húska, P. Poláčik and M. V. Safonov,
Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 711-739.
doi: 10.1016/j.anihpc.2006.04.006. |
[10] |
G. E. Hutchinson,
The paradox of the plankton, Am. Nat., 95 (1961), 137-145.
doi: 10.1086/282171. |
[11] |
S.-B. Hsu and Y. Lou,
Single species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.
doi: 10.1137/100782358. |
[12] |
H. Ishii and I. Takagi,
Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. Math. Biol., 16 (1982/83), 1-24.
doi: 10.1007/BF00275157. |
[13] |
D. Jiang, Y. Lou, K.-Y. Lam and Z. Wang,
Monotonicity and global dynamics of a nonlocal two-species phytoplankton model, SIAM J. Appl. Math., 79 (2019), 716-742.
doi: 10.1137/18M1221588. |
[14] |
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. |
[15] |
M. J. Ma and C. H. Ou,
Existence, uniqueness, stability and bifurcation of periodic patterns for a seasonal single phytoplankton model with self-shading effect, J. Differential Equations, 263 (2017), 5630-5655.
doi: 10.1016/j.jde.2017.06.029. |
[16] |
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. |
[17] |
J. Mierczyński,
Globally positive solutions of linear parabolic PDEs of second order with robin boundary conditions, J. Math. Anal. Appl., 209 (1997), 47-59.
doi: 10.1006/jmaa.1997.5323. |
[18] |
R. Peng and X.-Q. Zhao,
A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species, J. Math. Biol., 72 (2016), 755-791.
doi: 10.1007/s00285-015-0904-1. |
[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. |
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