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April  2021, 26(4): 1783-1795. doi: 10.3934/dcdsb.2020361

Competitive exclusion in phytoplankton communities in a eutrophic water column

Robert Stephen Cantrell 1, and  King-Yeung Lam 2,,

1. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA
2. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author: K.-Y. Lam

Received  August 2020 Revised  October 2020 Published  April 2021 Early access  December 2020

Fund Project: RSC is partially supported by NSF grant DMS-1853478; KYL is partially supported by NSF grant DMS-1853561

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.

Keywords: phytoplankton, competition, nonlocal reaction-diffusion systems, asymptotic behavior, multiple species.
Mathematics Subject Classification: Primary: 35B51, 35K57, 47H07; Secondary: 92D25.
Citation: Robert Stephen Cantrell, King-Yeung Lam. Competitive exclusion in phytoplankton communities in a eutrophic water column. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1783-1795. doi: 10.3934/dcdsb.2020361
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]

[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. 

[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]

[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. 

[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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