American Institute of Mathematical Sciences

Advanced
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.

[1]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114
[2]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385
[3]

Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105
[4]

Ana Carpio, Gema Duro. Explosive behavior in spatially discrete reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 693-711. doi: 10.3934/dcdsb.2009.12.693
[5]

Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415
[6]

Chiu-Ju Lin. Competition of two phytoplankton species for light with wavelength. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 523-536. doi: 10.3934/dcdsb.2016.21.523
[7]

Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471
[8]

Jean-Jacques Kengwoung-Keumo. Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation. Mathematical Biosciences & Engineering, 2016, 13 (4) : 787-812. doi: 10.3934/mbe.2016018
[9]

Shuichi Jimbo, Yoshihisa Morita. Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4013-4039. doi: 10.3934/dcds.2021026
[10]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65
[11]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[12]

Juraj Kováč, Václav Klika. Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (9) : 2553-2579. doi: 10.3934/dcdss.2022133
[13]

Yihong Du, Mingxin Wang, Meng Zhao. Two species nonlocal diffusion systems with free boundaries. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1127-1162. doi: 10.3934/dcds.2021149
[14]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[15]

Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 823-853. doi: 10.3934/dcdsb.2009.11.823
[16]

Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329
[17]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391
[18]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407
[19]

Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170
[20]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (309)
  • HTML views (117)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy