April  2021, 26(4): 2115-2132. doi: 10.3934/dcdsb.2020359

Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column

1. 

School of Mathematics and Computer Sciences, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

2. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan

3. 

Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

* Corresponding author: Feng-Bin Wang

(This paper is dedicated to the seventieth birthday of Professor Sze-Bi Hsu)

Received  August 2020 Revised  September 2020 Published  December 2020

This paper analytically investigates a nonlocal system of reaction-diffusion-advection equations modeling the competition of two phytoplankton species for a limiting nutrient and light in a water column, where dead phytoplankton species can get recycled back into the system as a resource for growth. The threshold dynamics of the single population model is first established. Then the utilization of abstract persistence theory enables us to show that two species population system admits a coexistence steady state and the system is uniformly persistent if the trivial steady state and two global attractors on the boundary are all weak repellers.

Citation: Linfeng Mei, Feng-Bin Wang. Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2115-2132. doi: 10.3934/dcdsb.2020359
References:
[1]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasilinear problem modelling phytoplankton Ⅰ: Existence, SIAM J. Math. Anal., 40 (2008), 1419-1440.  doi: 10.1137/07070663X.  Google Scholar

[2]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasilinear problem modelling phytoplankton Ⅱ: Limiting profile, SIAM J. Math. Anal., 40 (2008), 1441-1470.  doi: 10.1137/070706641.  Google Scholar

[3]

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.  Google Scholar

[4]

Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.  doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[5]

U. EbertM. ArrayasN. TemmeB. Sommeojer and J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124.   Google Scholar

[6]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[7]

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.  Google Scholar

[8]

J. HuismanP. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, American Naturalist, 154 (1999), 46-67.   Google Scholar

[9]

J. HuismanN. N. Pham ThiD. K. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic deep chlorophyll, Nature, 439 (2006), 322-325.   Google Scholar

[10]

S. B. HsuF. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dyn. Differ. Equ., 23 (2011), 817-842.  doi: 10.1007/s10884-011-9224-3.  Google Scholar

[11]

S. B. HsuF. B. Wang and X.-Q. Zhao, A reaction-diffusion model of harmful algae and zooplankton in an ecosystem, J. Math. Anal. Appl., 451 (2017), 659-677.  doi: 10.1016/j.jmaa.2017.02.034.  Google Scholar

[12] J. T. O. Kirk, Light and photosynthesis in aquatic ecosystems, 2d ed., Cambridge University Press, Cambridge, 1994.   Google Scholar
[13]

C. A. Klausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007.   Google Scholar

[14]

C. A. KlausmeierE. Litchman and S. A. Levin, Phytoplankton growth and stoichimetry under multiple nutrient limitation, Limnol Oceangr, 49 (2004), 1463-1470.   Google Scholar

[15]

V. Liebig, J. Die Organische Chemie in Ihrer Anwendung auf Agrikultur und Physiologie, Friedrich Vieweg, Braunschweig, 1840. Google Scholar

[16]

E. LitchmanC. A. KlausmeierJ. R. MillerO. M. Schofield and P. G. Falkowski, Multi-nutrient, multi-group model of present and furture oceanic phytoplankton communities, Biogeoscience, 3 (2006), 585-606.   Google Scholar

[17]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[18]

H. NieS.-B. Hsu and J. Wu, Coexistence solutions of a competitive model with two species in a water column, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2691-2714.  doi: 10.3934/dcdsb.2015.20.2691.  Google Scholar

[19]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[20]

A. B. Ryabov and B. Blasius, A graphical theory of competition on spatial resource gradients, Ecology Letters, 14 (2011), 220-228.   Google Scholar

[21]

A. B. Ryabov and B. Blasius, Depth of the Biomass Maximum Affects the Rules of Resource Competition in a Water Column, Nonlinear Anal., 184 (2014), E132–E146. Google Scholar

[22]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.  Google Scholar

[23]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[24]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[25]

K. YoshiyamaJ. P. MellardE. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, American Naturalist, 174 (2009), 190-203.   Google Scholar

show all references

References:
[1]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasilinear problem modelling phytoplankton Ⅰ: Existence, SIAM J. Math. Anal., 40 (2008), 1419-1440.  doi: 10.1137/07070663X.  Google Scholar

[2]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasilinear problem modelling phytoplankton Ⅱ: Limiting profile, SIAM J. Math. Anal., 40 (2008), 1441-1470.  doi: 10.1137/070706641.  Google Scholar

[3]

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.  Google Scholar

[4]

Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.  doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[5]

U. EbertM. ArrayasN. TemmeB. Sommeojer and J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124.   Google Scholar

[6]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[7]

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.  Google Scholar

[8]

J. HuismanP. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, American Naturalist, 154 (1999), 46-67.   Google Scholar

[9]

J. HuismanN. N. Pham ThiD. K. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic deep chlorophyll, Nature, 439 (2006), 322-325.   Google Scholar

[10]

S. B. HsuF. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dyn. Differ. Equ., 23 (2011), 817-842.  doi: 10.1007/s10884-011-9224-3.  Google Scholar

[11]

S. B. HsuF. B. Wang and X.-Q. Zhao, A reaction-diffusion model of harmful algae and zooplankton in an ecosystem, J. Math. Anal. Appl., 451 (2017), 659-677.  doi: 10.1016/j.jmaa.2017.02.034.  Google Scholar

[12] J. T. O. Kirk, Light and photosynthesis in aquatic ecosystems, 2d ed., Cambridge University Press, Cambridge, 1994.   Google Scholar
[13]

C. A. Klausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007.   Google Scholar

[14]

C. A. KlausmeierE. Litchman and S. A. Levin, Phytoplankton growth and stoichimetry under multiple nutrient limitation, Limnol Oceangr, 49 (2004), 1463-1470.   Google Scholar

[15]

V. Liebig, J. Die Organische Chemie in Ihrer Anwendung auf Agrikultur und Physiologie, Friedrich Vieweg, Braunschweig, 1840. Google Scholar

[16]

E. LitchmanC. A. KlausmeierJ. R. MillerO. M. Schofield and P. G. Falkowski, Multi-nutrient, multi-group model of present and furture oceanic phytoplankton communities, Biogeoscience, 3 (2006), 585-606.   Google Scholar

[17]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[18]

H. NieS.-B. Hsu and J. Wu, Coexistence solutions of a competitive model with two species in a water column, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2691-2714.  doi: 10.3934/dcdsb.2015.20.2691.  Google Scholar

[19]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[20]

A. B. Ryabov and B. Blasius, A graphical theory of competition on spatial resource gradients, Ecology Letters, 14 (2011), 220-228.   Google Scholar

[21]

A. B. Ryabov and B. Blasius, Depth of the Biomass Maximum Affects the Rules of Resource Competition in a Water Column, Nonlinear Anal., 184 (2014), E132–E146. Google Scholar

[22]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.  Google Scholar

[23]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[24]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[25]

K. YoshiyamaJ. P. MellardE. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, American Naturalist, 174 (2009), 190-203.   Google Scholar

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