doi: 10.3934/dcdsb.2022104
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Bifurcation analysis of solutions to a nonlocal phytoplankton model under photoinhibition

School of Mathematics, Hunan University, Changsha 410082, China

Received  October 2021 Revised  April 2022 Early access June 2022

Fund Project: Yuan-Yuan Zhou is supported by NSFC No. 12171143

We investigate the effect of photoinhibition in a nonlocal reaction-diffusion-advection equation, which models the dynamics of a single phytoplankton species in a water column where the growth of the species depends solely on light. First, for $ k_0 = 0 $, we proved that system (1)-(3) forms a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution function. Second, local and global bifurcation theory are used to show that the model with photoinhibition possesses multiple steady-states with the change of parameter ranges.

Citation: Yuan-Yuan Zhou. Bifurcation analysis of solutions to a nonlocal phytoplankton model under photoinhibition. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022104
References:
[1]

R. S. Cantrell and K.-Y. Lam, On the evolution of slow dispersal in multi-species communities, SIAM J. Math. Anal., 53 (2021), 4933-4964.  doi: 10.1137/20M1361419.

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Ration. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.

[4]

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.

[5]

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.

[6]

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.

[7]

Y. DuS.-B. Hsu and Y. Lou, Multiple steady-states in phytoplankton population induced by photoinhibition, J. Differential Equations, 258 (2015), 2408-2434.  doi: 10.1016/j.jde.2014.12.012.

[8]

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.

[9]

U. EbertM. ArrayásN. TemmeB. Sommeijer and J. Huisman, Critical conditions for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. 

[10]

D. J. GerlaW. M. Wolf and J. Huisman, Photoinhibition and the assembly of light-limited phytoplankton communities, Oikos, 120 (2011), 359-368. 

[11]

E. W. Helbling, et al., UVR-induced photosynthetic inhibition dominates over DNA damage in marine dinoflagellates exposed to fluctuating solar radiation regimes, J. Exp. Mar. Biol. Ecol., 365 (2008), 96–102.

[12]

W. J. Henley, Measurement and interpretation of photosynthetic light-response curves in algae in the context of photoinhibition and diel changes, J. Phycol., 29 (1993), 729-739. 

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.

[14]

S.-B. HsuC.-J. LinC.-H. Hsieh and K. Yoshiyama, Dynamics of phytoplankton communities under photoinhibition, Bull. Math. Biol., 75 (2013), 1207-1232.  doi: 10.1007/s11538-013-9852-3.

[15]

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.

[16]

J. HuismanP. Van Oostveen and F. J. Wessing, Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms, Limnol. Oceanogr., 44 (1999), 1781-1787. 

[17]

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

[18]

J. HuismanP. Van Oostveen and F. J. Wessing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, Amer. Natur., 154 (1999), 46-67. 

[19]

H. Ishii and I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. Math. Biol., 16 (1982), 1-24.  doi: 10.1007/BF00275157.

[20]

H. Ishii and I. Takagi, A nonlinear diffusion equation in phytoplankton dynamics with self-shading effect, in: Mathematics in Biology and Medicine, Bari, 1983, in: Lecture Notes in Biomath., vol.57, Springer, Berlin, 1985, 66–71. doi: 10.1007/978-3-642-93287-8_9.

[21]

D. JiangK.-Y. LamY. Lou and Z.-C. Wang, Monotonicity and global dynamics of a nonlocal two-species phytoplankton model, SIAM J. Appl. Math., 79 (2019), 716-742.  doi: 10.1137/18M1221588.

[22] J. T. Kirk, Light and Photosynthesis in Aquatic Ecosystems, Cambridge University Press, Cambridge, UK, 1994. 
[23]

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

[24]

C. A. KlausmeierE. Litchman and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. 

[25]

T. KolokolnikovC. 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.

[26]

E. LitchmanC. A. KlausmeierJ. R. MillerO. M. Schofield and P. G. Falkowski, Multinutrient, multi-group model of present and future oceanic phytoplankton communities, Biogeosciences, 3 (2006), 585-606. 

[27]

M. Ma and C. 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.

[28]

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.

[29]

M. Morse, et al., Photosynthetic and growth response of freshwater picocyanobacteria are strain-specific and sensitive to photoacclimation, J. Plankton Res., 31 (2009), 349–357.

[30]

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.

[31]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[32]

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.

[33]

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

show all references

References:
[1]

R. S. Cantrell and K.-Y. Lam, On the evolution of slow dispersal in multi-species communities, SIAM J. Math. Anal., 53 (2021), 4933-4964.  doi: 10.1137/20M1361419.

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Ration. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.

[4]

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.

[5]

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.

[6]

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.

[7]

Y. DuS.-B. Hsu and Y. Lou, Multiple steady-states in phytoplankton population induced by photoinhibition, J. Differential Equations, 258 (2015), 2408-2434.  doi: 10.1016/j.jde.2014.12.012.

[8]

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.

[9]

U. EbertM. ArrayásN. TemmeB. Sommeijer and J. Huisman, Critical conditions for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. 

[10]

D. J. GerlaW. M. Wolf and J. Huisman, Photoinhibition and the assembly of light-limited phytoplankton communities, Oikos, 120 (2011), 359-368. 

[11]

E. W. Helbling, et al., UVR-induced photosynthetic inhibition dominates over DNA damage in marine dinoflagellates exposed to fluctuating solar radiation regimes, J. Exp. Mar. Biol. Ecol., 365 (2008), 96–102.

[12]

W. J. Henley, Measurement and interpretation of photosynthetic light-response curves in algae in the context of photoinhibition and diel changes, J. Phycol., 29 (1993), 729-739. 

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.

[14]

S.-B. HsuC.-J. LinC.-H. Hsieh and K. Yoshiyama, Dynamics of phytoplankton communities under photoinhibition, Bull. Math. Biol., 75 (2013), 1207-1232.  doi: 10.1007/s11538-013-9852-3.

[15]

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.

[16]

J. HuismanP. Van Oostveen and F. J. Wessing, Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms, Limnol. Oceanogr., 44 (1999), 1781-1787. 

[17]

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

[18]

J. HuismanP. Van Oostveen and F. J. Wessing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light, Amer. Natur., 154 (1999), 46-67. 

[19]

H. Ishii and I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics, J. Math. Biol., 16 (1982), 1-24.  doi: 10.1007/BF00275157.

[20]

H. Ishii and I. Takagi, A nonlinear diffusion equation in phytoplankton dynamics with self-shading effect, in: Mathematics in Biology and Medicine, Bari, 1983, in: Lecture Notes in Biomath., vol.57, Springer, Berlin, 1985, 66–71. doi: 10.1007/978-3-642-93287-8_9.

[21]

D. JiangK.-Y. LamY. Lou and Z.-C. Wang, Monotonicity and global dynamics of a nonlocal two-species phytoplankton model, SIAM J. Appl. Math., 79 (2019), 716-742.  doi: 10.1137/18M1221588.

[22] J. T. Kirk, Light and Photosynthesis in Aquatic Ecosystems, Cambridge University Press, Cambridge, UK, 1994. 
[23]

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

[24]

C. A. KlausmeierE. Litchman and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. 

[25]

T. KolokolnikovC. 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.

[26]

E. LitchmanC. A. KlausmeierJ. R. MillerO. M. Schofield and P. G. Falkowski, Multinutrient, multi-group model of present and future oceanic phytoplankton communities, Biogeosciences, 3 (2006), 585-606. 

[27]

M. Ma and C. 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.

[28]

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.

[29]

M. Morse, et al., Photosynthetic and growth response of freshwater picocyanobacteria are strain-specific and sensitive to photoacclimation, J. Plankton Res., 31 (2009), 349–357.

[30]

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.

[31]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[32]

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.

[33]

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

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