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January  2019, 39(1): 41-74. doi: 10.3934/dcds.2019003

Single phytoplankton species growth with light and crowding effect in a water column

Danfeng Pang , Hua Nie , and  Jianhua Wu

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

* Corresponding author: niehua@snnu.edu.cn

Received  August 2017 Revised  May 2018 Published  October 2018

We investigate a nonlocal reaction-diffusion-advection model which describes the growth of a single phytoplankton species in a water column with crowding effect. The longtime dynamical behavior of this model and the asymptotic profiles of its positive steady states for small crowding effect and large advection rate are established. The results show that there is a critical death rate such that the phytoplankton species survives if and only if its death rate is less than the critical death rate. In contrast to the model without crowding effect, our results show that the density of the phytoplankton species will have a finite limit rather than go to infinity when the death rate disappears. Furthermore, for large sinking rate, the phytoplankton species concentrates at the bottom of the water column with a finite population density. For large buoyant rate, the phytoplankton species concentrates at the surface of the water column with a finite population density.

Keywords: Reaction-diffusion-advection model, crowding effect, uniqueness, longtime dynamical behavior, asymptotic profile.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 92D25.
Citation: Danfeng Pang, Hua Nie, Jianhua Wu. Single phytoplankton species growth with light and crowding effect in a water column. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 41-74. doi: 10.3934/dcds.2019003
References:
[1]

K. R. Arrigo, D. H. Robinson and D. L. Worthen, et al, Phytoplankton community structure and the drawdown of nutrients and CO2 in the Southern Ocean, Science, 283 (1999), 365-367. http://science.sciencemag.org/content/283/5400/365. Google Scholar

[2]

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

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R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ, Interscience Publishers, New York, 1953.  Google Scholar

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E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

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E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

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E. N. Dancer and Y. Du, Positive solutions for a three-competition system with diffusion-Ⅰ. General existence results, Nonlinear Anal., 24 (1995), 337-357.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar

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G. R. DiTullio, J. M. Grebmeier, K. R. Arrigo, et al, Rapid and early export of Phaeocystis antarctica blooms in the Ross Sea, Antarctica, Nature, 404 (2000), 595-598. https://www.nature.com/articles/35007061. Google Scholar

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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

[9]

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

[10]

Y. Du, S. 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.  Google Scholar

[11]

U. Ebert, M. Arrayas, N. Temme, B. Sommeojer and J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. https://link.springer.com/article/10.1006/bulm.2001.0261. Google Scholar

[12]

P. G. Falkowski, R. T. Barber and V. Smetacek, Biogeochemical controls and feedbacks on ocean primary production, Science, 281 (1998), 200-206. http://science.sciencemag.org/content/281/5374/200. Google Scholar

[13]

I. Ghoberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators, Vol. I Birkhäuser-Basel, Basel, 1990.http://b-ok.xyz/book/461145/3b6523. doi: 10.1007/978-3-0348-7509-7.  Google Scholar

[14]

S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Math. Anal., 70 (2010), 2942-2974.  doi: 10.1137/100782358.  Google Scholar

[15]

J. Huisman, M. Arrayas, U. Ebert, et al, How do sinking phytoplankton species manage to persist?, Amer. Nat., 159 (2002), 245-254. https://www.journals.uchicago.edu/doi/abs/10.1086/338511. Google Scholar

[16]

J. Huisman, P. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, Amer. Nat., 154 (1999), 46-68.https://www.journals.uchicago.edu/doi/abs/10.1086/303220. Google Scholar

[17]

T. W. Hwang and F. B. Wang, Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 147-161.  doi: 10.3934/dcdsb.2013.18.147.  Google Scholar

[18]

D. H. Jiang, H. Nie and J. H. Wu, Crowding effects on coexistence solutions in the unstirred chemostat, Appl. Anal., 96 (2017), 1016-1046.  doi: 10.1080/00036811.2016.1171319.  Google Scholar

[19]

P. De Leenheer, D. Angeli and E. D. Sontag, A feedback perspective for chemostat models with crowding effects, Positive Systems, 167-174, Lect. Notes Control Inf. Sci., 294, Springer, Berlin, 2003. doi: 10.1007/978-3-540-44928-7_23.  Google Scholar

[20]

P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.  doi: 10.1016/j.jmaa.2006.02.036.  Google Scholar

[21]

H. Lin and F. B. Wang, On a reaction-diffusion system modeling the dengue transmission with nonlocal infections and crowding effects, Appl. Math. Comput., 248 (2014), 184-194.  doi: 10.1016/j.amc.2014.09.101.  Google Scholar

[22]

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.  Google Scholar

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2nd edition, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[24]

G. A. Riley, H. M. Stommel and D. F. Bumpus, Quantitative ecology of the plankton of the western North Atlantic, Bull. Bingham Oceanogr. Coll., 12 (1949), 1-169.   Google Scholar

[25]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[26]

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.  Google Scholar

[27]

M. X. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010. Google Scholar

[28]

X. Zeng, J. Zhang and Y. Gu, Uniqueness and stability of positive steady state solutions for a ratio-dependent predator-prey system with a crowding term in the prey equation, Nonlinear Anal. Real World Appl., 24 (2015), 163-174.  doi: 10.1016/j.nonrwa.2015.02.005.  Google Scholar

show all references

References:
[1]

K. R. Arrigo, D. H. Robinson and D. L. Worthen, et al, Phytoplankton community structure and the drawdown of nutrients and CO2 in the Southern Ocean, Science, 283 (1999), 365-367. http://science.sciencemag.org/content/283/5400/365. Google Scholar

[2]

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

[3]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ, Interscience Publishers, New York, 1953.  Google Scholar

[4]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[5]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

[6]

E. N. Dancer and Y. Du, Positive solutions for a three-competition system with diffusion-Ⅰ. General existence results, Nonlinear Anal., 24 (1995), 337-357.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar

[7]

G. R. DiTullio, J. M. Grebmeier, K. R. Arrigo, et al, Rapid and early export of Phaeocystis antarctica blooms in the Ross Sea, Antarctica, Nature, 404 (2000), 595-598. https://www.nature.com/articles/35007061. Google Scholar

[8]

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

[9]

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

[10]

Y. Du, S. 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.  Google Scholar

[11]

U. Ebert, M. Arrayas, N. Temme, B. Sommeojer and J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (2001), 1095-1124. https://link.springer.com/article/10.1006/bulm.2001.0261. Google Scholar

[12]

P. G. Falkowski, R. T. Barber and V. Smetacek, Biogeochemical controls and feedbacks on ocean primary production, Science, 281 (1998), 200-206. http://science.sciencemag.org/content/281/5374/200. Google Scholar

[13]

I. Ghoberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators, Vol. I Birkhäuser-Basel, Basel, 1990.http://b-ok.xyz/book/461145/3b6523. doi: 10.1007/978-3-0348-7509-7.  Google Scholar

[14]

S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Math. Anal., 70 (2010), 2942-2974.  doi: 10.1137/100782358.  Google Scholar

[15]

J. Huisman, M. Arrayas, U. Ebert, et al, How do sinking phytoplankton species manage to persist?, Amer. Nat., 159 (2002), 245-254. https://www.journals.uchicago.edu/doi/abs/10.1086/338511. Google Scholar

[16]

J. Huisman, P. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, Amer. Nat., 154 (1999), 46-68.https://www.journals.uchicago.edu/doi/abs/10.1086/303220. Google Scholar

[17]

T. W. Hwang and F. B. Wang, Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 147-161.  doi: 10.3934/dcdsb.2013.18.147.  Google Scholar

[18]

D. H. Jiang, H. Nie and J. H. Wu, Crowding effects on coexistence solutions in the unstirred chemostat, Appl. Anal., 96 (2017), 1016-1046.  doi: 10.1080/00036811.2016.1171319.  Google Scholar

[19]

P. De Leenheer, D. Angeli and E. D. Sontag, A feedback perspective for chemostat models with crowding effects, Positive Systems, 167-174, Lect. Notes Control Inf. Sci., 294, Springer, Berlin, 2003. doi: 10.1007/978-3-540-44928-7_23.  Google Scholar

[20]

P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.  doi: 10.1016/j.jmaa.2006.02.036.  Google Scholar

[21]

H. Lin and F. B. Wang, On a reaction-diffusion system modeling the dengue transmission with nonlocal infections and crowding effects, Appl. Math. Comput., 248 (2014), 184-194.  doi: 10.1016/j.amc.2014.09.101.  Google Scholar

[22]

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.  Google Scholar

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2nd edition, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[24]

G. A. Riley, H. M. Stommel and D. F. Bumpus, Quantitative ecology of the plankton of the western North Atlantic, Bull. Bingham Oceanogr. Coll., 12 (1949), 1-169.   Google Scholar

[25]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[26]

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.  Google Scholar

[27]

M. X. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010. Google Scholar

[28]

X. Zeng, J. Zhang and Y. Gu, Uniqueness and stability of positive steady state solutions for a ratio-dependent predator-prey system with a crowding term in the prey equation, Nonlinear Anal. Real World Appl., 24 (2015), 163-174.  doi: 10.1016/j.nonrwa.2015.02.005.  Google Scholar

Figure 1.  The bifurcation diagrams of positive steady states of (1)-(2) versus the death rate with $\beta = 0$ in (a) and $\beta>0$ in (b).
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Figure 2.  Vertical distributions of phytoplankton for large advection rates with crowding effect. Here we take a typical Michaelis-Menten form $g(I) = \frac{mI}{b+I}$ as the specific growth rate of phytoplankton, and choose the basic parameters of the species to be $D = 0.1, d = 0.2, I_0 = 1, k_0 = 1, k_1 = 0.1, m = 1, b = 1$. We further fix the parameter $\beta = 0.01$ in (a) and (b); $\beta = 0.05$ in (c) and (d); $\beta = 0.15$ in (e) and (f). The advection rates $\upsilon = 0.2, 0.5, 1, 5.$ in (a), (c) and (e) for the red, blue, green and black line respectively; the advection rates $\upsilon = -0.1, -0.5, -1, -3$ in (b), (d) and (f) for the red, blue, green and black line respectively. We observe that phytoplankton concentrates at the bottom or surface of water column with a finite population density for large advection rates.
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