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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

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.

Citation: Danfeng Pang, Hua Nie, Jianhua Wu. Single phytoplankton species growth with light and crowding effect in a water column. Discrete and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

D. H. JiangH. 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.

[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.

[20]

P. De LeenheerD. 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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[28]

X. ZengJ. 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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

D. H. JiangH. 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.

[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.

[20]

P. De LeenheerD. 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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[28]

X. ZengJ. 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.

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).
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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