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March  2016, 21(2): 607-620. doi: 10.3934/dcdsb.2016.21.607

Growth of single phytoplankton species with internal storage in a water column

Linfeng Mei 1, , Sze-Bi Hsu 2, and  Feng-Bin Wang 3,

1. 

Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007
2. 

Department of Mathematics, National Tsing Hua University, National Center of Theoretical Science, Hsinchu 300
3. 

Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333

Received  December 2014 Revised  August 2015 Published  November 2015

In this paper, we analyze a system modeling the growth of single phytoplankton populations in a water column, where population growth increases monotonically with the nutrient quota stored within individuals. We establish a threshold result on the global extinction and persistence of phytoplankton. Condition for persistence is shown to depend on the principal eigenvalue of a boundary value problem, which is related to the physical transport properties of the water column (i.e. the diffusivity and the sinking speed), nutrient uptake rate, and growth rate.
Keywords: threshold dynamics, spatial variations, internal storage, Steady states, a water column..
Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 92D2.
Citation: Linfeng Mei, Sze-Bi Hsu, Feng-Bin Wang. Growth of single phytoplankton species with internal storage in a water column. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 607-620. doi: 10.3934/dcdsb.2016.21.607
References:
[1]

J. V. Baxley and S. B. Robinson, Coexistence in the unstirred chemostat, Appl. Math. Computation, 89 (1998), 41-65. doi: 10.1016/S0096-3003(97)81647-5.  Google Scholar

[2]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189-203. doi: 10.1016/S0022-5193(80)80003-8.  Google Scholar

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A. Cunningham and R. M. Nisbet, Transient and oscillation in continuous culture, Mathematics in microbiology, 77-103, Academic Press, London, 1983.  Google Scholar

[4]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272. doi: 10.1111/j.1529-8817.1973.tb04092.x.  Google Scholar

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J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409-428. Google Scholar

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J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.  Google Scholar

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J. P. Grover, Is storage an adaptation to spatial variation in resource availability?, The American Naturalist, 173 (2009), E44-E61. doi: 10.1086/595751.  Google Scholar

[8]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, The American Naturalist, 178 (2011), E124-E148. doi: 10.1086/662163.  Google Scholar

[9]

J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227. doi: 10.1093/plankt/fbq070.  Google Scholar

[10]

J. P. Grover, S. B. Hsu and F. B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Mathematical Biosciences, 222 (2009), 42-52. doi: 10.1016/j.mbs.2009.08.006.  Google Scholar

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J. P. Grover, S. B. Hsu and F. B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, Journal of Mathematical Biology, 64 (2012), 713-743. doi: 10.1007/s00285-011-0426-4.  Google Scholar

[12]

J. P. Grover and F.-B. Wang, Dynamics of a model of microbial competition with internal nutrient storage in a flowing habitat, Applied Mathematics and Computation, 225 (2013), 747-764. doi: 10.1016/j.amc.2013.09.054.  Google Scholar

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S. B. Hsu, S. Hubbell and P. Waltman, Mathematical theoy for single nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030.  Google Scholar

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Berlin, New York, Springer, 1981.  Google Scholar

[15]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.  Google Scholar

[16]

P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientific and Technical, 1991.  Google Scholar

[17]

S. B. Hsu and T. H. Hsu, Competitive exclusion of microbial species for a single-limited resource with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617. doi: 10.1137/070700784.  Google Scholar

[18]

S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014.  Google Scholar

[19]

S. B. Hsu, L. Mei and F. B. Wang, On a nonlocal reaction-diffusion-advection system modelling the growth of phytoplankton with cell quota structure, J. Diff. Eqns., 259 (2015), 5353-5378. doi: 10.1016/j.jde.2015.06.030.  Google Scholar

[20]

S. B. Hsu, H. L. Smith and P. Waltman, Dynamics of competition in the unstirred chemostat, Canad. Appl. Math. Quart., 2 (1994), 461-483.  Google Scholar

[21]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044. doi: 10.1137/0153051.  Google Scholar

[22]

P. Maga 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

[23]

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

[24]

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

[25]

H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: The variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131. doi: 10.1137/S0036139993245344.  Google Scholar

[26]

H. L. Smith and P. E. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[27]

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

[28]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A., 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.  Google Scholar

[29]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

J. V. Baxley and S. B. Robinson, Coexistence in the unstirred chemostat, Appl. Math. Computation, 89 (1998), 41-65. doi: 10.1016/S0096-3003(97)81647-5.  Google Scholar

[2]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189-203. doi: 10.1016/S0022-5193(80)80003-8.  Google Scholar

[3]

A. Cunningham and R. M. Nisbet, Transient and oscillation in continuous culture, Mathematics in microbiology, 77-103, Academic Press, London, 1983.  Google Scholar

[4]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272. doi: 10.1111/j.1529-8817.1973.tb04092.x.  Google Scholar

[5]

J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409-428. Google Scholar

[6]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.  Google Scholar

[7]

J. P. Grover, Is storage an adaptation to spatial variation in resource availability?, The American Naturalist, 173 (2009), E44-E61. doi: 10.1086/595751.  Google Scholar

[8]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, The American Naturalist, 178 (2011), E124-E148. doi: 10.1086/662163.  Google Scholar

[9]

J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227. doi: 10.1093/plankt/fbq070.  Google Scholar

[10]

J. P. Grover, S. B. Hsu and F. B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Mathematical Biosciences, 222 (2009), 42-52. doi: 10.1016/j.mbs.2009.08.006.  Google Scholar

[11]

J. P. Grover, S. B. Hsu and F. B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, Journal of Mathematical Biology, 64 (2012), 713-743. doi: 10.1007/s00285-011-0426-4.  Google Scholar

[12]

J. P. Grover and F.-B. Wang, Dynamics of a model of microbial competition with internal nutrient storage in a flowing habitat, Applied Mathematics and Computation, 225 (2013), 747-764. doi: 10.1016/j.amc.2013.09.054.  Google Scholar

[13]

S. B. Hsu, S. Hubbell and P. Waltman, Mathematical theoy for single nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030.  Google Scholar

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Berlin, New York, Springer, 1981.  Google Scholar

[15]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.  Google Scholar

[16]

P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientific and Technical, 1991.  Google Scholar

[17]

S. B. Hsu and T. H. Hsu, Competitive exclusion of microbial species for a single-limited resource with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617. doi: 10.1137/070700784.  Google Scholar

[18]

S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014.  Google Scholar

[19]

S. B. Hsu, L. Mei and F. B. Wang, On a nonlocal reaction-diffusion-advection system modelling the growth of phytoplankton with cell quota structure, J. Diff. Eqns., 259 (2015), 5353-5378. doi: 10.1016/j.jde.2015.06.030.  Google Scholar

[20]

S. B. Hsu, H. L. Smith and P. Waltman, Dynamics of competition in the unstirred chemostat, Canad. Appl. Math. Quart., 2 (1994), 461-483.  Google Scholar

[21]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044. doi: 10.1137/0153051.  Google Scholar

[22]

P. Maga 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

[23]

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

[24]

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

[25]

H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: The variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131. doi: 10.1137/S0036139993245344.  Google Scholar

[26]

H. L. Smith and P. E. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[27]

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

[28]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A., 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.  Google Scholar

[29]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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