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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 and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[29]

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

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[29]

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

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